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+Finite-size excess-entropy scaling for simple liquids
+Mauricio Sevilla,1 Atreyee Banerjee,1 and Robinson Cortes-Huerto1, ∗
+1Max Planck Institute for Polymer Research, Ackermannweg 10, 55128, Mainz
+(Dated: January 13, 2023)
+We introduce and validate a finite-size two-body excess entropy integral equation.
+By using
+analytical arguments and computer simulations of prototypical simple liquids, we show that the
+excess entropy s2 exhibits a finite-size scaling with the inverse of the linear size of the simulation
+box. Since the self-diffusivity coefficient D∗ displays a similar finite-size effect, we show that the
+scaling entropy relation D∗ = A exp(αs2) also depends on the simulation box size. By extrapolating
+to the thermodynamic limit, we report values for the coefficients A and α that agree well with values
+available in the literature. Finally, we find a power law relation between the scaling coefficients for
+D∗ and s2, suggesting a constant viscosity to entropy ratio.
+I.
+INTRODUCTION
+Excess entropy (sexc), the difference between the en-
+tropy of a system and its ideal gas counterpart at the
+same temperature and density, is connected to the dy-
+namical properties of simple liquids (See Ref. [1] for a
+recent review). This observation was first reported by
+Rosenfeld [2], who showed that, for simple model liq-
+uids, reduced transport properties such as diffusivity,
+viscosity and thermal conductivity scale with the excess
+entropy as
+X∗ = A exp (αsexc) ,
+(1)
+with X∗ a dimensionless transport property and A and
+α parameters, independent of the thermodynamic state,
+determined by the interparticle potential.
+Following similar physical arguments and assuming that
+the major contribution to sexc comes from two-body
+terms, Dzugutov proposed a similar scaling relation be-
+tween self-diffusivity and a two-body approximation to
+the excess entropy s2, namely [3]
+D∗ = A exp(αs2) ,
+(2)
+with D∗ =
+D
+Γσ2r where D is the self-diffusion coeffi-
+cient, σr measures the linear size of the particles and
+Γ = 4σ2g(σr)ρ
+�
+πkBT
+m
+the collision frequency given by
+the Enskog theory [4] where g(σr) is the value of the ra-
+dial distribution function at a distance σr. In this case,
+a large variety of simple liquids satisfy Eq. (2) with the
+universal choice of parameters A = 0.049 and α = 1 [3].
+This excess entropy scaling has been widely validated for
+a large variety of simple [5–11] and molecular liquids[12–
+15], including specially water [16–19]. We also highlight
+that experimental studies have tested entropy scaling
+in somewhat challenging scenarios [20, 21], and the fact
+that Rosenfeld and Dzugutov relations are empirical but
+have been justified on theoretical grounds [22, 23]. Fur-
+thermore, the structure–dynamics connection in Eq. (2)
+∗ [email protected]
+has been proposed as a tool to investigate the relation
+between dynamical properties of computational models
+at different resolutions, [24], which is now routinely con-
+sidered in the context of coarse-grained models [25, 26].
+Transport properties exhibit implicit size effects due
+to the finite size of the simulation box and the use
+of periodic boundary conditions (PBC) [27–29].
+In
+the particular case of the reduced self-diffusion coeffi-
+cient D∗, given a cubic simulation box of linear size L,
+D∗ ≡ D∗(L) takes the form [27, 30–33] (See Figure 1)
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+1/L
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+D ∗
+kBT = 0.7ϵ
+kBT = 0.85ϵ
+kBT = 1ϵ
+kBT = 2ϵ
+kBT = 3ϵ
+kBT = 4ϵ
+kBT = 5ϵ
+kBT = 6ϵ
+kBT = 7ϵ
+Figure 1.
+Reduced self-diffusion coefficient D∗ as a function
+of the inverse of the box linear size 1/L for a Lennard-Jones
+liquid with density ρσ3
+LJ = 0.864 in the range of tempera-
+tures kBT = [0.7ϵ, 7ϵ].
+D∗(L) = D∗∞ − δ
+L ,
+(3)
+with δ =
+kBT ζ
+6πηΓσ2r with ζ ≈ 2.837297 and η the system’s
+viscosity. In the thermodynamic limit (TL), namely, in
+the limit L → ∞, the self-diffusion coefficient takes the
+value D∗∞.
+Given the finite-size scaling of D∗, we expect that
+Eq. (2) also depends on the size of the simulation
+box.
+Recent computational studies investigating en-
+tropy scaling for liquid water using ab initio molecu-
+lar dynamics simulations [34] emphasise the relevance
+of this remark. In this case, the systems under consid-
+eration are rather small, and finite-size effects become
+arXiv:2301.05063v1  [cond-mat.soft]  12 Jan 2023
+
+2
+increasingly important.
+In this paper, we investigate the finite-size scaling of
+Eq. (2) by focusing on implicit and explicit finite-size
+effects present on the two-body excess entropy s2. We
+find that s2 obeys a finite-size scaling relation similar
+to D∗, which implies that the universal parameters A
+and α in Eq. (2) also depends on the size of the simu-
+lation box. Finally, and perhaps more interestingly, our
+results indicate that a power law relates the finite-size
+scaling coefficients of D∗ and s2, suggesting a constant
+viscosity/entropy ratio [35–38].
+The paper is organised as follows:
+In Section II we
+present the model and computational details. We show
+that s2 is ensemble invariant and that the only relevant
+finite-size effect comes from using finite integration do-
+mains in Section III. In Section IV, we introduce and
+validate a finite-size version of s2. We then present the
+finite-size scaling of the Dzugutov relation (Eq. (2)) in
+Section V. Finally, we conclude and provide our outlook
+in Section VI.
+II.
+COMPUTATIONAL DETAILS
+We investigate the excess entropy scaling for liquids
+whose potential energy is described by a 12–6 Lennard–
+Jones potential truncated, with cutoff radius rc/σLJ =
+2.5, and shifted. The parameters ϵ, σLJ and m, define
+the energy, length and mass units, respectively. All the
+results are expressed in LJ units with time σLJ(m/ϵ)1/2,
+temperature ϵ/kB and pressure ϵ/σ3
+LJ.
+In the follow-
+ing, we identify σr of Eq. (3) with σLJ. We consider
+cubic simulation boxes with linear sizes in the interval
+L/σLJ = [5, 50], with fixed density ρσ3
+LJ = 0.864. The
+systems are equilibrated at temperatures in the interval
+kBT = [0.7ϵ, 7.0ϵ], enforced with a Langevin thermostat
+with damping coefficient γ(σ(m/ϵ)1/2) = 1.0. We equili-
+brate the samples for 10×106 molecular dynamics (MD)
+steps using a time step of δt/(σLJ(m/ϵ)1/2) = 10−3, fol-
+lowed by additional 10 × 106 MD steps on the NVE
+ensemble to verify that the temperature does not de-
+viate substantially from the target value. Production
+runs span 10 × 106 MD steps. All the simulations have
+been performed with the LAMMPS simulation package
+[39].
+III.
+IMPLICIT AND EXPLICIT FINITE-SIZE
+EFFECTS
+In this section, we identify which finite-size effects are
+expected to affect the calculation of the excess entropy.
+We start with the definition of excess entropy for an
+N–particle system with respect to the ideal gas:
+sexc = S − SIG
+NkB
+= S2 + S3 + · · ·
+NkB
+,
+(4)
+with kB the Boltzmann constant. In the following, we
+focus on two-body contributions, which mostly amount
+to 80–90% of the overall value of the excess entropy for
+simple liquids. [40, 41] In particular, we have [42, 43]
+s2 = − ρ
+2V
+�
+V
+�
+V
+dr1 dr2 [g(r) ln g(r) − (g(r) − 1)] ,
+(5)
+with s2 =
+S2
+NkB the two-body excess entropy per particle.
+By taking the thermodynamic limit and assuming that
+the liquid is homogeneous and isotropic, we obtain the
+familiar expression
+s∞
+2 = −2πρ
+� ∞
+0
+dr r2 [g(r) ln g(r) − (g(r) − 1)] .
+(6)
+When performing molecular dynamics simulations, we
+usually consider systems with a finite number of parti-
+cles, typically not large enough to reach the thermody-
+namic limit. Furthermore, when evaluating the double
+integral in Eq. (5) we need to consider that the volume V
+is finite. For such a reason, and following the strategy
+used to compute the compressibility equation [44, 45]
+and the Kirkwood-Buff integrals [46–48] in computer
+simulations, we define a finite–size two–body excess en-
+tropy evaluated in a subvolume V of a system with a
+total number of particles N0 and the volume V0
+s2(V ; N0) = − ρ
+2V
+�
+V
+�
+V
+dr1 dr2 [g(r; N0) ln g(r; N0)
+−(g(r; N0) − 1)] ,
+(7)
+with g(r; N0) the finite-size RDF. The asymptotic cor-
+rection to the finite-size RDF, given by the difference in
+the thermodynamic ensemble, gives [49–54]
+g(r; N0) = g(r) − χ∞
+T
+N0
+(8)
+with χ∞
+T = ρkBTκT , and κT being the isothermal com-
+pressibility in the thermodynamic limit. We write the
+integrand in Eq. (7) as
+g(r; N0) ln g(r; N0) ≈ g(r) ln g(r)
+− χ∞
+T
+N0
+(1 + ln g(r))
+g(r; N0) − 1 = g(r) − 1 − χ∞
+T
+N0
+,
+(9)
+where in the first line in the previous expression, we
+have neglected terms of the order O
+�
+1
+N 2
+0
+�
+.
+The two
+contributions χ∞
+T
+N0 cancel out exactly. The contribution
+χ∞
+T
+N0 ln g(r) can be neglected by assuming a large number
+of particles (there is no V/V0 contribution, only 1/V0,
+hence, we can neglect it). This indicates that the two-
+body excess entropy is ensemble invariant, consistent
+with the result reported Ref. [55, 56]. We thus rewrite
+Eq. (7) as
+s2(V ) = − ρ
+2V
+�
+V
+�
+V
+dr1 dr2 [g(r) ln g(r)
+−(g(r) − 1)] .
+(10)
+
+3
+The volume V is finite and embedded into the vol-
+ume V0.
+The integration domains can be rearranged
+as
+�
+V
+�
+V (· · · ) =
+�
+V
+�
+V0(· · · ) −
+�
+V
+�
+V0−V (· · · ).
+Using
+a similar argument as the one used to calculate the
+finite-size compressibility [57] and Kirkwood-Buff inte-
+grals [46], the term
+�
+V
+�
+V0(· · · ) gives s∞
+2
+and the term
+�
+V
+�
+V0−V (· · · ) scales as 1/L with L = V 1/3 the linear
+size of the cubic simulation box. Thus,
+s2(L) = s∞
+2 + σ
+L ,
+(11)
+with σ a constant that depends on intensive thermody-
+namic quantities only. In the following section, we intro-
+duce a method to compute s2(L) and verify its scaling
+behaviour with the linear size of the simulation box.
+To finish this section, we compare our results with
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+R
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+sR
+2
+Kirkwood-Buff
+Information
+sR
+2  Eq. 12
+Figure 2.
+Plot of the two contributions, Kirkwood-Buff
+(g(r; N0) − 1) and Information (g(r; N0) ln g(r; N0)), to the
+truncated integral sR
+2 for a system of linear size L/σLJ = 35
+at kBT = 2.0ϵ. It is apparent that the two terms oscillate
+out-of-phase for small values of R, and their sum converges
+to s∞
+2
+when R → ∞.
+the usual truncation of Eq. (6) up to a cutoff radius R,
+namely
+sR
+2 = −2πρ
+� R
+0
+dr r2 [g(r; N0) ln g(r; N0)
+−(g(r; N0) − 1)] .
+(12)
+We use this truncated integral to verify numerically
+that ensemble finite-size contributions cancel out al-
+most exactly. [11] For a system of size L/σLJ = 35 at
+kBT = 2.0ϵ, we separate the g(r; N0) − 1, Kirkwood-
+Buff, and the g(r; N0) ln g(r; N0), Information, contri-
+butions and plot them as a function of the truncation
+radius R (See Figure 2). Both integrals diverge for large
+values of R, Kirkwood-Buff to infinity and Information
+to minus infinity, which signals a clear ensemble finite-
+size effect. However, these two finite-size contributions
+balance each other, and the sum of the two integrals
+converges to s∞
+2 for R >> 1. Due to this error cancella-
+tion, the truncation Eq. (12) gives s∞
+2 even for relatively
+small simulation boxes, and its finite-size dependence
+has been commonly overlooked in the literature.
+IV.
+FINITE-VOLUME EXCESS ENTROPY
+Based on previous work on finite-size isothermal com-
+pressibility [58] and Kirkwood-Buff integrals [59], we de-
+fine a finite-volume two-body excess entropy as follows.
+s2(V ) = − ρ
+2V
+� �
+dr1 dr2 R(r1) R(r2) h(r) ,
+(13)
+with R(r) a step function that defines the finite integra-
+tion subdomain, being equal to one inside and to zero
+outside the volume V [58]. The function h(r) is defined
+as
+h(r) = g(r) ln g(r) − (g(r) − 1) .
+(14)
+We write the double integral of s2(V ) in Fourier space
+and include the periodicity of the simulation of the box
+in h(r) explicitly. Thus
+s2(V ) = −
+ρ
+2(2π)3V
+�
+dk ˜R(k) ˜R(−k) ˜hPBC(k) ,
+(15)
+where [58]
+˜hPBC(k) =
+�
+nx,ny,nz
+e−k·snx,ny,nz ˜h(k) ,
+(16)
+with ˜h(k) the Fourier transform of h(r) and snx,ny,nz =
+(nx Lx, ny Ly, nz Lx) a vector specifying the system’s
+periodic images such that nx,y,z takes integer values. In
+the following, we consider a cubic simulation box with
+Lx = Ly = Lz = L. As before [59], we choose |nx| ≤ 1,
+|ny| ≤ 1 and |nz| ≤ 1 to compute Eq. (16). Finally,
+we assume a homogeneous and isotropic fluid such that
+˜h(k) = ˜h(k) with k =
+√
+k · k.
+To validate our approach, we verify that Eqs (15) and
+(12) converge to the same value in the thermodynamic
+limit.
+To this aim, we consider a system with linear
+size L/σLJ = 50 at kBT = 2.0ϵ, compute the RDF and
+evaluate the truncated integral Eq. (12). According to
+Eq. (11), implicit finite-size effects are the most relevant
+in this case. Hence, by considering a sufficiently large
+simulation box, the large R limit of Eq. (12) converges
+to the TL value. We present this result in Fig. 3 (black
+solid curve). To evaluate Eq. (15), we take the RDF
+from the simulation box with linear size L/σLJ = 20
+and perform the Fourier transform procedure described
+above to obtain ˜h(k). It is apparent, as expected, that
+with explicit PBC, the finite-size s2 gives the TL value
+(red dashed curve). Instead, by removing PBC, there
+is a significant deviation from the TL value that we at-
+tribute to the 1/L dependence in Eq. (11) (blue solid
+curved).
+We verify this 1/L dependence in the finite-size s2.
+In Figure 4, we plot the result of sR
+2 , Eq. (12), as a
+
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+R/L
+2.0
+1.5
+1.0
+0.5
+0.0
+s2
+Single Integral
+Double Integral (No PBC)
+Double Integral (PBC)
+Figure 3. Running s2 as a function of the ratio R/L for the
+case L/σLJ = 5 at kBT = 2.0ϵ. The black line corresponds
+to the truncation Eq. 12, and the red and blue curves are the
+result of Eq. (15) including (|nx| ≤ 1, |ny| ≤ 1 and |nz| ≤ 1)
+and not including (|nx| = |ny| = |nz| = 0) PBC, respec-
+tively. By including PBC, the integral Eq. (15) converges to
+the thermodynamic limit.
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+R/L
+2.4
+2.2
+2.0
+1.8
+s2
+Single integral
+Double Integral (No PBC)
+Double Integral
+Figure 4. s2 as a function of the inverse of the simulation
+box size L for systems at kBT = 2.0ϵ.
+The black trian-
+gles are calculated with the truncated integral (Eq. (12)),
+the red triangles and blue squares were calculated with the
+double integral (Eq. (15)) including and excluding PBC, re-
+spectively.
+function of 1/L (black inverted triangles). There, it is
+apparent that the integral converges when the linear size
+of the system is L/σLJ > 10. The result of using s2(V ),
+Eq. (15), with explicit PBC, always converges to the
+TL value (red triangles), regardless of the linear size of
+the system. More interestingly, by removing PBC from
+Eq. (15), we observe a clear linear dependence with 1/L
+(blue squares). Furthermore, by extrapolating this be-
+haviour (blue dashed line) to the axis 1/L = 0, we ob-
+tain a linear extrapolation to s∞
+2 . This result completes
+the validation of both, Eqs (11) and (15).
+V.
+FINITE-SIZE EXCESS-ENTROPY SCALING
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+1/L
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+−s2
+kbT = 0.7ϵ
+kbT = 0.85ϵ
+kbT = 1ϵ
+kbT = 2ϵ
+kbT = 3ϵ
+kbT = 4ϵ
+kbT = 5ϵ
+kbT = 6ϵ
+kbT = 7ϵ
+Figure 5.
+-s2 as a function of 1/L for a LJ system at
+ρσ3
+LJ = 0.864 and different temperatures. All data points
+were obtained with the RDF for the system of linear size
+L/σLJ = 20 and using Eq. (15) without PBC.
+In this section, we investigate the finite-size effects of
+the self-diffusivity entropy scaling, Eq. (2). To this aim,
+we verify that the scaling of s2 with 1/L is valid in a
+wide temperature range. We present these results for
+a LJ system with density ρσ3
+LJ = 0.864 in the range of
+temperatures kBT = [0.7ϵ, 7ϵ]. The results in Figure 5
+indicate that the 1/L scaling is apparent for all temper-
+atures considered here.
+We now collect all our data to investigate the scaling
+of Eq. (2) with the simulation box size. The result is
+presented in Figure 6 where the diffusion constant D∗
+is plotted against −s2. A clear trend with system size
+emerges, indicating that Eq. (2) remains valid even for
+the smallest simulation boxes considered and showing
+that the parameters A and α are also size dependent.
+By extrapolating D∗ and −s2 to the limit 1/L → 0,
+we obtain the TL values given by the black empty tri-
+angles that well agree with the reference scaling pro-
+vided by Eq. (2) (black dashed line). Indeed, we report
+A∞ = 0.048 ± 0.001 and α∞ = 1.000 ± 0.013 in the TL,
+in good agreement with the value originally estimated
+in Ref. [3].
+Finally, we investigate the relation between the coef-
+ficients δ and σ of the finite-size scaling of D∗ and s2,
+respectively. In Figure 7, we plot σ as a function of δ
+and observe a power law relation of the form σ = aδb
+with a = 1.256 ± 0.118 and b = −0.513 ± 0.020.
+VI.
+SUMMARY AND OUTLOOK
+We define a finite-size two-body excess entropy s2(L)
+integral equation with L the linear size of the simulation
+box. Using analytical arguments and simulations of a
+prototypical Lennard-Jones liquid at different densities
+and temperatures, we show that s2(L) = s∞
+2 +σ/L with
+
+5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+−s2
+10-4
+10-3
+10-2
+D ∗
+Reference
+5.0σ
+10.0σ
+15.0σ
+20.0σ
+25.0σ
+30.0σ
+35.0σ
+40.0σ
+50.0σ
+Figure 6.
+Reduced self-diffusion coefficient D∗ as a func-
+tion of −s2 for different system sizes and temperatures. The
+empty black triangles show the thermodynamic limit values
+for −s2 and D∗.
+10-2
+10-1
+100
+δ
+100
+101
+σ
+fit
+Data
+Figure 7. Coefficients σ(T) as a function of δ(T) for all the
+temperatures considered here. We used a power law σ = aδb
+to fit the data.
+σ a constant that depends on intensive thermodynamic
+quantities.
+Given the well-know finite-size scaling of
+the self-diffusivity, D∗(L) = D∗∞ − δ/L, we show that
+the universal scaling relation between entropy and
+diffusion D∗ = A exp (αs2) also exhibits a finite-size
+dependence and, by extrapolating to the TL, report
+A = 0.048 ± 0.001 and α = 1.000 ± 0.013, in good
+agreement with values reported in the literature.
+Fi-
+nally, and perhaps more interestingly, we show that the
+scaling coefficients σ and δ of s2 and D∗, respectively,
+are related by a somewhat simple power law σ = aδb
+with a = 1.256 ± 0.118 and b = −0.513 ± 0.020.
+The finite-size scaling of s2 can be rationalised in terms
+of the thermodynamics of small systems [60, 61]. In par-
+ticular, the statistical mechanics of a few model small
+systems in confinement has been derived recently [62].
+The authors have shown that given the high surface
+area–to–volume ratio of small systems, thermodynamic
+properties include surface contributions. In the case of
+entropy, these contributions include 1/L terms with L,
+the linear size of the system. In this context, we feel
+that the finite-size entropy scaling investigated here
+might play a role in understanding the non-equilibrium
+thermodynamics of confined, small systems [63].
+The power law relation between the scaling coeffi-
+cients of self-diffusion and two-body excess entropy
+is somewhat intriguing.
+On the one hand, the size
+scaling in the self-diffusion appears as a consequence
+of the conservation of linear momentum [30]. On the
+other hand, the finite-size scaling in the two-body
+entropy results from a surface contribution due to the
+confinement of the system [62]. Admittedly, we do not
+have a satisfactory explanation for this connection.
+Nevertheless, we point out that the ratio δb/σ = 1/a
+might be related to a constant viscosity/entropy ratio.
+Indeed, δ is inversely proportional to the system’s
+viscosity, and a simple dimensional analysis tells us
+that σ has units of entropy times length. Interestingly,
+string theory methods have been used to conjecture
+that, for fluids in equilibrium, the viscosity to entropy
+density ratio has a lower bound at ℏ/4πkB [35] with
+ℏ the reduced Planck constant.
+This relation, tested
+for various fluid systems [36–38], has been originally
+derived by considering that the entropy density of a
+black hole is proportional to the surface to volume
+ratio of its event horizon, i.e. a 1/L contribution. We
+find this connection fascinating, and, in our opinion, it
+deserves further investigation.
+ACKNOWLEDGMENTS
+We are grateful to Kurt Kremer for his insightful dis-
+cussions. We also thank Denis Andrienko for his critical
+reading of the manuscript. R.C.-H. gratefully acknowl-
+edges funding from SFB-TRR146 of the German Re-
+search Foundation (DFG). Simulations have been per-
+formed on the THINC cluster at the Max Planck Insti-
+tute for Polymer Research and the COBRA cluster at
+the Max Planck Computing and Data Facility.
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+arXiv:2301.04636v1  [math.CO]  11 Jan 2023
+Unavoidable structures in infinite tournaments
+Alistair Benford∗
+Louis DeBiasio†
+Paul Larson‡
+January 12, 2023
+Abstract
+We prove a strong dichotomy result for countably-infinite oriented graphs; that
+is, we prove that for all countably-infinite oriented graphs G, either (i) there is a
+countably-infinite tournament K such that G ̸⊆ K, or (ii) every countably-infinite
+tournament contains a spanning copy of G.
+Our characterization implies a corre-
+sponding result for transitive acyclic oriented graphs (i.e. strict partial orders). We
+also consider an extension of the above result to uncountable oriented graphs. Finally,
+we consider a problem of a slightly different nature; that is, which oriented graphs are
+guaranteed to appear in tournaments on N with a sufficient density of forward edges.
+1
+Introduction
+An oriented graph G is an anti-symmetric directed graph (that is, if (u, v) ∈ E(G), then
+(v, u) ̸∈ E(G)). For other standard terminology, see Section 1.2.
+A finite oriented graph is unavoidable if there exists a positive integer N such that
+G is subgraph of every tournament on at least N vertices. In the case of finite oriented
+graphs G, it is easy to see that G is unavoidable if and only if G is acyclic. Indeed, an
+oriented graph G on n vertices is acyclic if and only if G is a subgraph of the transitive
+tournament on n vertices, and it is well-known (see [11]) that every tournament on at
+least 2n−1 vertices contains a transitive tournament of order n. This leads to the following
+natural definition. For a finite acyclic oriented graph G, let ⃗r(G) be the smallest integer
+N such that every tournament on N vertices contains a copy of G. While there were some
+earlier sporadic results, the systematic study of the quantitative problem of minimizing
+⃗r(G) began with [20] and [21]. A few of the major results are as follows: for the transitive
+tournament ⃗Kn on n vertices, 2n/2 ≤ ⃗r( ⃗Kn) ≤ 2n−1 [11], for every oriented path P on
+n ≥ 8 vertices, ⃗r(P) = n [16], and for every oriented tree T on n vertices (for sufficiently
+large n), ⃗r(T) ≤ 2n − 2, and this is best possible for certain oriented trees T [18] (for a
+refinement of this result, see [9], [4], [5]).
+The goal of this paper is to extend the notion of unavoidability to infinite oriented
+graphs. We say that a countably-infinite oriented graph G is unavoidable if G is a subgraph
+of every countably-infinite tournament, otherwise we say that G is avoidable. It turns out
+∗School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK. [email protected]
+†Department of Mathematics, Miami University, Oxford, OH. [email protected]. Research sup-
+ported in part by NSF grant DMS-1954170.
+‡Department of Mathematics, Miami University, Oxford, OH. [email protected].
+1
+
+that in the infinite case, it is not true that an oriented graph G is unavoidable if and only if
+G is acyclic. So our first goal is to characterize which countably-infinite oriented graphs are
+unavoidable. Having done that, the next goal is to get quantitative results for unavoidable
+oriented graphs along the lines of the results mentioned above. For instance, motivated
+by recent Ramsey-type results regarding monochromatic subgraphs in edge-colorings of
+KN ([8], [7], [6], [19], [1]), it would be natural try to prove that there exists d > 0 such
+that for every countably-infinite unavoidable oriented tree T and every tournament K on
+N, there is an embedding φ : T → K such that φ(V (T)) ⊆ N has upper density at least d.
+So it is perhaps surprising that we prove the following result which both character-
+izes unavoidable oriented graphs and proves that all such countably-infinite unavoidable
+oriented graphs are unavoidable in a very strong sense (in a way which makes the quan-
+titative question mentioned above irrelevant). We say that an countably-infinite oriented
+graph G is strongly unavoidable if G is a spanning subgraph of every countably-infinite
+tournament.
+Theorem 1.1. Let G be an countably-infinite oriented graph. The following are equiva-
+lent:
+(i) G is acyclic, locally-finite, and has no infinite directed paths.
+(ii) G is unavoidable.
+(iii) G is strongly unavoidable.
+A strict partial order P = (V, ≺) is a relation ≺ on a set V which is anti-reflexive, anti-
+symmetric, and transitive. Defined in this way, every strict partial order P = (V, ≺) is an
+acyclic oriented graph. While it is not the case that every acyclic oriented graph is a strict
+partial order, there is an equivalence relation on acyclic oriented graphs, where G ∼ F
+if and only if the transitive closures of G and F are isomorphic, where the equivalence
+classes correspond to strict partial orders. Note that if G is acyclic, locally-finite, and has
+no infinite directed paths, then the transitive closure of G is a strict partial order in which
+every element is comparable to finitely many others (note that we aren’t using the phrase
+“locally-finite” in the context of partial orders since this seems to have a different meaning
+in the literature) and every strict partial order in which every element is comparable to
+finitely many others is acyclic, locally-finite, and has no infinite directed paths. So we
+have the following corollary.
+Corollary 1.2. Let P = (V, ≺) be an countably-infinite strict partial order. The following
+are equivalent:
+(i) Every element in P is comparable to finitely many others.
+(ii) P is unavoidable.
+(iii) P is strongly unavoidable.
+1.1
+Organization of the paper
+We will prove Theorem 1.1 (i)⇔(ii) in Section 2.1 and Theorem 1.1 (i)⇒(iii) in Section
+2.2 (note that Theorem 1.1 (iii)⇒(ii) just follows from the definition). In Section 3, we
+will consider an extension of Theorem 1.1 to the uncountable case. In Section 4 we will
+2
+
+consider a variant on the original problem where we ask if we can force a tournament K to
+contain an infinite forward directed path (which is avoidable by Theorem 1.1) by requiring
+that the lower density of forward edges in K is large enough.
+Since the paper consists of three distinct parts each with a distinct proper subset of
+authors, we point out that the results in Section 2 were obatined by Benford and DeBiasio,
+the results in Section 3 were obtained by DeBiasio and Larson, and the results in Section
+4 were obtained by Benford.
+1.2
+Notation
+An (oriented) graph is locally-finite if every vertex is incident with finitely many edges.
+A directed cycle on n vertices is the oriented graph ⃗Cn with V ( ⃗Cn) = {x1, . . . , xn} and
+E( ⃗Cn) = {(x1, x2), . . . , (xn−1, xn), (xn, x1)}. Say that an oriented graph is acyclic if it
+contains no directed cycles. A directed path is an orientation of a finite, one-way-infinite,
+or two-way-infinite path having the property that there are no vertices of out-degree 2
+or in-degree 2. The infinite directed path with exactly one vertex of in-degree 0 is called
+the infinite forward directed path and the infinite directed path with exactly one vertex of
+out-degree 0 is called the infinite backward directed path.
+Say that an oriented graph G is weakly-connected if the underlying graph (i.e. the
+symmetric closure of G) is connected.
+Given oriented graphs H and G, an embedding of H into G, denoted φ : H → G is an
+injection φ : V (H) → V (G) with the property that if (u, v) ∈ E(H), then (φ(u), φ(v)) ∈
+E(G). We say H is a subgraph of G, denoted H ⊆ G, if there exists an embedding of H
+into G. We say that H is a spanning subgraph of G if there exists a surjective embedding
+of H into G.
+Given an oriented graph G and S ⊆ V (G), we let G[S] be the subgraph induced by
+S. Given v ∈ V (G), we let G − v = G[V (G) \ {v}], and more generally, given S ⊆ V (G),
+we let G − S = G[V (G) \ S]. If G is an oriented graph and (u, v) ∈ E(G), then we say
+that v is an out-neighbor of u, and that u is an in-neighbor of v. Given v ∈ V (G), the
+out-neighborhood of v, written N +(v), is the set of out-neighbors of v in V (G), and the
+in-neighborhood of v, written N −(v) is the set of in-neighbors of v in V (G). Throughout,
+we use + and − interchangeably with ‘out’ and ‘in’ respectively. For each ⋄ ∈ {+, −}, the
+⋄-degree of v in G is d⋄(v) = |N ⋄(v)| and the common ⋄-neighborhood of a set X ⊆ V (G)
+is N ⋄(X) = �
+v∈X N ⋄(v).
+Given an acyclic oriented graph G and u ∈ V (G), let
+Γ+(u) = {v ∈ V (G) : there exists a directed path from u to v}
+and let
+Γ−(u) = {v ∈ V (G) : there exists a directed path from v to u}
+(equivalently, Γ⋄(u) is the ⋄-neighborhood of u in the transitive, reflexive closure of G).
+Given a strict total order τ = (V, ≺), let Kτ be the tournament on V where (u, v) ∈
+E(Kτ) if and only if u ≺ v. We write τ ∗ to be the converse of τ; that is, τ ∗ = (V, ≻). We
+say that Kτ and Kτ ∗ are the transitive tournaments of type-τ.
+Throughout the paper, we assume the axiom of choice whenever necessary. We use
+the von Neumann definition of ordinals where an ordinal is the strictly well-ordered set
+3
+
+of all smaller ordinals. Thus, given an ordinal λ, the definition of Kλ and Kλ∗ is given
+in the previous paragraph. As is standard, we let ω be the first infinite ordinal. Given a
+cardinal κ, we view κ as the smallest ordinal of cardinality κ. We let κ+ be the smallest
+cardinal greater than κ, and we let 2κ be the cardinality of the power set of κ.
+We say that a cardinal κ is regular if it cannot be written as the union of fewer than
+κ many sets each of cardinality less than κ. If κ is not regular, we say that it is singular.
+Given an ordinal α, the cofinality of α, denote cof(α), is the smallest cardinality of a
+cofinal subset of α (S ⊆ α is cofinal if for all x ∈ S, there exists a ∈ α such that x < a).
+With this terminology, we may equivalently say that κ is singular if and only if cof(κ) < κ.
+2
+Countably-infinite oriented graphs
+2.1
+Characterizing unavoidable oriented graphs
+The following lemma can essentially be found in [13, 2.15.1] (we give the more general
+statement later as Lemma 3.7). However, since this special case is likely folklore and the
+proof is straightforward, we give it here.
+Lemma 2.1. Let G be a countably-infinite oriented graph. G is acyclic, locally-finite, and
+has no infinite directed paths if and only if G ⊆ Kω and G ⊆ Kω∗.
+Proof. First note that if G has a cycle, then G ̸⊆ Kω and G ̸⊆ Kω∗. If G has a vertex
+of infinite in-degree, then G ̸⊆ Kω, and if G has a vertex of infinite out-degree, then
+G ̸⊆ Kω∗. If G has an infinite directed path with the first vertex having out-degree 0,
+then G ̸⊆ Kω, and if G has an infinite directed path with the first vertex having in-degree
+0, then G ̸⊆ Kω∗.
+Now suppose G is acyclic, locally-finite, and has no infinite directed paths. Let (vi)i∈ω
+be an enumeration of V (G). For all vi ∈ V (G), let f(vi) = max{j : vj ∈ Γ−(vi)} and note
+that by the assumptions on degrees and the fact that there are no infinite directed paths
+we have that f(vi) is finite.
+We will produce an embedding φ : G → Kω as follows: Let i0 ∈ ω be minimum such
+that vi0 has in-degree 0 in G and set φ(vi0) = 0. On step j ≥ 1, let ij ∈ ω be minimum
+such that vij has in-degree 0 in G − {vi0, . . . , vij−1} and set φ(vij) = j.
+In the resulting embedding we have the property that for all i ∈ ω, φ−1(i) has no
+in-neighbors v with φ(v) > i and thus we have the desired embedding, provided that
+V (G) = dom φ. However, this holds because for all v ∈ V (G), we will assign a value for
+φ(v) by step f(vi).
+The following lemma shows that it is possible to use a Ramsey-type result to get a
+result about transitive tournaments.1 We state it in a form which is more general than
+what is needed for this section because we will make reference to it again later in a more
+general setting. While this folklore result surely appears in the literature, we include a
+proof for completeness.
+1On the surface, it is strictly stronger because it is possible to order the vertices of the tournament K
+and have a copy of Kτ or Kτ∗ which doesn’t obey the ordering.
+4
+
+Lemma 2.2. Let σ = (S, <σ) be a strict total order. If for every 2-coloring of the pairs
+of elements in S, there exists a monochromatic copy of the strict total order τ = (T, <τ),
+then for every tournament K of cardinality |S| we have Kτ ⊆ K or Kτ ∗ ⊆ K.
+Proof. Let K be a tournament of cardinality |S| and take an arbitrary bijection φ :
+V (K) → S. For (u, v) ∈ E(K), if φ(u) <σ φ(v), color (u, v) red, and if φ(u) >σ φ(v),
+color (u, v) blue. By the assumption, there is a monochromatic copy of τ. If the copy is
+red, then we have Kτ ⊆ K, and if the copy is blue, then we have Kτ ∗ ⊆ K.
+Theorem 2.3. Let G be a countable oriented graph. The following are equivalent:
+(i) G is acyclic, locally-finite, and has no infinite directed paths.
+(ii) G ⊆ Kω and G ⊆ Kω∗.
+(iii) G is unavoidable.
+Proof. (i)⇔(ii): This is Lemma 2.1.
+(ii)⇒(iii): Suppose G ⊆ Kω and G ⊆ Kω∗ and let K be a countably-infinite tour-
+nament. By Ramsey’s theorem and Lemma 2.2 (with σ = τ = ω) we have Kω ⊆ K or
+Kω∗ ⊆ K. Either way, we have G ⊆ K.
+(iii)⇒(ii): If G ̸⊆ Kω or G ̸⊆ Kω∗, then G is avoidable.
+2.2
+Unavoidable oriented graphs are strongly unavoidable
+Given an countably-infinite acyclic weakly-connected oriented graph G, a ±-partition of
+G is a partition {Ci : i ∈ N} of V (G) such that the following properties hold:
+A1 For all i ∈ N, Ci is finite and non-empty.
+A2 For all (u, v) ∈ E(G), there exists i ∈ N such that {u, v} ⊆ Ci ∪ Ci+1.
+A3 If i is odd, then every vertex in Ci has in-degree 0 to Ci−1 ∪ Ci+1, and if i is even,
+then every vertex in Ci has out-degree 0 to Ci−1 ∪ Ci+1.
+A4 If i is odd, then there exists a vertex in Ci with in-degree 0 in G, and if i is even
+there exists a vertex in Ci with out-degree 0 in G.
+If i is odd, we say that Ci has type +, and if i is odd, we say that Ci has type −.
+Likewise one can define a ∓-partition by switching every instance of in/out in the
+above definition. We note that a similar definition for finite oriented trees was given by
+Dross and Havet [9].
+Lemma 2.4. Let G be a countably-infinite oriented graph.
+If G is weakly-connected,
+acyclic, locally-finite, and has no infinite directed paths, then for every vertex v of in-
+degree 0, G has a ±-partition with C1 = {v}, and for every vertex v of out-degree 0, G
+has a ∓-partition with C1 = {v}.
+Proof. Since G is acyclic and has no infinite directed paths, the set of vertices with in-
+degree 0 is non-empty; let v be a vertex of in-degree 0 and set C1 = {v}. For even i ≥ 1,
+let Ci =
+��
+v∈Ci−1 Γ+(v)
+�
+\ (Ci−1 ∪ Ci−2), and for odd i ≥ 1, let Ci =
+��
+v∈Ci−1 Γ−(v)
+�
+\
+(Ci−1 ∪ Ci−2). Note that since G is locally-finite, and has no infinite directed paths, each
+5
+
+Ci is finite. In addition, because G is weakly-connected, {Ci : i ∈ N} is a partition of
+V (G), and each Ci is non-empty. Therefore, A1 holds.
+Suppose, for some i < j, that (u, v) ∈ E(G) with u ∈ Ci and v ∈ Cj. Then, we must
+have that i is odd (else v ∈ Ci) and j = i + 1. On the other hand, if (v, u) ∈ E(G) is such
+that u ∈ Ci and v ∈ Cj for some i < j, then we must have that i is even (else v ∈ Ci) and
+j = i + 1. Therefore, we deduce that A2 and A3 hold.
+Finally, note that since Ci is finite and G is acyclic, G[Ci] has a vertex ui of in-degree
+0 in G[Ci] and a vertex vi of out-degree 0 in G[Ci]. Thus, by A3, if i is even, then vi has
+out-degree 0 in G, and if i is odd, then ui has in-degree 0 in G. Therefore, A4 holds, and
+{Ci : i ∈ N} is a ±-partition with C1 = {v}.
+Likewise by switching every instance of in/out in the above proof, we get that for every
+vertex v of out-degree 0, G has a ∓-partition with C1 = {v}.
+Theorem 2.5. Let G be a countably-infinite oriented graph. If G is acyclic, locally-finite,
+and has no infinite directed paths, then G is a spanning subgraph of every countably-infinite
+tournament.
+Proof of Theorem 2.5. Suppose G is acyclic, locally-finite, and has no infinite directed
+paths.
+If G is not weakly-connected, we can make it so while maintaining the three
+properties (say by choosing a vertex vi from each component Hi of G and adding an anti-
+directed path on v1, v2, . . . ). Let K be a countably-infinite tournament and let (ui)i∈N be
+an enumeration of V (K). Define ∗1, ∗2, ∗3, . . . inductively by
+∗i =
+
+
+
++
+if
+��i−1
+j=1 N ∗j(uj)
+�
+∩ N +(ui) is infinite,
+−
+otherwise.
+Let V + = {ui ∈ V (K) : ∗i = +} and let V − = {ui ∈ V (K) : ∗i = −}. The key property
+is that for all ⋄, ∗ ∈ {+, −} and all finite non-empty subsets X ⊆ V ⋄ and Y ⊆ V ∗,
+N ⋄(X) ∩ N ∗(Y ) is infinite. (A more standard approach to assigning the ∗i would have
+been to choose an ultrafilter on N and let ∗i = ⋄ iff N ⋄(ui) is in the ultrafilter. We note
+that our assignment of ∗i without the use of ultrafilters is inspired by the proof of [19,
+Lemma 3.4].)
+If ∗1 = +, then we choose a vertex v1 ∈ V (G) with in-degree 0 and apply Lemma 2.4
+to get a ±-partition {Ci : i ∈ N} of G with C1 = {v1}. If ∗1 = −, then we choose a vertex
+v1 ∈ V (G) with out-degree 0 and apply Lemma 2.4 to get a ∓-partition {Ci : i ∈ N} of
+G with C1 = {v1}. We may suppose without loss of generality that ∗1 = + and thus we
+choose a vertex v1 ∈ V (G) with in-degree 0 and apply Lemma 2.4 to get a ±-partition
+{Ci : i ∈ N} of G with C1 = {v1}. Finally, define
+⋄i =
+�
++
+if i is odd,
+−
+if i is even
+and note that ⋄i simply describes the type of the set Ci.
+We construct a sequence i1 ≤ i2 ≤ . . ., growing an embedding φ : G[∪i∈[ij]Ci] → K as
+we do so, such that following properties hold for every j ∈ N.
+B1 {u1, . . . , uj} ⊆ φ(∪i∈[ij]Ci), and
+6
+
+B2 φ(Cij) ⊆ V ⋄ij .
+If such a sequence exists, then by B1, the resulting embedding φ : G → K proves the
+theorem.
+We initially set i1 = 1 and φ(v1) = u1. Then, given ij−1 and φ : G[∪i∈[ij−1]Ci] → K
+satisfying B1 and B2, we proceed as follows.
+If uj ∈ φ(∪i∈[ij−1]Ci), then set ij = ij−1 (trivially, B1 and B2 are satisfied). Otherwise,
+by B2 we have that Uj−1 := N ∗j(uj)∩N ⋄ij−1(φ(Cij−1)) is infinite. If Uj−1∩V + is infinite,
+set ij to be the smallest integer at least ij−1 +5 with ⋄ij = + (i.e. the smallest odd integer
+at least ij−1 + 5). Otherwise, Uj−1 ∩ V − is infinite and we set ij to be the smallest integer
+at least ij−1 + 5 with ⋄ij = − (i.e.
+the smallest even integer at least ij−1 + 5).
+We
+now embed the acyclic finite subgraph G[Cij−1+1 ∪ . . . ∪ Cij] into the infinite tournament
+K[{uj} ∪ (Uj−1 ∩ V ⋄ij )] in such a way that if ∗j = ⋄ij−1, then we will choose a vertex
+vj ∈ Cij−1+2 which only has ∗j-neighbors and embed vj to uj, and if ∗j ̸= ⋄ij−1, then we
+will choose a vertex vj ∈ Cij−1+3 which only has ∗j-neighbors and embed vj to uj. Thus
+B1 is satisfied. Also note that by construction, every vertex in Cij is embedded into V ⋄ij ,
+so B2 is satisfied.2
+3
+Uncountable oriented graphs
+Let κ be a cardinal and let G be an oriented graph with |V (G)| ≤ κ. We say that G is
+κ-unavoidable if G is contained in every tournament K with |V (K)| = κ, otherwise we
+say that G is κ-avoidable. We say that G is strongly κ-unavoidable if G is a spanning
+subgraph of every tournament K with |V (K)| = κ.
+In Section 2.1 we determined that a countable acyclic oriented graph G is ℵ0-unavoidable
+if and only every vertex in G has degree less than ℵ0 and G has no infinite directed paths.
+If κ > ℵ0, it is natural to wonder if any acyclic oriented graphs G with vertices of infi-
+nite degree or infinite directed paths are κ-unavoidable. As it turns out, we cannot relax
+the condition on directed paths at all, but we can potentially relax the condition on the
+degrees of the vertices relative to κ.
+Observation 3.1. Let κ be an infinite cardinal and let G be an acyclic oriented graph.
+If G has an infinite directed path or a vertex of degree κ, then G is κ-avoidable. More
+specifically, we have the following:
+(i) If G has a vertex of in-degree κ or an infinite backward directed path, then G ̸⊆ Kκ.
+(ii) If G has a vertex of out-degree κ or an infinite forward directed path, then G ̸⊆ Kκ∗.
+Proof. Suppose G has an infinite directed path or a vertex of degree κ. Note that Kκ
+has no infinite backward directed paths and every vertex in Kκ has in-degree less than κ.
+Likewise Kκ∗ has no infinite forward directed paths and every vertex in Kκ∗ has out-degree
+less than κ. Thus G ̸⊆ Kκ or G ̸⊆ Kκ∗.
+2In the above paragraph, it is instructive to have a specific example, so suppose φ(Cij−1) ⊆ V + (i.e.
+ij−1 is odd), uj ∈ V −, and (N −(uj)∩N +(φ(Cij−1))∩V − is infinite. In this case we would set ij = ij−1 +5
+(note that ij is even), embed a vertex from Cij−1+3 (ij−1 + 3 is also even) with in-degree 0 to uj and
+embed the rest of Cij−1+1 ∪ · · · ∪ Cij into (N −(uj) ∩ N +(φ(Cij−1)) ∩ V −. Note that since ij is even and
+φ(Cij) ⊆ V −, B2 is satisfied.
+7
+
+This leads to the following question which attempts to generalize Theorem 1.1.
+Question 3.2 (Naive version). Let κ be an infinite cardinal and let G be an oriented graph
+with |V (G)| = κ. Are the following equivalent?
+(i) G is acyclic, has no infinite directed paths, and every vertex has degree less than κ.
+(ii) G is κ-unavoidable.
+(iii) G is strongly κ-unavoidable.
+Note that (iii)⇒(ii) just holds by definition and Observation 3.1 proves (ii)⇒(i).
+We answer Question 3.2 positively in the case when κ = ℵ1. In fact, we prove something
+slightly more general.
+Theorem 3.3. Let κ be an uncountable cardinal and let G be an acyclic oriented graph
+with no infinite directed paths and |V (G)| = κ. If every vertex in G has degree at most
+ℵ0, then G is strongly κ-unavoidable.
+On the other hand, we answer Question 3.2((i)⇒(ii)) negatively in the case when κ is
+a singular cardinal. Essentially, this boils down to the fact that if κ is a singular cardinal,
+then it is possible for every vertex in G to have degree less than κ, but in the transitive
+closure of G some vertex has degree κ.
+Example 3.4. Let κ be a singular cardinal. There exists an acyclic oriented graph with
+|V (G)| = κ such that G has no infinite directed paths and every vertex has degree less than
+κ, but G ̸⊆ Kκ and G ̸⊆ Kκ∗.
+Proof. Let ⟨λα : α < cof(κ)⟩ be an increasing cofinal sequence of regular cardinals less
+than κ.
+Let X = {xα : α < cof(κ)} be a set of vertices such that each xα has a unique set
+of λα in-neighbors and let Z = {zα : α < cof(κ)} be a set of vertices such that each zα
+has a unique set of λα out-neighbors. Then add a vertex y such that N −(y) = X and
+N +(y) = Z. Call the resulting oriented graph G. Note that y has in-degree and out-degree
+equal to cof(κ) < κ and every other vertex has in-degree and out-degree at most λα < κ
+for some α < cof(κ). If say G ⊆ Kκ, then since Kκ is transitive, the transitive closure,
+⃗G, of G satisfies ⃗G ⊆ Kκ (likewise if G ⊆ Kκ∗). However, in ⃗G it is the case that y has
+in-degree and out-degree equal to κ. So by Observation 3.1, we have that ⃗G ̸⊆ Kκ and
+⃗G ̸⊆ Kκ∗, a contradiction.
+While Question 3.2 may still be true when restricted to regular cardinals or to transitive
+acyclic oriented graphs (i.e. strict partial orders), Example 3.4 leads us to the following
+refinement of Question 3.2.
+Question 3.5 (Quantitative version). Let κ be an infinite cardinal and let G be an oriented
+graph with |V (G)| = µ ≤ κ. For which cardinals λ < κ is the following true?
+(i) If G is acyclic, has no infinite directed paths, and every vertex has degree at most λ,
+then G is κ-unavoidable.
+(ii) If µ = κ and G is acyclic, has no infinite directed paths, and every vertex has degree
+at most λ, then G is strongly κ-unavoidable.
+8
+
+3.1
+Embedding into transitive tournaments
+To prepare for the proof of Theorem 3.3, we begin with a few preliminary results.
+The first result is another variant on Szpilrajn’s extension theorem (see the comment
+before Lemma 2.1) which is essentially equivalent to the fact that every well-founded
+partial order of cardinality κ can be extended to a well-order of cardinality κ. A sketch
+of a proof can be found in [13, 2.9.2], but we give a full proof for completeness.
+Proposition 3.6. Let κ be an infinite cardinal and let G be an acyclic oriented graph
+with |V (G)| ≤ κ. If G has no infinite backward directed paths, then there exists an ordinal
+β of cardinality κ such that G ⊆ Kβ. Likewise if G has no infinite forward directed paths,
+then there exists an ordinal β of cardinality κ such that G ⊆ Kβ∗.
+Proof. We define a function h from G to the ordinals where h(x) = sup{h(w) + 1 : w ∈
+Γ−(x)} for all x ∈ V (G). Given an ordinal α, let Vα = {x ∈ V (G) : h(x) = α}. Let
+h(P) = {α : Vα ̸= ∅} (which is an ordinal) and set β = h(P). (Note that β = h(P) must
+satisfy |β| ≤ κ, as otherwise G has more than κ-many non-empty levels, which means
+|V (G)| > κ, a contradiction.)
+Now for all α < β, let (vα
+γ )γ<|Vα| be an enumeration of Vα. Then let φ : G → Kβ be
+an injection such that φ(vα′
+γ′ ) < φ(vα
+γ ) if and only if α′ < α, or α′ = α and γ′ < γ.
+The following lemma can essentially be found in [13, 2.15.1] (where it is attributed to
+Milner and Pouzet).
+Lemma 3.7. Let κ be an infinite cardinal and let G be an acyclic transitive oriented graph
+(i.e. a strict partial order) with |V (G)| ≤ κ. G ⊆ Kκ and G ⊆ Kκ∗ if and only if G has
+no infinite directed paths and every vertex in G has degree less than κ.
+Now we have the following corollary which is a generalization of Lemma 2.1.
+Corollary 3.8. Let λ < κ be infinite cardinals. Let G be an acyclic oriented graph with
+no infinite directed paths and |V (G)| ≤ κ.
+(i) If κ is a regular cardinal and every vertex in G has degree less than κ, then G ⊆ Kκ
+and G ⊆ Kκ∗.
+(ii) If every vertex in G has degree at most λ, then G ⊆ Kκ and G ⊆ Kκ∗.
+Proof. Suppose first that κ is regular. If every vertex in G has degree less than κ, then
+the transitive closure of G has the same property. Thus we can apply Lemma 3.7 to the
+transitive closure of G.
+Now let κ be a cardinal (regular or singular). If every vertex in G has degree at most
+λ, then the transitive closure of G has the same property. Since λ < κ, we can apply
+Lemma 3.7 to the transitive closure of G.
+3.2
+Embedding into general tournaments
+To give some context for the next result, note that it is known by a result of Laver (see
+[15]) that there is a tournament of cardinality ℵ1 which doesn’t contain any transitive
+subtournament of cardinality ℵ1. On the other hand a result of Baumgartner and Hajnal
+9
+
+[2] shows that the next best thing is true; that is, for every countable ordinal α and every
+2-coloring of the pairs in ω1, there is a monochromatic copy of α, which by Lemma 2.2
+implies that every tournament of cardinality ℵ1 contains Kα or Kα∗. (Note that while
+Baumgartner and Hajnal use some deep set-theoretic ideas to establish their result in
+ZFC, Galvin [14] later gave a direct combinatorial proof in ZFC.)
+Theorem 3.9 (Baumgartner-Hajnal; Galvin). Let K be an uncountable tournament. For
+every countable ordinal α we have Kα ⊆ K or Kα∗ ⊆ K.
+We now use Proposition 3.6 and Theorem 3.9 to obtain the following key lemma.
+Lemma 3.10. Let K be an uncountable tournament. If G is a countable acyclic oriented
+graph with no infinite directed paths, then for all v ∈ V (K) there exists an embedding
+φ : G → K such that v ∈ φ(V (G)).
+Proof. Let v ∈ V (K).
+Either v is incident with uncountably many out-edges or un-
+countably many in-edges.
+If it is the former, let u ∈ V (G) with in-degree 0 and set
+K′ = K[N +(v)]. If it is the latter, let u ∈ V (G) with out-degree 0 and set K′ = K[N −(v)].
+Now in either case set φ(u) = v. By Proposition 3.6, there exists a countable ordinal α
+such that G − u ⊆ Kα and G − u ⊆ Kα∗. By Theorem 3.9, Kα ⊆ K′ or Kα∗ ⊆ K′
+and thus we can embed G − u in K′ which gives us an embedding φ : G → K such that
+v ∈ φ(V (G)).
+Finally, we use Lemma 3.10 to obtain the main result.
+Proof of Theorem 3.3. Suppose G is an acyclic oriented graph with |V (G)| = κ such that
+G has no infinite directed paths and every vertex has degree at most ℵ0. Let K be a
+tournament with |V (K)| = κ and let (vα)α∈κ be an enumeration of V (K). Note that
+since every vertex in G has countable degree, G is not weakly-connected; in fact, every
+weakly-connected component of G is countable (and thus there are κ-many components).
+Let (Gα)α∈κ be an arbitrary κ-ordering of the components of G.
+Now we construct a surjective embedding φ : G → K as follows. For all α ∈ κ, if
+vα ̸∈ ran(φ), let K(α) = K − ran(φ) and apply Lemma 3.10 to embed Gα into K(α) such
+that vα ∈ φ(V (Gα)). Since every vertex vα is eventually mapped to, this completes the
+proof.
+Finally, we wrap up this section with some comments regarding possible extensions
+of Theorem 3.3. A classical result of Erd˝os [10] implies that for all infinite cardinals λ,
+every tournament K of cardinality (2λ)+ contains Kλ+ or K(λ+)∗. So our above result
+generalizes to give the following.
+Theorem 3.11. Let κ and λ be infinite cardinals such that κ ≥ (2λ)+ and let G be an
+acyclic oriented graph with no infinite directed paths and |V (G)| = κ. If every vertex of
+G has degree at most λ, then G is strongly κ-unavoidable.
+Proof. Every vertex in G having degree at most λ implies that every component of G has
+cardinality at most λ. Thus every component of G embeds into Kβ and Kβ∗ for some
+ordinal β < λ+ by Proposition 3.6. Now we complete the proof analogously to the proof
+of Theorem 3.3 (using the result of Erd˝os mentioned above in place of Theorem 3.9).
+10
+
+Note that under the generalized continuum hypothesis (GCH), Erd˝os’ result says that
+for all infinite cardinals κ = λ+, every tournament K of cardinality κ contains Kλ or Kλ∗.
+If it were true that that for every infinite cardinal κ and every ordinal β < κ, every tour-
+nament K of cardinality κ contains Kβ or Kβ∗, this would answer Question 3.2 positively
+in the case that κ is a successor cardinal. However, it is not the case that ZFC+GCH
+implies that every infinite cardinal κ and every ordinal β < κ, every tournament K of
+cardinality κ contains Kβ or Kβ∗ (see [12]). So as it stands, in ZFC+GCH, the best thing
+we can say is that Theorem 3.11 implies that Question 3.5(ii) has a positive answer when
+κ = λ++.
+4
+Forward edge density
+We saw in Section 2.1 that if we are given a countably-infinite oriented graph G such that G
+is acyclic, locally-finite, and has no infinite directed paths, then for every countably-infinite
+tournament K, we have G ⊆ K. However, if we drop one or more of these conditions on
+G, what conditions could we impose on the tournament K to still ensure G ⊆ K? One
+approach to this question is to consider the density of the forward (or backward) edges in
+the tournament K.
+For an infinite tournament K on N, define, for each n,
+d+(K[n]) = |{(i, j) ∈ E(K) : 1 ≤ i < j ≤ n}|
+�n
+2
+�
+,
+and define the forward density of K to be
+d+(K) = lim inf
+n→∞ d+(K[n]).
+Given an oriented graph G, define −→ρ (G) to be the smallest ρ ∈ [0, 1] such every tournament
+K on N with d+(K) > ρ contains a copy of G. By Theorem 2.3, we know that if G is
+acyclic, locally-finite, and has no infinite directed paths, then −→ρ (G) = 0. On the other
+hand, if G contains a cycle, a vertex of infinite in-degree, or a backward oriented infinite
+path, then G does not appear in Kω, and so we have −→ρ (G) = 1. In addition, if G contains
+a vertex of infinite out-degree, then we also have −→ρ (G) = 1, due to the existence of
+tournaments with forward density 1 in which every vertex has finite out-degree (such as
+the tournament on N with forward edges given by {(i, j) : j ≤ 2i}).
+The only oriented graphs then left to consider are those which are acyclic, locally-finite,
+and contain a forward oriented infinite path (but no backward oriented infinite path). A
+natural case to consider then is to determine the value of −→ρ (P), where P is the forward
+oriented infinite path. Notably, we have −→ρ (P) strictly between 0 and 1.
+Proposition 4.1. Let P be the infinite forward directed path. Then, −→ρ (P) = 3/4.
+Note that obtaining a lower bound of −→ρ (P) ≥ 1/2 is easy. Let Ik = [k!] \ [(k − 1)!]
+for k ≥ 2, and let K be the tournament on N with all edges with both endpoints some Ik
+oriented forward and all other edges oriented backward. Then, d+(K) = 1/2, but, because
+K ∼= Kω∗, K contains no copy of P.
+11
+
+Proving an upper bound strictly below 1, and also sharpening the lower bound, is more
+involved. To do this, we will reduce the problem to finding the smallest possible forward
+density of a large class of P-free tournaments on N, defined as follows. Given an ordinal
+λ and an injection f : N → λ, define Kf∗ to be the tournament on N with
+E(Kf∗) = {(i, j) : f(i) > f(j)}.
+Because λ is well ordered, no tournament defined in this way contains a copy of P. In
+addition, the forward density of Kf∗ can be easily related to the density of inversions of
+f. Precisely, if we define, for sets A, B ⊆ N,
+If[A, B] = |{(i, j) ∈ A × B : i < j and f(i) > f(j)}|
+and also If[A] = If[A, A] and If[n] = If[[n]], then we have
+d+(Kf∗) = lim inf
+n→∞ If[n]/
+�n
+2
+�
+.
+(1)
+The quantity lim infn→∞ If[n]/
+�n
+2
+�
+can be thought of as an infinite analogue of the inversion
+number of a permutation (see, for example, [17]), especially in the case where we have a
+bijection f : N → ω.
+We note that the earlier P-free tournament showing −→ρ (P) ≥ 1/2 can be realized as
+Kf∗ for some appropriate f : N → ω. In fact, given any P-free tournament K on N, we
+can construct an injection h : N → ω1 such that d+(Kh∗) ≥ d+(K). From this we obtain
+the following correspondence.
+Lemma 4.2. Define
+C = sup
+�
+lim inf
+n→∞ If[n]/
+�n
+2
+�
+: f : N → ω1 is an injection
+�
+.
+Then −→ρ (P) = C.
+Proof. Because −→ρ (P) ≥ d+(Kf∗) for any injection f : N → ω1, we have, by (1), that
+−→ρ (P) ≥ C. Therefore it is enough to show that for any P-free tournament K on N there
+is an injection f : N → ω1 with d+(Kf∗) ≥ d+(K), and hence
+−→ρ (P) = sup {d+(K) : K is P-free}
+≤ sup {d+(Kf∗) : f : N → ω1 is an injection}
+(1)
+= C.
+So suppose K is a P-free tournament on N. For i, j ∈ N, say that j is a forward out-
+neighbor of i if i < j and i →K j. Define A0 = ∅. Given an ordinal α, define Aα+1 ⊇ Aα
+to be the set of x ∈ N such that every forward out-neighbor of x is in Aα. If α is a limit
+ordinal, define Aα = ∪β<αAβ. Define ∂Aα = Aα+1 \ Aα. Let λ be the smallest ordinal
+with ∂Aλ = ∅, and note that λ is well defined and λ < ω1 (because we can inject λ → N
+by sending α < λ to the smallest x ∈ N with x ∈ ∂Aα). If Aλ ̸= N, then K contains a
+copy of P (any xi /∈ Aλ has a forward out-neighbor xi+1 /∈ Aλ). Therefore, we have a
+partition N = ∪α<λ∂Aα. Given i ∈ N, let α(i) < λ be the unique ordinal with i ∈ ∂Aα(i).
+Define an injection f : N → ω · λ < ω1 by f(i) = (i, α(i)). Note that, if i < j and i →K j,
+then α(i) > α(j), and hence f(i) > f(j). Therefore, d+(K) ≤ d+(Kf∗).
+12
+
+With Lemma 4.2, determining the value of −→ρ (P) is now equivalent to this natural
+question of what is the maximum ‘inversion density’ that can be attained by an injection
+f : N → ω1. Thus, Proposition 4.1 is now an immediate consequence of the following
+theorem.
+We do not present the proof here as it involves some technical details, but
+instead refer the reader to [3].
+Theorem 4.3.
+(i) If f : N → ω1 is an injection, then lim infn→∞ If[n]/
+�n
+2
+�
+≤ 3/4.
+(ii) There is an injection g : N → ω such that lim infn→∞ Ig[n]/
+�n
+2
+�
+= 3/4.
+There are still other interesting problems regarding the forward density required to
+ensure a copy of an infinite oriented graph.
+Problem 4.4. Let G be the family of countably-infinite acyclic oriented graphs which are
+locally-finite, have no infinite backward directed paths, but do have an infinite forward
+directed path.
+(i) Given G ∈ G, determine −→ρ (G).
+(ii) For which d ∈ [3
+4, 1] does there exist G ∈ G with −→ρ (G) = d?
+5
+Acknowledgements
+We thank Trevor Wilson for pointing out the reference [2].
+References
+[1] J. Balogh and A. Lamaison. Ramsey upper density of infinite graph factors. arXiv
+preprint arXiv:2010.13633, 2020.
+[2] J. Baumgartner and A. Hajnal. A proof (involving Martin’s axiom) of a partition
+relation. Fundamenta Mathematicae, 78(3):193–203, 1973.
+[3] A. Benford. On the appearance of oriented trees in tournaments. PhD thesis, Uni-
+versity of Birmingham, in preparation.
+[4] A. Benford and R. Montgomery. Trees with few leaves in tournaments. Journal of
+Combinatorial Theory, Series B, 155:141–170, 2022.
+[5] A. Benford and R. Montgomery.
+Trees with many leaves in tournaments.
+arXiv
+preprint arXiv:2207.06384, 2022.
+[6] J. Corsten, L. DeBiasio, A. Lamaison, and R. Lang. Upper density of monochromatic
+infinite paths. Advances in Combinatorics, page 10810, 2019.
+[7] J. Corsten, L. DeBiasio, and P. McKenney. Density of monochromatic infinite sub-
+graphs II. arXiv preprint arXiv:2007.14277, 2020.
+[8] L. DeBiasio and P. McKenney. Density of monochromatic infinite subgraphs. Com-
+binatorica, 39(4):847–878, 2019.
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+[9] F. Dross and F. Havet. On the unavoidability of oriented trees. Electronic Notes in
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+[10] P. Erd˝os. Some set-theoretical properties of graphs. Compositio Math, 2:463–470,
+1935.
+[11] P. Erd˝os and J. W. Moon. On sets of consistent arcs in a tournament. Canadian
+Mathematical Bulletin, 8(3):269–271, 1965.
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+Mathematische Annalen, 325(3):583–623, 2003.
+[13] R. Fra¨ıss´e. Theory of relations, Revised edition, volume 145. North-Holland, Ams-
+terdam, 2000.
+[14] F. Galvin. On a partition theorem of Baumgartner and Hajnal. In Infinite and finite
+sets (Colloq., Keszthely, 1973; dedicated to P. Erd˝os on his 60th birthday), Vol. II,
+Colloq. Math. Soc. J´anos Bolyai, Vol. 10, pages 711–729. North-Holland, Amsterdam,
+1975.
+[15] F. Galvin and S. Shelah. Some counterexamples in the partition calculus. Journal of
+Combinatorial Theory, Series A, 15(2):167–174, 1973.
+[16] F. Havet and S. Thomass´e. Oriented hamiltonian paths in tournaments: A proof of
+Rosenfeld’s conjecture. Journal of Combinatorial Theory, Series B, 78(2):243–273,
+2000.
+[17] D. E. Knuth. The Art of Computer Programming, Volume 3: (2nd Ed.) Sorting and
+Searching. Addison Wesley Longman Publishing Co., Inc., USA, 1998.
+[18] D. K¨uhn, R. Mycroft, and D. Osthus. A proof of Sumner’s universal tournament
+conjecture for large tournaments.
+Electronic Notes in Discrete Mathematics, 38,
+2010.
+[19] A.
+Lamaison.
+Ramsey
+upper density
+of
+infinite graphs.
+arXiv preprint
+arXiv:2003.06329, 2020.
+[20] N. Linial, M. Saks, and V. S´os.
+Largest digraphs contained in all tournaments.
+Combinatorica, 3:101–104, 1983.
+[21] M. Saks and V. S´os. On unavoidable subgraphs of tournaments. Proc. 6th Hung.
+Comb. Coll. Eger, Coll. Math. Soc. J. Bolyai, 37, 1984.
+14
+

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+arXiv:2301.02619v1  [cs.CR]  28 Oct 2022
+Review of Cookie Synchronization Detection Methods
+Jake Smith
+University of California, Davis
+Email: [email protected]
+January 9, 2023
+1
+Abstract
+The research community has deemed cookie syn-
+chronization detection
+an
+inherently
+challenging
+task [1, 2, 3]. Studies aiming to identify cookie syn-
+chronizations often share high-level design choices,
+but
+deviate
+amongst
+low-level
+implementations.
+For example, the majority of studies label a cookie
+synchronization iff a user identifier is shared with a
+third party; however, there is a lack of consistency
+among implementations,
+such as party relations
+or identifier
+value definitions,
+or whether such
+definitions are even included. This review intends to
+provide a record of established methods and promote
+standardization of methods choice in future work.
+CCS
+Concepts:
+Web protocol security;
+Net-
+work privacy and anonymity; Surveillance.
+Keywords: cookie synchronization; cookie match-
+ing; tracking; cookies; methods.
+2
+Introduction
+The sharing of user browsing information is neces-
+sary for the Internet advertising and tracking indus-
+tries to serve targeted ads [4, 5, 6, 7], perform cross-
+device tracking [8], and sell user information [6, 7,
+9]. Browser cookies are a standard container for user
+browsing data, and the sharing of first party cookies
+with third parties is restricted by the Same-Origin
+policy [10] to protect user privacy [9, 11, 12]. Cookie
+synchronization is used to bypass the Same-Origin
+policy and share first party cookies with third par-
+ties to support the advertising and tracking ecosys-
+tem [9, 11, 13]. Cookie synchronization is defined by
+a variety of terms in the research community, such as
+cookie matching, cookie linking, cookie leaking, and
+ID syncing.
+3
+Background
+3.1
+Browser Cookies and User Identi-
+fiers
+Cookies are key=value pairs set on a user’s browser
+to bring state to the HTTP protocol and provide ses-
+sion management, user personalization, and tracking
+functionality.
+Browser cookies can be set by the Set-Cookie
+header of HTTP responses [12,
+14,
+15] or the
+document.cookie operation of JavaScript embedded
+in a visited website [16].
+Cookie synchronization involves the sharing of
+cookie values that can uniquely identify a user (i.e.
+the cookie value is unique to one user). This review
+defines such cookie values as identifiers.
+Methods
+used to define and label identifiers are discussed in
+Section 7.2.
+3.2
+Party Relations
+First party cookies are set by a user requested do-
+main, and third party cookies are set by an entity
+(i.e. domain or parent organization) other than the
+domain requested.
+1
+
+3.3
+Cookie Synchronization
+Cookie synchronization is defined as the sharing of a
+first or third party identifier with another third party,
+which can be initiated by an embedded third party
+resource, third party redirect, or the first party itself
+[13, 17, 9].
+3.4
+How is Cookie Synchronization
+Performed?
+Assume
+a
+user
+is
+browsing
+website1.com and
+website2.com, and there exists tracking entities
+tracker1.com and tracker2.com who both set iden-
+tifiers on the user’s browser, ABC and 123, re-
+spectively.
+The user later visits website3.com,
+which has an embedded resource from tracker1.com
+that initiates a GET request to tracker1.com.
+tracker1.com responds with a 3XX redirect in-
+structing the user’s browser to issue another re-
+quest to tracker2.com,
+with the identifier for
+tracker1.com (ABC), placed in the parameters of the
+requested URL1. tracker2.com is now able to link its
+identifier (123) with tracker1.com’s identifier (ABC)
+[9, 11, 13].
+3.5
+User Privacy Erosion
+Cookie synchronization allows a third party to re-
+construct portions of a user’s browsing history by re-
+ceiving the visited first party site in the Referer field
+of a GET request header [13, 18]. Websites visited
+over TLS are not exempt from this history leakage, as
+plaintext HTTP requests to third parties share URLs
+requested using HTTPS [11, 13, 19].
+As a tracker
+learns more third party identifiers for a single user,
+it can reconstruct a larger portion of her browsing
+history [13].
+Cookie respawning methods such as evercookie
+[20] can enable third parties to re-identify users after
+clearing browser cookies. A respawned identifier can
+be re-synced with a tracker, effectively eliminating a
+user’s ability to delete browser cookies. This enables
+1Additional locations to share identifiers are discussed in
+Section 7.2.
+third parties to track users and join browsing histories
+across browser refreshes [13, 3].
+Server-to-server user data merges are facilitated by
+cookie synchronization. Separate tracker data-sets of
+known user information can be combined by linking
+respective identifiers for each tracker [13, 3].
+3.6
+Cookie Synchronization and the
+Advertising Industry
+Advertising companies are motivated to collect as
+much user information as possible in order to serve
+the most targeted ads; Bashir et.
+al.
+[4] report
+Demand-Side Platforms (DSPs) place higher bids to
+serve users whom they have more information about.
+Cookie synchronization enables this information ac-
+quisition by sharing user browsing data and linking
+tracker databases, which enables ad targeting based
+on web history [13]. Papadopoulos et. al. [13] report
+ad related domains are the most prevalent entities
+involved in cookie synchronization, participating in
+75% of all synchronizations and acquiring as much as
+90% of all identifiers synced.
+4
+Related Work
+As early as 2014, Olejnik et. al. [6] showed how ad-
+vertisers use cookie synchronization in real-time bid-
+ding (RTB) to reconstruct and share browsing his-
+tory.
+HTTP traffic and browser cookies were col-
+lected from 100 real users browsing more than 70 sites
+each. After 70 site visits, a user experienced on av-
+erage 100 cookie synchronizations with 30 domains
+involved.
+Acar
+et.
+al.
+[3]
+investigated
+the
+ef-
+fect Firefox’s privacy settings {Allow Third Party
+Cookies,
+Allow All Cookies but Do Not Track,
+Block Third-Party Cookies} have on the number
+of cookie synchronization a user encounters. Multi-
+ple crawls of the Alexa top 3,000 sites were performed
+with browser cookies logged. When third party cook-
+ies were allowed, 596 identifiers were synced over
+407 unique first parties, with 323 third parties in-
+volved. Selecting Do Not Track only decreased the
+number of domains involved in cookie synchroniza-
+2
+
+tion by 2.9% and identifiers shared by 2.6%. When
+third party cookies were blocked, this decreased the
+number of identifiers synced to 353 over 321 first par-
+ties, with 129 third parties involved. They report 3
+instances of respawned cookies being synced over two
+3,000 site crawls.
+Papadopoulos et. al. [13] investigated the preva-
+lence of cookie synchronization events in mobile web
+traffic. The study collected 850 mobile users HTTP
+traffic for 12 months.
+263,635 cookie synchroniza-
+tions were detected over 179M total requests, with
+22,329 identifiers shared; 91.996% of the shared iden-
+tifiers were located in URL parameters, 3.705% in
+the Referer URL, and 3.771% in the URL path. The
+study reports 5% of identifiers set in TLS sessions be-
+ing leaked over plain HTTP, as well as the websites
+visited in the Referer field.
+Brookman et.
+al.
+[8] examined the extent of
+cross-device tracking visible to an end-user, includ-
+ing cookie synchronizations. The study crawled the
+Alexa top 100 websites four times each. They report
+106 unique third parties syncing identifiers with 210
+other third parties.
+Englehardt et. al. [2] performed an extensive anal-
+ysis of online tracking using their open source crawler,
+OpenWPM. They collected web traffic and browser
+cookies from two crawls of the top 10K Alexa web-
+sites. They report the majority of common third par-
+ties embedded in websites participating in cookie syn-
+chronization: 45 of the top 50, 85 of the top 100, and
+157 of the top 200.
+Papadopoulos et.
+al.
+[11] investigated TLS pri-
+vacy breaches facilitated by cookie synchronization,
+specifically the sharing of websites visited and iden-
+tifiers set over HTTPS. The top 12K Alexa websites
+were crawled, with 440K HTTP(S) requests logged.
+They report 89,479 HTTP(S) syncing requests (i.e.
+HTTP redirects sharing an identifier) occurring from
+32% of the crawled domains; 17,171 unique iden-
+tifiers were shared with 733 unique domains.
+Of
+the 8,398 websites visited over TLS, 2,317 websites
+were involved in cookie synchronization. Most criti-
+cally, these TLS websites conducted 2,879 cookie syn-
+chronizations with non-TLS websites and leaked 174
+HTTPS visits over plaintext.
+They report 1 in 13
+TLS-supported websites performing cookie synchro-
+nization over HTTP.
+Urban et. al. [21] performed a longitudinal study
+documenting the effects of the General Data Pro-
+tection Regulation (GDPR) on cookie synchroniza-
+tion rates in the European Union (EU). 12 measure-
+ments were performed, with one occurring a month
+before the GDPR going into effect (May 2018), and
+the rest performed each month after. Each measure-
+ment instrumented 400 individual browsing profiles
+(i.e. unique browsing instances). The measurements
+each crawled an average of 8.5K domains, totalling
+over 2.5M requests over the year. After the legisla-
+tion’s passing in May 2018, they report an immediate
+drop in the number of cookie synchronizations per
+month (∼510) in relation to the pre-GDPR measure-
+ment (898); a year later, this number decreased to
+∼480 cookie synchronization per month. The number
+of third parties conducting cookie synchronizations
+per month also decreased from ∼12K to ∼10.2K. The
+number of involved third parties per month gradually
+recovered over the year to ∼12K. The study claims
+“cookie synchronization is still used in practice, but
+its extent is significantly reduced and still declining”
+in the EU [21]. This claim is not supported by the
+results of later studies conducted in the EU by Fouad
+et. al. [17] and Papadogiannakis et. al [9].
+Fouad et. al. [17] investigated the role of 1x1 pixel
+images and other embedded content types in initi-
+ating cookie synchronization. They conducted two
+crawls of the Alexa top 10k domains, and successfully
+crawled 8,744 domains. They report 34.36% of track-
+ing was initiated by scripts, 23.34% by pixels, 20.01%
+by text/html, 8.54% by large images, and 4.32% by
+application or JSON. Of the 8,744 websites crawled,
+67.96% were involved in cookie synchronization, with
+17,425 third parties involved. Third party identifiers
+were shared with other third parties in 22.73% of web-
+sites with 1,263 unique partners.
+Sanchez-Rola et. al. [19] conducted a large scale
+crawl of the Tranco top 1M most accessed domains
+list to reconstruct the cookie ecosystem, clarifying
+known roles and defining novel ones involved in the
+creation and sharing of cookies.
+They define the
+ghost cookie, which is created by an embedded third
+party script on a first party website that sets a first
+party cookie.
+The study claims the existence of a
+3
+
+ghosted cookie decreases a first party’s control over
+the cookies their web-page sets on a browser. They
+report 8.97M cookie synchronization across 387K
+websites, with the most common sender and receiver
+relationship (48%) being the own sender to own re-
+ceiver (i.e.
+a first party ghost cookies shared with
+the third party who embedded the script). 52.4% of
+domains experience at least one cookie synchroniza-
+tion or cookie value overwriting event. Reflecting the
+results of Papadopoulos et. al. [11, 13], 37.71% of
+cookies synchronized over HTTP were created in a
+TLS session.
+Papadogiannakis et. al. [9] investigated whether
+third party trackers respect cookie consent banner
+choices {No Action, Reject All Cookies, Accept
+All Cookies}.
+Their data-set was derived from
+the Tranco top 850K sites and successfully crawled
+27,953
+domains
+containing
+a
+Consent
+Manage-
+ment Platform (CMP). They specify two types of
+cookie synchronization relationships.
+They define
+a First-Party ID Leak if a first party identifier
+is shared with a third party, and a Third-Party
+ID Synchronization if a third party identifier is
+shared with a third party. When the user takes No
+Action, 52.88% and 24.03% of websites conduct
+First-Party ID Leaking
+and
+Third-Party ID
+Synchronization, respectively.
+When Rejecting
+All Cookies, 56.41% and 26.20% of websites con-
+duct First-Party ID Leaking and Third-Party
+ID Synchronization, respectively.
+5
+Purpose
+This review intends to document the variety of meth-
+ods employed to detect cookie synchronization. All
+studies under review must log HTTP data and label
+cookie synchronizations from the collected network
+traffic.
+6
+Data-set Collection Methods
+Crawled Data-set: Web crawlers instrumented in-
+clude OpenWPM [8, 12, 21, 17, 2, 1], Chromium-
+based crawlers [4,
+19,
+22,
+23],
+Selenium-based
+crawlers [11, 3, 24], or custom crawlers [9].
+User Data Collection:
+To collect the HTTP
+traffic of real users, study-specific browser plugins are
+installed on a user’s browser [6, 7, 13, 14].
+Henceforth, the term user will refer to the browser
+instance instrumented, regardless of whether the
+study collected crawled or real user data.
+7
+Labeling
+Cookie
+Synchro-
+nizations by Shared Identi-
+fiers
+7.1
+Shared Identifier Heuristic
+The majority of cookie synchronization detection
+methods draw inspiration from the shared identi-
+fier heuristic proposed by Olejnik et. al. [6]. This
+method labels a cookie synchronization iff an identi-
+fier is shared in a HTTP request’s URL parameters
+to an entity other than the entity who set the cookie
+(i.e. a third party) [6, 7, 17, 14]. An entity can be
+defined as either a domain or the parent organization
+of a domain.
+Related methods build on this heuristic by addi-
+tionally extracting identifiers shared with third par-
+ties from the URL path of requests [9, 13, 25],
+Referer URL of requests2 [9, 11, 13, 2, 25, 3], redi-
+rect Location URL [3], nonstandard request and
+redirect headers [12], or POST request bodies [9].
+7.2
+Extracting
+Identifiers
+from
+Browser Cookies
+7.2.1
+What Defines an Identifier?
+A cookie set on a user’s browser is an identifier iff
+the cookie’s value can identify a specific user (i.e. the
+value is mapped to only one user). These identifying
+cookie values and the entities who set them are stored
+to later detect instances of identifiers shared in HTTP
+2As
+of
+November
+2020,
+the
+HTTP
+Referrer-Policy
+default
+directive
+has
+been
+updated
+to
+strict-origin-when-cross-origin to only share the origin
+of a request.
+This prevents identifiers from being shared in
+the path and querystring [26].
+4
+
+traffic.
+This method confirms that a cookie value
+shared with a third party can uniquely identify the
+user who initiated the third party request [7, 8, 9, 11,
+12, 13, 3, 21, 17, 19, 2, 14].
+7.2.2
+Extracting Browser Cookies
+To create the set of all cookies set on a user’s browser,
+cookie values are extracted from the Set-Cookie
+header of HTTP responses [12, 14, 15, 13] or Cookie
+header of HTTP requests [12, 27].
+Solomos
+et.
+al.
+[1]
+use
+OpenWPM’s
+javascript instrument [2] to log cookie values set
+by JavaScript embedded in visited web pages.
+7.2.3
+User Identifier Filtering
+The following restrictions are used to filter identi-
+fier cookie values from the original browser cookie set.
+Value
+Length
+Restrictions:
+Identifiers often
+have minimum length requirements: cookie values
+> 10 characters [6, 7, 13, 25], > 8 characters [12],
+> 7 characters [21, 2, 14], and > 5 characters [9].
+Of studies that provide identifier length restrictions,
+only one provides an upper bound: ≤ 100 characters
+[2].
+Value Character Quality Restrictions:
+Iden-
+tifiers can be extracted based on character values.
+Studies that set character value restrictions only
+extract cookie values consisting of alphanumeric
+characters and other common characters [2, 17, 12].
+Common character values include [-,
+, =], with =
+indicating a key=value pair [2]. Fouad et. al. [17]
+also consider the comma and period and exclude the
+equals sign.
+Delimiter
+Parsing:
+To
+extract
+consecutive
+identifier strings bounded by known characters,
+cookie values can be parsed (i.e. split) at these com-
+mon delimiters. All studies that split consecutively
+shared identifiers consider [&, ;] to be delimiters,
+except Ghosh et.
+al.
+[5] who consider the colon
+rather than semicolon [9, 12, 3, 21, 17, 2, 14, 25, 13].
+Similarity
+Measurement:
+Identifiers
+can
+be
+extracted by uniqueness.
+All studies extracting
+identifiers based on string entropy use the Rat-
+cliff/Obershelp
+Algorithm
+[28]
+with
+a
+provided
+maximum similarity score: eliminate cookie values
+> 66% similar to another cookie value [2], > 33%
+similar [8, 3, 25], or not provided [21].
+Multiple Values Set for a Key=Value Pair:
+Falahraster et. al. [14] and Urban et. al. [21] ex-
+clude any cookie value extracted from a key=value
+pair containing more than one value.
+Key=Value Pairs with Dynamic Values: Cookie
+values can be eliminated if the key’s value changes
+over the course of a crawl or user browsing session
+[3, 2, 25].
+Keyword
+Filtering:
+Papadogiannakis et.
+al.
+[9] use a manually curated list of keywords to elim-
+inate cookie values containing dates, timestamps,
+regions, locale, URLs, prevalent keywords, consent
+information (e.g.
+values of the keys euconsent,
+eupubconsent,
+cmpconsnent,
+cmpiab), or end in
+common file extensions.
+Filtering Non-Unique
+Strings:
+Studies with
+access to multiple cookie data-sets from multiple
+crawls or user browsing sessions can eliminate cookie
+values present for multiple crawls or users [13, 21,
+17, 14].
+Session
+Cookie
+Values:
+Session cookies are
+deleted at the end of a browsing session and their
+values can be eliminated [27]. Studies that eliminate
+session cookies examine the Expires and Max-Age
+attributes [27] and eliminate values associated with
+cookies lacking an expiration date [11, 13] or expire
+earlier than a specified future date: earlier than 90
+days [2] or 30 days [3].
+5
+
+7.3
+Detecting
+Identifiers
+Shared
+in
+HTTP Traffic
+7.3.1
+Labeling Requests to First or Third
+Parties
+Studies that label the party relation of (referrer, re-
+quest) pairs only label identifiers shared in requests
+to third parties [6, 7, 8, 9, 11, 12, 13, 21, 17, 19, 2, 1,
+14, 25].
+Parent Organization Mapping: Domain names
+can be mapped to parent organizations using DNS
+whois records and blacklists [11, 13, 14, 25] or the
+WhoTracks.me database [19, 29].
+To resolve do-
+main names obfuscated by CNAME cloaking [30],
+Sanchez et. al. [19] use the NextDNS blocklist [31]
+to resolve these cloaked domains to known trackers;
+tldExtract [32] is then used to determine the private
+suffix of each domain; private suffixes are mapped
+to parent organizations using the Disconnect [33],
+WhoTracks.me [29], and webxray [34] lists.
+String Matching: Domain name string matching
+is also common, with matches indicating a first party
+and mismatches indicating a third party [6, 7, 8, 9].
+Englehardt et. al. Case Study: Englehardt et.
+al. [2] label request party relations using the Mozilla
+Public Suffix list [35]; iff the landing page’s domain
+name and public suffix (not including subdomains)
+do not match a request’s domain name and public
+suffix, the request is labeled as to a third party.
+7.3.2
+HTTP Identifier Sharing Locations
+The research community has examined the following
+HTTP elements for instances of shared identifiers
+using exact string matching, with matches indicating
+a cookie synchronization.
+HTTP GET Requests: URL query parameters
+[6, 3, 2, 8, 11, 17, 13, 25, 9, 12], URL path3 [3, 2, 8,
+11, 13, 25, 9, 12], Referer URL4 [2, 11, 13, 25, 9, 3],
+3Studies who report examining URLs–without specifying
+which elements–are assumed to check both the path and
+querystring.
+4As
+of
+November
+2020,
+the
+HTTP
+Referrer-Policy
+default
+directive
+has
+been
+updated
+to
+strict-origin-when-cross-origin to only share the origin
+and non-standard headers [12].
+HTTP Redirects: Location URL [3, 2, 25] and
+non-standard headers [12].
+HTTP POST Requests: Request bodies [9].
+7.3.3
+Papadopoulos et. al. Shared Identifier
+Labeling Case Study
+Papadopoulos et. al. [13, 11] implemented a distinct
+method of detecting instances of shared identifiers
+over two cookie synchronization studies.
+Rather than using string matching to label in-
+stances of shared identifiers, they first extract all ID-
+looking strings from GET request URL paths, query
+parameters, and Referer headers.
+An ID-looking
+string is defined by the same qualities used for fil-
+tering identifiers from browser cookies {Section 7.2}.
+The study stores detected ID-looking strings in
+a hashtable with the receiving domain.
+If an ID-
+looking string is seen for the first time in an HTTP
+element, the string is added to the hashtable with the
+requested domain. If an ID-looking string is seen for
+at least the second time, all requests carrying it are
+labeled as an ID-sharing event.
+Cookie synchronizations are labeled from the ID-
+sharing event set; iff an ID-looking string present in
+an ID-sharing event matches a known identifier, the
+ID-sharing event is labeled a cookie synchronization.
+8
+Alternative Cookie Synchro-
+nization Detection Methods
+8.1
+Decision Tree Classifier
+of En-
+crypted
+Identifier
+Synchroniza-
+tion
+Papadopoulos et.
+al.
+[13] trained a decision tree
+model to detect cookie synchronizations of encrypted
+identifiers. The model does not consider the presence
+of a shared, known identifier when classifying cookie
+synchronizations.
+of a request.
+This prevents identifiers from being shared in
+the path and querystring [26].
+6
+
+The study assumes an equal distribution of HTTP
+traffic feature variability between cookie synchro-
+nization of non-encrypted and encrypted identi-
+fiers.
+The training and testing sets were labeled
+by non-encrypted cookie synchronizations detected
+using the study’s shared identifier heuristic.
+The
+features selected include requested entity name,
+type of entity {Content, Social, Advertising,
+Analytics, Other}, URL parameter names, loca-
+tion of hashed identifier {URL parameter, URL
+path, Referer URL parameter}, HTTP status code,
+browser type, and number of parameters.
+8.2
+Labeling Cookie Synchronizations
+in Retargeted Ad Serving Infor-
+mation Flows
+Bashir et. al. [4] collect the resource inclusion chain
+for all websites crawled.
+At a high level, a cookie
+synchronization is labeled iff an auction is held by the
+publisher-side and requests between the Supply-
+Side Platforms (SSP) of the chain directly include a
+resource.
+The
+study
+defines
+the
+following
+terminology.
+Personas are individually created to represent 90
+unique categories of shoppers by browsing specific
+products on e-commerce sites. These categories are
+used to later compare with the qualities of retargeted
+ads for each persona. A publisher-side resource
+chain serves a retargeted ad to a user’s browser. pub
+is the root node’s publisher domain. d is the last en-
+tity in a chain and serves the ad. s denotes a SSP.
+shop is the e-commerce site domain of the retargeted
+ad.
+Cookie synchronizations are labeled iff s and d are
+adjacent at the end of a chain, d observes the persona
+at shop, and a request from s to d (or d to s) is
+present in a chain prior to the retargeted ad being
+served [4].
+8.3
+Labeling
+Tracker
+to
+Tracker
+Cookie
+Synchronizations
+with
+Pre-Existing Data-sets
+Bashir et.
+al.
+[22] and Solomos et.
+al.
+[1] la-
+bel any (tracker, tracker) referrer-request pair as a
+cookie synchronization iff the pair is present on a list
+of known cookie synchronizing third parties [4, 13].
+9
+Acknowledgements
+The author would like to thank Dr. Zubair Shafiq and
+Dr. Katie Rodger for their technical and expository
+insights.
+References
+[1]
+Konstantinos Solomos et al. “Clash of the
+trackers: Measuring the evolution of the on-
+line tracking ecosystem”. In: arXiv preprint
+arXiv:1907.12860 (2019).
+[2]
+Steven Englehardt and Arvind Narayanan.
+“Online tracking: A 1-million-site measurement
+and analysis”. In: Proceedings of the 2016 ACM
+SIGSAC conference on computer and commu-
+nications security. 2016, pp. 1388–1401.
+[3]
+Gunes Acar et al. “The Web Never Forgets”. In:
+Computer and Communications Security, ACM
+(2014), pp. 674–689.
+[4]
+Muhammad Ahmad Bashir et al. “Tracing in-
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+targeted ads”. In: 25th USENIX Security Sym-
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+doi: 10.14722/ndss.2014.23270.
+[7]
+Michalis Pachilakis et al. “YourAdvalue: Mea-
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+bankrupting user privacy”. In: Proceedings of
+the ACM on Measurement and Analysis of
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+tracking in the post-cookie era: How websites
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+pp. 2130–2141.
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+https://www.w3.org/Security/wiki/Same_Origin_Policy.
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+Panagiotis Papadopoulos, Nicolas Kourtellis,
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+the (synced) cookie monster breached my en-
+crypted vpn session”. In: Proceedings of the
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+https://www.usenix.org/conference/usenixsecurity22/presentation/iqbal.
+[13]
+Panagiotis Papadopoulos, Nicolas Kourtellis,
+and Evangelos Markatos. “Cookie synchroniza-
+tion: Everything you always wanted to know
+but were afraid to ask”. In: The World Wide
+Web Conference. 2019, pp. 1432–1442.
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+Marjan Falahrastegar et al. “Tracking personal
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+surement. Springer. 2016, pp. 30–41.
+[15]
+Set-cookie - http: MDN. Aug. 2022. url:
+https://developer.mozilla.org/en-US/docs/Web/HTTP/Headers/Set-Cookie.
+[16]
+Document.cookie
+-
+web
+apis:
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+Sept.
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+url:
+https://developer.mozilla.org/en-US/docs/Web/API/Document/cookie.
+[17]
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+pixels”.
+In:
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+Referer
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+2022.
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+https://developer.mozilla.org/en-US/docs/Web/HTTP/Headers/Referer.
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+[23]
+Eric
+Bidelman.
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+started
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+chrome.
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+[25]
+Pushkal Agarwal et al. “Stop tracking me bro!
+differential tracking of user demographics on
+hyper-partisan websites”. In: Proceedings of
+The Web Conference 2020. 2020, pp. 1479–
+1490.
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+https://developer.mozilla.org/en-US/docs/Web/HTTP/Headers/Referrer-Policy.
+[27]
+Using
+HTTP
+cookies
+-
+http:
+MDN.
+Sept.
+2022.
+url:
+https://developer.mozilla.org/en-US/docs/Web/HTTP/Cookies.
+8
+
+[28]
+Paul
+Black.
+Ratcliff/Obershelp
+pattern
+recognition.
+2021.
+url:
+https://xlinux.nist.gov/dads/HTML/ratcliffObershelp.html.
+[29]
+Bringing transparency to online tracking. url:
+https://whotracks.me/.
+[30]
+Romain Cointepas. CNAME Cloaking, the dan-
+gerous disguise of third-party trackers. 2019.
+[31]
+NextDNS.
+url:
+https://github.com/nextdns.
+[32]
+John-Kurkowski.
+John-Kurkowski/tldextract:
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+separates
+a
+URL’s
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+main,
+domain,
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+ing
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+suffix
+list
+(PSL).
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+https://github.com/john-kurkowski/tldextract.
+[33]
+Disconnectme
+-
+Overview.
+url:
+https://github.com/disconnectme.
+[34]
+Tim Libert. Timlib/webxray: WebXray is a
+tool for analyzing webpage traffic and con-
+tent, extracting legal policies, and identifying
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+https://github.com/timlib/webXray.
+[35]
+Public
+suffix
+list.
+url:
+https://publicsuffix.org/.
+9
+

5NE0T4oBgHgl3EQfvgGE/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='02619v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='CR]  28 Oct 2022  Review of Cookie Synchronization Detection Methods  Jake Smith  University of California, Davis  Email: jsssmit@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='edu  January 9, 2023  1  Abstract  The research community has deemed cookie syn-  chronization detection  an  inherently  challenging  task [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Studies aiming to identify cookie syn-  chronizations often share high-level design choices,  but  deviate  amongst  low-level  implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  For example, the majority of studies label a cookie  synchronization iff a user identifier is shared with a  third party;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' however, there is a lack of consistency  among implementations,  such as party relations  or identifier  value definitions,  or whether such  definitions are even included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' This review intends to  provide a record of established methods and promote  standardization of methods choice in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  CCS  Concepts:  Web protocol security;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Net-  work privacy and anonymity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Keywords: cookie synchronization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' cookie match-  ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' tracking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' cookies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  2  Introduction  The sharing of user browsing information is neces-  sary for the Internet advertising and tracking indus-  tries to serve targeted ads [4, 5, 6, 7], perform cross-  device tracking [8], and sell user information [6, 7,  9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Browser cookies are a standard container for user  browsing data, and the sharing of first party cookies  with third parties is restricted by the Same-Origin  policy [10] to protect user privacy [9, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Cookie  synchronization is used to bypass the Same-Origin  policy and share first party cookies with third par-  ties to support the advertising and tracking ecosys-  tem [9, 11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Cookie synchronization is defined by  a variety of terms in the research community, such as  cookie matching, cookie linking, cookie leaking, and  ID syncing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  3  Background  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='1  Browser Cookies and User Identi-  fiers  Cookies are key=value pairs set on a user’s browser  to bring state to the HTTP protocol and provide ses-  sion management, user personalization, and tracking  functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Browser cookies can be set by the Set-Cookie  header of HTTP responses [12,  14,  15] or the  document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='cookie operation of JavaScript embedded  in a visited website [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Cookie synchronization involves the sharing of  cookie values that can uniquely identify a user (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  the cookie value is unique to one user).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' This review  defines such cookie values as identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Methods  used to define and label identifiers are discussed in  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2  Party Relations  First party cookies are set by a user requested do-  main, and third party cookies are set by an entity  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' domain or parent organization) other than the  domain requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='3  Cookie Synchronization  Cookie synchronization is defined as the sharing of a  first or third party identifier with another third party,  which can be initiated by an embedded third party  resource, third party redirect, or the first party itself  [13, 17, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='4  How is Cookie Synchronization  Performed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Assume  a  user  is  browsing  website1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com and  website2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com, and there exists tracking entities  tracker1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com and tracker2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com who both set iden-  tifiers on the user’s browser, ABC and 123, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  The user later visits website3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com,  which has an embedded resource from tracker1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com  that initiates a GET request to tracker1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  tracker1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com responds with a 3XX redirect in-  structing the user’s browser to issue another re-  quest to tracker2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com,  with the identifier for  tracker1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com (ABC), placed in the parameters of the  requested URL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' tracker2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com is now able to link its  identifier (123) with tracker1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com’s identifier (ABC)  [9, 11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='5  User Privacy Erosion  Cookie synchronization allows a third party to re-  construct portions of a user’s browsing history by re-  ceiving the visited first party site in the Referer field  of a GET request header [13, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Websites visited  over TLS are not exempt from this history leakage, as  plaintext HTTP requests to third parties share URLs  requested using HTTPS [11, 13, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  As a tracker  learns more third party identifiers for a single user,  it can reconstruct a larger portion of her browsing  history [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Cookie respawning methods such as evercookie  [20] can enable third parties to re-identify users after  clearing browser cookies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' A respawned identifier can  be re-synced with a tracker, effectively eliminating a  user’s ability to delete browser cookies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' This enables  1Additional locations to share identifiers are discussed in  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  third parties to track users and join browsing histories  across browser refreshes [13, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Server-to-server user data merges are facilitated by  cookie synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Separate tracker data-sets of  known user information can be combined by linking  respective identifiers for each tracker [13, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='6  Cookie Synchronization and the  Advertising Industry  Advertising companies are motivated to collect as  much user information as possible in order to serve  the most targeted ads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Bashir et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [4] report  Demand-Side Platforms (DSPs) place higher bids to  serve users whom they have more information about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Cookie synchronization enables this information ac-  quisition by sharing user browsing data and linking  tracker databases, which enables ad targeting based  on web history [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Papadopoulos et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [13] report  ad related domains are the most prevalent entities  involved in cookie synchronization, participating in  75% of all synchronizations and acquiring as much as  90% of all identifiers synced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  4  Related Work  As early as 2014, Olejnik et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [6] showed how ad-  vertisers use cookie synchronization in real-time bid-  ding (RTB) to reconstruct and share browsing his-  tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  HTTP traffic and browser cookies were col-  lected from 100 real users browsing more than 70 sites  each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' After 70 site visits, a user experienced on av-  erage 100 cookie synchronizations with 30 domains  involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Acar  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [3]  investigated  the  ef-  fect Firefox’s privacy settings {Allow Third Party  Cookies,  Allow All Cookies but Do Not Track,  Block Third-Party Cookies} have on the number  of cookie synchronization a user encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Multi-  ple crawls of the Alexa top 3,000 sites were performed  with browser cookies logged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' When third party cook-  ies were allowed, 596 identifiers were synced over  407 unique first parties, with 323 third parties in-  volved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Selecting Do Not Track only decreased the  number of domains involved in cookie synchroniza-  2  tion by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='9% and identifiers shared by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' When  third party cookies were blocked, this decreased the  number of identifiers synced to 353 over 321 first par-  ties, with 129 third parties involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' They report 3  instances of respawned cookies being synced over two  3,000 site crawls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Papadopoulos et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [13] investigated the preva-  lence of cookie synchronization events in mobile web  traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' The study collected 850 mobile users HTTP  traffic for 12 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  263,635 cookie synchroniza-  tions were detected over 179M total requests, with  22,329 identifiers shared;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='996% of the shared iden-  tifiers were located in URL parameters, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='705% in  the Referer URL, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='771% in the URL path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' The  study reports 5% of identifiers set in TLS sessions be-  ing leaked over plain HTTP, as well as the websites  visited in the Referer field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Brookman et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [8] examined the extent of  cross-device tracking visible to an end-user, includ-  ing cookie synchronizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' The study crawled the  Alexa top 100 websites four times each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' They report  106 unique third parties syncing identifiers with 210  other third parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Englehardt et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [2] performed an extensive anal-  ysis of online tracking using their open source crawler,  OpenWPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' They collected web traffic and browser  cookies from two crawls of the top 10K Alexa web-  sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' They report the majority of common third par-  ties embedded in websites participating in cookie syn-  chronization: 45 of the top 50, 85 of the top 100, and  157 of the top 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Papadopoulos et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [11] investigated TLS pri-  vacy breaches facilitated by cookie synchronization,  specifically the sharing of websites visited and iden-  tifiers set over HTTPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' The top 12K Alexa websites  were crawled, with 440K HTTP(S) requests logged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  They report 89,479 HTTP(S) syncing requests (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  HTTP redirects sharing an identifier) occurring from  32% of the crawled domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 17,171 unique iden-  tifiers were shared with 733 unique domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Of  the 8,398 websites visited over TLS, 2,317 websites  were involved in cookie synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Most criti-  cally, these TLS websites conducted 2,879 cookie syn-  chronizations with non-TLS websites and leaked 174  HTTPS visits over plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  They report 1 in 13  TLS-supported websites performing cookie synchro-  nization over HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Urban et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [21] performed a longitudinal study  documenting the effects of the General Data Pro-  tection Regulation (GDPR) on cookie synchroniza-  tion rates in the European Union (EU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 12 measure-  ments were performed, with one occurring a month  before the GDPR going into effect (May 2018), and  the rest performed each month after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Each measure-  ment instrumented 400 individual browsing profiles  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' unique browsing instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' The measurements  each crawled an average of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='5K domains, totalling  over 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='5M requests over the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' After the legisla-  tion’s passing in May 2018, they report an immediate  drop in the number of cookie synchronizations per  month (∼510) in relation to the pre-GDPR measure-  ment (898);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' a year later, this number decreased to  ∼480 cookie synchronization per month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' The number  of third parties conducting cookie synchronizations  per month also decreased from ∼12K to ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' The  number of involved third parties per month gradually  recovered over the year to ∼12K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' The study claims  “cookie synchronization is still used in practice, but  its extent is significantly reduced and still declining”  in the EU [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' This claim is not supported by the  results of later studies conducted in the EU by Fouad  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [17] and Papadogiannakis et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Fouad et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [17] investigated the role of 1x1 pixel  images and other embedded content types in initi-  ating cookie synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' They conducted two  crawls of the Alexa top 10k domains, and successfully  crawled 8,744 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' They report 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='36% of track-  ing was initiated by scripts, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='34% by pixels, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='01%  by text/html, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='54% by large images, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='32% by  application or JSON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Of the 8,744 websites crawled,  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='96% were involved in cookie synchronization, with  17,425 third parties involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Third party identifiers  were shared with other third parties in 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='73% of web-  sites with 1,263 unique partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Sanchez-Rola et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [19] conducted a large scale  crawl of the Tranco top 1M most accessed domains  list to reconstruct the cookie ecosystem, clarifying  known roles and defining novel ones involved in the  creation and sharing of cookies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  They define the  ghost cookie, which is created by an embedded third  party script on a first party website that sets a first  party cookie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  The study claims the existence of a  3  ghosted cookie decreases a first party’s control over  the cookies their web-page sets on a browser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' They  report 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='97M cookie synchronization across 387K  websites, with the most common sender and receiver  relationship (48%) being the own sender to own re-  ceiver (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  a first party ghost cookies shared with  the third party who embedded the script).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='4% of  domains experience at least one cookie synchroniza-  tion or cookie value overwriting event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Reflecting the  results of Papadopoulos et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [11, 13], 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='71% of  cookies synchronized over HTTP were created in a  TLS session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Papadogiannakis et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [9] investigated whether  third party trackers respect cookie consent banner  choices {No Action, Reject All Cookies, Accept  All Cookies}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Their data-set was derived from  the Tranco top 850K sites and successfully crawled  27,953  domains  containing  a  Consent  Manage-  ment Platform (CMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' They specify two types of  cookie synchronization relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  They define  a First-Party ID Leak if a first party identifier  is shared with a third party, and a Third-Party  ID Synchronization if a third party identifier is  shared with a third party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' When the user takes No  Action, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='88% and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='03% of websites conduct  First-Party ID Leaking  and  Third-Party ID  Synchronization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  When Rejecting  All Cookies, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='41% and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='20% of websites con-  duct First-Party ID Leaking and Third-Party  ID Synchronization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  5  Purpose  This review intends to document the variety of meth-  ods employed to detect cookie synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' All  studies under review must log HTTP data and label  cookie synchronizations from the collected network  traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  6  Data-set Collection Methods  Crawled Data-set: Web crawlers instrumented in-  clude OpenWPM [8, 12, 21, 17, 2, 1], Chromium-  based crawlers [4,  19,  22,  23],  Selenium-based  crawlers [11, 3, 24], or custom crawlers [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  User Data Collection:  To collect the HTTP  traffic of real users, study-specific browser plugins are  installed on a user’s browser [6, 7, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Henceforth, the term user will refer to the browser  instance instrumented, regardless of whether the  study collected crawled or real user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  7  Labeling  Cookie  Synchro-  nizations by Shared Identi-  fiers  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='1  Shared Identifier Heuristic  The majority of cookie synchronization detection  methods draw inspiration from the shared identi-  fier heuristic proposed by Olejnik et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' This  method labels a cookie synchronization iff an identi-  fier is shared in a HTTP request’s URL parameters  to an entity other than the entity who set the cookie  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' a third party) [6, 7, 17, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' An entity can be  defined as either a domain or the parent organization  of a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Related methods build on this heuristic by addi-  tionally extracting identifiers shared with third par-  ties from the URL path of requests [9, 13, 25],  Referer URL of requests2 [9, 11, 13, 2, 25, 3], redi-  rect Location URL [3], nonstandard request and  redirect headers [12], or POST request bodies [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2  Extracting  Identifiers  from  Browser Cookies  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='1  What Defines an Identifier?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  A cookie set on a user’s browser is an identifier iff  the cookie’s value can identify a specific user (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' the  value is mapped to only one user).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' These identifying  cookie values and the entities who set them are stored  to later detect instances of identifiers shared in HTTP  2As  of  November  2020,  the  HTTP  Referrer-Policy  default  directive  has  been  updated  to  strict-origin-when-cross-origin to only share the origin  of a request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  This prevents identifiers from being shared in  the path and querystring [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  4  traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  This method confirms that a cookie value  shared with a third party can uniquely identify the  user who initiated the third party request [7, 8, 9, 11,  12, 13, 3, 21, 17, 19, 2, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2  Extracting Browser Cookies  To create the set of all cookies set on a user’s browser,  cookie values are extracted from the Set-Cookie  header of HTTP responses [12, 14, 15, 13] or Cookie  header of HTTP requests [12, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Solomos  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [1]  use  OpenWPM’s  javascript instrument [2] to log cookie values set  by JavaScript embedded in visited web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='3  User Identifier Filtering  The following restrictions are used to filter identi-  fier cookie values from the original browser cookie set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Value  Length  Restrictions:  Identifiers often  have minimum length requirements: cookie values  > 10 characters [6, 7, 13, 25], > 8 characters [12],  > 7 characters [21, 2, 14], and > 5 characters [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Of studies that provide identifier length restrictions,  only one provides an upper bound: ≤ 100 characters  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Value Character Quality Restrictions:  Iden-  tifiers can be extracted based on character values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Studies that set character value restrictions only  extract cookie values consisting of alphanumeric  characters and other common characters [2, 17, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Common character values include [-,  , =], with =  indicating a key=value pair [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Fouad et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [17]  also consider the comma and period and exclude the  equals sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Delimiter  Parsing:  To  extract  consecutive  identifier strings bounded by known characters,  cookie values can be parsed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' split) at these com-  mon delimiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' All studies that split consecutively  shared identifiers consider [&, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='] to be delimiters,  except Ghosh et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [5] who consider the colon  rather than semicolon [9, 12, 3, 21, 17, 2, 14, 25, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Similarity  Measurement:  Identifiers  can  be  extracted by uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  All studies extracting  identifiers based on string entropy use the Rat-  cliff/Obershelp  Algorithm  [28]  with  a  provided  maximum similarity score: eliminate cookie values  > 66% similar to another cookie value [2], > 33%  similar [8, 3, 25], or not provided [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Multiple Values Set for a Key=Value Pair:  Falahraster et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [14] and Urban et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [21] ex-  clude any cookie value extracted from a key=value  pair containing more than one value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Key=Value Pairs with Dynamic Values: Cookie  values can be eliminated if the key’s value changes  over the course of a crawl or user browsing session  [3, 2, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Keyword  Filtering:  Papadogiannakis et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [9] use a manually curated list of keywords to elim-  inate cookie values containing dates, timestamps,  regions, locale, URLs, prevalent keywords, consent  information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  values of the keys euconsent,  eupubconsent,  cmpconsnent,  cmpiab), or end in  common file extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Filtering Non-Unique  Strings:  Studies with  access to multiple cookie data-sets from multiple  crawls or user browsing sessions can eliminate cookie  values present for multiple crawls or users [13, 21,  17, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Session  Cookie  Values:  Session cookies are  deleted at the end of a browsing session and their  values can be eliminated [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Studies that eliminate  session cookies examine the Expires and Max-Age  attributes [27] and eliminate values associated with  cookies lacking an expiration date [11, 13] or expire  earlier than a specified future date: earlier than 90  days [2] or 30 days [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='3  Detecting  Identifiers  Shared  in  HTTP Traffic  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='1  Labeling Requests to First or Third  Parties  Studies that label the party relation of (referrer, re-  quest) pairs only label identifiers shared in requests  to third parties [6, 7, 8, 9, 11, 12, 13, 21, 17, 19, 2, 1,  14, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Parent Organization Mapping: Domain names  can be mapped to parent organizations using DNS  whois records and blacklists [11, 13, 14, 25] or the  WhoTracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='me database [19, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  To resolve do-  main names obfuscated by CNAME cloaking [30],  Sanchez et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [19] use the NextDNS blocklist [31]  to resolve these cloaked domains to known trackers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  tldExtract [32] is then used to determine the private  suffix of each domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' private suffixes are mapped  to parent organizations using the Disconnect [33],  WhoTracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='me [29], and webxray [34] lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  String Matching: Domain name string matching  is also common, with matches indicating a first party  and mismatches indicating a third party [6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Englehardt et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Case Study: Englehardt et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [2] label request party relations using the Mozilla  Public Suffix list [35];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' iff the landing page’s domain  name and public suffix (not including subdomains)  do not match a request’s domain name and public  suffix, the request is labeled as to a third party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2  HTTP Identifier Sharing Locations  The research community has examined the following  HTTP elements for instances of shared identifiers  using exact string matching, with matches indicating  a cookie synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  HTTP GET Requests: URL query parameters  [6, 3, 2, 8, 11, 17, 13, 25, 9, 12], URL path3 [3, 2, 8,  11, 13, 25, 9, 12], Referer URL4 [2, 11, 13, 25, 9, 3],  3Studies who report examining URLs–without specifying  which elements–are assumed to check both the path and  querystring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  4As  of  November  2020,  the  HTTP  Referrer-Policy  default  directive  has  been  updated  to  strict-origin-when-cross-origin to only share the origin  and non-standard headers [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  HTTP Redirects: Location URL [3, 2, 25] and  non-standard headers [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  HTTP POST Requests: Request bodies [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='3  Papadopoulos et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Shared Identifier  Labeling Case Study  Papadopoulos et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [13, 11] implemented a distinct  method of detecting instances of shared identifiers  over two cookie synchronization studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Rather than using string matching to label in-  stances of shared identifiers, they first extract all ID-  looking strings from GET request URL paths, query  parameters, and Referer headers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  An ID-looking  string is defined by the same qualities used for fil-  tering identifiers from browser cookies {Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  The study stores detected ID-looking strings in  a hashtable with the receiving domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  If an ID-  looking string is seen for the first time in an HTTP  element, the string is added to the hashtable with the  requested domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' If an ID-looking string is seen for  at least the second time, all requests carrying it are  labeled as an ID-sharing event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Cookie synchronizations are labeled from the ID-  sharing event set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' iff an ID-looking string present in  an ID-sharing event matches a known identifier, the  ID-sharing event is labeled a cookie synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  8  Alternative Cookie Synchro-  nization Detection Methods  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='1  Decision Tree Classifier  of En-  crypted  Identifier  Synchroniza-  tion  Papadopoulos et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [13] trained a decision tree  model to detect cookie synchronizations of encrypted  identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' The model does not consider the presence  of a shared, known identifier when classifying cookie  synchronizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  of a request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  This prevents identifiers from being shared in  the path and querystring [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  6  The study assumes an equal distribution of HTTP  traffic feature variability between cookie synchro-  nization of non-encrypted and encrypted identi-  fiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  The training and testing sets were labeled  by non-encrypted cookie synchronizations detected  using the study’s shared identifier heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  The  features selected include requested entity name,  type of entity {Content, Social, Advertising,  Analytics, Other}, URL parameter names, loca-  tion of hashed identifier {URL parameter, URL  path, Referer URL parameter}, HTTP status code,  browser type, and number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2  Labeling Cookie Synchronizations  in Retargeted Ad Serving Infor-  mation Flows  Bashir et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' [4] collect the resource inclusion chain  for all websites crawled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  At a high level, a cookie  synchronization is labeled iff an auction is held by the  publisher-side and requests between the Supply-  Side Platforms (SSP) of the chain directly include a  resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  The  study  defines  the  following  terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Personas are individually created to represent 90  unique categories of shoppers by browsing specific  products on e-commerce sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' These categories are  used to later compare with the qualities of retargeted  ads for each persona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' A publisher-side resource  chain serves a retargeted ad to a user’s browser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' pub  is the root node’s publisher domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' d is the last en-  tity in a chain and serves the ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' s denotes a SSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  shop is the e-commerce site domain of the retargeted  ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Cookie synchronizations are labeled iff s and d are  adjacent at the end of a chain, d observes the persona  at shop, and a request from s to d (or d to s) is  present in a chain prior to the retargeted ad being  served [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='3  Labeling  Tracker  to  Tracker  Cookie  Synchronizations  with  Pre-Existing Data-sets  Bashir et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [22] and Solomos et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [1] la-  bel any (tracker, tracker) referrer-request pair as a  cookie synchronization iff the pair is present on a list  of known cookie synchronizing third parties [4, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  9  Acknowledgements  The author would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Zubair Shafiq and  Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Katie Rodger for their technical and expository  insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  References  [1]  Konstantinos Solomos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' “Clash of the  trackers: Measuring the evolution of the on-  line tracking ecosystem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' In: arXiv preprint  arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='12860 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [2]  Steven Englehardt and Arvind Narayanan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  “Online tracking: A 1-million-site measurement  and analysis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' In: Proceedings of the 2016 ACM  SIGSAC conference on computer and commu-  nications security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 1388–1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [3]  Gunes Acar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' “The Web Never Forgets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' In:  Computer and Communications Security, ACM  (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 674–689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [4]  Muhammad Ahmad Bashir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' “Tracing in-  formation flows between ad exchanges using re-  targeted ads”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' In: 25th USENIX Security Sym-  posium (USENIX Security 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 481–  496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [5]  Arpita Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' “To match or not to match:  Economics of cookie matching in online adver-  tising”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' In: ACM Transactions on Economics  and Computation (TEAC) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 1–  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  7  [6]  Lukasz Olejnik, Minh-Dung Tran, and Claude  Castelluccia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  “Selling  off  privacy  at  auc-  tion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' In: Proceedings 2014 Network and Dis-  tributed System Security Symposium (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='14722/ndss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='23270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [7]  Michalis Pachilakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' “YourAdvalue: Mea-  suring  advertising  price  dynamics  without  bankrupting user privacy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' In: Proceedings of  the ACM on Measurement and Analysis of  Computing Systems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='3 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 1–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [8]  Justin Brookman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='  Priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Enhancing  Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content=' In: Pro-  ceedings of the Web Conference 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='  [10]  Same  origin  policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  url:  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='w3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='org/Security/wiki/Same_Origin_Policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content=' “Exclusive: How  the (synced) cookie monster breached my en-  crypted vpn session”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' In: Proceedings of the  11th European Workshop on Systems Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='  [12]  Umar  Iqbal  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  “Khaleesi:  Breaker  of  Advertising  and  Tracking  Request  Chains”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  In:  31st  USENIX Security  Sym-  posium  (USENIX  Security  22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  Boston,  MA:  USENIX  Association,  Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  2022,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content=' isbn: 978-1-939133-31-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' url:  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='usenix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='org/conference/usenixsecurity22/presentation/iqbal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [13]  Panagiotis Papadopoulos, Nicolas Kourtellis,  and Evangelos Markatos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' “Cookie synchroniza-  tion: Everything you always wanted to know  but were afraid to ask”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' In: The World Wide  Web Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content=' In: International  conference on passive and active network mea-  surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' url:  https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='mozilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='org/en-US/docs/Web/HTTP/Headers/Set-Cookie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='  Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='  url:  https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='mozilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='org/en-US/docs/Web/API/Document/cookie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='  url:  https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='mozilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='seleniumhq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='gov/dads/HTML/ratcliffObershelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [29]  Bringing transparency to online tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' url:  https://whotracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='me/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [30]  Romain Cointepas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' CNAME Cloaking, the dan-  gerous disguise of third-party trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [31]  NextDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  url:  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com/nextdns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [32]  John-Kurkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  John-Kurkowski/tldextract:  Accurately  separates  a  URL’s  subdo-  main,  domain,  and  public  suffix,  us-  ing  the  public  suffix  list  (PSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  url:  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='com/john-kurkowski/tldextract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [33]  Disconnectme Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  url:  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='  [34]  Tim Libert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content=' Timlib/webxray: WebXray is a  tool for analyzing webpage traffic and con-  tent, extracting legal policies, and identifying  the companies which collect user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+page_content='com/timlib/webXray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  [35]  Public  suffix  list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='  url:  https://publicsuffix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
+page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfvgGE/content/2301.02619v1.pdf'}
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+1 
+ 
+MIS: A Multi-Identifier Management and Resolution System Based on Consortium 
+Blockchain in Metaverse 
+Han Wang1, Hui Li1, Abla Smahi1, Yao Yao1, Shuo-Yen Robert Li2 
+1 School of Electronic and Computer Engineering, Peking University, Shenzhen, 518055, China 
+2 University of Electronic Science & Technology of China, Chengdu, 611731, China 
+As digital resources become diverse in the metaverse, a DNS-like system is necessary for management and resolution. However, the 
+legacy DNS for the TCP/IP architecture was designed with security vulnerabilities and trust risks due to centralized management issues. 
+Although there are several DNS alternatives based on the blockchain, manage only a single type of identifiers or separate identity and 
+other types of identifiers, making it impossible for sub-metaverses to coexist and interconnect. This paper proposes the first-ever 
+consortium blockchain System that simultaneously manages Multiple types of Identifiers (MIS) in the metaverse. We opt for a consortium 
+blockchain due to its efficiency, enhanced security and manageability properties. MIS is first portrayed as a four-tier architecture whereby 
+on-chain data is lightweight and compressed to save on storage while accelerating reading and writing operations. The resource data of 
+sub-metaverses exists heterogeneously in the storage tier to alleviate migration. Then we introduce the key functions of the multi-
+identifiers comprising the registration, resolution, and inter-translation. Finally, MIS is implemented on two testbeds and is accessible 
+online as an open-source system. The first testbed consists of 4 physical servers located in the UK and Malaysia while the second is made 
+up of 200 virtual machines (VMs) spread over 26 countries across all 5 continents on Google Cloud. Experiments indicate that MIS 
+provides efficient reading and writing performance that meets the requirements of traditional legacy DNS. It also shows that the latency 
+of MIS outperforms public blockchain-based workarounds.  
+Additional Keywords and Phrases: Metaverse, Blockchain, Domain Name System, Identifier Management 
+1 INTRODUCTION 
+Neil Stephenson proposed the concept of metaverse in his 1992 science fiction novel named Snow Crash [1]. A computer-
+generated universe that exists side by side with the real world is what is meant when the prefix "meta" is added to the term 
+"universe" [2]. Since its inception, the term metaverse has been described in a variety of ways, such as a second life [3], 
+3D virtual worlds [4], and life-logging [5]. The metaverse is now often seen as completely immersive, hyper spatiotemporal, 
+and self-sustaining virtual shared world [6]. As shown in Figure 1, users of the physical world access the metaverse as 
+avatars in the virtual world. Users can generate content, or mint Non-Fungible Tokens (NFTs), interact with other avatars 
+and link up with digital twins of real objects, virtual objects, applications, and other accessible things. The virtual world is 
+composed of a series of interconnected sub-metaverses [7]. The avatar of the user can access a variety of applications from 
+each sub-metaverse, like gaming, social dating, virtual museums, and concerts. 
+Driven by realistic demand and the development of emerging technologies, the metaverse is recognized as the evolving 
+paradigm for the ongoing development of the next-generation Internet [8], and has attracted global attention. In fact, there 
+are currently more than 160 pertinent projects, and according to Citi’s estimations, this industry can be worth $13 trillion 
+by 2030 [9]. Facebook, Microsoft, Tencent, Nvidia and many other tech behemoths have announced their foray into the 
+metaverse. In particular, Facebook has rebranded itself "meta" and is devoted to building the next large-scale service that 
+can be integrated into metaverse [10]. 
+The metaverse is intended to connect everything in the world, including the digital twins of physical entities and systems, 
+the avatars of users, as well as the vast amounts of data that comes with them. Therefore, the resources in the metaverse 
+include various identities, contents, services and the data that goes with them, making up the key components of the virtual 
+
+2 
+metaverse. We believe that in order to improve the user experience, a DNS-like system for managing and resolving 
+resources plays a vital role as the infrastructure of the metaverse. 
+The DNS was originally widely used to solve the problem that IP addresses on the conventional TCP/IP network 
+architectures are not human-friendly. The legacy DNS maintains the mapping between domain names and IP addresses 
+and provides users with resolution services. The root servers, top-level domain (TLD), and authoritative servers are all 
+arranged in a distributed and hierarchical manner (i.e., from top to bottom) in the traditional DNS. The distributed structure 
+solves the Single Point of Failure (SPoF) problem to some extent, and the hierarchical cache strategy speeds up the 
+resolution process. 
+DNS is a centralized system in terms of root zone administration even if DNS servers are distributed. In other words, 
+DNS servers operate in reference to a central authority. More precisely, the recursive resolution of domain names is 
+ultimately determined by the root zone, which is overseen by the Internet Assigned Numbers Authority (IANA) and the 
+Internet Corporation for Assigned Names and Numbers (ICANN). The legacy DNS, on the other hand, is susceptible to 
+Denial of Service (DDoS) attacks because of its centralized hierarchical structure. Therefore, in order to avoid 
+centralization risks in resolving massive resources, the architecture of DNS alternative system should be decentralized, 
+especially in the metaverse [11]. 
+ 
+Figure 1: The architecture of the metaverse. 
+
+Infrastructure (network, computation, storage, blockchain platform, :..)
+processingandmanagement
+ofresourcedata
+Virtual World
+Sub-Metaverse 1
+Sub-Metaverse 2
+Sub-Metaverse 3
+Content
+Metaverse
+Service
+Avatar
+Interaction
+User3 
+Blockchain, the underlying technology of Bitcoin, is a popular decentralized technology [12]. With the advent of 
+blockchain platforms like Ethereum, blockchain applications have gone far beyond cryptocurrencies. In essence, a 
+blockchain is a distributed ledger. The technology is nowadays impacting a variety of applications such as the Internet of 
+Things (IoT) [13], intelligent manufacturing, supply chains etc.. There is a consensus that the blockchain is one of the core 
+technologies in the metaverse [14]. In regard to blockchain-enabled metaverse applications, the hash-chained blocks and 
+Merkle trees provide cryptographic assurance for identifiers of resources, such as identities, contents, services and IP 
+addresses, in append-only ledgers, preventing the original data records from being tampered with. In addition, consensus 
+protocols help to fairly generate and efficiently deliver ledger entries among multiple entities concurrently, thereby solving 
+the problems related to trusted third parties. Due to the properties of immutability, decentralization, and privacy protection 
+in trustless distributed environments, blockchain can be a promising solution to the DNS alternatives in the metaverse. 
+The first blockchain-based DNS system, Namecoin [15], supports registration, update, and ownership transfers similar 
+to traditional DNS but on a Bitcoin fork. It was designed to be a more general name-value system rather than an alternative 
+to legacy DNS. This was also the first Bitcoin-driven solution to square the Zooko’s triangle [16], long-standing and 
+persisting problem of creating a naming system that is simultaneously secure, decentralized, and human-friendly. On the 
+basis of Namecoin. Blockstack [17] integrates DNS and Public-Key Infrastructure (PKI) and develops the so-called 
+"virtualchain layer" to achieve good portability. The first actual naming system that is based directly on Bitcoin is said to 
+be Blockstack. However, the most functional system, to date, is Ethereum Name Service (ENS) [18]. Unlike Ethereum, 
+ENS places more of an emphasis on name resolution than identity management. Other initiatives, including DNSChain 
+[19] and Emercoin [20], were built on other underlying technologies that were comparable to Namecoin or Blockstack. 
+These initiatives have only enhanced social or economic aspects that are beyond the purview of this paper. 
+All of the aforementioned DNS alternative systems are built on top of public blockchains. However, although Bitcoin, 
+the first-generation public blockchain, is still the largest and most actively maintained blockchain application, many 
+challenges arise when implementing a secure and efficient DNS alternative system in metaverse: 
+ 
+First, the efficiency of public blockchains is low, especially those that adopt non-deterministic consensus such as 
+Nakamoto consensus. Increasing the number of transactions in a block or reducing the difficulty of mining can 
+lead to throughput gains, but also to security vulnerabilities [21]. 
+ 
+Second, small public blockchains are vulnerable to attacks. DNS alternative systems that are based on such 
+blockchains should expand their network scales, or use technologies like virtualchain to switch to large public 
+blockchains. However, even on Bitcoin, attackers who have 25% of the computing power may be able to launch 
+a 51% attack that threatens the system security [22]. 
+ 
+Last but not least, the compliance of identifiers is difficult to censor. The nodes of the public blockchain are 
+permissionless, so it is difficult to perform compliance checks on identifiers and their corresponding resource data 
+during consensus. Similarly, illegal identifiers and data are difficult to seize. 
+The consortium blockchain introduces constrained permission and authorization, weakening the public blockchain's 
+decentralized features while assuring that core nodes are regulated and trusted. The deterministic consensus algorithm is 
+usually used in the consortium blockchain to improve efficiency and reduce energy consumption. It is worth noting that, 
+unlike traditional centralized architectures, the core positions within the consortium blockchain are not administratively 
+specified. Such positions are obtained using the results of an election algorithm. Therefore, in our practice, adopting a 
+consortium blockchain is a promising approach for striking a compromise between efficiency and security. 
+Few works have attempted to integrate DNS with consortium blockchain. For instance, in order to obtain high 
+credibility of domain name resolution results, DNSTSM [23] maintains DNS cache resources on a consortium blockchain. 
+
+4 
+Such systems merely provide incremental improvement on the legacy DNS framework without solving its centralization 
+problem. TD-Root [24] proposed a trustworthy decentralized DNS root management architecture based on a permissioned 
+blockchain [25]. Ho-Kyung Yang et al. [26] also proposed a solution to manage content identifiers in Named Data Network 
+(NDN) environment. 
+On the other hand, with the development of new ecology and application platforms, the metaverse gradually evolves 
+into a human-centric collection of sub-metaverse covering different types of resources [27]. Therefore, identity identifiers 
+that uniquely identify entities are the core of all identifiers. It has a tight bind with blockchain, which supports identity 
+management by default. By establishing trusted digital identity identifiers based on cryptographic means in the metaverse, 
+no matter how the resource data changes, it can be traced back to its associated encrypted address. That is, the only anchor 
+in the metaverse is the identities for all entities. As a result, combining the trusted identity and transactions on the 
+underlying stack, it is feasible to manage multiple types of identifiers (multi-identifiers) in the blockchain-based DNS 
+alternative system in the metaverse. 
+However, the majority of blockchain-based DNS alternative systems manage only a single type of identifiers, such as 
+ENS; or manage multi-identifiers separately. For example, Blockstack creates a new namespace in addition to domain 
+name services to provide PKI and identity management, which is arguably not enough. The relationship between these 
+multi-identifiers and how to manage them cohesively within a system must thus be considered by DNS alternative systems. 
+In this paper, we implemented MIS, a Multi-Identifier System based on the consortium blockchain, and made it publicly 
+available (i.e., open source) [28]. MIS is part of the application layer of Multi-Identifier Network (MIN) architecture, in 
+which the complete addressing and routing protocol stack of multi-identifiers has been developed on the network layer 
+[29]. MIS records a global state of multi-identifiers, including identity, content, service, space, IP address, and domain 
+names. Identity is the unique anchor for the various identifiers. Each individual and organization user (collectively referred 
+to as user in this paper) and network device, only when owning an identity identifier, is allowed to apply for other types of 
+identifiers. These users, for example, can apply for content identifiers that bind data to published media resources. They 
+can also register service identifiers to provide various services. The following is a summary of our contributions: 
+ 
+We are the first to suggest the MIS consortium blockchain, which concurrently manages multiple Identifiers 
+across several sub-metaverses. Similar to DNS, MIS offers registration, update, revocation, and resolution 
+services to metaverse applications for a single type of identifier.  
+ 
+We design unified data structure for different sorts of identifiers in the metaverse by integrating identifier types 
+in both present and future networks. Additionally, we build the identifier space that allows one or more identifier 
+types in order to enable the unified management of multi-identifiers in various sub-metaverses. The relationships 
+of identifier spaces are maintained to realize inter-translation between identifiers. 
+ 
+We present a lightweight scheme for on-chain data, so as to speed up reading and writing the append-only 
+operation logs of multi-identifiers. 
+ 
+To test the functionality and performance of MIS, we implemented MIS on two testbeds. By doing so, we set up 
+two testbeds, one with 80 nodes on four physical servers in the UK and Malaysia, and the other with 200 nodes 
+spread over 26 countries across all five continents using Google Cloud. 
+The remainder of this paper is organized as follows. In Section 2, we introduce related works of the blockchain-based 
+DNS alternatives in the metaverse. Then, the identifier and identifier space in MIS are formally defined in Section 3. We 
+give the design details of MIS in Section 4. Section 5 shows experiment results on two testbeds related to the performance 
+of reading and writing identifiers. Section 6 summarizes the differences between MIS and other existing DNS alternatives. 
+We conclude our work in Section 7. 
+
+5 
+2 RELATED WORK 
+This section outlines the work related to building a blockchain-based DNS alternative system in the metaverse. In this 
+paper, the term identifier means the identification information of various resource data in the network. 
+2.1 Resource Management in Metaverse 
+Resources such as identities, contents and services are generated and exchanged in the metaverse to form a dynamic virtual 
+world. For example, users or avatars trade land parcels and equipment in the decentralized virtual world called 
+Decentraland [30]. They can also construct their own buildings and social games. In Cryptovoxels [31], users or avatars 
+are able to purchase land and build virtual shops and art galleries, where digital assets are displayed and traded. Sandbox 
+[32] developed a blockchain-based game system that maintains users' ownership of digital lands and contents. In these 
+projects, transaction details of resource data are recorded on the Ethereum blockchain to guarantee belonging identities. 
+As a result, the reliable identity and resource data are essential to the continued existence of virtual worlds. 
+2.1.1 Management of Identity Identifiers 
+In the metaverse, a class of objects is mapped from the real world to the virtual world. For example, Haihan Duan et al. 
+[33] implemented the CUHKSZ Metaverse, a virtual copy of the physical campus of the Chinese University of Hong Kong, 
+Shenzhen. The consortium blockchain FISCO-BCOS is adopted as the underlying framework due to the low cost and 
+regulability. It is necessary to identify the objects so that they can be discovered, addressed and accessed on the network. 
+The other class of objects is produced natively in the virtual world, which also requires identity identifiers for management. 
+At present, identifiers are commonly adopted in the IoT field, including Bar Code, Quick Response (QR) Code, Radio 
+Frequency Identification (RFID), etc. Considering the inherent characteristics of immersion and interoperability in the 
+metaverse, identity identifiers can also combine biological, spatio-temporal and other attribute information [34]. 
+Aleksandar Jovanović et al. [35] designed a multi-dimension authentication scheme based on facial recognition, fingerprint, 
+voice recognition, message, and mail. An access control system was then implemented in the metaverse platform VoRtex 
+for identity management. The identity data, however, was located on a cloud server with centralization risks. Zijian Bao et 
+al. [36] proposed a three-layer IoT security architecture based on blockchain to support functions of identity authentication, 
+access control and privacy protection, but with long read and write time. Mohammed Amine Bouras et al. [37] further 
+improved the efficiency of identity management in IoT systems and implemented it on the Hyperledger Fabric platform. 
+Yongjun Song et al. [38] proposed a smart city standard model that provides identity management services based on 
+blockchain, but this is only a rudimentary mode that lacks technical details. 
+2.1.2 Data Storage and Management 
+A natural problem in the metaverse is how to protect data of users and avatars [39]. If ownership and provenance are not 
+determined, the data will not be protected as digital assets. On the other hand, the virtual world is essentially an enormous 
+amount of resource data. Due to the limited network ability, it is impossible to manage such giant data in a centralized 
+cloud server [40]. InterPlanetary File System (IPFS) [41] is a well-known decentralized data management solution. Each 
+data file is uniquely identified as content for addressing, preventing the same data files from being uploaded to the network. 
+Based on IPFS and blockchain cloud storage, GooDData developed an efficient and domain-aware decentralized storage 
+solution called GooDData File System (GDFS) [42]. However, the above solutions back up data multiple times. At the 
+macro level, storage costs have not decreased, and have even increased in total. Storj Decentralized Cloud Storage (DCS) 
+
+6 
+[43], a blockchain-based data storage platform, uses erasure codes to store data, thereby reducing the overall storage space. 
+If a data chunk is unusable, modified or inaccessible, it can be reconstructed with other available chunks. 
+In order to save on-chain costs and improve scalability, the data storage of the metaverse system usually uses a 
+combination of on-chain and off-chain solutions. Blockchain nodes only store the necessary metadata and the index of off-
+chain data, through which the associated data can be quickly located. Solutions for off-chain storage include centralized 
+data centers, decentralized cloud storage and edge storage, IPFS, etc. CryptoPunks stores the hash value of the metadata 
+and media data on the blockchain, as well as integrated NFT pictures on a centralized server [44]. The integrity and 
+reliability of pictures can be verified by comparing their hash values. However, the centralized server exposes this content 
+resource, that is, the NFT pictures, to security threats such as loss and DDoS attacks. In Decentraland, ownership and other 
+tradable information of NFTs are written on the Ethereum blockchain. At the same time, the user location and scenario 
+status for real-time interaction are stored on off-chain end devices or non-core edge servers [45]. Similarly, in Sandbox, 
+the transaction data of digital assets is stored on the Ethereum blockchain, while the associated media data and uncasted 
+digital assets are on IPFS and Amazon's S3 cloud servers, respectively [33]. 
+From a data perspective, the above sub-metaverse projects balance cost, performance and reliability on their own. 
+Although the metadata of the resources are all stored on the Ethereum blockchain, it is difficult for these projects to 
+communicate due to their bidding relationships. In this paper, we propose MIS to manage resource data, which integrates 
+multiple sub-metaverses with different types of identifiers on the blockchain. In addition, the complete resource data is 
+still heterogeneously kept off-chain, which helps to solve the problem of sub-metaverses’ data migration. 
+2.2 Blockchain-based DNS Alternatives 
+Blockchains are divided into public, private and consortium blockchains. This categorization is based on technical 
+characteristics of complete decentralization, centralization and partial decentralization/multi-decentralization, respectively. 
+Current research on blockchain-based DNS alternative systems is mainly considering public and consortium blockchains. 
+2.2.1 Solutions Based on Public Blockchains 
+In DNS alternative systems, Bitcoin is now the most popular underlying public blockchain. On a fork of Bitcoin, Namecoin 
+[15] enables name-value storage and resolution. Therefore, a name can only be 64 characters long at most in order to 
+prevent the blockchain from growing rapidly. Instead of utilizing Namecoin, Blockstack [17] implements a virtualchain 
+on top of the Bitcoin main chain to expand storage capacity and improve security. D3NS [46] integrates a distributed hash 
+table and domain name ownership implementation based on Bitcoin. All of the aforementioned DNS alternative systems 
+adopted Nakamoto consensus and hence, they inherited a slow and expensive registration process. HandShake [47] adopts 
+Bitcoin-NG [21] to improve throughput. 
+Since adding new features to Bitcoin is challenging [17], Ethereum has been introduced as a programmable and flexible 
+solution that can extend new functions directly via smart contracts. ENS [18] is an extended smart contracts-enabled on-
+chain DNS mapping human-readable names to machine-readable Ethereum addresses. Stacks 2.0 [48] also incorporates 
+general-purpose smart contracts into its original Blockstack, to overcome some of Bitcoin’s limitations. Additionally, 
+BlockZone’s [49] improved Practical Byzantine Fault Tolerance (PBFT) [50] consensus mechanism provides more 
+efficient name resolution. 
+
+7 
+2.2.2 Solutions Based on Consortium Blockchains 
+Researchers are motivated to work on the cache security of the legacy DNS since the core nodes of the consortium 
+blockchain are regarded as reliable. DNSTSM [23] implements a secure DNS cache to prevent cache poisoning attacks. It 
+also introduces identity-based access control to prevent potential threats from unknown nodes. Likewise, BlockDND [51], 
+DecDNS [52], and BlockONS [53] suggested generating a unique hash of the original data and storing it on the blockchain 
+to avert cache poisoning. TD-Root [24] introduced a trust value and penalty mechanism to eliminate the risks of cache 
+poisoning and trust risks. 
+A small number of studies concentrated on decentralizing the domain name servers leveraging consortium blockchain 
+in addition to incrementally optimizing the classic DNS cache. Yantao Shen et al. [54] provide DNS service on a 
+permissioned blockchain by building a TLDChain to save computing power. ConsortiumDNS [55] introduces a domain 
+name service based on a hierarchical consortium blockchain. To alleviate storage limitations of blockchain, a 3-tier 
+architecture was designed to separate the data and operation of domain name transactions. However, ConsortiumDNS 
+showed a very low throughput urging the need for a consensus algorithm with less data and higher efficiency, such as 
+Hotstuff [56] or Parallel Proof of Vote (PPoV) [57]. 
+3 FORMAL DEFINITIONS 
+Before designing the architecture of MIS, it is important to first define the unified form of MIS identifiers. Then, the formal 
+definition of the identifier space is given.  
+3.1 MIS Identifier 
+According to current metaverse applications, the main resources are identity, content, service, space, and their associated 
+data. We analyze the identifier types in traditional and future networks. Just as IPv4 address works at the network layer of 
+traditional network architectures, identity, content, service, and space identifiers are meaningful in future networks [58-
+61]. In addition, the domain name is also considered as an identifier type in MIS. We define the MIS identifier as follows. 
+Definition 1: (Identifier). The identifier 𝑖𝑗 = 𝑡𝑦𝑝𝑒𝑗: 𝑖𝑑𝑒𝑛𝑡𝑖𝑓𝑒𝑟_𝑛𝑎𝑚𝑒 of a user or device is defined as a combination 
+of two string, where 𝑡𝑦𝑝𝑒𝑗(𝑗 = 0,1,2, … , 𝑘)  represents the type of identifiers. The actual identifier is written in 
+𝑖𝑑𝑒𝑛𝑡𝑖𝑓𝑖𝑒𝑟_𝑛𝑎𝑚𝑒. 
+Table 1 compares the form of identifiers in their original network architectures with MIS, using random examples. 
+Table 1: Examples of MIS identifiers. 
+Identifier Type 
+Example 
+Original 
+MIS 
+Identity 
+e98a32e6175bbd375 
+type0:e98a32e6175bbd375 
+Content 
+/metaverse_sub1/002.mp4 
+type1:/metaverse_sub1/002.mp4 
+Service 
+/metaverse_sub2/web 
+type2:/metaverse_sub2/web 
+Geographical Location 
+(113.97,22.59) 
+type3:(113.97,22.59) 
+Hyperbolic Coordinate 
+(ln2,5π/6) 
+type4:(ln2,5π/6) 
+IPv4 Address 
+192.168.234.1 
+type5:192.168.234.1 
+Domain Name 
+metaverse.sub3.com 
+type6:metaverse.sub3.com 
+
+8 
+3.2 Identifier Space 
+We construct an identifier space by considering identifiers and nodes. The formal definition of identifier space is given in 
+this section. 
+Definition 2: (Identifier Type Set). 𝐼 = {𝑖0, 𝑖1, 𝑖2, … } represents the set of all the identifier types in the network. Where, 
+𝑖0 is the identity of users or devices, which is the most basic and indispensable identifier. {𝑖1, 𝑖2, … } optionally refer to 
+multi-identifiers such as content 𝑖1, service 𝑖2, geographic location 𝑖3, and IPv4 address 𝑖5. In particular, 𝐼𝑘 is a subset of 𝐼 
+and if we take k=2 as an example, then 𝐼2 = {𝑖0, 𝑖1}. 
+Definition 3: (Node Set). 𝑉 is the set of all nodes in the network, including end devices, hosts, routers, and switches. 
+Definition 4: (Identifier Space). 𝐶𝐼𝑘 = (𝐼𝑘, 𝑉𝐼𝑘) is a 2-tuple. 𝐼𝑘 represents the identifier type set in 𝐶𝐼𝑘, and 𝑉𝐼𝑘 is a 
+subset of node set 𝑉. 𝐶𝐼𝑘 constitutes an identifier space when the following conditions are met: 
+ (1) The identifiers and nodes in 𝐶𝐼𝑘 are defined in the network. That is, 𝐼𝑘 ⊆ 𝐼 and 𝑉𝐼𝑘 ⊆ 𝑉. 
+ (2) The identifier types set in 𝐶𝐼𝑘 must contain the identity. That is, 𝑖0 ∈ 𝐼𝑘. 
+ (3) All nodes in 𝐶𝐼𝑘 own all the identifier types it contains. That is, ∀𝑣 ∈ 𝑉𝐼𝑘 and 𝑖𝑗 ∈ 𝐼𝑘, 𝑣 owns 𝑖𝑗. 
+ (4) Any node owning all the identifier types in the set 𝐼𝑘 belongs to 𝐶𝐼𝑘. That is, if ∃𝑣 ∈ 𝑉 and ∀𝑖𝑗 ∈ 𝐼𝑘, 𝑣 owns 𝑖𝑗, 
+then 𝑣 ∈ 𝑉𝐼𝑘. 
+Figure 2 illustrates graphically how the identifier space is divided in the network. each circle indicates the node and its 
+identifier types. Identity (𝑖0) is the basic identifier that all nodes own, so the entire network is the identity identifier space.  
+ 
+Figure 2: A schematic diagram of identifier space division. 
+
+io,i1
+io,
+Content
+i2,i3
+io,i1
+io,i1
+Identifier Space
+io,i2
+io,
+io,i1
+1
+i2,i3
+io,i1,
+Geographic
+i2,is
+io,
+io,i1,
+Location
+io,i1
+1
+io,is
+i2,i3
+i2,i5
+Identifier
+IPv4 Address
+11
+Identifier Space
+Space
+io,ji2
+io,
+io,i1
+io,i1
+i2,i3
+10?
+10?
+i2,i5
+io,is
+i2,i5
+io,i2
+io,i2
+io
+io,i2
+Service
+io,
+Identifier Space
+i2,i3
+io
+io,i2
+io,i2
+L
+Identity Identifier Space9 
+4 DESIGN OF MIS 
+For the conditions of diverse sub-metaverses to coexist, MIS realizes the simultaneous management of identifiers and their 
+associated data from different sources on a consortium blockchain. This section describes the 4-tier architecture of MIS in 
+details. The managing and resolving functions of multi-identifiers are also implemented. 
+4.1 MIS Tiers 
+MIS is designed based on the consortium blockchain to essentially maintain a global state of multi-identifiers and the 
+associated resource data among distributed nodes. As depicted in Figure 3, MIS has a 4-tier architecture. Candidate nodes 
+compete for the privilege related to the block generation process to become core nodes. The lightweight consensus 
+algorithm is run among core nodes, logging state changes of resources in the form of transactions. Meanwhile, the metadata 
+and complete data of resources are indexed hierarchically and eventually distributed off-chain. Such architecture provides 
+good extensibility, and a loosely-coupled relationship between the different tiers. Therefore, at some point in the future, it 
+will be feasible to upgrade only one tier without altering the the operating logic of the other tiers. 
+ 
+Figure 3: MIS’s 4-tier architecture. 
+4.1.1 Tier 1: Network Tier for Blockchain Nodes 
+Consortium blockchain nodes occupy the lowest tier. Nodes responsible for managing the identifiers at the same level 
+constitute the core network. There are three rights in it, which require different processing of messages and can be permuted 
+and combined arbitrarily. That is, a core node has one or more rights. 
+
+Sub-Metaverse 1
+Sub-Metaverse 2
+Sub-Metaverse 3
+Tier 4:
+Metadata Servers
+Storage Servers
+Storage Tier
+(4) Write metadata
+with hash(X+i) as
+address, where iis
+MIS identifier
+(3) Write resource data and return metadata
+MIS
+Username Table
+MIS Identifier Table
+Processor(2)Write global state tables
+X
+Identity Digest of resource data
+Metadata for User
+/hash(X+ldentity)
+Information Table
+Tier 3:
+Index Tier
+x
+(1) Read operations in transactions
+Y
+OP
+OP
+COLD
+h
+h.+l
+WARM
+w+1
+HOT
+None blocks are cached by nodes.
+I Blocks are cached by some nodes.
+Blocks are cached by all nodes.
+Tier 2:
+Lightweight
+Consensus Tier
+block body
+encode
+data chunk 1
+parity chunk 1
+Each node stores a chunk.
+None nodes store chunks.
+data chunk 2
+parity chunk 2
+io,is
+io,is
+Tier 1:
+Network Tier for
+io,1s
+Blockchain Nodes
+io,is
+iosis10 
+(1) Bookkeeping: The bookkeeper node has the right to write transactions. The bookkeeper receives the transaction 
+proposals and saves them into its local pool. After starting consensus round, the bookkeeper selects part of the transactions 
+as a set 𝑇𝑥_𝐿𝑖𝑠𝑡 = {𝑡1, 𝑡2, ⋯ , 𝑡𝑡𝑜𝑡𝑎𝑙} and, together with the block height ℎ, timestamp 𝑇𝑠, and the hash of the previous 
+block 𝑃𝑟𝑒_𝐻𝑎𝑠ℎ, it generates the signed message 〈𝑃𝑟𝑒𝑝𝑎𝑟𝑒 − 𝑇𝑥_𝐿𝑖𝑠𝑡〉. The prepare message 〈𝑃𝑟𝑒𝑝𝑎𝑟𝑒 − 𝑇𝑥_𝐿𝑖𝑠𝑡〉 is 
+then broadcasted to the consortium blockchain network for collective voting. If approved, the set of transactions 𝑇𝑥_𝐿𝑖𝑠𝑡 
+will be appended in the highest block. 
+(2) Voting: The voter node has the right to vote. The voter votes on the transaction sets 𝑇𝑥_𝐿𝑖𝑠𝑡𝑠 from bookkeepers in 
+the consensus round. If some bookkeepers are malicious or crashed, the voter cannot collect all the sets. In this case, it 
+needs to wait for a threshold time before voting partially on the received sets. Specifically, the voter checks them one by 
+one, voting 1 for passed, −1 for failed, and 0 for unreceived. Further, the set of votes 𝑉𝑜𝑡𝑒_𝐿𝑖𝑠𝑡 of all the received 
+transaction sets is compressed into a long integer and signed, and is then sent to the aggregator together with the digests 
+𝑇𝑋_𝐻𝑎𝑠ℎ as the vote message 〈𝑉𝑜𝑡𝑒〉.  
+(3) Aggregating: The aggregator node has the right to count votes. The aggregator collects and aggregates long integer 
+𝑉𝑜𝑡𝑒_𝐿𝑖𝑠𝑡𝑠 from voters. Considering that both bookkeepers and voters are Byzantine, the aggregator similarly has a finite 
+wait. As soon as the threshold time is reached, the aggregator will start to count the received votes as 𝑉𝑜𝑡𝑒_𝑟𝑒𝑠𝑢𝑙𝑡. The 
+transaction set that gets approval or disapproval from more than 2/3 of all voters is considered to be determinably valid or 
+invalid. Otherwise, the aggregator will request missing votes. In addition, since all the original votes are signed by voters, 
+the aggregator compress the votes and signatures (Section 4.1.2). The commit message 〈𝐶𝑜𝑚𝑚𝑖𝑡〉, including 𝑉𝑜𝑡𝑒_𝑟𝑒𝑠𝑢𝑙𝑡, 
+compressed votes and aggregate signature, is then generated and broadcasted across the network. 
+Aggregation is the most important right of nodes in the network. Because only one node acts as the aggregator in a 
+consensus round, the system services will be seriously affected if the aggregator fails. Therefore, malicious nodes should 
+be prevented from having this privilege as much as possible. In our current implementation, the aggregator is periodically 
+chosen among good voters. 
+(4) No right: The ordinary node in the consortium blockchain network without rights do not participate in the consensus 
+process. This node has however the ability to request or receive the transaction set and the block header in a way similar 
+to the bookkeeper and the voter. When receiving a transaction set 𝑇𝑥_𝐿𝑖𝑠𝑡 from a bookkeeper, nodes need to check the 
+identifiers in it, as well as the bookkeeping right and the signature of the bookkeeper. When receiving the commit message 
+〈𝐶𝑜𝑚𝑚𝑖𝑡〉 from the aggregator, nodes should check whether all the transaction sets 𝑇𝑥_𝐿𝑖𝑠𝑡𝑠 mentioned in the block 
+header 𝐵𝐻 have been received. The missing sets will be obtained by requesting them from the neighbors. Finally, they 
+assemble the complete block, and store it based on the timeline strategy (Section 4.1.2). 
+4.1.2 Tier 2: Lightweight Consensus Tier 
+The lightweight consensus tier is located above the network tier for consortium blockchain nodes. MIS uses PPoV [57], 
+an efficient Byzantine Fault Tolerance (BFT) consensus algorithm, for writing and verifying operation transactions such 
+as registration, update, revocation, validity period extension, and ownership transfer to the blockchain. The consensus 
+process is described in Section 4.1.1. The advantage of using PPoV in a DNS alternative system is that blockchains are 
+increasingly seen as the underlying communication channel for announcing state changes. The consensus can only serve 
+for instrumental ordering. PPoV consensus separates a voting right to enable voters to vote for transactions in the 
+blockchain according to their own rules. As a result, the consensus results will be more reasonable. 
+
+11 
+However, all votes and signatures are stored as proof in the block header, thus consuming a large amount of storage 
+space. Another consideration is that each bookkeeper signs its generated transaction set. We lightweight the PPoV 
+consensus algorithm in three aspects. 
+The first is, to solve the problem of large signatures, we use the Boneh-Lynn-Shacham (BLS) signature scheme [62] 
+instead of the Elliptic Curve Digital Signature Algorithm (ECDSA) to aggregate signatures. Another optimization is to 
+redesign the structure of PPoV blocks, as shown in Figure 4, to reduce the size of on-chain data.  
+ 
+Block Header 
+Height 
+Pre-hash 
+Timestamp 
+Vote_result 
+Votes 
+Voters’ Aggregate Signature 
+Bookkeepers’ 
+Aggregate 
+Signature 
+Merkle Root_0+Tamestamp_0 
+… 
+Merkle Root_h+Tamestamp_h 
+Block Body 
+Tx_Lists{0,…,h} 
+Figure 4: Redesigned lightweight block structure of PPoV. 
+The third is to reduce redundancy. Generally speaking, each node has to store the full data, which causes storage stress 
+for the nodes. However, the frequency of querying old blocks is low. Assume that there are 𝑛 nodes in the consortium 
+blockchain, no more than 𝑓 of which are Byzantine. Given that data in the block header is accessed frequently by the nodes, 
+we only compress actual transactions in the block body using (𝑛 − 2𝑓, 2𝑓) Reed-Solomom (RS) code [63], an erasure 
+code. It has the advantage of recovering all the original data chunks with only (𝑛 − 2𝑓) chunks, which is suitable for 
+blockchain with Byzantine nodes.  
+Since transactions in the newly generated block are frequently accessed, compression of this part of the data will lead 
+to low query efficiency. Therefore, we propose a timeline strategy for lightweight storage. All blocks are divided by heights 
+into hot, warm, and cold states, with the newest block being hot and the oldest being cold. 
+(1) Hot blocks are frequently accessed, so each node should cache a copy. Let us take the length of hot space as 𝐿ℎ and 
+the latest block height as ℎ𝑚𝑎𝑥, then the blocks with the heights in the range of (ℎ𝑚𝑎𝑥 − 𝐿ℎ, ℎ𝑚𝑎𝑥] are in a hot state. 
+(2) Warm blocks are between hot and cold blocks, and they have a certain probability of being accessed. Let the length 
+of warm space be 𝐿𝑤, so that blocks with heights in the range of (ℎ𝑚𝑎𝑥 − 𝐿ℎ − 𝐿𝑤, ℎ𝑚𝑎𝑥 − 𝐿ℎ] are in a warm state. When 
+the state of a block changes from hot to warm, each node first encodes the warm block to (𝑛 − 2𝑓) data chunks {𝐶}𝑖=1
+𝑛−2𝑓 
+and 2𝑓 parity chunks {𝐶}𝑖=𝑛−2𝑓+1
+𝑛
+. Then, the block header 𝐵𝐻 and one of the chunks are uniquely stored by the node. If 
+the chunk fails, the node can request other arbitrary (𝑛 − 2𝑓) chunks to decode to the original block.  
+In order to reduce the probability of decoding, we assign some nodes to cache the warm blocks. Suppose a block is 
+cached on 𝑛𝑤 nodes. The decoding process is triggered only when all the 𝑛𝑤 caches are down.  
+We believe that the probability of a warm block being accessed decreases over time, and therefore the number of caches 
+should decrease. The latest warm block is cached on (2𝑓 + 1) nodes, because the number of failed caches is no more than 
+2𝑓, and transactions can be queried without decoding. The oldest warm block is cached only on one node to ensure that it 
+does not take up extra storage space. The relationship between cache quantity 𝑛𝑤 and block height ℎ is that 𝑛𝑤 (ℎ) =
+ ⌊(ℎ − ℎ𝑚𝑎𝑥 + 𝐿ℎ + 𝐿𝑤)  
+𝐿𝑤
+2𝑓
+⁄
+⌋ +  1. 
+(3) Cold blocks have a low probability of being accessed, and their heights belong to the range [0, ℎ𝑚𝑎𝑥 − 𝐿ℎ − 𝐿𝑤]. 
+At this stage, nodes only hold block headers and chunks generated in the warm space and remove caches. 
+
+12 
+4.1.3 Tier 3: Index Tier 
+The index layer maintains the global state for MIS, including who owns which identifiers and how to find off-chain 
+resource data associated with these identifiers. 
+An object in the metaverse owns multiple identifiers, which are dynamically added, updated or deleted as the network 
+changes. We use a non-relational database that holds a username table and multiple MIS identifier tables by key-value 
+pairs. A username record corresponds to a MIS identifier table, including the binding records of multi-identifiers (only 
+identity is mandatory) as well as hashes for the username and identifier (as the metadata address for associated resources). 
+Using the user's identity identifier as an example, the user information table stores data associated with such an identifier. 
+The MIS processor first finds the identity registration operation of username 𝑋 in the blockchain. After that, it adds a 
+username record 𝑋 to the username table and creates a new MIS identifier table of 𝑋 with identifier record 𝐼𝑑𝑒𝑛𝑡𝑖𝑡𝑦. It 
+also stores the address of metadata as ℎ𝑎𝑠ℎ(𝑋 + 𝐼𝑑𝑒𝑛𝑡𝑖𝑡𝑦) and the digest information of resource data. Then the MIS 
+processor writes the user information table to the storage servers and the metadata into the metadata servers (Section 4.1.4), 
+so that the data separation of identity identifier and associated data can be achieved for fast indexing and verifying. Note 
+that for resource data that already exists in the sub-metaverse, only the metadata needs to be migrated. 
+In MIS, each object must have an identity identifier. An object’s identity is uniquely identified in the virtual world of 
+metaverse so that this identity plays the role of "passport number". In addition, for other types of identifiers, identity serves 
+as a unified base identifier and an anchor point. 
+4.1.4 Tier 4: Storage Tier 
+The storage tier is the top-most, which stores the complete off-chain resource data associated with multi-identifiers. 
+Different storage systems, devices and replica strategies are applicable. For resources of different systems and 
+characteristics, the MIS processor stores and manages metadata uniformly in distributed metadata servers. The metadata 
+contains the storage location of the actual resources of data or services and some verification information. In this storage 
+mode, users are able to read off-chain metadata based on the address ℎ𝑎𝑠ℎ(𝑈𝑠𝑒𝑟𝑛𝑎𝑚𝑒 + 𝐼𝑑𝑒𝑛𝑡𝑖𝑓𝑖𝑒𝑟) to find all resources, 
+and verify it through the digest data in the global state tables. For large data resources, data blocks will be read 
+simultaneously to improve efficiency. In addition, the metadata servers should have all types of identifiers so that all users 
+are guaranteed access to it. 
+4.2 Multi-Identifier Management and Resolution System 
+MIS uses its 4 tiers to implement a complete DNS alternative system of multi-identifiers. Its main functions are registration, 
+update and revocation. Resolution of a single type of identifiers and inter-translation across identification spaces are also 
+supported. In addition, MIS is compatible with the legacy DNS. 
+4.2.1 Registration, Update, and Revocation 
+Identity is the basic identifier in MIS. The identity of the device is assigned, while the user needs to register. Before 
+registering an identity, the user first generates a pair of keys with appropriate strength. To register a public key as an 
+identity, a user initiates an identity registration request, which includes the username, identity, Time-to-Live (TTL), 
+registration fee, timestamp, signature, and other information listed in Table 2. It is then submitted to the bookkeeper of the 
+identifier space. The user can select any bookkeeper for such identifier type, because the entire network is an identity space. 
+In view of cybersquatting, an auction method is applied. When the bookkeeper receives the request, it checks the format 
+and content, including but not limited to checking whether duplicate valid identifiers already exist in the global state tables. 
+
+13 
+If successful, the bookkeeper will preferentially encapsulate requests with high registration fees into transactions and put 
+them into the local pool for lightweight consensus. When the consensus is passed, the MIS processor will extract usernames 
+and other required fields of newly valid identities from registration transactions and write the global state tables. If the 
+AboutMe field is also filled out, it is also extracted and stored as an off-chain user information table. 
+After the user get an identity successfully, it can register other multi-identifiers, such as content and service. For the 
+identifiers within the validity period, only the owner can be allowed to operate on it, for example, update or revoke. Expired 
+or revoked identifiers can also be re-registered. The operation process is similar to identity registration. It is noted that all 
+operations are performed only after achieving the consensus in the consortium blockchain. 
+4.2.2 Resolution 
+When the network holds only one identifier space, identifier resolution is the basic service provided by MIS as DNS. When 
+the identifier is resolved in this case, users can access the storage server or the associated resource. The specific resolution 
+process is as follows. 
+Step 1: The user queries the global state tables of MIS to obtain the address of metadata ℎ𝑎𝑠ℎ(𝑈𝑠𝑒𝑟𝑛𝑎𝑚𝑒 +
+𝐼𝑑𝑒𝑛𝑡𝑖𝑓𝑖𝑒𝑟) and digest information of the associated resource. 
+Step 2: According to the metadata address, the user accesses the metadata server and receives the metadata file as a 
+response. This file includes the location of storage servers and other metadata information. 
+Step 3: After receiving the metadata file, the user requests the specific resource associated with the identifier from the 
+network. The process of requesting resources includes push and pull methods, both of which have been integrated into 
+MIR. It is important to note that with pull transport, the user does not actually access the storage server location recorded 
+in the metadata file. 
+Table 2: Field in an identity registration request. 
+Field Name 
+Data Type 
+Description 
+Username 
+String 
+*Required. 
+Identity_ 
+Identifier 
+String 
+*Required. 
+The format is “type0:”+“public key of user” 
+AboutMe 
+String 
+Optional, resource data.  
+Individual information includes the real name, phone number, ID/passport number, 
+and fingerprint/face/iris data.  
+Organization information includes the organization code, postal address, legal 
+representative, and bank account. 
+Device information includes the manufacturing plant and machine number. 
+Hash_Identity_Identifier 
+String 
+*Required, the address of the metadata. 
+Digest 
+String 
+Optional, digest of resource data. 
+TTL 
+Int 
+*Required, the valid period of the identity identifier. 
+Fee 
+Float 
+*Required. The registration fee can be zero. 
+Timestamp 
+Double 
+*Required, the generation time of the identity registration request. 
+Signature 
+String 
+*Required. The user signs with the private key. 
+. 
+
+14 
+Step 4: When the user receives a resource associated with the identifier, it verifies the integrity according to the digest 
+information. If verified, the identifier will be considered to be resolved. Otherwise, return to Step 3 and continue the request. 
+Although IP address identifiers in MIS are different from IP addresses in the traditional network, MIS is still compatible 
+with legacy DNS. This is because the domain name is defined in MIS as an identifier type 6, which is bound to website 
+resources. The IP addresses mapped to domain names are stored in the associated metadata file. In addition, we register 
+the special usernames in the form of 𝐷𝑁𝑆_𝑐𝑎𝑐ℎ𝑒: 𝑖(𝑖 = 1,2,3, … ) in MIS to cache DNS resources. 
+When a user or device wants to resolve a domain name, it queries the MIS identifier table linked by the username record 
+(𝐷𝑁𝑆_𝑐𝑎𝑐ℎ𝑒: 𝑖). If the query fails, the MIS processor will forward the query request in a typical form to the DNS server 
+and cache the DNS response into the metadata file which is associated with the domain name identifier in the global state 
+tables for future queries. 
+Given that domain names are actually controlled by DNS servers rather than users or devices in MIS, the caching 
+process does not wait for consensus unlike the registration of identities and other identifiers. In other words, the credibility 
+of the resolution service for domain names depends on the DNS itself. 
+4.2.3 Inter-translation 
+In addition to TCP/IP networks, there are many future networks with different types of identifiers (and corresponding 
+communication modes) to satisfy new functions or requirements. For example, content identifiers have advantage in the 
+acquisition of data resources, such as videos, page resources. Service identifiers improve the flexibility, loose coupling and 
+reusing properties of services. Identity and location identifiers are suitable for mobile devices that constantly switch 
+locations. 
+Therefore, resource providers in the metaverse are encouraged to apply for identifiers and publish resources as much 
+as possible to form multiple identifier spaces. Through the inter-translation service, MIS supports users to resolve all types 
+of identifiers and obtain resource data. Figure 5 illustrates the detailed process. 
+In Figure 5, we assume that the identity identifier space 𝐶0 holds identity (𝑖0). In addition to the basic identity (𝑖0), 𝐶1 
+also holds IPv4 address (𝑖5) and domain name (𝑖6), 𝐶2holds identity (𝑖0) and content (𝑖1). Metadata servers own all types 
+of identifiers (𝑖0, 𝑖1, 𝑖5, 𝑖6). A normal individual end user 𝐸 has only the required identity and is in the identifier space 𝐶0. 
+In case user 𝐸 wants to resolve the domain name (𝑡𝑦𝑝𝑒6:𝑚𝑒𝑡𝑎𝑣𝑒𝑟𝑠𝑒. 𝑠𝑢𝑏3. 𝑐𝑜𝑚) of the identifier space 𝐶1, it first 
+queries in the global state tables in MIS. If fails, the MIS processor will perform the resolution process (Section 4.2.2) and 
+response the user 𝐸  with the inter-translating result, including the username (𝐷𝑁𝑆_𝑐𝑎𝑐ℎ𝑒: 1)  and its identity 
+(𝑡𝑦𝑝𝑒0: 04𝑑9806𝑒𝑐30𝑑𝑎𝑐7𝑒5). At this point, the domain name (𝑡𝑦𝑝𝑒6: 𝑚𝑒𝑡𝑎𝑣𝑒𝑟𝑠𝑒. 𝑠𝑢𝑏3. 𝑐𝑜𝑚) has a form that can be 
+recognized in the identifier space 𝐶0 . Then, the user 𝐸  requests the metadata server 𝑀  for the IPv4 address 
+(𝑡𝑦𝑝𝑒5: 142.251.42.228) in the metadata file. Finally, it communicates to the storage server 𝑆1  with the identity 
+(𝑡𝑦𝑝𝑒0: 04𝑑9806𝑒𝑐30𝑑𝑎𝑐7𝑒5) and IPv4 address (𝑡𝑦𝑝𝑒5:142.251.42.228) for routing, so as to obtain the associated 
+resource for the domain name  (𝑡𝑦𝑝𝑒6:𝑚𝑒𝑡𝑎𝑣𝑒𝑟𝑠𝑒. 𝑠𝑢𝑏3. 𝑐𝑜𝑚).  
+Likewise, user 𝐸  can resolve the content (𝑡𝑦𝑝𝑒1:/𝑚𝑒𝑡𝑎𝑣𝑒𝑟𝑠𝑒_𝑠𝑢𝑏1/002. 𝑚𝑝4) of the identifier space 𝐶2 . The 
+difference between this process and IPv4 address of 𝐶1 is that the data of identity and IPv4 address is obtained in push 
+mode, while the data of content is obtained in pull mode. Therefore, the resolution process involves changing the 
+communication mode. MIN has realized a communication mode that supports both push and pull semantics in the protocol 
+stack. Simply speaking, one or more edge routers are set up in each identifier space, which are responsible for processing 
+packets sent to other identifier spaces, including recognizing identifier types and changing semantics. 
+
+15 
+In conclusion, the inter-translation service enables the existence of the same resource data in multiple forms (identifiers) 
+and the accessibility for users of different identifier spaces. 
+5 EXPERIMENT 
+We evaluate the performance of MIS to demonstrate that it can operate at competitive rates. The reading and writing time 
+of metadata and resource data depend on the off-chain storage system. The time cost becomes a deterministic load for MIS.  
+We implement MIS in Golang [28]. We also established a testbed for MIS on 4 physical servers located in UK and 
+Malaysia. Each server has two 8-core CPUs and 10 Gbps network bandwidth, and contains 20 consortium nodes. A 
+consortium node is simulated as representative of a country or top-level organization, and has bookkeeping and voting 
+rights. The aggregation permission is held by each node in turn. We encapsulate no more than 100,000 transactions into a 
+block. The bookkeepers sign all transactions and blocks using the BLS signature scheme. 
+The average response time for multi-identifier resolution requests is shown in Figure 6. For 20,000 registered identifiers, 
+the average time required to query one in MIS is only 34.035 milliseconds, and this meets the actual query performance of 
+the legacy DNS (48 milliseconds, according to the method in [64]).  
+In our experimental blockchain, users randomly send writing operation requests for registering, updating and revoking 
+multi-identifiers to consortium nodes at an average rate of 1,000 per second. The time spent in each phase of writing 
+operations is shown in Figure 7. We obtained an average latency of 349.79 milliseconds in total. The spikes at block heights 
+ 
+Figure 5: An example of inter-translation. 
+. 
+
+MIS
+MIS Identifier Table
+Username Table
+type0:e98a32e6175bbd375
+Digest of data
+Metadata address
+M
+type5:142.251.42.228
+.
+.
+xers
+(3)DNS
+LegacyDNS
+S_1
+type0:9a7678c2da4c4587b
+.
+.
+response
+S_2
+type1:/metaverse_sub1/002.mp4
+.
+...
+DNS_cache:1
+type0:04d9806ec30dac7e5
+(2)Quer)
+(4)
+metaverse.sub3.com
+type6:metaverse.sub3.com
+...
+(1)Query
+(5)Inter-traslate as
+typeo:metaver
+DNS cache;type0:
+Co
+se.sub3.com
+00000000000000000
+(1)Query
+(2)Inter-translate as
+typel:/metaverse
+S 2;type0:9a7678c2d
+S1
+sub1/002.mp4
+a4c4587b
+(associated with IPv4 addresses
+Storage Servers
+(7)Request and (S)Verify
+E
+(4)Request and (5)Verify
+resources associated with
+resources associated with
+and domain names)
+typeo:metaverse.sub3.com
+typel:/metaverse_sub1/002.mp4
+(oReguest
+: (3)Request
+metadata with
+Imetadata with
+DNS_cache;type6:
+Is2;type0:9a76
+(associated with contents)
+metaverse.sub3.com
+78c2da4c4587b
+CΛ
+C2
+Metadata Servers
+(associated with all types of identifiers)16 
+of 146, 169, 303, 361, 460, and 526 are caused by unstable bandwidth speeds between servers in different regions. In 
+addition, the main time cost is spent on storing blocks into the disc (presented as Storage Phase in Figure 7), in an average 
+of 284.23 milliseconds. This is limited by the speed at which the CPU reads and writes memory. The consensus phase, 
+including propose, vote and commit phases, takes only 65.56 milliseconds, accounting for 18.74% of the total latency. This 
+ 
+Figure 7: The writing performance of MIS on the 80-node testbed in UK and Malaysia. 
+ 
+ 
+Figure 6: The response time of resolving multi-identifiers. 
+
+600
+ProposePhase
+VotePhase
+CommitPhase
+StoragePhase
+X: 526
+Y:597.8
+X: 169
+Y: 492.4
+X: 361
+500
+Y: 480.5
+400
+Time (milliseconds)
+300
+200
+X: 146
+X: 303
+X: 460
+100
+Y: 81.45
+Y: 82.92
+Y: 80.69
+0
+100
+200
+300
+400
+500
+Block Height35
+30
+34.035
+27.715
+25
+20
+20.006
+15
+11.237
+10
+Average
+5
+3.759
+01000
+5000
+10000
+15000
+20000
+Numberof Multi-identifiers17 
+is because, the communication bandwidth between nodes within the server was considered as infinite on the testbed. 
+However, performance will be largely network-bound in practice.  
+In order to verify the writing performance of MIS more realistically, we simulate a globally distributed deployment of 
+our MIS on the Google Cloud. There are 200 VMs located in 26 countries from 5 continents in our simulation environment, 
+guaranteeing that all countries can participate in the management of multiple identifier spaces. 6 to 9 VMs are deployed in 
+each country as TLD consortium blockchain nodes for MIS. In total, there are 16 VMs in North America, 64 in South 
+America, 58 in Europe, 46 in Asia and 16 in Oceania. Each VM is configured with 2 vCPUs and 4GB memory. In addition, 
+we use speedtest [65] to measure the bandwidths of VMs in each country. The average uplink bandwidth is 897.84Mbit/s 
+and the downlink is 1525.60Mbit/s. Other parameters are the same as the above 80-node testbed. 
+We evaluate the average latency in each phase of writing operations by gradually increasing the nodes number from 50, 
+100, 150 and 200 respectively, as shown in Figure 8. Although it takes a little longer in the consensus phase as the number 
+of consortium nodes increases, the consensus time is still acceptable. This can be referred to the deterministic consensus 
+used in MIS. For example, the original Blockstack overlays Bitcoin with latency of 600 seconds, whereas MIS latency is 
+even less than 10 seconds. 
+ 
+Figure 8: The writing performance of MIS on the testbed located in 26 countries from 5 continents. 
+6 COMPARATIVE DISCUSSION 
+In order to acquire decentralized DNS alternatives, public and consortium blockchains are two feasible means. Bitcoin is 
+the largest, most secure, and most actively maintained blockchain, and is the most suitable public blockchain for carrying 
+decentralized applications and services [17]. However, when used for identifier management, Bitcoin still poses several 
+challenges, as described below.  
+6.1.1 Challenge 1: Limited Efficiency 
+The consensus algorithms of public blockchains face a serious efficiency barrier [66]. In Bitcoin, the time target for the 
+block generation interval is conservatively set at 10 minutes with a block size limit of 1MB. For typical transaction sizes 
+of 240 Bytes, only 7 transactions can be processed per second. 
+Essentially, the problem with the Nakamoto consensus is that the consensus process relies on transaction/block 
+synchronization. This encourages many proposals to adopt parallel block generation with synchronization. One idea is to 
+elect a leader or committee [21, 67-69] before generating blocks. Another is to recognize miners’ work on non-longest 
+
+10000
+ProposePhase
+9487.43
+VotePhase
+8000
+CommitPhase
+7794.22
+Time (milliseconds)
+StoragePhase
+6582.03
+6000
+4000
+2435.50
+2000
+0
+50
+100
+150
+200
+NumberofNodes18 
+chains [70-72]. Consortium blockchain usually adopt deterministic consensus algorithms, most of which are derived from 
+the BFT algorithm. The validity of blocks relies on message exchange rather than nodes’ proof of work so that the 
+generation time is reduced. Although the BFT algorithm provides good performance for small-scale applications, it does 
+not suit large-scale applications due to its high communication cost and poor scalability. Speculative BFT [73] is proposed 
+to reduce the communication complexity to O(N) in the ideal case. However, it requires higher communication complexity 
+when replacing the leader, making it inapplicable to practical blockchain networks. Further, Proof of Vote (PoV) [57] and 
+Hotstuff are proposed with linear consensus decision and view change. Actually, all peers need to validate each transaction 
+in a decentralized system. Therefore, O(N) is the lower bound in terms of communication complexity. Although the leader 
+or committee election can optimize consensus in both public and consortium blockchain, the actual effects are different. 
+In a completely decentralized public blockchain, any node is free to join or leave the network. To ensure fairness, majority 
+of nodes needs to participate in the consensus. Therefore, the computing capacity and bandwidth requirement of nodes 
+should not be highly set. On the contrary, in a consortium blockchain, nodes are censored for access, thus the governing 
+agency is in a position to put forward higher requirements on them. For example, core nodes must use commercial servers 
+with strong computing and security capabilities, or be organized to a specific topology. Especially for DNS alternative 
+systems that provide identifier services, efficient consortium blockchains are more suitable than public blockchains. In 
+MIS's current implementation, the core TLD nodes are relatively stable even though they belong to different organizations. 
+In order to provide reliable service and reduce the probability of network partition, we adopt the extensible physical 
+connection mode of the multi-dimensional hypercube. 
+6.1.2 Challenge 2: Final Consistency 
+Nakamoto consensus nodes that solve the puzzle can generate a subsequent of the same block, which leads to forks. The 
+longest chain rule is adopted to choose the main chain, so that only one of these conflicting blocks with the same height is 
+valid. However, forks can turn strong consistency to a final consistency over time. 
+Allowing forking is essentially a balance of three properties in the CAP (Consistency, Availability, and Partition 
+tolerance) trilemma [74]. Specifically, in the distributed theory, when the network topology is split into components by 
+missing or failed links, a partition happens such that nodes in one component (partition) cannot communicate with nodes 
+in another. Under this unfortunate circumstance, the distributed system designers are enforced to choose either availability 
+or consistency [75]. Andrew Lewis-Pye et al. [76] have further proved an analog of the CAP trilemma for blockchains 
+arguing that no system is both adaptive and has finality in the unsized and partially synchronous network. The BFT-based 
+algorithms used by consortium blockchains reach consensus without forks. More than 2/3 of core nodes deterministically 
+commit the block in a round through 2 or 3 phases of message exchange. Therefore, these algorithms satisfy strong 
+consistency, rather than final consistency as public blockchain consensus.  
+6.1.3 Challenge 3: Limits on Storage Capacity 
+Nodes locally store a copy of all the on-chain data to independently process transactions and blocks. However, the endless 
+growth of blockchain poses a great challenge to the storage capacity of nodes.  For example, by July 10, 2022, it requires 
+406.05GB of disk space to copy all the Bitcoin data [77]. Bitcoin designs light nodes as a lightweight approach, which 
+only store block headers and verify transactions by downloading blocks from full nodes. Since nodes without sufficient 
+storage space can only become light nodes instead of full nodes, the decentralization and security of Bitcoin has been 
+objectively influenced. 
+
+19 
+The EC-based cooperative storage schemes [78, 79], including MIS, use erasure code to encode and decode data. 
+Erasure code was originally a coding scheme for wireless channel transmission, and it was later used in distributed storage 
+system to reduce duplications. The EC-based cooperative schemes reduce the on-chain data stored on nodes in a fixed 
+proportion only by modifying the reading and writing processes of data incrementally. This makes such schemes suitable 
+for both public and consortium blockchains. 
+We conclude that these challenges have been solved by MIS using consortium blockchain technologies. Table 3 below 
+presents a comparative summary between MIS and different existing DNS alternative systems. 
+Table 3: Comparison between MIS and existing DNS alternatives. 
+DNS 
+Alternatives 
+Type of 
+Blockchain 
+Consensus 
+Consistency 
+Multi-identifier 
+Management 
+Compatibility of 
+Legacy DNS 
+Consensus 
+Latency (s) 
+Namecoin 
+Public 
+Nakamoto 
+Consensus 
+Final 
+No 
+No 
+2400 
+Blockstack 
+Public 
+Nakamoto 
+Consensus 
+Final 
+No 
+No 
+600 
+ENS 
+Public 
+Nakamoto 
+Consensus 
+Final 
+No 
+No 
+13.38 
+DNSTSM 
+Consortium 
+Kafka 
+Final 
+No 
+Yes 
+0.095 
+TD-Root 
+Consortium 
+C-RAND 
+Strong 
+No 
+Yes 
+<350 
+MIS (ours) 
+Consortium 
+PPoV 
+Strong 
+Yes 
+Yes 
+<10 
+7 CONCLUSION 
+In this paper, we propose MIS, the first-ever multi-identifier management and resolution system of metaverse based on 
+consortium blockchain. In order to separate on-chain and off-chain data, MIS is designed as a 4-tier architecture with the 
+assumption that the future metaverse would continue to persistently evolve using a range of sub-metaverses and different 
+sorts of resources. Resource data can be stored heterogeneously off-chain, alleviating the problem of data migration of sub-
+metaverses. As a DNS alternative, MIS eliminates the centralization of DNS management and resolution of the TCP/IP 
+architecture. Instead, it manages multiple identifier spaces and provides inter-translation services. Several innovative 
+improvements are also made to the size of data to reduce the storage pressure on consortium blockchain nodes. We have 
+deployed two testbeds in 26 countries from 5 continents worldwide. Our experimental results show that MIS outperforms 
+the legacy DNS and Blockstack in terms of latency. Finally, it should be mentioned that MIS has been released online as 
+an open-source management system for the MIN architecture. 
+ACKNOWLEDGMENTS 
+This work was supported by the Guangdong Province Research and Development Key Program [grant number 
+2019B010137001]; Basic Research Enhancement Program of China [grant number 2021-JCJQ-JJ-0483]; Shenzhen 
+Research Programs [grant number GXWD20201231165807007-20200807164903001]; [JCYJ20210324122013036]; 
+[JCYJ20190808155607340]; China Environment for Network Innovation (CENI) GJFGW No.[2020]386, SZFGW 
+[2019]261; the National Keystone Research and Development Program of China [grant number 2017YFB0803204]; ZTE 
+Funding [grant number 2019ZTE03-01] HuaWei Funding [grant number TC20201222002].  
+
+20 
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+Photon Bose-Einstein condensation and lasing in semiconductor cavities
+Aurelian Loirette-Pelous and Jean-Jacques Greffet
+Universit´e Paris-Saclay, Institut d’Optique Graduate School,
+CNRS, Laboratoire Charles Fabry, 91127, Palaiseau, France
+(Dated: January 31, 2023)
+Photon Bose-Einstein condensation and photon thermalisation have been largely studied with
+molecular gain media in optical cavities. Their observation with semiconductors has remained elusive
+despite a large body of experimental results and a very well established theoretical framework. We
+use this theoretical framework as a convenient platform to revisit photon Bose-Einstein condensation
+in the driven dissipative regime and compare with the lasing regime. We discuss the thermalisation
+figures of merit and the different experimental procedures to asses thermalization. We compare the
+definitions of lasing and condensation thresholds. Finally, we explore the fluctuations of the system
+and their relation to the different regimes.
+I.
+INTRODUCTION
+In
+2010,
+experiments
+by
+Klaers
+et
+al.
+[1,
+2]
+identified and demonstrated Bose-Einstein condensation
+of photons, a new light emission regime.
+While this
+regime share with lasing the macroscopic occupation of
+one mode, cavity photons are in near-thermodynamic
+equilibrium.
+As a direct consequence, cavity modes
+occupation follow a Bose-Einstein (BE) distribution and
+condensation is forced in the lowest energy cavity mode.
+At first glance, Bose-Einstein condensation (BEC)
+with photons seems to be impossible.
+On the one
+hand, lasers are usually thought to operate far from
+equilibrium.
+On the other hand,
+in the so-called
+blackbody radiation, equilibrium between photons is
+reached due to walls acting as a reservoir, but the null
+chemical potential precludes condensation. Actually, a
+suitable gain material such as pumped dyes molecules or
+semiconductors can act as a reservoir providing a photon
+chemical potential [3]. Thermalization of the photon gas
+with such a reservoir is made possible with a high-Q
+cavity, when the number of absorption-emission cycles
+made by a photon before leaving the cavity becomes
+large.
+Furthermore, the cavity introduces a band gap
+in the photon dispersion relation so that a lowest energy
+state can be defined for a given band. These ingredients
+are sufficient to ensure BE condensation of photons at
+room temperature in the weak coupling regime [2].
+In the last decade, the pioneering experiments [1, 2]
+in a dye-filled microcavity triggered a large amount of
+works in similar devices in order to understand further
+this new regime and its properties. An important issue
+has been to clarify the similarities and differences with
+the lasing regime. While the overall crossover from the
+standard out-of-equilibrium lasing phase to the BEC one
+has been shown to be quite smooth [4], some features
+of BEC have appeared.
+At equilibrium, the emission
+spectrum follows a BE distribution, and condensation
+occurs into the lowest energy cavity modes.
+When
+thermalization breaks down, major spectral alterations
+have been observed, ranging from deformation of the
+thermal tail [1] to lasing in excited modes and multimode
+lasing [5–7]. Early experiments investigating the second-
+order coherence in the BEC regime evidenced large
+fluctuations g(2)(0) = 2 even far above the condensation
+threshold [8]. This thermal behaviour suggests a closest
+resemblance of a photon BEC to a pumped blackbody
+than to a standard laser.
+As a consequence, first
+order temporal coherence is also delayed to above-
+threshold excitation [9].
+In recent years, the question
+about the difference between BEC and lasing has been
+renewed due to the emergence of nanostructured cavity
+mirrors enabling to realize complex potentials for light
+[10–14].
+Indeed,
+in these systems,
+controlling the
+thermalization enables, for example, the study of vortices
+formation and annihilation [15–17], or to envision analog
+simulation with synchronized arrays of out-of-equilibrium
+condensates [18–20].
+Still, in the quest for these new
+applications, we observe that several aspects of the
+problem have been overlooked so far.
+We list several
+of them in the next paragraphs.
+We first note that BE condensation of photons
+has
+been
+observed
+in
+dye-filled
+microcavities
+[2,
+6,
+21]
+and
+plasmonic
+nanoparticles
+arrays
+[22],
+and
+erbium–ytterbium
+co-doped
+fiber
+cavities
+[23].
+Alternatively, semiconductors have received much less
+attention up to now, in spite of being a very common and
+versatile active medium. In particular, photon BEC in
+semiconductor-based devices is not fully recognized yet.
+This is surprising in many respects. On the experimental
+one, spectral signatures hinting at thermalization and
+BEC of photons has been observed early in a VCSEL
+designed for polariton physics [24, 25].
+More recently,
+similar features have been observed in a commercial
+VCSEL [26], suggesting that BEC (or near-equilibrium
+BEC) of photons could be more common than it is
+usually thought.
+A high absorption/emission cycles
+number before cavity loss has also
+been
+reported
+in a quantum-well photonic crystal laser [27], while
+not interpreted as BEC. On the theoretical side, the
+possibility of a chemical potential for photons has been
+historically demonstrated on a semiconductor example
+[3].
+Furthermore, simple and accurate models of gain
+and lasing in semiconductors are available so that this
+system is a very good playground to explore the physics
+of photon Bose-Einstein condensation and lasing.
+arXiv:2301.13073v1  [cond-mat.quant-gas]  30 Jan 2023
+
+2
+Second,
+finding
+a
+clear
+signature
+of
+photon
+thermalization
+is
+not
+an
+obvious
+task.
+On
+the
+experimental side,
+the analysis of emission spectra
+is often compared with a Bose-Einstein function.
+On
+the theoretical side, a dimensionless number quantifying
+the
+degree
+of
+thermalization
+has
+been
+introduced
+theoretically by some authors.
+A simple connection
+between these two approaches is still lacking.
+Third, the connection between lasing and condensation
+is not fully understood.
+While a clear threshold
+is observed in both cases, its exact positions differs.
+This may impact the interpretation of the observed
+phenomena.
+Hence, there is a need to compare the
+definitions of thresholds from laser and from equilibrium
+BE condensates physics.
+Fourth,
+intensity
+correlations
+are
+often
+used
+to
+distinguish coherent light from stochastic light.
+It is
+interesting to revisit condensation and lasing by studying
+fluctuations. While many results have been reported, the
+role of the degree of thermalization and the role of the
+β-factor of the cavity have not been fully discussed so
+that it is difficult to draw final conclusions.
+In
+this
+paper,
+we
+take
+advantage
+of
+the
+well-
+developped formalism to describe gain in semiconductors
+to analyse all these issues.
+In the next section, we
+present a simple unified theory of equilibrium and non-
+equilibrium condensation of photons in a semiconductor-
+based cavity. While similar to the pioneering model by
+Kirton and Keeling [28, 29] for dye-filled microcavities,
+we show that our model provides a straightforward
+interpretation of the photons chemical potential.
+We
+then
+derive
+a
+generalized
+BE
+distribution
+in
+the
+driven-dissipative regime and exhibit a dimensionless
+number that characterizes quantitatively the degree of
+thermalization.
+We discuss some of its properties and
+clarify the connection with other dimensionless numbers
+such as cooperativity, Knudsen number and optical
+thickness.
+In this new framework, we show how to
+revisit some lasing features such as gain clamping and
+inversion, and discuss the selection of the lasing mode.
+An extended definition of the equilibrium condensation
+threshold is also introduced for nonequilibrium systems,
+and compared to the standard lasing threshold definition.
+In Section 3, we discuss several observables to evaluate
+to which extent a device is thermalized.
+Equipped
+with the explicit form of the degree of thermalization
+introduced in the previous section, we can revisit the
+typical experimental situations. In particular, we show
+that the most common practice, consisting in studying
+the emission spectrum, should be used with caution.
+We finally focus on the second-order coherence in
+Section 4.
+We tackle the thermalisation issue by
+calculating analytically the intensity autocorrelation
+function g(2)(0) as a function of the β-factor and the
+degree of thermalization.
+II.
+EQUILIBRIUM AND NON-EQUILIBRIUM
+CONDENSATION OF PHOTONS IN A
+SEMICONDUCTOR-BASED CAVITY
+In this section, we first summarize basic forms of
+the emission and absorption rate in a semiconductor.
+We then use this formalism to recover the equilibrium
+number of photons per mode in a lossless cavity.
+We
+finally compare this case to the one of a lossy cavity with
+gain operating in the so-called driven-dissipative regime.
+This approach enables us to discuss in a very simple
+framework (i) the thermalization regime, introducing a
+degree of thermalization in a very systematic way, (ii)
+the connection between condensation and lasing and (iii)
+the definitions of their respective threshold.
+A.
+Model of semiconductor gain medium in a
+cavity
+Throughout this paper, we will focus on a piece of
+semiconductor placed in a cavity.
+We assume finite
+extension of the cavity so that photonics modes are
+spectrally discretized. We index them with l = 0, 1, 2...nc
+corresponding to increasing energies.
+The various
+particle exchange pathways between the gain medium,
+the modes and the environment are shown on Fig. 1 (a).
+In the cavity, photons in the l-th mode can be created
+or annihilated by the gain medium at the rates Rl
+em
+for spontaneous emission, Rl
+emN l for stimulated emission
+and Rl
+absN l for absorption, where N l is the number of
+photons in the mode l.
+Alternatively, radiative cavity
+losses occurs at the rate κlN l.
+In the semiconductor,
+excited electrons are created at the rate Rin through
+pumping (indistinctly electrical or optical) Conversely,
+relaxation can occur through the above-depicted emission
+in the cavity modes, through spontaneous emission into
+vacuum modes at the rate Rvac
+em , or through non-radiative
+relaxation pathways (for example Auger effect) at the
+rate Rnr.
+In contrast with dye molecules, explicit forms of Rl
+em
+and Rl
+abs can be derived for semiconductors.
+Here
+we focus on an intrinsic direct bandgap semiconductor,
+indifferently
+2
+or
+3-dimensional,
+and
+follow
+usual
+approximations [30].
+As sketched on Fig.
+1 (b),
+the conduction and heavy-hole valence band [31] are
+described by the isotropic dispersion Ec(k) and Ev(k)
+respectively, were k stands for the wavevector modulus.
+Assuming that only vertical interband transitions are
+possible, a transition involving a photon in the mode l
+with energy El requires an electron and a hole with the
+same wavevector ⃗kl so that Ec(kl) − Ev(kl) = El. We
+also assume that the ground cavity mode energy is higher
+than the gap energy E0 > Egap.
+Interestingly, conduction electrons and valence holes
+close to the gap edges can be well described as free
+particles with an effective mass m∗
+c/v, which leads to
+the simple parabolic band model Ec/v(k) = E0
+c/v ±
+
+3
+Gain medium
+Cavity 
+mode 𝑙
+𝑅𝑖𝑛
+𝜅𝑙𝑁𝑙
+𝑅𝑎𝑏𝑠
+𝑙
+𝑁𝑙
+𝑅𝑒𝑚
+𝑙
+𝑁𝑙
+𝑅𝑒𝑚
+𝑙
+𝑅𝑒𝑚
+𝑣𝑎𝑐
+Pumping
+Cavity loss
+Absorption
+Stim. 
+emission
+Spont. 
+emission
+Spont. 
+emission into
+vacuum modes
+a)
+𝑅𝑛𝑟
+Non-radiative
+relaxation
+0
+k l
+k=|k|
+E 0
+v
+E 0
+c
+Ec(k l)
+Ev(k l)
+E
+E l
+E gap
+Ec(k)=E 0
+c +
+2k 2
+2m *
+c
+Ev(k)=E 0
+v
+2k 2
+2m *
+v
+b)
+0
+1
+2
+1 Occupation
+probability
+E
+c
+v
+fFD(E,T,
+c)
+fFD(E,T,
+v)
+FIG. 1. Scheme of the system and notations. Panel (a): flux
+of particles between the gain medium, the cavity, and the
+environment. Panel (b): semiconductor band structure as a
+function of the wavevector modulus (left) and distribution of
+the electrons in each band (right). See the main text for a
+detailed description.
+ℏ2k2
+2m∗
+c/v with E0
+c/v the energy minimum/maximum of the
+conduction/valence band and ℏ the Planck constant.
+Analytical expressions for El(kl) can be derived, as well
+as for the density of states in each band ρc/v(k) and
+the joint density of state ρJ(k) associated to the vertical
+transitions [30].
+Next,
+we
+assume
+that
+the
+bands
+are
+in
+local
+thermodynamic equilibrium characterized by a Fermi-
+Dirac distribution with a common temperature T and
+local chemical potentials µc and µv, the so-called quasi-
+Fermi levels. In the case of electrical pumping, we have
+µc − µv = eV where V is the applied voltage.
+As
+the voltage increases, µc increases (resp. µv decreases)
+from the Fermi-level, so that their difference µc − µv is
+controlled. It is also possible to define quasi-Fermi levels
+under optical pumping.
+In this context, the spontaneous emission, stimulated
+emission and absorption rates for the mode l can be
+written respectively as [30]:
+Rl
+em = glfF D(Ec(kl), T, µc)[1 − fF D(Ev(kl), T, µv)]
+(1)
+and
+Rl
+abs = glfF D(Ev(kl), T, µv)[1−fF D(Ec(kl), T, µc)], (2)
+where gl is a pumping-independent transition rate
+and fF D(E, T, µ) = 1/[exp ( E−µ
+kBT ) + 1] is the Fermi-
+Dirac distribution with E the electron or hole energy,
+kB the Boltzmann constant and µ a quasi-Fermi levels.
+The microscopic expression of gl is given in Appendix
+A. The right hand side of Eq.
+(1) expresses that
+emission is proportional to the probability of finding an
+electron at the right energy in the conduction band and
+a corresponding hole in the valence band, and conversely
+for absorption in Eq. (2).
+Finally, we define the fraction of spontaneous emission
+into the mode l through the generalized β-factor:
+βl =
+Rl
+em
+Rvac
+em + �
+j Rj
+em
+.
+(3)
+In a single cavity mode context, this dimensionless
+number
+characterizes
+the
+emission
+regime.
+A
+macroscopic laser corresponds to β → 0 while a nanolaser
+corresponds to β → 1.
+Indeed due to large (resp.
+small) mode volume, a macroscopic (resp. nano-) laser
+is characterized by a low (resp.
+high) Purcell factor,
+so that spontaneous emission into the numerous vacuum
+modes (the mode l) is dominant. Reducing the volume
+further tends to reduce the cavity mode number. Still the
+cavity modes spacing and number can also be adjusted
+e.g.
+through engineering of the mirrors curvature for
+Fabry-Perot-like cavities. Hence the quantities Rvac
+em and
+�
+l Rl
+em can be partially tuned independently.
+B.
+Photon BEC in a lossless cavity with gain
+Bose-Einstein
+condensation
+is
+a
+property
+of
+an
+ensemble
+of
+bosons
+in
+thermodynamic
+equilibrium.
+Quantitatively, thermodynamic equilibrium means that
+a state at energy E is occupied according to a Bose-
+Einstein distribution 1/[exp( E−µ
+kBT ) − 1] where µ is the
+chemical potential. Condensation may occur when the
+chemical potential approaches the ground state energy
+µ → E0.
+In
+the
+blackbody
+radiation,
+photons
+reach
+a
+thermodynamic equilibrium due to walls acting as a
+reservoir.
+This equilibrium is characterized by a null
+photon chemical potential. Remarkably, Wurfel showed
+that it is possible to introduce a photon chemical
+potential when dealing with stationary systems with gain
+[3].
+We reproduce here the reasoning for clarity.
+We
+
+4
+start by assuming a perfectly lossless cavity, i.e. κl = 0.
+In the steady-state regime, the balance between the
+spontaneous and stimulated emission processes and the
+absorption in the l-th photonic mode yields:
+Rl
+em + Rl
+emN l = Rl
+absN l,
+(4)
+where N l is the number of photons in the l-th mode. It
+is readily seen that the photon number only depends on
+the ratio between the absorption rate and the emission
+rate Rl
+abs/Rl
+em. Given Eqs. (1),(2) which assumes that
+the gain medium is in local thermodynamic equilibrium,
+simple algebra allows to recover the Van Roosbroeck-
+Shockley relation [32]:
+Rl
+abs
+Rlem
+= exp
+�El − µ
+kBT
+�
+,
+(5)
+where µ = µc − µv. From Eqs (4) and (5), it follows
+that the photon number in the mode l is given by:
+N l =
+1
+exp
+� El−µ
+kBT
+�
+− 1
+,
+(6)
+namely a Bose-Einstein distribution with temperature
+T and a chemical potential defined as the quasi-Fermi
+levels splitting. In the absence of pumping, the chemical
+potential is null and we recover the blackbody radiation
+distribution with the temperature of the semiconductor
+at equilibrium.
+Finally, beyond the semiconductor model used here,
+we emphasize the key role of local thermodynamic
+equilibrium in each band under pumping to derive this
+result. Indeed, this appears as a sufficient condition on
+the gain medium to reach photons BEC. In particular,
+this explains why eq.
+(5) can be written similarly
+for dyes molecules in terms of emission and absorption
+cross sections, a formula known as the Kennard-Stepanov
+relation [33–35] (sometimes also called the Neporent-
+McCumber relation,
+see Ref.
+[36] and references
+therein).
+To summarize, the number of photons in
+a non-lossy cavity filled with a gain medium in local
+thermodynamic equilibrium can be described by a Bose-
+Einstein distribution with a non-zero chemical potential.
+C.
+The driven-dissipative regime of a lossy cavity
+with gain: Lasing or BEC ?
+We now consider a cavity coupled to the environment
+through the loss rates κl > 0. Such a system composed
+of a gain medium and a cavity with radiative losses is
+usually considered to be a laser.
+A natural question
+then arises: what is the difference between Bose-Einstein
+condensation and lasing ?
+We repeat the analysis of the previous section using
+the same assumptions and notations, now accounting
+for cavity losses so that the system is in the driven-
+dissipative regime.
+The balance equation (4) becomes
+Rl
+em +Rl
+emN l = (Rl
+abs +κl)N l. The steady-state photon
+number in the mode l can then be cast in the form [30]:
+N l =
+Rl
+em
+κl − (Rlem − Rl
+abs).
+(7)
+In this last equation, the quantity Rl
+em −Rl
+abs is better
+known as the net gain rate of the mode l. Hence, this
+simple model recovers that the mode l starts to lase as the
+net gain compensates the radiative losses. So far, we have
+isolated a mode and computed its occupation number by
+expressing the balance between gain and losses.
+This
+approach is at first glance at odds with the study of
+the population of different modes in an equilibrium
+system.
+Nevertheless, we now cast this laser equation
+in a form that mimicks Eq.(6). Upon factorization by
+Rl
+em and inserting the relation (5) in eq. (7), we find the
+alternative form [37]:
+N l =
+1
+exp
+�
+El−µ
+kBT
+�
+[1 + Kln(T, µ)] − 1
+,
+(8)
+where
+Kl
+n(T, µ) =
+κl
+Rl
+abs(T, µ)
+(9)
+is a dimensionless number often called Knudsen
+number in the context of transport phenomena and
+Boltzmann equation. The Knudsen number is given by
+the ratio of the absorption time
+1
+Rl
+abs by a characteristic
+time of the cavity, the residence time of a photon in
+the cavity
+1
+κl .
+Hence, in the regime where a photon
+undergoes a large number of absorption and emission
+cycles during the residence time, the Knudsen number
+is small and the distribution (8) approaches the BE
+distribution of a non-lossy cavity. In other words, the
+large number of absorption and emission events enables
+the photons to thermalize with the semiconductor acting
+as a reservoir.
+The Knudsen number appears to be
+the natural quantity that quantifies how thermalized is
+a mode. Importantly, note that a Knudsen number is
+associated to each mode, it is not a global quantity. We
+stress that some modes may be thermalized while others
+are not.
+As a conclusion of this section, it is clear from eq.
+(8) that Bose-Einstein condensation of photons is a
+particular regime of lasing, in which (i) Eq. (5) is satisfied
+for the gain medium and (ii) the Knudsen number is small
+for all the modes to ensure that they are all thermalized.
+In the remaining of this work, we will use ”lasing” to refer
+indistinctly to Bose-Einstein condensation or standard
+out-of-equilibrium lasing. In addition, Eq. (8) provides
+an alternative point of view to interpret lasing. Indeed,
+
+5
+while Eq. (7) provides a good description of single mode
+lasing in a system with significant losses and gain, we
+anticipate that Eq. (8) will be more suited to the study
+of multimode phenomena in the thermalized regime.
+D.
+Knudsen number, thermalization degree,
+optical thickness, cooperativity and photon number
+at transparency
+In the last section, we have introduced the Knudsen
+number Kl
+n of a mode l as the absorption time divided
+by the residence time in the cavity.
+It takes small
+values in the thermalized regime.
+Its inverse, that we
+note Dl, was called thermalization degree in Ref. [6] or
+thermalization coefficient in Ref.
+[38].
+Its key role in
+photon Bose-Einstein condensation had been suggested
+[1] and identified [39] in early papers.
+Here, we have
+shown how it appears naturally from laser rates equation
+in the context of an equilibrium distribution perturbed
+by the introduction of cavity losses. Let us now discuss
+alternative physical interpretations of the thermalization
+degree. We first note that it can be viewed as the effective
+cavity length Ll = c/κl divided by the absorption mean
+free path ll
+abs = c/Rl
+abs.
+With this point of view,
+which is often used to discriminate between diffusive
+regime and ballistic regime in transport phenomena, we
+identify the degree of thermalization with the optical
+thickness Ll/ll
+abs = Dl.
+Second, we remind that the
+optical thickness is proportional to the cooperativity
+C(Na). This quantity had been initially introduced to
+characterize the absorption of a photon by an ensemble of
+Na atoms in a cavity in the context of non-linear optics in
+a cavity [40]. It is currently used as a measure of the light-
+matter interaction in cavity quantum electrodynamics
+(CQED) [41]. Finally, the thermalization degree has been
+interpreted historically in laser physics as the photon
+number at transparency [42]. Here this follows from Eq.
+(7) when Rl
+em = Rl
+abs.
+Interestingly, this suggests to
+reinterpret some experiments featuring a high photon
+number at transparency as Bose-Einstein condensation
+of photons, see for example Ref. [27] for a semiconductor
+laser in a photonic crystal cavity.
+E.
+Lasing mode in the BEC picture
+In the previous sections, we showed that the laser
+equation (8) giving the mode photon number has the
+structure of a Bose-Einstein distribution apart from a
+correction term given by 1 + Kl
+n(T, µ). Hence, we can
+revisit the lasing transition in terms of Bose-Einstein
+distribution.
+We start with the laser point of view given by Eq. (7).
+In this framework, lasing in the mode l occurs as the gain
+rate saturates when it approaches the loss rate (Rl
+em −
+Rl
+abs) → κl. This is called gain clamping. In addition,
+finite losses require positive gain, that is, population
+inversion of the corresponding transition [43].
+We now place in the perspective of the generalized
+Bose-Einstein distribution.
+We start by writing (8) in
+a slightly different form [37]:
+N l =
+1
+exp
+� El−µl
+eff (T,µ)
+kBT
+�
+− 1
+,
+(10)
+where
+we
+have
+introduced
+an
+effective
+chemical
+potential µl
+eff(T, µ) = µ − kBT log[1 + Kl
+n(T, µ)]. Here,
+we stress that this form enables to use the Bose-
+Einstein distribution which is an equilibrium concept
+in the nonequilibrium driven-dissipative regime.
+The
+effective chemical potential is composed of a term µ
+which accounts for the gain and a term −kBT log[1+Kn]
+which accounts for the losses. The usual condition for
+Bose-Einstein condensation in the mode l is then directly
+generalized as:
+µl
+eff(T, µ) → El.
+(11)
+Here, the increase of the pump power is interpreted as
+increasing the quasi-Fermi levels splitting. Therefore µ
+converges toward a fixed value µclp defined as the solution
+of the implicit equation:
+µclp − kBT log[1 + Kl
+n(T, µclp)] = El.
+(12)
+This saturation of µ corresponds to gain clamping
+in the BEC point of view.
+In this last equation,
+the correction term is always negative.
+Hence, the
+quasi-Fermi levels splitting must exceed the transition
+energy El to trigger lasing.
+This corresponds to
+population inversion.
+It highlights the importance to
+distinguish between the quasi-Fermi levels splitting and
+the effective chemical potential, since only the latter can
+be interpreted as the photon chemical potential.
+We now focus on a multimode system. The usual laser
+textbook picture is the following [30, 44, 45]: the gain
+curve is taken to be a bell-shaped function of frequency,
+while the frequency dependence of the mirrors losses is
+neglected. Lasing is thus expected to occur in the cavity
+mode with largest gain.
+This picture is at odds with
+the one of ideal Bose-Einstein condensation, which is
+expected to occur in the ground cavity mode.
+We now revisit this issue using the Bose-Einstein
+picture given by Eq.
+(12).
+In the present multimode
+situation, each mode l defines a different clamping
+value of the quasi-Fermi levels splitting, that we note
+µl
+clp. Single mode lasing takes place in the mode with
+the smallest µl
+clp.
+To gain further insight, we assume
+Kl
+n(µl
+clp) ≈ Kl
+n(El).
+The clamped quasi-Fermi levels
+splitting of each mode l is simply given by:
+µl
+clp ≈ El + kBT log[1 + Kl
+n(T, El)].
+(13)
+
+6
+Interestingly,
+this expression is composed of two
+competing terms: on one hand, the mode energy favors
+lasing in low energy modes;
+on the other hand, it
+depends on the Knudsen number and favors lasing
+in highly thermalized modes.
+Therefore, without the
+second contribution coming from the cavity losses, we
+would recover the usual condensation on the ground
+mode. In practice, lasing in a mode above the ground
+mode is thus the signature of a system in which the
+modes have very different thermalization degrees. This
+discussion highlights that thermalization is primarily a
+modal property and not a system property. Indeed, as
+explained in Sec. II B, thermalization occurs between a
+mode and the reservoir, rather than between modes.
+Finally, we note that some authors used lasing in
+the ground mode versus an excited mode as a criterion
+to distinguish between BE condensation and out-of-
+equilibrium lasing [38, 46]. While lasing in an excited
+mode is indeed a signature of nonequilibrium operation,
+Eq. (13) shows that condensation in the ground mode is
+only the signature of a Knudsen number slowly varying
+from one mode to another, regardless of its absolute
+amplitude.
+F.
+Condensation versus lasing threshold
+In the previous section, we showed how to interpret
+the
+mode
+selected
+for
+lasing
+within
+a
+generalized
+Bose-Einstein condensation approach,
+stressing that
+condensation and lasing are two faces of the same coin.
+As a next step, it is natural to compare the definitions
+used for lasing threshold and for condensation threshold.
+We first remind the lasing threshold definition. Many
+different criteria can be used to characterize lasing [47,
+48]. Here, we consider the widely used condition based
+on an input/output curve. On Fig. 2 (a), the number
+of photons N j in the cavity is plotted as a function of
+the injection rate of excited carriers, that we note Rin.
+On a linear scale, N j turns suddenly from sublinear to
+linear on a small pumping range. The threshold is defined
+as the input rate of excited carriers Rin,LAS when the
+linear slope is continued down to 0 output rate (see the
+dashed blue line on Fig. 2 (a)). This input rate is equal
+to the value of the losses, evaluated at gain clamping.
+Indeed, close to clamping, stimulated emission funnel
+all additional photons in the lasing mode.
+The losses
+are due to different mechanisms: the leakage through
+non-lasing cavity modes with rate �
+l̸=j κlN l(µj
+clp), the
+emission into vacuum modes (Rvac
+em (µj
+clp)) and other non-
+radiative charge carrier relaxation processes (Rnr(µj
+clp)).
+The lasing threshold is thus given by:
+Rin,LAS = Rnr(µj
+clp)+Rvac
+em (µj
+clp)+
+�
+l̸=j
+κlN l(µj
+clp). (14)
+We now focus on the condensation threshold definition.
+In the literature on BEC in thermodynamic equilibrium,
+the BEC threshold is defined by the equality between
+the total number of particles and the number of particles
+in the excited states in the condensed phase [49]. First,
+note that in photons BEC experiments, the number N l
+of photons in a mode l cannot be measured directly. Still,
+the driven-dissipative regime enables to derive it from the
+measured flux κlN l and the knowledge of the loss rate κl.
+Second, note that in essence, this definition relies on the
+same idea as for a laser: beyond threshold, all additional
+photons will go to the condensed phase. As shown on Fig.
+2 (b), the condensation threshold is extracted graphically
+in a similar fashion as for the laser threshold when
+plotting the number of photons in the condensed mode
+versus the total number of photons in the cavity. The
+total number of photons in the cavity at threshold is then
+given by the sum of non-condensing modes population at
+clamping, namely �
+l̸=j N l(µj
+clp) = N tot
+BEC.
+Still, this
+procedure differs from the lasing threshold definition, as
+it is based on a number of photons in a cavity and not
+on a comparison of fluxes of input carriers and emitted
+photons. In particular, the nonradiative losses and the
+radiative losses into vacuum modes are not taken into
+account. Hence, the BEC definition leads to a smaller
+value of the threshold for the quasi-Fermi levels splitting
+than the lasing condition.
+The difference is not very
+large when the β factor is close to 1 but may be very
+large when emission into vacuum modes dominates. This
+is illustrated in Figure 2 (c) where it is seen that the
+thresholds can differ by orders of magnitude (in term of
+photons in the lasing mode).
+To conclude, the choice of using the BEC or laser
+threshold has to be conducted carefully, as they can
+take very different values.
+To guide this choice, one
+should note that for the lasing threshold, both the
+input and emitted power must be monitored, while
+the emitted power spectrum is sufficient to determine
+the condensation one.
+As already encountered in
+the previous sections, this suggests that other than
+making a real difference between BEC and lasing, the
+”condensation” point of view is a framework suited to the
+study of the multimode character of the system, while the
+”lasing” one rather focus on its driven-dissipative aspect.
+III.
+EXPERIMENTAL ASSESSMENT OF
+THERMALIZATION
+In the previous section, we made a clear distinction
+between BEC and lasing using the thermalization degree
+of the modes. However, the thermalization degree cannot
+be measured directly. Indeed, it is proportionnal to the
+absorption rate Rl
+abs, but only the net absorption rate
+Rl
+abs − Rl
+em is given by a transmission measurement.
+In this section, we aim at finding observable quantities
+that depend sharply on the thermalization degree,
+enabling its assessment. We first analyze the emission
+spectrum under homogenenous pumping, which is the
+most common experimental practice,
+and find that
+
+7
+Rin,LAS/ j
+Rin/ j
+0
+N j
+LAS
+N j
+a)
+Laser input-ouput curve
+(Rin
+Rin,LAS)/ j fit
+N tot
+BEC
+N tot
+0
+N j
+BEC
+N j
+b)
+Photon-Photon curve
+(N tot
+N tot
+BEC) fit
+10
+1
+101
+103
+105
+107
+109
+Rin/ 0
+10
+1
+101
+103
+105
+107
+109
+N 0
+c)
+R vac
+em /
+l
+j
+lN l =0
+0 =2.5×10 2
+R vac
+em /
+l
+j
+lN l =2.4×102
+0 =10 4
+R vac
+em /
+l
+j
+lN l =2.5×106
+0 =10 8
+Condensation threshold
+Laser threshold
+FIG. 2. Panel (a): schematic input-output laser curve (red
+line) on a linear scale. The dashed blue line is a linear fit of
+the laser curve. Its intersection with the N j = 0 axis defines
+the laser pumping threshold Rin,LAS.
+The corresponding
+lasing mode photon number at threshold is noted N j
+LAS.
+Panel (b): schematic photon-photon curve of a multimode
+driven-dissipative BEC condensing in the mode j (red line)
+on a linear scale.
+N tot = �
+l N l is the total number of
+photons in the cavity.
+The dashed blue line is a linear fit
+of the BEC curve.
+Its intersection with the N j = 0 axis
+defines the BEC threshold N tot
+BEC. The corresponding lasing
+mode photon number at threshold is noted N j
+BEC.
+Panel
+(c): comparison of the lasing mode photon number at laser
+and BEC thresholds, on input-output curves corresponding to
+different rates of spontaneous emission into vacuum modes. A
+constant cavity modes spacing is assumed so that their energy
+reads El = E0(1 + 0.001 × l), with E0 = 1.271 eV. κl and gl
+are assumed constant over the modes, with a ratio gl/κl = 10.
+This enforces lasing in the ground mode. Non-radiative losses
+are neglected Rnr = 0.
+To help considering the value of
+Rvac
+em (µj
+clp)/ �
+l̸=j κlN l(µj
+clp), the corresponding value of β0
+is given in the legend.
+Other parameters are compiled in
+Appendix C.
+this method may not be reliable.
+We then discuss
+spectral and spatial measurements under inhomogeneous
+pumping. We finally discuss the influence of band-filling
+on the thermalization degree.
+A.
+Spectrum analysis
+In an ideally thermalized system, we saw in Section
+II B that the mode occupation follows a Bose-Einstein
+distribution N l = 1/[exp( El−µ
+kBT ) − 1]. At low occupation
+numbers,
+the classical regime is recovered,
+namely,
+the BE distribution reduces to a Maxwell-Boltzmann
+distribution N l
+≈ exp(− El−µ
+kBT ).
+Hence, a common
+practice to prove thermalization consists in looking for
+a linear decay on a semilogarithmic plot of the spectrum
+[24, 26, 50, 51]. Here, we compare this approach with the
+characterization based on the Knudsen number.
+In the classical regime, the generalized BE distribution
+eq. (8) becomes:
+N l ≈
+exp(− El−µ
+kBT )
+1 + Kln
+.
+(15)
+It is readily seen that an exponential decay of the
+cavity photons spectrum is observed in two cases: (i) the
+Knudsen number of all the modes is much lower than
+1, and (ii) the Knudsen number is constant over the
+modes, whatever its value. In the second case, despite
+an exponential behaviour of the spectrum, the Knudsen
+number may take values ≳ 1 indicating a non thermalized
+system.
+Beyond the classical regime, it is noteworthy that this
+issue persists in the quantum degenerate regime. Indeed,
+according to Eq. (10), the generalized BE distribution
+with constant Knudsen number Kn simplifies in an
+equilibrium BE distribution with the effective chemical
+potential µeff
+= µ − kBT log[1 + Kn] [37].
+All in
+all, it means that spectrum analysis with homogeneous
+pumping in order to quantify the thermalization may
+not be reliable. In particular, we note in Appendix B
+that devices featuring a large, planar and homogeneously
+pumped cavity are likely to feature a nearly constant
+Knudsen number. This may explain the BE-like spectra
+observed in optically [24, 25] and electrically [26] pumped
+large area VCSELs.
+B.
+Inhomogeneous pumping
+An interesting signature of thermalization can be
+observed when using an inhomogeneous pumping with
+a beam or injection area much smaller than the cavity.
+Indeed, the pumped part of the gain medium emits
+photons isotropically through spontaneous emission.
+These photons can be reabsorbed efficiently everywhere
+in a thermalized system.
+As a consequence,
+the
+gain is homogeneous in the cavity despite a localized
+pumping.
+To describe this effect, it is necessary to
+include additional rate equations describing locally the
+gain medium population [38, 39, 52].
+While this goes
+far beyond the scope of the present work, we give a
+hint of the complexity of this case by writing how the
+
+8
+photon occupation number is modified.
+The balance
+equation (4) with the losses κl for a mode l has
+to be integrated over the gain medium volume (also
+called active volume) Vact, namely
+�
+Vact d3⃗r
+�
+Rl
+em(⃗r ) +
+Rl
+em(⃗r )N l�
+= N l �
+Vact d3⃗r
+�
+Rl
+abs(⃗r ) + κl/Vact
+�
+where the
+rates are now defined locally.
+In particular, the local
+Knudsen number is Kl
+n(⃗r) = κl/[Rl
+abs(⃗r )Vact].
+The
+photon number in the mode l then becomes:
+N l =
+1
+exp(
+El
+kBT )
+�
+exp( µ(⃗r )
+kBT )
+�l
+�
+1 +
+�
+Kln(⃗r )
+�l�
+− 1
+,
+(16)
+where ⟨A(⃗r )⟩l =
+�
+Vact d3⃗r Rl
+abs(⃗r )A(⃗r )/
+�
+Vact d3⃗r Rl
+abs(⃗r )
+is a spatial average weighted and normalized by the
+absorption rate.
+While the global distribution still
+appears as a generalized BE distribution, additional
+complexity is brought by the spatial average.
+In
+particular,
+the
+weighting
+by
+the
+local
+absorption
+now gives a modal dependence to the quasi-Fermi
+levels splitting.
+An enhanced sensitivity to imperfect
+thermalization is thus expected. Experimentally, it was
+reported in Ref.
+[1] a departure from the ideal BE
+distribution of high energy modes occupation, while the
+thermalization degree was tuned down. Given the small
+extension of the optical pump used compared to the
+large extension of these high energy modes, this is in
+good qualitative agreement with our considerations. A
+similar observation has also been made in Ref. [22] for
+plasmon-polaritons.
+Beside spectrum analysis, we eventually mention two
+other types of measurements that reveal efficiently
+the thermalization of the system with inhomogeneous
+pumping. The first consists in measuring the size of the
+condensate as a function of the size of the pumping beam.
+When the system is well thermalized, the condensate
+size is invariant, while it follows the size of the spot in
+the opposite case. This type of measurement has been
+reported [53]. In the same fashion, the spatial position
+of the condensate in a trap can be compared with the
+position of the pump beam. As the pump is moved away
+from the center of the trap, the longer condensation keeps
+occurring in the center, the higher the thermalization
+rate. This measurement has been reported in Ref. [1, 4].
+C.
+Thermalization and saturation at high pumping
+In
+the
+last
+subsection,
+we
+discussed
+how
+the
+thermalization
+of
+a
+system
+can
+be
+probed
+with
+inhomogeneous pumping.
+Noteworthy, this has been
+done as if the thermalization of a mode was a general
+quantity, independent on the pumping strength. Here,
+we discuss how the thermalization evolves as the system
+is driven toward the degenerate regime through strong
+pumping.
+The key issue is simple:
+thermalization is
+ensured by absorption and reemission; if the gain medium
+is highly pumped and approaches saturation, absorption
+is reduced and hence thermalization decreases.
+1.00
+1.05
+1.10
+1.15
+1.20
+1.25
+1.30
+1.35
+ (eV)
+10
+1
+100
+101
+102
+103
+D l
+gl/ l =1000
+gl/ l =100
+gl/ l =10
+gl/ l =2
+=E l
+( clp,D l
+clp)
+FIG. 3.
+Variation of the thermalization degree Dl of a
+mode l as a function of the quasi-Fermi levels splitting µ,
+for various ratio gl/κl. The colored dots indicate clamping as
+defined in Eq. (12), at the quasi-Fermi levels splitting µclp
+and the thermalization degree Dl
+clp = Dl(µclp). The vertical
+dark dashed line indicates transparency, namely µ = El. The
+energy of the mode is El = 1.271 eV. Other parameters are
+compiled in Appendix C.
+Inserting Eq. (2) into Eq. (9), the dependence of the
+thermalization degree on the quasi-Fermi levels (that is
+pumping) reads:
+Dl = gl
+κl fF D(Ev(kl), T, µv)[1−fF D(Ec(kl), T, µc)]. (17)
+On Fig. 3, we show the evolution of the thermalization
+degree of a mode l as a function of the quasi-Fermi
+levels splitting µ [54] for various gl/κl.
+At low
+pumping, filling of the conduction band (and accordingly
+depletion of the valence band) is negligible so that Dl =
+gl/κl. When increasing the quasi-Fermi levels splitting,
+the thermalization degree decreases significantly.
+At
+clamping (colored dot), the fall is about a multiplication
+factor 1/5 at high gl/κl, and more than 1/10 at low gl/κl.
+In the first case, corresponding to a well thermalized
+mode, clamping occurs right over transparency (dark
+dashed vertical line). In a two-level system, transparency
+corresponds to an occupation probability of 1/2 of the
+upper and lower level, so that the product of the levels
+occupation is 1/4.
+Here, the slightly different value is
+due to the asymmetry of the bands of our semiconductor
+model (see Appendix C). In the low mode thermalization
+case, a large inversion population is needed for lasing,
+that occurs well above transparency.
+The occupation
+probability of the conduction band is then much greater
+than 1/2, and conversely for the valence band. Hence,
+the degree of thermalization is significantly decreased
+compared to the near-equilibrium case.
+In
+summary,
+reliable
+assessments
+of
+the
+system
+thermalization should be made in the degenerate regime
+
+9
+due to this dependence of the thermalization degree
+dependence on pumping.
+IV.
+INTENSITY FLUCTUATIONS: ARE THEY
+A BEC SIGNATURE ?
+In the previous part, we showed that the spectrum is a
+quantity that can reveal the thermalization of the system,
+but which needs to be analyzed and probed with care. In
+this section, we investigate the intensity fluctuations as
+an alternative observable to distinguish between the BEC
+and the out-of-equilibrium laser regimes.
+A.
+Context
+In the textbook picture of out-of-equilibrium lasing,
+coherence sets up right at the lasing threshold [30].
+Above threshold, the intensity fluctuations are ruled
+by Poissonian statistics resulting in a second order
+correlation function at zero-time delay g(2)(0) = 1. On
+the contrary, earlier works on intensity fluctuations in the
+BEC regime predicted [55] and then measured [8] super-
+Poissonian statistics for light well-above condensation
+threshold.
+This thermal regime,
+characterized by
+g(2)(0) = 2, was found to extend deeply in the condensed
+phase, before the crossover to the usual Poissonian light
+was recovered.
+This ask the question whether large
+fluctuations are a signature of BEC.
+While the picture described in the last paragraph
+suggests studying the fluctuations according to the
+thermalization degree,
+other parameter have to be
+taken
+into
+account.
+In
+Refs.
+[8,
+55],
+it
+has
+been
+pointed
+out
+that
+the
+reservoir
+size
+has
+an
+important influence on fluctuations. For large reservoirs,
+the
+gain
+medium
+can
+be
+loosely
+thought
+as
+an
+infinite reservoir, recovering grand-canonical ensemble
+conditions. The large condensed mode photon number
+fluctuations, comparable to its mean value even above
+condensation threshold, are then identified to the so-
+called grand-canonical fluctuation catastrophe [56]. On
+the contrary, fluctuations become limited when the
+reservoir excitations number is smaller than the mean
+photon number.
+Besides the role of the volume, it has been shown that
+the β-factor has a strong influence on the fluctuations
+for micro- and nano-lasers [57, 58]. While macroscopic
+lasers with low β−factor show the usual steep crossover
+between thermal and Poissonian light at lasing threshold,
+in high-β devices the crossover is slow and occurs well
+above threshold. Knowing that the perfect equilibrium
+approach of Ref. [55] assumed a β = 1 cavity, this rather
+suggests that intensity fluctuations could be independent
+on the system thermalization, at least in the nanolaser
+limit.
+In summary, assessing the role of thermalization on
+fluctuations requires to carefully control both the effect of
+the size of the reservoir and the β-factor. With that many
+degrees of freedom, it is a theoretical challenging task.
+Methods to calculate the photon number distribution
+like master equations for the lasing mode photon number
+[29, 55, 59, 60] or stochastic rate equations [61–64] can
+hardly been solved numerically. In the next subsections,
+we proceed to a simpler investigation, focusing only on
+the second-order coherence at zero-time delay g(2)(0).
+We calculate this quantity by studying the small photon
+number deviations over the steady state in the Langevin
+approach. Interestingly, we note that for this quantity,
+this approach showed good agreement with a more
+rigorous stochastic rate equations model [61].
+Thus,
+this allows to get accurate and analytical insights for an
+observable easily accessible experimentally.
+B.
+Second-order coherence at zero delay time
+We base our investigation on the dynamical evolution
+equation of the photon number N j(t) in the cavity mode
+j that is lasing. For simplicity, we neglect the influence of
+other cavity modes on the system dynamics, and of non-
+radiative losses. Hereafter, we omit the mode superscript
+N j = N. Following the notations of Fig. 1 (a) for the
+various exchange pathways between the cavity, the gain
+medium and the environment, the dynamical evolution
+equation for N(t) is given by:
+dN
+dt = −κN +[Rem(Ne)−Rabs(Ne)]N +Rem(Ne), (18)
+where Ne(t) is the number of excited electrons in the
+gain medium.
+We switched from the variables µc and
+µv to Ne for simplicity. In semiconductors gain media,
+the excited electrons dynamics is usually commensurable
+with the one of the cavity photons [30].
+Hence the
+corresponding evolution equation of Ne(t) must be taken
+into account. According to Fig. 1, it yields:
+dNe
+dt
+= Rin −[Rem(Ne)−Rabs(Ne)]N − Rem(Ne)
+β
+, (19)
+where we used that Rvac
+em = (1/β − 1)Rem as follows
+from Eq. (3). In the following, β will be assumed to be
+independent on pumping, as usual in laser physics [30].
+All in all, these 2-coupled rate equations correspond to
+a standard class-B model broadly used to describe the
+dynamics of most semiconductor single mode lasers [30].
+We now note the steady-state solutions of equations
+(18),(19) as Nss, Ne,ss, respectively. We also introduce
+the small deviations δN(t), δNe(t), with |δN| ≪ Nss
+and |δNe| ≪ Ne,ss.
+We then linearize Eqs.
+(18),(19)
+to first order in these parameters. The noise due to the
+quantization of the emission, absorption, pumping and
+loss process is finally added to each equation through
+the respective stochastic terms Fp, Fe.
+We obtain the
+following coupled Langevin equations:
+
+10
+dδN
+dt
+= −γppδN + γpeδNe + Fp
+(20)
+and
+dδNe
+dt
+= −γepδN − γeeδNe + Fe,
+(21)
+where
+we
+have
+defined
+the
+short-hands
+γpp
+=
+−[Rem(Ne,ss) − Rabs(Ne,ss)] + κ, γpe = Nss∂Ne[Rem −
+Rabs](Ne,ss) + ∂NeRem(Ne,ss), γep
+=
+[Rem(Ne,ss) −
+Rabs(Ne,ss)],
+γee
+=
+Nss∂Ne[Rem − Rabs](Ne,ss) +
+(1/β)∂NeRem(Ne,ss). In addition, the stochastic terms
+verify the usual correlations properties ⟨Fx(t1)Fy(t2)⟩ =
+2Sxyδ(t1 − t2) with x, y
+∈
+(p, e) [30],
+where the
+expressions of the Sxy are given in Appendix D.
+In this linearized Langevin approach, the second-order
+intensity correlation g(2)(0) = ⟨N(0)[N(0) − 1]⟩/N 2
+ss is
+given by:
+g(2)(0) = 1 −
+1
+Nss
++ ⟨δN(0)2⟩
+N 2ss
+.
+(22)
+A detailed expression is then obtained by Fourier
+transforming (20),(21), so that the problem can be
+reformulated into a matricial form which is easy to invert.
+The full result, well-known in the literature [30, 61, 65],
+is given in Appendix D. Simple asymptotic expressions
+can be written for limiting values of some parameters, as
+discussed in the next subsection.
+C.
+Results
+We first focus on the usual macroscopic laser limit
+β → 0. From the full expression in Appendix D, simple
+algebra show that the second-order coherence at zero
+delay time reduces to:
+g(2)(0) = 1 +
+1
+1 +
+� Nss
+NLAS
+�2 ,
+(23)
+where NLAS is the photon number at lasing threshold.
+The coherence threshold is defined at g(2)(0) = 1.5
+corresponding to N = NLAS. Going straight to the point,
+this equality between the coherence and laser thresholds
+does not allow for a distinction between standard laser
+and photons BEC in this limit. Indeed here the crossover
+from thermal to Poissonian statistics always occurs at
+lasing threshold, regardless of the thermalization degree.
+We now focus on the opposite, ”nanolaser” limit β →
+1, where most of the spontaneous emission goes into the
+single cavity mode. The second-order coherence at zero
+delay time now follows the asymptotic behaviour [55, 57,
+58, 63, 65]:
+g(2)(0) = 1 +
+1
+1 +
+� Nss
+NCO
+�2 ,
+(24)
+where the coherence threshold is now given by NCO ≈
+�
+Rem(Nss=∞)
+∂Ne[Rem−Rabs](Nss=∞)
+�1/2
+. It is seen that NCO presents
+no explicit dependence on the thermalization degree.
+Coming back to our initial question, we conclude that
+the study of intensity fluctuations through the second-
+order coherence at zero delay time does not provide a
+mean to distinguish between the out-of-equilibrium laser
+and the BE condensation regimes.
+Finally, we discuss the consequence of this conclusion
+on the grand canonical fluctuation catastrophe. In the
+nanolaser limit, the coherence threshold is not given
+by the laser nor the BEC threshold. In particular, for
+realistic parameter values, the coherence threshold is
+shifted to much stronger pumping values than the laser
+threshold [57, 58, 65] and the BEC threshold [55]. Hence,
+there is a lasing/BEC regime with large fluctuations
+between these two thresholds. It is possible to attribute
+this regime to grand canonical fluctuations.
+Indeed,
+it has been shown [8] that the coherence threshold
+square N 2
+CO corresponds to an effective number of excited
+carriers in the gain medium.
+Therefore, in the range
+between the condensation and the coherence thresholds,
+the gain medium is large compared to the photon gas and
+can be considered to be an infinite reservoir. Remarkably,
+we find that the concept of grand canonical fluctuations
+is not restricted to equilibrium BE condensation but can
+be extended to non-equilibrium systems.
+V.
+CONCLUSION
+To summarize, we have explored the photon Bose-
+Einstein condensate regime for semiconductors in a
+cavity.
+Owing to the explicit form of the gain for
+semiconductors and the extensive body of knowledge for
+semiconductors lasers, this system is a very convenient
+playground which provides a theoretical framework to
+discuss both lasing and condensation.
+Using the
+Van Roosbroek-Schockley relation, we have shown that
+the photon Bose-Einstein condensation in the driven
+dissipative regime is a particular case of the lasing regime.
+The theoretical framework also enables to compare the
+definitions of threshold used either for condensation
+or for lasing.
+A Knudsen number emerges naturally
+from the analysis to characterize thermalization.
+We
+have discussed its close connection with other quantities
+introduced in different contexts such as thermalisation
+degree, optical thickness and cooperativity.
+Equipped
+with
+this
+theoretical
+figure
+of
+merit
+to
+quantify
+thermalization, we have analysed different experimental
+procedures to assess thermalization and put forward their
+strengths and limitations.
+Finally, we have explored
+the connection between the intensity fluctuations and
+
+11
+the emission regime.
+Large fluctuations are a priori
+expected to be a signature of the grand canonical regime
+typical of the equilibrium condensation. However, using
+a Langevin analytical model of the fluctuations in the
+driven-dissipative regime, we showed that the coherence
+threshold does not depend on the thermalization degree,
+both for large and small β-factors.
+In this paper, we have explored the stationary regime
+of a single BEC. The semiconductor platform appears to
+be a very fruitful playground to study BEC physics. An
+interesting direction for future work is to revisit in the
+BEC regime recent results obtained with semiconductor
+cavities such as topological lasers [66–68], chiral emission
+[69], nonlinearities [70] including superfluidity [71]. The
+platform is also well suited to further explore the
+dynamical behavior of BEC [72]. Also, the analysis of
+the fluctuations has revealed an interesting regime for
+micro and nanolasers above the lasing threshold and
+below the coherence threshold which can be viewed as
+grand-canonical fluctuations in non-equilibrium systems.
+This calls for more detailed studies of this phenomenon
+in the framework of open systems. It may provide new
+experimental platforms for the study of nonequilibrium
+statistical phenomena.
+ACKNOWLEDGMENTS
+We are grateful to Gian Luca Lippi for helpful
+discussions.
+This work is supported by the French
+National Agency (ANR) (ANR-17-CE24-0046). J.-J.G.
+acknowledges the support of Institut Universitaire de
+France (IUF).
+Appendix A: Microscopic expression of gl
+According to the Fermi golden rule, it is shown that gl
+can be cast into the form [30]:
+gl = 2π
+ℏ [ℏΩ]2ρJVactΓl,
+(A1)
+where ℏΩ is the projected light-matter coupling
+Hamiltonian between a single vertical transition and a
+plane wave, Vact is the volume of the gain medium (also
+called active medium) and Γl is the overlap integral
+between the gain medium and the cavity mode.
+The
+spatial structure of the mode electric field is thus fully
+contained in Γl which is thus mode dependent.
+Appendix B: Thermalization degree in devices
+featuring a large, planar and homogeneously
+pumped cavity
+Plugging Eq.
+(2) into Eq.
+(9), the thermalization
+degree has the form:
+Dl = gl
+κl fF D(Ev(kl), T, µv)[1 − fF D(Ec(kl), T, µc)],
+(B1)
+that is a pump-independent term gl/κl and a pump
+dependent term corresponding to the product of the
+Fermi-Dirac distributions.
+According to the Appendix A, the pump-independent
+term
+reads
+gl/κl
+=
+2π
+ℏ [ℏΩ]2ρJVactΓl/κl.
+In
+semiconductor devices featuring a large and planar
+cavity, the measurable spectrum typically extends over
+∼ 40 meV due to detection angle limitation and high
+refractive index material [26].
+The variations of the
+transition matrix element Ωl, the joint density of states
+ρJ and the mirror loss rate κl are negligible in this range
+[30]. Also, as the in-plane part of the modes is nearly a
+plane wave, the overlap integral Γl is constant. Hence,
+the pump-independent part of the thermalization degree
+gl/κl should thus be constant over the modes to a good
+approximation.
+We now focus on the pump dependent term, which
+describes the saturation of the absorption through
+pumping (see Section III C). Rigorously speaking, this
+term has always some dependence on the modes, since
+saturation of the absorption is greater for low energy
+modes.
+Still, it is showed on Fig.
+4 that the impact
+of this dependence is almost unnoticeable on the emitted
+spectrum for values of g/κ as low as ∼ 4. Noting that
+lasing is prevented if g/κ < 1 [73], the range over which
+the absorption saturation term has significant influence is
+quite narrow. As a conclusion, devices featuring a large,
+planar and homogeneously pumped cavity are expected
+to be well described by a constant Knudsen number over
+the modes.
+1.30
+1.35
+1.40
+1.45
+1.50
+E (eV)
+100
+101
+102
+103
+104
+105
+106
+Normalized occupation
+Gen. BE dis. (E,
+=1.3073eV,g/ =4)
+Fit by BE dis. (E,
+fit =1.2709eV)
+FIG. 4. Best fit of a generalized BE distribution ( Eq. (8))
+by an ideal BE distribution ( Eq. (6)). The fit parameter
+of the ideal BE distribution is the quasi-Fermi levels splitting
+µfit.
+gl and κl are assumed constant over all modes with
+g/κ = 4.
+All curves are normalized to 1 at E = 1.50 eV.
+Other parameters are compiled in Appendix C.
+
+12
+Appendix C: Model parameters values in figures
+We take parameter values representative of the VCSEL
+used in Ref.
+[26].
+The semiconductor gain medium
+consists of InGaAs quantum wells. We take Egap = 1.215
+eV, m∗
+c = 0.059 × me, m∗
+v = 0.37 × me where me
+is the electron mass [30].
+The hole mass corresponds
+to the valence band heavy-hole mass.
+Contribution of
+transitions with other valence bands is neglected.
+We
+assume room temperature operation T = 300 K.
+Appendix D: Full expression of g(2)(0)
+According to the treatment and notations of Sec.
+IV B, the full expression of the second-order intensity
+correlations at zero-time delay writes [30, 61, 65]:
+g(2)(0) = 1 −
+1
+Nss
++ γ2
+peSee + γpeγee(2Sp e) + (γpeγep + γppγee + γ2
+ee)Spp
+(γpp + γee)(γpeγep + γppγee)N 2ss
+,
+(D1)
+where 2See = 2Spp = 2Rem[Nss + 1] and 2Sep =
+−Rem[2Nss + 1] + Nss[Rem − Rabs].
+This expression can be expanded more explicitely as
+g(2)(0) = 1 −
+1
+Nss
++
+�
+1
+1 + Nss/Nβ + N 2ss/N 2
+CO
+�
+×
+�
+1
+1 + N 2ss/N 2
+LAS
+�
+×
+�
+1 +
+N 2
+ss
+N 2
+LAS
++ Nss
+Nβ
++
+� Nss
+NCO
+�2�
+∂NeRem
+∂Ne[Rem − Rabs] +
+Nss
+N 2
+LAS
++ κRem
+g
+��
+(D2)
+where Nβ ≈
+�
+1
+β − 1
+�−1
+Rem(Nss=∞)
+∂NeRem(Nss=∞) and NCO ≈
+�
+Rem(Nss=∞)
+∂Ne[Rem−Rabs](Nss=∞)
+�1/2
+.
+In the limit β → 1 (resp.
+β → 0), the first (resp. second) term between brackets
+dominates while the product of the other terms ≈ 1. The
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+in 1d topological active arrays, Physical review letters
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+Carlon
+Zambon,
+P.
+St-Jean,
+M.
+Mili´cevi´c,
+A. Lemaˆıtre,
+A. Harouri,
+L. Le Gratiet,
+O. Bleu,
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+controlling the emission chirality of microlasers, Nature
+Photonics 13, 283 (2019).
+[70] P.
+Hamel,
+S.
+Haddadi,
+F.
+Raineri,
+P.
+Monnier,
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+043814 (2020).
+[73] This
+follows
+from
+the
+gain
+clamping
+condition
+[Rl
+em(µclp) − Rl
+abs(µclp)]
+=
+κl. The left hand side
+of this expression is bounded by the full inversion value
+of the gain [Rl
+em(µclp → ∞) − Rl
+abs(µclp → ∞)] = gl.
+Losses that exceeds this bounds prevents gain clamping
+to occur, and thus lasing. Hence, the condition for lasing
+reformulates in gl/κl > 1.
+

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@@ -0,0 +1,3 @@
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+oid sha256:228037adb5680ae24a121dfc8258551d4a3837e0f38ea3d3ea3ac5665446d0a4
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CdA0T4oBgHgl3EQfAf80/content/tmp_files/2301.01962v1.pdf.txt

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+Astronomy & Astrophysics manuscript no. aanda
+©ESO 2023
+January 6, 2023
+Effect of stellar rotation on the development of post-shock
+instabilities during core-collapse supernovae
+A.-C. Buellet1, T. Foglizzo1, J. Guilet1, E. Abdikamalov2
+1 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France
+2Department of Physics and Energetic Cosmos Laboratory, Nazarbayev University, Astana 010000, Kazakhstan
+e-mail: [email protected]
+Received ??, 2022; accepted ??,2022
+ABSTRACT
+Context. The growth of hydrodynamical instabilities is key to trigger a core-collapse supernova explosion during the phase of stalled
+accretion shock, immediately after the birth of a proto-neutron star (PNS). Stellar rotation is known to affect the standing accretion
+shock instability (SASI) even for small rotation rates, but its effect on the onset of neutrino-driven convection is still poorly known.
+Aims. In this paper, we assess the effect of stellar rotation on SASI when neutrino heating is taken into account, and the effect of
+rotation on the convective instability when neutrino heating is higher. The interplay of rotation with these two instabilities affects the
+frequency of the mode m = 2 which can be detected with gravitational waves at the onset of a supernova explosion.
+Methods. We use a linear stability analysis to study the dynamics of the accreting gas in the equatorial plane between the surface of
+the PNS and the stationary shock. We explore rotation effects on the relative strength of SASI and convection by considering a large
+range of specific angular momenta, neutrino luminosities and mass accretion rates.
+Results. The nature of the dominant non-axisymmetric instability developing in the equatorial postshock region depends on both
+the convection parameter χ and the rotation rate. Equatorial convective modes with χ ≳ 5 are hampered by differential rotation. At
+smaller χ, however, mixed SASI-convective modes with a large angular scale m = 1, 2, 3 can take advantage of rotation and become
+dominant for relatively low rotation rates at which centrifugal effects are small. For rotation rates exceeding ∼ 30% of the Keplerian
+rotation at the PNS surface, the growth rate of the dominant mode depends weakly on the rate of neutrino heating which highlights
+the arising of a new instability regime. Similarly, the frequency of this mode is surprisingly independent of the heating rate, with a
+strong prograde spiral m = 2 dominating over a large parameter range which is favourable to the production of gravitational waves.
+For any heating rate, a simple linear relation exists between the oscillation frequency of the dominant mode and the specific angular
+momentum of the accreted gas.
+Conclusions. Three different regimes of postshock instabilities can be distinguished depending on the rotation rate. For small rotation
+rates (less than 10% of the Keplerian rotation at the PNS surface), differential rotation has a quadratic stabilising or destabilising
+effect on equatorial purely convective modes and a linear destabilising effect on SASI. Intermediate rotation rates (10 to 30% of the
+Keplerian rotation) lead to the emergence of mixed SASI/convection/rotation modes involving large angular scales. Finally, strong
+rotation erases the influence of the buoyancy and heating rate on the instability. This independency allows for a reduction of the
+parameter space, which is valuable for gravitational wave analysis.
+Key words. supernova – convection – rotation – hydrodynamics
+1. Introduction
+The death of massive stars begins with the collapse of their inner
+core, which forms a proto-neutron star (PNS). The outer core of
+the star bounces off and creates a shock wave that propagates
+outward, gradually losing energy, dissociating iron atoms until it
+stalls. The development of multidimensional instabilities during
+this phase of stalled shock impacts both the multimessenger sig-
+nature and the revival of the shock (Herant et al. 1994; Janka &
+Mueller 1996; Couch & O’Connor 2014; Takiwaki et al. 2016;
+Janka et al. 2016; Müller 2020; Burrows & Vartanyan 2021).
+Among them, the stationary accretion shock instability (SASI)
+(Blondin et al. 2003; Blondin & Mezzacappa 2006) can gener-
+ate shock oscillations and contribute to pushing the shock fur-
+ther up (Scheck et al. 2008; Marek & Janka 2009; Hanke et al.
+2013). The magnitude of this effect depends on the concurrent
+growth of the neutrino-driven convection, which can generate
+turbulence in the post-shock region (Abdikamalov et al. 2015;
+Radice et al. 2016). Different paths to explosions dominated ei-
+ther by neutrino-driven convection or SASI may occur depend-
+ing on the precise physical conditions determined by the progen-
+itor structure (Müller et al. 2012; Murphy et al. 2013; Fernández
+et al. 2014) and the magnitude of pre-collapse turbulence asym-
+metries (Müller et al. 2017).
+The development of neutrino-driven convection and/or SASI
+leaves clear signatures in gravitational waves (Murphy et al.
+2009; Kuroda et al. 2016; Andresen et al. 2017). The eigenfre-
+quencies deduced from the perturbative study of the accretion
+flow can be recognised in the gravitational wave signal despite
+the nonlinear development of instabilities (Torres-Forné et al.
+2018, 2019b) and can be used to reconstruct the flow struc-
+ture (Torres-Forné et al. 2019a; Sotani & Takiwaki 2020; Sotani
+et al. 2021). These asteroseismic properties can help constrain
+the equation of state at nuclear densities (Kuroda et al. 2016;
+Sotani et al. 2017).
+The neutrino signal is also a precious messenger carrying
+direct information on the frequency of large-scale shock oscilla-
+Article number, page 1 of 16
+arXiv:2301.01962v1  [astro-ph.HE]  5 Jan 2023
+
+A&A proofs: manuscript no. aanda
+tions induced by SASI, currently detectable for a galactic super-
+nova (Tamborra et al. 2013).
+The competition between SASI and convection seems to vary
+from one numerical code to another (Hanke et al. 2013; Ott et al.
+2018; O’Connor et al. 2018; Glas et al. 2019). A detailed un-
+derstanding of the instabilities mechanisms can help interpret
+the results of numerical simulations and guide the exploration
+of the parameter space. SASI results from the interaction of
+pressure and advected perturbations of entropy and vorticity be-
+tween the shock with the surface of the PNS (Foglizzo et al.
+2007; Foglizzo 2009; Fernández & Thompson 2009; Guilet &
+Foglizzo 2012). The convective instability in the post-shock re-
+gion is driven by neutrinos emitted by the cooling PNS: in the re-
+gion where the absorption of neutrino energy exceeds the losses
+by neutrino emission, the heating by neutrino absorption cre-
+ates a negative entropy gradient favourable to convection (Bethe
+1990). The growth of this neutrino-driven convection requires a
+strong enough neutrino heating such that the buoyancy timescale
+is shorter than one third of the advection timescale across this re-
+gion (Foglizzo et al. 2006). Above a critical heating rate, the fun-
+damental oscillatory mode of SASI becomes the purely growing
+convective instability that dominates the dynamics of the flow
+(Yamasaki & Yamada 2007; Fernández et al. 2014).
+Most core-collapse simulations and theoretical work neglect
+the impact of rotation, which is therefore not well known (Suwa
+et al. 2010; Chatzopoulos et al. 2016; Fujisawa et al. 2019; Pow-
+ell & Müller 2020). Rotation is however thought to play an im-
+portant role in at least a small fraction of core-collapse super-
+novae with more extreme properties, such as superluminous su-
+pernovae (Woosley 2010; Inserra et al. 2013) or hypernovae and
+long gamma-ray bursts (Woosley 1993; Metzger et al. 2011). It
+is furthermore possible that rotation plays a less extreme role in
+a larger fraction of core-collapse supernovae. Theoretical mod-
+els of stellar evolution constrained by the efficient transport of
+angular momentum inferred from asteroseismic observations of
+red giants (Cantiello et al. 2014), and the observations of pulsar
+spins (Popov & Turolla 2012) suggest that the majority of su-
+pernova explosions originate from slowly rotating stellar cores
+(< 0.5 rad/s at 1000km).
+Rotation rates up to 2 rad/s are more exceptional but com-
+monly considered to explain extreme events such as hypernovae,
+superluminous supernovae and GRBs. In this regime, the con-
+vective dynamo (Thompson & Duncan 1993; Raynaud et al.
+2020, 2022; Masada et al. 2022; White et al. 2022) and the mag-
+netorotational instability (Akiyama et al. 2003; Obergaulinger
+et al. 2009; Guilet & Müller 2015; Guilet et al. 2022; Reboul-
+Salze et al. 2021, 2022) are expected to amplify efficiently the
+magnetic field of the PNS. Raynaud et al. (2020) predicted that
+magnetar-like magnetic fields can be generated by the strong
+field branch of the convective dynamo, which takes place for spe-
+cific angular momentum larger than 4 × 1015 cm2/s. This thresh-
+old corresponds to an angular frequency as low as 0.4 rad/s at
+1000 km in the progenitor. The extraction of rotational energy
+with such a strong magnetic field can lead to strong magnetorota-
+tional explosions (e.g. Burrows et al. 2007; Takiwaki et al. 2009;
+Kuroda et al. 2020; Obergaulinger & Aloy 2021; Bugli et al.
+2021).
+Beside the generation of magnetic fields, rotation can have
+several other effects: the centrifugal force affects the shape of
+the neutrinosphere; the differential rotation modifies the devel-
+opment of SASI and convective instabilities; differential rotation
+can also generate a new instability. These multiple effects are
+often studied in the hydrodynamical approximation because the
+dynamo timescale is uncertain.
+The centrifugal force diminishes the action of gravity and
+results in a larger radius of the neutrinosphere in the equatorial
+plane. The equatorial decrease in neutrino luminosity and neu-
+trino mean energy is not favourable to the explosion according
+to axisymmetric simulations (Marek & Janka 2009). The lower
+equatorial temperature favours neutrino heating in the polar re-
+gion and a bipolar explosion (Suwa et al. 2010).
+Stellar rotation can be favourable to the development of a
+vigorous, prograde, spiral SASI mode dominated by large an-
+gular scales l = 1, 2 when neutrino absorption is neglected
+(Blondin & Mezzacappa 2007). This destabilising effect of dif-
+ferential rotation exists even for slow rotation with a negligible
+centrifugal contribution, as shown by perturbative analyses (Ya-
+masaki & Foglizzo 2008; Walk et al. 2022) and confirmed by nu-
+merical simulations (Kazeroni et al. 2017; Blondin et al. 2017).
+However, the driving mechanism of this rotational destabilisa-
+tion is not understood yet (Walk et al. 2022).
+Classical studies of the effect of rotation on convection con-
+sidered a rotation axis aligned with gravity. In a viscous fluid
+with thermal diffusion, the critical Rayleigh number defining
+the onset of thermal convection is increased by rotation (Chan-
+drasekhar 1961; Rossby 1969; Wedi et al. 2021). The few stud-
+ies that considered the impact of rotation in the equatorial plane
+did not involve radial advection, which is crucial in the super-
+nova case (Feudel & Feudel 2021). According to the Solberg-
+Høiland criterion, the development of axisymmetric convection
+can be stabilised by rotation if the specific angular momentum
+increases outward (Endal & Sofia 1978). In axisymmetric simu-
+lations of stellar core-collapse, this effect produces less vigorous
+convective motions in the equatorial plane, and results in later-
+time explosions which are weaker at the equator than at the poles
+(Fryer & Heger 2000). These 2D results seemed to be confirmed
+in 3D for the fastest spinning progenitors (Fryer & Warren 2004)
+(Ω ∼ 4.1 rad/s at 1000 km), in a regime where centrifugal effects
+can be dominant at reducing both the effective gravity and the
+neutrino luminosity in the equatorial plane, and thus a reduced
+buoyancy compared to the polar region.
+Estimating the effect of modest rotation in the equatorial re-
+gion of post-shock convection is less obvious when centrifugal
+effects are small, remembering that inward accretion produces
+a uniform profile of specific angular momentum. With a rota-
+tion rate of Ω ∼ 1.3 rad/s at 1000 km, core-collapse simulations
+showed an earlier onset of neutrino-driven convection that pro-
+duced a stronger explosion in the equatorial plane, earlier than
+the non-rotating case (Nakamura et al. 2014).
+Even a modest amount of rotational kinetic energy T com-
+pared to the potential energy |W| can trigger a spiral instability
+known as “low-T/|W| instability” in the interior of isolated neu-
+tron stars (Shibata et al. 2002; Watts et al. 2005; Passamonti &
+Andersson 2015). The mechanism of this instability relies on
+the extraction of energy and angular momentum from internal
+regions rotating faster than the spiral pattern towards external re-
+gion rotating slower (Cairns 1979; Saijo & Yoshida 2006). This
+instability has been observed in 3D simulations of stellar core-
+collapse, where it enhances the energy transport from the PNS
+to the shock and can lead to stronger explosions (Ott et al. 2005;
+Cerdá-Durán et al. 2007; Takiwaki et al. 2016, 2021).
+The interplay of centrifugal effects, SASI, convection and the
+low-T/|W| instability can be difficult to disentangle in numerical
+simulations considering the diversity of progenitors and numer-
+ical approximations (Ott et al. 2008). Our current understanding
+relies on a very sparse sampling of this diversity. During the col-
+lapse of a 27M⊙ progenitor, the dynamic of the shock is driven
+by SASI and the convection for small rotation rates, and domi-
+Article number, page 2 of 16
+
+A.-C. Buellet, T. Foglizzo, J. Guilet, E. Abdikamalov: Stellar rotation and post-shock instabilities during core-collapse
+nated by the low-T/|W| instability for high enough rotation rates
+(Ω = 2 rad/s) (Takiwaki et al. 2021). The enhancement of SASI
+by differential rotation can compensate for the loss of neutrino
+energy due to the centrifugal force during the collapse of a 15M⊙
+progenitor (Summa et al. 2018). The gravitational wave (GW)
+analysis of these simulations can help identify the physical pro-
+cesses, such as the enhancement of SASI for high rotation rates
+(Andresen et al. 2019). The 3D simulations of stationary accre-
+tion by Iwakami et al. (2014) considered a range of rotation rates,
+mass accretion rates, and neutrino luminosities. The structure of
+the dominant instability and the observed patterns were classi-
+fied according to three main categories: spiral, buoyant bubbles,
+or spiral and buoyant bubbles. However, the simultaneous varia-
+tion of both rotation and neutrino luminosity made it difficult to
+disentangle their respective effects.
+To have a better understanding of the effect of each parameter
+on the growth of instabilities, we propose to use a linear analy-
+sis, varying the rotation and the heating rate separately. Doing
+this, we will be able to disentangle the effect of these parameters
+in the linear regime. By focusing on the accretion region above
+the surface of the PNS, we do not include the interaction with
+the low-T/|W| instability developing inside the PNS. The aim of
+this paper is thus to study the impact of rotation on the onset of
+the convective instability and its interplay with SASI, when both
+the neutrino heating and the rotation rate are varied. In Sect. 2,
+we detail the numerical set-up and define the stationary and per-
+turbed flows. We study in Sect. 3 the effect of rotation on SASI,
+convection and their interplay. Finally, we focus in Sect. 4 on the
+features that might be observable in a gravitational wave signal
+coming from an exploding supernova.
+2. Methods
+2.1. Numerical setup
+To study the growth of the convective and SASI instabilities, we
+solve numerically the system of perturbed equations correspond-
+ing to an idealised model of a perfect gas, in spherical geometry
+restrained to the equatorial plane, using the coordinates (r, φ).
+The parameters we vary are the reference shock radius rsh0 ob-
+tained without neutrino heating, dissociation and rotation, the
+rate ε of nuclear dissociation across the shock, the neutrino lumi-
+nosity and the specific angular momentum J reaching the surface
+of the PNS.
+2.1.1. Stationary flow
+In spherical geometry, the system of stationary equations de-
+scribing the conservation of mass, the entropy profile S (r) and
+the profile of the Bernoulli parameter in the equatorial plane are:
+∂
+∂r
+�
+ρvr2�
+=
+0,
+(1)
+∂S
+∂r
+=
+L
+Pv,
+(2)
+∂
+∂r
+�v2
+2 + J
+2r2 +
+c2
+γ − 1 − GM
+r
+�
+=
+L
+ρv,
+(3)
+where G is the universal gravity constant and M is the mass of
+the PNS, v the radial velocity and c the sound speed in the gas
+with pressure P and density ρ. The self-gravity of the infalling
+matter is neglected compared to that of the PNS. Non-adiabatic
+heating and cooling processes are described by a local function
+L ≡ Lh +Lc, as in Fernández et al. (2014). The cooling function
+Lc used in Houck & Chevalier (1992) is a parametric function
+of P and ρ:
+Lc = −Ac ρβ−αPα.
+(4)
+We use α = 3/2 and β = 5/2 as in Blondin & Mezzacappa
+(2006), Foglizzo et al. (2007), Yamasaki & Foglizzo (2008), Fer-
+nández & Thompson (2009), Fernández et al. (2014), Guilet &
+Foglizzo (2012) and Blondin et al. (2017). The normalisation
+constant Ac describes the micro-physical processes responsible
+for the cooling by neutrino emission. Its value is determined so
+that the radial velocity vanishes at the surface rPNS of the PNS.
+Using dimensional analysis, we express Ac as a function of the
+PNS radius rPNS, the surface gravity GM/r2
+PNS and the mass ac-
+cretion rate ˙M:
+Ac = ˜Ac × ˙M1−β
+������
+GM
+r2
+PNS
+������
+1−α+ β
+2
+r
+5β
+2 −2−α
+PNS
+,
+(5)
+where ˜Ac is a dimensionless quantity. The value of ˜Ac sets the
+value of the shock radius without dissociation, rotation, or heat-
+ing. ˜Ac does not vary when these parameters are changed. The
+heating function Lh is expressed as
+Lh = Ah
+ρ
+r2 .
+(6)
+The normalisation constant Ah, which is proportional to the neu-
+trino luminosities and the neutrino opacities per unit mass, is
+varied as a free parameter in our model. The heating and cooling
+functions are effective in the post-shock region and are turned
+off above the shock, as in Fernández et al. (2014).
+In order to study the influence of stellar rotation on the
+growth of the instabilities, we assume a constant specific angular
+momentum J. Its dimensionless measure j is defined using the
+Keplerian specific angular momentum at the PNS surface
+j ≡
+J
+(GMrPNS)1/2 .
+(7)
+We note that j2 measures the ratio of the centrifugal and gravita-
+tional forces at the PNS radius:
+J2
+r3
+PNS
+r2
+PNS
+GM = j2.
+(8)
+In our analysis, we vary j from 0 to 0.5 which corresponds to
+a maximum centrifugal force equal to 25% of the gravitational
+force at the PNS boundary. The angular momentum at 50% of
+the Keplerian rotation can be expressed as
+J
+1.5 × 1016cm2/s
+=
+j
+0.5
+� rPNS
+50km
+M
+1.4M⊙
+�1/2
+,
+(9)
+=
+Ω
+1.5rad/s
+�
+r
+1000km
+�2
+,
+(10)
+where Ω in the second equation is the angular frequency in the
+progenitor at a reference radius r. For reference, the strong mag-
+netic field mentioned in Raynaud et al. (2020) could be generated
+for j ≳ 0.13. Our hydrodynamical study neglecting magnetic
+fields assumes that dynamo processes are too slow to interfere.
+Our input parameters and normalisation are similar to Fer-
+nández et al. (2014), with the addition of stellar rotation. The
+dissociation parameter ε corresponds to the fraction of kinetic
+energy of the incoming matter that is used to photo-dissociate
+Article number, page 3 of 16
+
+A&A proofs: manuscript no. aanda
+the iron nuclei. ε is expressed in units of the specific kinetic en-
+ergy v2
+ff0/2 associated to free-fall at the reference radius rsh0. The
+Mach number is imposed so that M1 = 5 above the shock, with-
+out heating, rotation, or dissociation. The Bernoulli parameter
+is set to zero above the shock, and the adiabatic index is set to
+γ = 4/3. The default values of the shock radius and the disso-
+ciation rate are rsh0 = 5rPNS and ε = 0. For each figure, the
+value rsh0 = 5rPNS is used and the only exception is the use of
+rsh0 = 3.2rPNS for comparison in Figs. 1 and 5. The value of
+rsh specified in each figure corresponds to the shock radius tak-
+ing into account the parameters (Ah, ε, J). The shock radius de-
+creases when dissociation increases. As the rotation or the heat-
+ing rate Ah is increased, the shock radius increases (blue curve in
+Fig. 1) and the corresponding Mach number immediately above
+the shock decreases.
+Distances are normalised with either the PNS radius rPNS or
+the reference shock radius rsh0, densities with the density imme-
+diately above the shock ρ10, and velocities with the free-fall ve-
+locity vff0 at rff0. These units are chosen for (Ah, ε, J) = (0, 0, 0)
+and they do not vary when (Ah, ε, J) are varied. With these units
+Ah = ˜Ahv3
+ff0rsh0,
+(11)
+where ˜Ah is a dimensionless parameter characterising the heating
+rate. Another way to measure the heating rate is to use the χ
+parameter defined in Foglizzo et al. (2006) as
+χ ≡
+� rsh
+rg
+N(r)dr
+|v| ,
+(12)
+where rg is the gain radius above which absorption of neutrino
+energy exceeds the losses by neutrino emission, and N(r) is the
+local Brunt-Väisälä frequency:
+N ≡
+�γ − 1
+γ
+g∇S
+� 1
+2
+.
+(13)
+In this equation, the gravity term g was not corrected by the cen-
+trifugal force for the sake of simplicity, which is acceptable be-
+cause of the low value of the ratio j2 of those two forces. Indeed,
+Eq. (8) shows that even for our rapid rotation rate, the ratio of
+those forces ∼ 7% in the middle of the post-shock region.
+Figure 1 indicates that the gain radius varies little with the
+strength of neutrino heating or the dissociation rate when the
+heating rate is high enough. The gain radius is located near the
+shock for small values of χ, approaches twice the PNS radius
+for χ ∼ 2 and remains constant for χ ≫ 2 while the radius of
+the stationary shock is pushed further out by the strong heating
+of the matter. For χ ≳ 2, the gain radius stalls at ∼ 2rPNS and
+the size of the gain region increases proportionally to the shock
+radius.
+The stronger the dissociation, the lower the energy after the
+shock. This leads to lower velocities of sound and matter than in
+the case without dissociation. As a result, the integrated intensity
+of the cooling function needed to decelerate across the shocked
+region is reduced. As ˜Ac is determined with ε = 0, the size of the
+shocked region decreases when the dissociation rate increases,
+as observed in Fig. 1. The impact of rotation is barely visible in
+this figure with j = 0.08 because of the quadratic dependence of
+the centrifugal force with the rotation frequency (= j2 < 1%, see
+Eq. (8)).
+However, we will see in the next section that this modest
+rotation is large enough to have significant effects on SASI and
+the convective instability.
+Fig. 1. Evolution of the shock (blue) and gain (red) radii as a function of
+χ. Unless stated otherwise, the values of the parameters are rsh0 = 5rPNS,
+ε = 0 and j = 0. For the moderate value of angular momentum j = 0.08,
+the stationary flow is almost unaffected by rotation. The case (j, ε) =
+(0, 0) for rsh0 = 3.2rPNS is plotted in dashed lines for comparison with
+the case including dissociation. As expected, the shock radius increases
+with χ and the introduction of rotation increases the shock radius when
+the cooling normalisation Ac is conserved.
+2.1.2. Perturbed flow
+For the perturbed system, we use the same set of perturbed vari-
+ables as in Yamasaki & Foglizzo (2008). δ f, δh, δS , and δq are
+defined as follows:
+δ f
+≡
+vδv + J
+r δvφ +
+2c
+γ − 1δc − δq,
+(14)
+δh
+≡
+δv
+v + δρ
+ρ ,
+(15)
+δS
+≡
+2
+γ − 1
+δc
+c − δρ
+ρ ,
+(16)
+δq
+≡
+δ
+��
+L
+ρvdr
+�
+.
+(17)
+The time and angular dependence of the perturbations are as-
+sumed to follow the form exp (−iωt + imφ). Their radial depen-
+dence is then governed by the same differential equations as
+Eqs. (6-11) in Yamasaki & Foglizzo (2008) with kz = 0, despite
+the different geometry and the inclusion of neutrino heating in
+the local function L:
+d
+dr
+δ f
+ω
+=
+ic2
+v �1 − M2�
+�
+M2
+�
+δh − ω′
+c2
+δ f
+ω
+�
+(18)
++
+�
+1 + (γ − 1) M2� δS
+γ − δq
+c2
+�
+,
+dδh
+dr
+=
+iω
+v �1 − M2�
+�
+µ2 ω′
+c2
+δ f
+ω − M2δh − δS + δq
+c2
+�
+,
+(19)
+dδS
+dr
+=
+iω′
+v δS + δ
+� L
+Pv
+�
+,
+(20)
+dδq
+dr
+=
+iω′
+v δq + δ
+� L
+ρv
+�
+,
+(21)
+where
+ω′
+≡
+ω − mJ
+r2 ,
+(22)
+µ2
+≡
+1 − m2
+r2
+c2
+ω′2
+�
+1 − M2�
+.
+(23)
+Article number, page 4 of 16
+
+10
+j= O,ε=,rsho =5rpNS
+shock radius
+- j= 0.08
+8
+-- j = 0.4
+rsho=3.2rpNS
+=0.3
+6
+r[rpNs]
+4
+2
+gain radius
+2
+4
+6
+8
+10
+xA.-C. Buellet, T. Foglizzo, J. Guilet, E. Abdikamalov: Stellar rotation and post-shock instabilities during core-collapse
+In these equations ω′ corresponds to the Doppler shifted value
+of ω and is thus a function of radius. Note that this effect of
+rotation is linear with respect to Ω, whereas the centrifugal force
+is proportional to Ω2 as discussed above. A linear dependence
+allows for strong effects of rotation for slow rotation, as will be
+discussed later.
+The boundary conditions at the shock are expressed using
+conservation laws across a perturbed shock:
+δfsh
+ω
+=
+iv1∆ζ
+�
+1 − vsh
+v1
+�
+,
+(24)
+δhsh
+=
+−i ω′
+vsh
+∆ζ
+�
+1 − vsh
+v1
+�
+,
+(25)
+δS sh
+=
+iω′v1
+c2
+sh
+∆ζ
+�
+1 − vsh
+v1
+�2
+− Lsh − L1
+ρshvsh
+∆ζ
+c2
+sh
+(26)
++
+�
+1 − vsh
+v1
+� ∆ζ
+c2
+sh
+������
+2v1vsh
+rsh
++ J2
+r3
+sh
++ GM
+r2
+sh
+������ ,
+δqsh
+=
+−Lsh − L1
+ρshvsh
+∆ζ,
+(27)
+where ∆ζ is the shock displacement, the indices “1” and “sh”
+refer to the values above and below the shock, respectively.
+The only difference with the boundary conditions (12-15) in Ya-
+masaki & Foglizzo (2008), who assumed a cylindrical symme-
+try, is the factor 2 implied by our choice of spherical-equatorial
+geometry, in the term 2v1vsh/rsh in Eq. (26).
+We use the same lower boundary condition at the PNS sur-
+face, where the radial velocity vanishes. We use the Newton-
+Raphson method to find the eigenvalues of this system when ε,
+M1, χ, and rsh (or ˜Ac) are given. This method determines the per-
+turbation growth rate (ωi) and frequency (ωr) for each azimuthal
+number m.
+In order to identify the effect of the four parameters
+(ε, rsh0, J, χ) on the growth of instabilities, we select a mode and
+follow its evolution when the parameters are varied. For a given
+value of m, we study the evolution of each harmonic without ro-
+tation and select the overtone with the highest growth rate, which
+happens to be the fundamental mode as obtained by Yamasaki &
+Yamada (2007). We follow the properties of this fundamental
+mode when parameters (ε, rsh0, J, χ) are varied. When rotation
+exceeds j ∼ 0.15, we noticed that some higher harmonics can be
+slightly more unstable than the fundamental mode, but this effect
+is marginal.
+3. Results
+3.1. SASI and the convective instability without rotation
+Figure 2 shows the effect of the heating parameter χ on the
+growth rate and the frequency of the fundamental mode corre-
+sponding to several azimuthal numbers m = 1 to 5. Similarly
+to Yamasaki & Yamada (2007) and Fernández et al. (2014), for
+each value of m, the instability is oscillatory (like SASI) for mod-
+est heating and purely growing (like the convective instability)
+for stronger heating. The most unstable perturbations in the os-
+cillatory regime have a large angular scale corresponding to a
+small azimuthal number m = 1, as expected for SASI. For each
+mode, the SASI frequency decreases abruptly to zero when the
+heating rate is increased. The transition to a mode with zero fre-
+quency corresponds to a steep increase of the growth rate. The
+azimuthal scale of the dominant convective modes is smaller (i.e.
+Fig. 2. Evolution of the modes as a function of χ parameter. Two in-
+stability domains can be identified: SASI (ωr �0) for modest neutrino
+heating and the convective instability (ωr =0) for stronger heating.
+larger m, here m = 4) than SASI modes. The heating rate corre-
+sponding to the transition between SASI and convection is ∼7%
+smaller than expected by Fernández et al. (2014). This may be
+due to their use of an entropy cut-off in the heating/cooling func-
+tion, exp − (S/S min)2, where S min is the entropy at the PNS sur-
+face. Except for this small quantitative difference, the general
+behaviour is the same as in Fernández et al. (2014).
+To study the influence of the parameters ε and J on the
+growth of the convective instability (Fig. 3), we fixed the value
+of χ = 4 such that the dominant mode is convective for every
+dissociation rates. We compare the value mnum corresponding to
+the most unstable mode obtained numerically, to the expected
+value estimated analytically by Foglizzo et al. (2006):
+mana = π
+2
+rsh + rg
+rsh − rg
+.
+(28)
+This geometric formula is based on the assumption that convec-
+tive cells have a circular shape, radially centred in the gain re-
+gion. It neglects the geometry of the star and the fact that the
+local value of the Brunt-Väisälä frequency is highest near the
+gain radius and the width of its radial profile is inferior to the
+gain region size. As a result, the limiting scale for the convective
+cells is smaller than the whole gain region and comparable to the
+size of the most buoyant region, as can be seen in Fig. 10. This
+may explain the offset between the blue triangles and the blue
+curve in Fig. 3, indicating that mana over-estimates the angular
+size of the dominant convective scale.
+The magnitude of the increase of mnum with ε in Fig. 3 is
+consistent with the increase predicted by Eq. (28) based on the
+increase of the size of the gain region visible in Fig. 1.
+We also varied rsh0 between 2 and 4.5 times the PNS radius
+by changing the cooling intensity ˜Ac and adapting the pre-shock
+Mach number M1 such that M1 = 5 for rsh = 5rPNS, ε = 0. For
+this variable, the agreement between mnum and mana is similar to
+the one obtained in Fig. 3 without dissociation.
+Article number, page 5 of 16
+
+E=0,j=0,rsho = 5rpNS
+4
+[Vsh/rsh]
+2
+0
+3
+-4
+5
+m=1
+432
+[Vsh/rsh]
+m=2
+m=3
+m=4
+m=5
+31
+0
+0
+1
+2
+3
+4
+5
+6
+7
+XA&A proofs: manuscript no. aanda
+Fig. 3. Influence of the dissociation on the azimuthal number of the
+most unstable fundamental mode mnum (triangles) at χ = 4 for three
+different values of the rotation rate. The numerical values of the most
+unstable modes (triangle) are compared to the analytical predictions of
+Eq. (28) (dashed lines) as a function of the dissociation rate. We al-
+low for quarter-integer values of mnum for an easier comparison with
+Eq. (28).
+3.2. Influence of stellar rotation
+In this part, we characterise the dependence of the growth rate
+ωi and the oscillation frequency ωr on the rotation rate for both
+SASI and the convective instability.
+3.2.1. Effect of rotation and heating on SASI
+The growth rate of SASI is expected to increase approximately
+linearly with the rotation rate, and the angular scale of the dom-
+inant mode is expected to diminish when the rotation rate in-
+creases. The corotation radius, defined as
+rcorot ≡
+�mJ
+ωr
+�1/2
+,
+(29)
+is also expected to increase with the rotation rate. These re-
+sults were obtained without neutrino heating using a perturbative
+analysis and numerical simulations both in cylindrical geometry
+(Yamasaki & Foglizzo 2008; Kazeroni et al. 2017) and in spher-
+ical geometry (Blondin et al. 2017; Walk et al. 2022).
+Our perturbative analysis in spherical equatorial geometry,
+shown in Fig. 4, confirms these results for χ = 0 and measures
+the impact of neutrino heating for χ = 2. The lower plot in Fig. 4
+shows the same dependence of the corotation radius on the rota-
+tion rate as in Yamasaki & Foglizzo (2008). We note that increas-
+ing the rotation rate introduces a corotation radius rcorot > rPNS
+associated to the mode m = 2 before the mode m = 1. This hier-
+archy is conserved when neutrino heating is taken into account.
+In addition, in Yamasaki & Foglizzo (2008), the frequencies for
+which rcorot = rPNS are fp2 ∼ 0.15kHz and fp1 ∼ 0.25kHz for
+m = 2 and m = 1, respectively. These values are very close to
+our results, where fp2 ∼ 0.17kHz and fp1 ∼ 0.28kHz.
+The growth rate of the mode m = 1, dominant without rota-
+tion, becomes inferior to the growth rate of the mode m = 2 with
+higher rotation rates. The slope of the m = 2 mode in the upper
+plot in Fig. 4 is indeed steeper than that of the m = 1 mode. We
+find that this behaviour is robust when neutrino heating is taken
+into account, as demonstrated in Fig. 4 for χ = 2.
+Fig. 4. Evolution of the growth rate ωi in units of vsh/rsh (top), and the
+corotation radius in units of rPNS (bottom) as a function of the rotation
+rate, measured as in Yamasaki & Foglizzo (2008) with fp, the pulsar
+frequency at 10 km. The azimuthal numbers m = 1 (blue) and m = 2
+(red) are displayed for χ = 0 (solid line), and χ = 2 (dash dotted line).
+To allow for a comparison with Fig. 1 of Yamasaki & Foglizzo (2008),
+we considered the same parameter rsh = 5rPNS with a strong adiabatic
+shock.
+In the absence of dissociation and heating (solid lines in
+Fig. 4), the observed transition with the mode m = 2 dominat-
+ing the instability for j ∈ [0.3, 0.5] is consistent with the results
+of Blondin et al. (2017) where the mode m = 2 dominates for
+j ∼ 0.36 (corresponding to h = 0.115 in their units). When neu-
+trino heating is taken into account, Fig. 4 shows that the tran-
+sition from m = 1 to m = 2 requires a slightly larger amount
+of rotation. On the other hand, this transition requires a smaller
+amount of rotation when the shock radius is smaller or when dis-
+sociation is taken into account (Fig. 5). The influence of disso-
+ciation can be explained by its effect on the advection timescale.
+For a given shock radius (kept constant in Fig. 5), the mat-
+ter velocity decreases with ε because of the larger compression
+through the shock. As a consequence, the advection timescale
+rsh/vsh increases with ε and the frequency of the SASI modes
+decreases. A slower rotation is therefore sufficient to be compa-
+rable to the SASI frequency, which may explain the steeper effect
+of rotation. This interpretation is confirmed by the bottom panel
+of Fig. 5 where the rotation rate has been normalised using the
+advection timescale. This renormalisation indeed collapsed the
+curves with and without dissociation, at least for the small rota-
+tion rates. Note, however, that this argument cannot explain the
+dependence on the shock radius, which remains mysterious.
+3.2.2. Effect of rotation on the convective instability
+We consider in Fig. 6 the effect of the heating rate on the eigen-
+frequency of the mode m = 4 for different rotation rates. The
+mode m = 4 was chosen because it is the first mode to ex-
+hibit a convective instability when j = 0. The most striking fea-
+ture regarding the oscillation frequency (lower plot in Fig. 6),
+compared to Fig. 2, is the loss of a sharp transition between an
+oscillatory instability for modest neutrino-heating and a purely
+Article number, page 6 of 16
+
+6
+j=0
+j=0.08
+j=0.25
+5
+analytical
+4
+m
+3
+2
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0j(%)
+0
+10
+20
+30
+40
+50
+2.5
+W; [Vsh/rsh]
+2
+m=1
+1.5
+1
+m=2
+3 0.5
+0
+X=0
+3
+rcorot[rpNs]
+X=2
+2
+1
+0
+0.0
+0.5
+1.0
+1.5
+2.0
+fp (kHz)A.-C. Buellet, T. Foglizzo, J. Guilet, E. Abdikamalov: Stellar rotation and post-shock instabilities during core-collapse
+Fig. 5. Influence of rotation on the growth rate of SASI with dissoci-
+ation (solid lines) and without dissociation (dashed lines). The bottom
+panel presents the rotation rate normalised with a proxy for the advec-
+tion timescale from the shock to the PNS calculated for (χ, j) = (0, 0).
+Other parameters are χ = 0 and rsh (j = 0) = 3.2rPNS.
+growing instability for strong enough neutrino-heating. This ef-
+fect blurs the distinction between SASI and convection, which
+becomes difficult for j ≳ 0.1 as will be discussed in the next
+section. In this section, we focus on the regime of slow rotation
+( j ≲ 0.1), in which the two instabilities can be still be distin-
+guished.
+Even with modest angular momentum, the differential na-
+ture of rotation implies that convective motions are going to be
+sheared in addition to being entrained in rotation with a fre-
+quency intermediate between the shock and PNS rotational fre-
+quency. The region most unstable to buoyant motions, i.e. the
+radius rBV corresponding to the largest Brunt-Väisälä frequency
+N(r) (Eq. 13) is expected to contribute most to defining the phase
+frequency of the convective mode, i.e. the rotation frequency of
+the frame in which the convective pattern is stationary. We com-
+pare in Fig. 7 rBV with the corotation radius rcorot defined by
+Eq. (29). For small rotation rates ( j ≲ 0.1), we remark in Fig. 7
+that rcorot is close to rBV as anticipated for a purely convective
+mode.
+Figure 8 illustrates the effect of rotation on the growth rate
+of convective modes at χ = 6.5, which is significantly larger
+than the marginal stability threshold of the most unstable mode
+(χ = 3.5 − 4). For both dissociation rates (ε = 0 and ε = 0.3),
+the differential rotation reduces the growth rate of the convec-
+tive instability, in particular for the smaller scale modes. The ad-
+verse effect on the convective instability is less abrupt for larger
+Fig. 6. Evolution as a function of χ of the growth rate ωi (top) and
+the frequency ωr (bottom) of the fundamental mode, with an azimuthal
+number m = 4 for different values of the rotation rate.
+scale modes, which might be interpreted by a smaller effect of
+the shearing due to differential rotation between the shock and
+the PNS. As a result, larger angular scales become dominant
+for high rotation rates. At the largest scales (m = 1 for ϵ = 0,
+m = 1 − 2 for ϵ = 0.3), rotation can on the contrary have a ben-
+eficial effect on the growth rate, which will be discussed in the
+next paragraph and section.
+Fig. 9 shows the effect of rotation on the heating rate at
+marginal stability, expressed in terms of χmarg. This threshold
+for unstable convective modes decreases with rotation for all
+m considered, thus showing a destabilising effect on convection
+near its marginal stability. Such a destabilising effect is surpris-
+ing and stands in striking contrast to the adverse effect of rota-
+tion at χ = 6.5 discussed in the last paragraph. We propose that
+this effect can be interpreted by the mixed nature of the modes
+near the transition between SASI and convection. In this inter-
+pretation, the destabilising effect of rotation on convection is a
+remnant of the rotational destabilisation of SASI in the regime
+where the convective mode retains some characteristics of SASI
+near the transition between the two instabilities. The behaviour
+of a mode with respect to rotation would then be expected to
+depend on how close the heating rate is to the SASI/convection
+transition. Indeed, Fig. 9 shows that the heating rate at marginal
+stability χmarg decreases faster for smaller m when rotation is in-
+creased. In the absence of rotation, when m increases, the transi-
+tional χtrans from SASI to convection decreases (see Fig. 2) while
+the marginal stability depends little on the azimuthal number.
+As a result, the difference χtrans − χmarg increases with m, and
+the residual effect of SASI decreases. This highlight the mixed
+SASI/convection state of the modes near χtrans for small rotation
+rates.
+3.2.3. Mixed SASI-convective mode induced by rotation
+For intermediate rotation rates (0.1 < j < 0.3), the behaviour of
+the modes is modified by rotation to such an extent that it be-
+comes impossible to distinguish clearly between convective and
+Article number, page 7 of 16
+
+X=0.0
+m=1
+3
+m=2
+m=3
+2
+m=4
+W; [Vsh/rsh]
+1
+0
+-1
+0
+10
+20
+30
+40
+50
+j(%)E=0,m=4,rsho = 5rpNS
+4
+[4S1/S^] !
+2
+0
+j= 0%
+3-2
+j=10%
+j= 2%
+j=20%
+j= 5%
+j=30%
+-4
+10
+8
+Wr [Vsh/rsh]
+6
+4
+2
+00
+1
+2
+3
+4
+5
+6
+7
+XX=0.0
+=0.3
+3
+m=0
+2
+W; [Vsh/rsh]
+1
+0
+0
+0.3
+0.6
+0.9
+1.2
+1.5
+J [rsho * Vsho]A&A proofs: manuscript no. aanda
+Fig. 7. Evolution of the corotation radius rcorot as a function of the rota-
+tion rate for ε = 0 (top) and ε = 0.3 (bottom). The corotation radius of
+the most unstable mode is represented with a blue line (numbers above
+the line specify the corresponding azimuthal number m). For compari-
+son, grey lines show the PNS radius rPNS, the gain radius rgain, the radius
+of the Brunt-Väisälä frequency maximum rBV and the shock radius rsh.
+With both values of the dissociation, the corotation radius of the most
+unstable mode is close to rBV for slow rotation and moves away for
+higher rotation rates ( j > 30%).
+SASI modes. As will be described in this section, the modes in
+this regime retain characteristics of both instabilities and should
+be understood as mixed SASI/convection/rotation modes.
+In order to gain a deeper understanding of the transition be-
+tween the SASI dominated and the buoyancy dominated mode
+including rotation, we show in Fig. 10 the entropy structure of
+the eigenfunction for the modes m = 1, 2, 4, 5 for three values of
+the angular momentum j = 0, 0.1 and 0.4 and χ ∼ 6. When there
+is no rotation (first column in Fig. 10), the entropy structure of
+SASI (first line) is uniformly spread in the radial direction as a
+spiral pattern. It is clearly distinct from the structure of convec-
+tion, whose perturbations are more localised in the vicinity of the
+gain radius and do not display any spiral pattern. In the second
+column of this figure, we note that the m = 1 spiral SASI pattern
+is little affected by a moderate rotation with j = 0.1 (compare left
+and middle columns). This can be interpreted by the fact that for
+such moderate rotation, the corotation radius is close to the PNS
+surface. For the same rotation rate, the differential rotation shears
+the convective cells with m = 2 − 5. This leads to spiral struc-
+tures that are smaller but similar to a SASI pattern. However, the
+convective nature of the mode is still visible by the localisation
+Fig. 8. Growth rate evolution depending on the rotation rate for several
+modes at χ = 6.5 with ε = 0 (top panel) and ε = 0.3 (bottom panel). For
+j = 0, rsh(ε = 0) = 7.8rPNS and rsh(ε = 0.3) = 4.1rPNS. For this heating
+rate, non axisymmetric convective modes are quenched by rotation. In
+the absence of rotation, all modes are above the transition from SASI to
+convection, except for the mode m = 1 in the bottom panel.
+Fig. 9. Impact of rotation on the parameter χmarg characterising the ad-
+vective stabilisation of the convective mode. We consider several az-
+imuthal numbers m and ε = 0.3. The modes m = 1 and m = 2 are not
+convective in this region and are not shown. The curves are cut when the
+mode is unstable for all heating rates and χmarg does not exist anymore.
+Article number, page 8 of 16
+
+X =6.5
+4.0
+m=0
+3.5
+3.0
+2.5
+2.0
+3
+1.5
+1.0
+m=1
+m=2
+0.5
+m=3
+m=4
+0.0
+0
+15
+30
+45
+j(%)= 0.3
+4.0
+m=3
+m=4
+3.5
+m=5
+3.0
+m=6
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0
+5
+10
+15
+20
+j(%)E=0, X=6.5
+rsh
+8
+6
+[rpNs]
+rcorot, m = 2
+rcorot, most unstable
+4
+1
+4
+3
+rBV.
+2
+gain
+rpNS
+0
+0
+10
+20
+30
+40
+50
+j(%)E=0.3, X=6.5
+rsh
+4
+3
+3
+2
+4
+3
+5
+2
+rBV
+rgain
+1
+rpNS
+rcorot, m = 2
+rcorot, most unstable
+0
+0
+10
+20
+30
+40
+50
+j(%)X =6.5
+4.0
+3.5
+3.0
+2.5
+2.0
+E=0.3
+1.5
+3
+m=1
+1.0
+m=2
+m=3
+0.5
+m=4
+m=5
+0.0
+0
+15
+30
+45
+j(%)A.-C. Buellet, T. Foglizzo, J. Guilet, E. Abdikamalov: Stellar rotation and post-shock instabilities during core-collapse
+Fig. 10. Spatial structure of the eigenmode entropy perturbation in the equatorial plane, for ε = 0.3 and χ ∼ 6. Each row corresponds to a
+value of m = (1, 2, 4, 5) (from top to bottom) and each column to a rotation rate with j = (0, 0.1, 0.4) (from left to right). In magenta and cyan, we
+represented the gain and corotation radii, respectively. The diagonal sequence of framed plots corresponds to the most unstable modes. Coordinates
+x and y are expressed in rPNS unit. The deformation induced by rotation is stronger for smaller scale modes.
+of the perturbations near the gain radius. These similar structures
+can be explained if we consider that the mode is not purely con-
+vective, but becomes a mixed mode SASI/convection/rotation.
+A global panorama of the parameter space is shown in
+Fig. 11 where we vary χ and j for three values of the dissoci-
+ation parameter ε = 0, 0.3 and 0.5 (upper to lower rows). The
+four columns represent the growth rate and oscillation frequency
+of the most unstable mode, its azimuthal number m and the coro-
+tation radius rcorot. The main effect of dissociation is to diminish
+the radius of the stationary shock and thus the size of the gain
+region and the timescale of advection.
+For small values of j, we can see a sharp jump in frequency
+and m when neutrino heating exceeds the threshold χ ∼ 4 cor-
+responding to the transition from SASI to the convective insta-
+Article number, page 9 of 16
+
+j=0%, X=6.0, m = 1.0
+i=40%, X=5.9, m = 1.0
+4
+4
+4321
+10°
+３
+3
+2
+2
+10-1
+gain
+1
+1
+y
+0
+0
+y
+0
+y
+0
+-1
+-1
+-1
+Fgain
+-2
+-2
+-2
+-10-1
+Tcorot
+-3
+-3
+-3
+-100
+-41
+¥34
+-4-3-2-1 0 1 2 3
+4
+x
+X
+X
+j=40%, x=5.9, m = 2.0
+j=0%, X=6.0, m = 2.0
+j=10%, X=6.0, m = 2.0
+4
+4
+100
+3
+3
+2
+2
+10-1
+gair
+1
+1
+１
+0
+y
+0
+y
+0
+>
+-1
+-1
+Fgain
+Fgain
+-2
+-2
+Fcorot
+-10-1
+-3
+-3
+73
+-10°
+F4-3 -2
+１2
+-1
+1
+2
+3 
+4
+¥34
+-4-3-2-1 0
+１2
+4
+0
+x
+X
+X
+j=0%, X=6.0, m = 4.0
+j=10%, X=6.0, m = 4.0
+j=40%, X=5.9, m = 4.0
+41
+4
+10°
+3
+3
+３
+2
+2
+２１
+10-1
+aai
+1
+1
+0
+0
+0
+y
+y
+-1
+-1
+-1
+-2
+Fgain
+-2
+-10-1
+col
+-3
+-3
+-4
+-10°
+4.
+44-3-2-1 0
+4-3-2-1 0
+-4-3-2-1 0 1 2
+1
+2
+3
+¥3
+4
+X.
+x
+j=0%, X=6.0, m = 5.0
+j=40%, X=5.9, m = 5.0
+j=10%, X=6.0, m = 5.0
+4
+4
+4
+10°
+３
+３２１
+2
+10-1
+gais
+1
+1
+0
+0
+y
+0
+0
+y
+y
+-1
+-1
+-1
++gain
+-2
+-10-1
+-3
+-100
+4
+4
+2
+3
+=4 -3 -2-1
+4
+4-3-2
+-1
+-1 
+0
+4
+0
+1
+2
+3 
+0
+2
+3
+4
+X
+X
+XA&A proofs: manuscript no. aanda
+Fig. 11. From left to right, maps of the growth rate ωi, frequency ωr, azimuthal number m and corotation radius of the most unstable fundamental
+mode with three dissociation rates ε = 0 (top row), 0.3 (middle row) and 0.5 (bottom row). Above the magenta dashed line, the growth rate of
+equatorial perturbations decreases with rotation. In the two left columns, the white lines show where the corotation radius equals rPNS and rgain. In
+the two right columns, the black lines highlight the variation of the dominant azimuthal number m (numbers).
+Article number, page 10 of 16
+
+ = 0.0, rsh(j= 0,X= 0) = 5.00rpNS
+8
+4
+4
+7
+6
+1
+1
+3
+3
+5
+、
+2
+2
+X4
+3
+rcorot-
+rot
+2
+=rpNS
+1
+=gain
+1
+2
+2
+0
+ε = 0.3, rshj= 0,X= 0) = 3.25rpNs
+8
+7
+6
+5
+41
+3
+54
+5
+3
+X4
+3
+Fcorot
+rcorot
+t = [pNS
+=rgain
+1
+2
+1
+2
+2
+３
+3
+1
+0.
+ = 0.5, rsh(= 0,X= 0) = 2.76rpNS
+8
+7
+6
+5
+5
+4
+6514
+3
+3
+X4
+3
+Icorot =IpNS
+rcorot = fpNS
+1
+2
+1
+Igain
+2
+2
+1
+25
+50 0
+25
+50 0
+25
+50 0
+25
+50
+j(%)
+j(%)
+j(%)
+j(%)
+012345
+5
+1015
+i2345
+0
+0
+1
+2
+W;[Vsh/rsh]
+Wr[Vsh/rsh]
+m
+rcorot[ rpNs]A.-C. Buellet, T. Foglizzo, J. Guilet, E. Abdikamalov: Stellar rotation and post-shock instabilities during core-collapse
+bility as the most unstable mode. However, as rotation increases,
+the frequency of the convective instability increases until its fre-
+quency is similar to the SASI frequency. At angular momentum
+jcrit ∼ 0.08 − 0.12, the jump disappears and it becomes impos-
+sible to distinguish clearly convective from SASI modes: mixed
+modes appear.
+The appearance of mixed modes can also be recognised in
+the evolution of the corotation radius shown in Fig. 7. Above a
+critical rotation rate, the corotation radius indeed starts to move
+away from the Brunt-Väisälä radius, where it is located for a
+convective mode. We remark that rcorot starts to shift outwards
+for rotation rates exceeding j ∼ 10%, indicating a lower fre-
+quency than expected for a purely convective mode. This slower
+oscillation frequency compared to the rotation rate of the most
+buoyant layer suggests that some physical process associated to
+rotation and/or SASI contributes significantly to the instability
+mechanism. However, the change of the azimuthal number of
+the dominant mode (see below) keeps the corotation close to the
+Brunt-Väisälä radius, indicating that convection still plays a role
+in the instability mechanism.
+We remark in the 3rd column of Fig. 11 that the azimuthal
+number m of the most unstable mode decreases with the rotation
+rate for χ > 4, while it increases for χ < 4. The change of dom-
+inant convective scale induced by rotation is not predicted by
+Eq. (28): comparing the magenta and blue triangles in Fig. 3 in-
+dicates a shift to larger angular scales induced by a moderate an-
+gular momentum j = 0.08, despite a negligible change in the size
+of the gain region. For this small rotation rate, the shortcoming of
+Eq. (28) is not surprising when considering the rotation-induced
+distortion of the convective cells in Fig. 10 and the residual influ-
+ence of SASI. The rotational stabilisation of perturbations with
+a large azimuthal wave number m can be understood as a conse-
+quence of their strong shear (Eq. 22) which induces a short radial
+wavelength 2πv/ω′ for advected perturbations near the PNS, and
+thus a phase mixing particularly visible in the lower right corner
+of Fig. 10.
+Fig. 12. Evolution of the critical rotation rate jcrit depending on the dis-
+sociation using two methods. This critical rotation corresponds to the
+apparition of the mixed-modes. The higher the dissociation, the later
+the transition. For the blue points, we used the discontinuity end of the
+dominant m at χ ∼ 4. For the magenta points, we used the advection
+timescale τadv from the shock to the gain radius for m = 4 and χ ∼ 4.
+The third column of Fig. 11 can be used to better constrain
+the domain of existence of these mixed modes. For small rota-
+tion rates, the azimuthal number contours converge to a line at
+χ ∼ 4. It corresponds to the transition from purely SASI modes
+to purely convective modes, across which the azimuthal number
+and frequency of the dominant mode is discontinuous. Above
+this boundary, the high m modes dominate, and their frequency
+depends linearly on the rotation rate. However, this frontier dis-
+appears for higher rotation rates, and the transition from small
+m to higher m is continuous. The value of the rotation rate jcrit
+when the frontier disappears slightly increases when the disso-
+ciation rate increases (blue points in Fig. 12). This behaviour
+can be quantitatively reproduced if we assume that mixed modes
+appear when the frequency of the convective mode ≃ mΩ(rBV)
+matches the frequency of the corresponding SASI mode in the
+absence of rotation or heating ≃ 2π/τadv. This criterion defines a
+critical specific angular momentum
+Jmix = 2πr2
+BV
+mτadv
+,
+(30)
+which reproduces remarkably well the variation of
+jcrit
+when considering the mode m
+=
+4 (magenta points of
+Fig. 12). This is the signature of the growth of mixed modes
+SASI/convection/rotation.
+The first column in Fig. 11 shows the adverse effect of rota-
+tion on the growth rate of the convective instability for χ > 5, as
+discussed in Sect. 3.2.2. For each rotation rate, the magenta line
+highlights the value of χ above which the rotation has an adverse
+effect on the growth of convection in the equatorial plane. In this
+region, the dynamic of the flow will be dominated by axisym-
+metric convective modes in the plane (r, z), that are expected to
+be little affected by the rotation for j ∈ [0, 0.3]. This stands in
+contrast with the beneficial effect of rotation on the growth rate
+of SASI for χ < 3 − 4, as discussed in Sect. 3.2.1, where spiral
+modes are therefore expected to dominate the dynamics. Inter-
+estingly, rotation has a beneficial effect on the growth rate of
+the dominant non-axisymmetric convective modes for moderate
+convection parameters χ ≃ 4 − 5 (Fig. 11). The same behaviour
+can be observed for the m = 4 mode in the upper plot of Fig. 6,
+where the growth rate for χ ∼ 3 − 4 is significantly enhanced by
+rotation. It is also linked to the easier onset of convection with
+rotation shown by the decrease of χmarg in Fig. 9, as discussed
+in Sect. 3.2.2. All these observations suggest that the perturba-
+tion of the shock induced by the convective instability produces
+an advective-acoustic cycle favourable to enhance the convec-
+tive instability, as already suggested by the entropy structure in
+Fig. 10. The mode m = 1 in Fig. 8 calls for a particular attention,
+with a significant enhancement of the growth rate by rotation
+(this is also true for the m = 2 mode in the lower panel of the
+same figure) in a regime χ = 6.5 a priori dominated by convec-
+tion with an adverse effect of rotation. This peculiar behaviour
+of the m = 1−2 modes can also be explained by a mixed state of
+this mode with the advective-acoustic cycle interacting construc-
+tively with the convective mode. Indeed, for these low values of
+m the transition from SASI to convection takes place at a rather
+large value of χ (e.g. χ ≃ 6 for m = 1 in Fig. 2). As a conse-
+quence, mixed modes with a beneficial effect of rotation may be
+expected for larger values of χ close to this transition.
+Figs. 13 and 14 illustrate the effect of rotation on both insta-
+bilities in two configurations mainly differing by their shock ra-
+dius, induced by the dissociation parameter. The value χ = 4.5 is
+chosen in the region where the most unstable mode is convective
+and the growth rate increases with rotation (below the magenta
+line of the first column in Fig. 11). In the bottom panel of these
+figures, the oscillation frequency without rotation ( j = 0) reveals
+the convective (ωr = 0) or SASI (ωr � 0) nature of the modes.
+We can see that the convective frequency, enhanced by rotation,
+Article number, page 11 of 16
+
+13
+J = 2réy/(mTadv)
+12
+jcrit
+linear fit
+11
+10
+9
+8
+7
+6
+5
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+EA&A proofs: manuscript no. aanda
+Fig. 13. Growth rate (top) and frequency (bottom) evolution of several
+modes depending on the rotation rate j, for χ = 4.5, ε = 0 and rsh( j =
+0) = 6.8rPNS.
+Fig. 14. Same as Fig. 13 but for ε = 0.3 and rsh( j = 0) = 3.9rPNS.
+is similar to the SASI frequency for j ∈ [0.02, 0.08]. The top
+panels of this figure show that the effect of rotation varies de-
+pending on the nature of the instability. The growth rate of SASI
+modes is enhanced by the rotation, as in Yamasaki & Foglizzo
+(2008) and Blondin et al. (2017). For convective modes, the ef-
+fect of rotation depends on the proximity of χ to the transition
+value χtrans defining the transition of a mode between SASI and
+convection. Defining ∆χ ≡ χ − χtrans, the rotation can induce an
+increase of the growth rate, similar to the effect of rotation on
+SASI, if ∆χ < 1. Otherwise, the convective modes are quenched
+by rotation, as observed in Fig. 8. Comparing Figs. 13 and 14,
+the increase of the frequency and growth rate induced by rotation
+is stronger for ε = 0.3.
+Fig. 15. Evolution of the growth rate depending on the azimuthal num-
+ber m for several rotation rates and χ = 4.5.
+Figure 15 illustrates the complexity of the impact of rotation
+depending on the azimuthal number. The growth rate of SASI
+modes (m = 1, 2) is increased. However, the effect of rotation on
+convective modes depends on both the rotation and the azimuthal
+number. For m ∼ 3, the destabilising effect of rotation on the
+convective instability can be clearly seen by continuity with the
+increase of the SASI growth rate. For m ∈ [3, 5], the effect of
+rotation on the convection is similar to its effect on SASI, but
+with a smaller magnitude. For higher m ≳ 6 and low rotation
+rates, the rotation has an adverse effect and leads to a decrease
+of the growth rate. The effect changes for high rotation rates and
+will be discussed in Sect. 3.3.
+As a summary, the rotation smoothens the transition from
+SASI to convection, leading to a mixed mode phase. These
+mixed modes appear when the corotation radius moves away
+from the Brunt-Väisälä radius and exist up to j ∼ 30% where
+the properties of the modes starts to change.
+3.3. Strong rotation-induced instability
+The mixed state of the modes seems to disappear when the
+rotation becomes too strong. When the rotation rate exceeds
+j ∼ 20%, Fig. 6 shows that both the frequency and the growth
+rate of m = 4 become strikingly insensitive with respect to the
+heating rate when expressed in units of vsh/rsh. Rotation also
+changes the behaviour of the dominating m as a function of heat-
+ing. While the dominant m increases with χ for small and inter-
+mediate rotations, it decreases slightly with heating for higher
+rotation rates. Also, when considering the most unstable mode,
+the rotation seems to lessen the influence of the parameter χ on
+the frequency and growth rate of the dominant mode for j ≳ 0.3
+(Fig. 11). This suggests that the mode loses its convective nature
+and that the buoyancy driven by the entropy gradient no longer
+plays an important role in the strong rotation regime. Another
+indication that the role of buoyancy in the instability mechanism
+has been erased by rotation comes from the fact that the corota-
+tion radius moves definitively away from the Brunt-Väisälä ra-
+dius for high rotation rate j ≳ 30% (Fig. 7).
+Article number, page 12 of 16
+
+X =4.5, ε=0, rsho = 5rpNS
+4
+m=1
+m=3
+[Vsh/rsh]
+m=2
+m=4
+3
+2
+31
+0
+12
+Wr [Vsh/rsh]
+8
+4
+0
+0
+10
+20
+30
+40
+50
+j(%)X =4.5, E=0.3, rsho = 5rpNS
+4
+ [Vsh/ rsh]
+3
+m=1
+m=4
+31
+m=2
+m=5
+m=3
+0
+12
+Wr [Vsh/rsh]
+8
+4
+0
+0
+10
+20
+30
+40
+50
+j(%)E=0.3, X=4.0,rsho = 5rpNS
+3
+W; [Vsh/rsh]
+2
+1
+j=0%
+j=10%
+0
+j=2%
+j=15%
+j=4%
+j=20%
+j=6%
+j=30%
+j=8%
+j=50%
+-1
+1
+2
+3
+4
+5
+6
+7
+mA.-C. Buellet, T. Foglizzo, J. Guilet, E. Abdikamalov: Stellar rotation and post-shock instabilities during core-collapse
+The right column of Fig. 10 illustrates the structure of the
+modes in the strong rotation regime. The localisation around the
+Brunt-Väisälä radius, a characteristic of the convective instabil-
+ity that was still visible at intermediate rotation, is absent in this
+regime. The radial profile of entropy perturbations suggests that
+it is driven by the deformations of the shock rather than by the re-
+gion of maximum buoyancy. The spiral mode structure of SASI
+is also clearly perturbed by rotation due to the corotation radius,
+which is well inside the domain. The pattern is similar to the
+one obtained by Blondin et al. (2017) at high rotation rates with-
+out heating. In an adiabatic flow where entropy perturbations are
+simply advected, the entropy pattern would trace the flow lines
+and display a change of direction at the corotation radius since
+the pattern rotates faster than the outer flow and slower than the
+inner flow. In Fig. 10, a change of direction of the spiral pat-
+tern is clearly visible in the strong rotation case (right column).
+The radial location of the pattern extrema is, however, slightly
+above the corotation radius, which can be interpreted as due to
+the non-adiabatic heating/cooling functions.
+For these high rotation rates (j > 0.3), it seems that the
+modes are not mixed SASI/convection/rotation any more, but re-
+sult from a rotation-induced instability. Some characteristics of
+this instability can be inferred from the evolution of the growth
+rate. Comparing Figs. 13 and 14, we remark that when the modes
+properties do not depend on the heating rate, the dependence of
+the modes on the rotation rate seems to be stronger for increasing
+dissociation rates. In addition, the slope of this increase seems to
+be independent of the value of the azimuthal number. This be-
+haviour of the mode for high rotation rates can also be seen in
+Fig. 15. We notice that the rotation enhances the growth rate of
+every mode. In particular, for initially convective modes, this ef-
+fect is opposite to the rotation effect for low rotation rates.
+4. Expected consequences on the gravitational
+wave signature
+The mode frequencies computed with a linear analysis have been
+shown to reproduce well the frequencies present in the gravi-
+tational waves spectra in non-linear numerical simulations. We
+first focus on the properties of the fundamental mode m = 2
+which has a direct impact on the production of gravitational
+waves according to the quadrupole formula. We include the ef-
+fect of dissociation with ε ∼ 0.3 (Huete et al. 2018) and consider
+the typical range of shock radius rsh ∼ (3 − 7)rPNS.
+Figure 7 (bottom) shows that the mode m = 2 is dominant
+in the high rotation regime j ∈ [0.2, 0.5] for χ = 6.5. The mode
+m = 2 is actually dominant over a large domain of the parameter
+space ( j, χ) according to Fig. 11 (third column with ϵ = 0.3).
+Fig. 16 shows the impact of rotation on the m = 2 mode
+frequency for different values of χ. In the case of SASI modes
+(χ = 0 and χ = 2), the frequency increases almost linearly with
+the rotation rate for j varying from 0 to 0.5. The convective
+mode corresponding to χ = 6 has a more complex behaviour.
+For small rotation (j ≲ 0.1), the frequency is approximately pro-
+portional to the rotation rate similarly to the convective modes
+in Fig. 13 and 14. This behaviour can be interpreted with an ap-
+proximately constant corotation radius located between the gain
+radius and the radius where the Brunt-Väisälä frequency is max-
+imum (Fig. 7), as expected for the convective instability and vi-
+sualised by the grey-shaded region in Fig. 16. At higher rotation
+rates j > 0.2, the frequency separates from this linear trend, cor-
+responding to the corotation radius moving to larger radii (Fig. 7)
+when the mode takes a mixed nature. At still higher rotation fre-
+quencies ( j ≳ 0.3), the frequencies expressed in units of vsh/rsh
+Fig. 16. Frequency of the m = 2 fundamental mode as a function of the
+specific angular momentum j for several values of χ. The grey-shaded
+region illustrates the frequency range corresponding to a corotation ra-
+dius located between the gain radius and the maximum of the Brunt-
+Väisälä frequency.
+converge toward a single linear trend that does not depend on the
+heating rate. The effect of heating has been erased by rotation.
+This behaviour is consistent with the behaviour of the dominant
+modes seen earlier.
+Fig. 17. Maps, for varying heating and rotation rates, of the growth rate,
+the frequency and the corotation radius of the most unstable fundamen-
+tal mode between modes m = 1 and m = 2. In the second panel, we
+plotted 2ωr when m = 1 is dominant to account for the doubled fre-
+quency expected in the gravitational wave signal. In the third panel, the
+black line illustrates the frontier between the modes.
+To account for the possible signature of a dominant mode
+m = 1 on the gravitational wave signal, we include in Fig. 17 the
+doubled frequency of the mode m = 1 when it is more unstable
+than the m = 2 mode. Note that this frequency is higher than
+the frequency of the m = 2 mode. This remarkable property can
+be deduced from Figs. 5 and 6 in Foglizzo et al. (2007) where
+the frequency of the SASI mode m = 2 is always smaller than
+twice the frequency of the mode m = 1. We observe that the
+region of the parameter space dominated by the mode m = 1 is
+limited to moderate heating rates χ < 5.2 and modest rotation
+rates j < 15%.
+In the regime j ≥ 30%, an analytical form of the linear de-
+pendence illustrated by Fig. 16 would be useful to directly trans-
+Article number, page 13 of 16
+
+E = 0.3, rsh(j=0,X=0) = 3.2rpNS
+10
+8
+Wr[Vsh/ rsh]
+6
+4
+2
+0=X
+rotation freg. at gain radius
+X=2
+rotation freq. at Brunt-Vaisala radius
+X=6
+0
+0
+10
+20
+30
+40
+50
+j(%) = 0.3, rsh(j=0,X=0) = 3.2rpNS
+7
+6
+2
+5
+rcorot
+≥rPNS
+3
+2
+=rPNS
+1
+00
+25
+500
+25
+500
+25
+50
+j(%)
+j(%)
+j(%)
+1234
+¥36912
+1
+2
+W;[Vsh/rsh]
+Wr[Vsh/rsh]
+rcorot[rpNs]A&A proofs: manuscript no. aanda
+late the frequency observed in the gravitational wave signal into
+a relation between j and the advection frequency at the shock
+vsh/rsh.
+5. Discussion and conclusion
+In this paper, we studied a spherical toy model where we consid-
+ered the dynamics of the equatorial plane between the shock and
+the surface of the PNS. Thanks to a linear analysis, we computed
+unstable non-axisymmetric modes for varying rotation and heat-
+ing rates and different dissociation energies through the shock.
+This led to several discoveries for the behaviour of the convective
+instability with rotation and its interplay with SASI and rotation-
+induced instabilities.
+Relatively slow rotation rates (less than 10% of the Keplerian
+rotation at the PNS radius), for which centrifugal effects are very
+small, can have a strong and complex influence on both SASI
+and the convection instability:
+– In the presence of rotation, convective modes have an oscilla-
+tion frequency that corresponds to a corotation radius located
+close to the maximum of the Brunt-Väisälä frequency, where
+the convective engine is expected to be most powerful. The
+appearance of this non-vanishing oscillation frequency blurs
+somewhat the abrupt transition between SASI and convec-
+tion.
+– For large values of the convection parameter χ ≳ 5, the ro-
+tation hampers the growth of non-axisymmetric convective
+modes. The decrease in growth rate is more pronounced for
+larger m modes, which can be interpreted by the larger sta-
+bilising effect of the shear due to differential rotation. In this
+regime, axisymmetric modes would be expected to dominate
+the dynamics, as they are expected to be only slightly af-
+fected by rotation.
+– By contrast, for χ ≲ 5, where the advection has a strong
+influence on the growth of convection, rotation increases
+the growth rate of both SASI and the convective instabil-
+ity. The increase in the SASI growth rate with rotation was
+well known in the absence of heating (Yamasaki & Foglizzo
+2008; Kazeroni et al. 2017; Blondin et al. 2017) and is here
+shown to hold in the presence of neutrino heating. On the
+other hand, this behaviour was not expected for the convec-
+tive instability, which was generally thought to be hampered
+by rotation. The destabilising effect of rotation on convection
+in this regime leads to a decrease with rotation of the value
+of χ at marginal stability. We interpret this effect as a resid-
+ual influence of an advective-acoustic cycle which acts to
+reinforce convective motions close to the transition between
+convection and SASI.
+– We observe two different effects of rotation on the domi-
+nant scale, depending on the instability mechanism that dom-
+inates without rotation. For SASI, the dominant m increases
+with rotation (as in Yamasaki & Foglizzo 2008 and Blondin
+et al. 2017), while it decreases for the convective instabil-
+ity. The diminution of the dominant azimuthal number m as
+rotation is increased was not expected to occur as it does.
+Indeed, in our case, not only are large m disadvantaged by
+rotation, but there is a heating domain for which small m are
+advantaged by rotation.
+At moderate rotation rates (10 to 30% of the Keplerian ro-
+tation at the PNS surface), the modes are modified by rota-
+tion to such an extent that it becomes impossible to distinguish
+clearly between convective and SASI modes. When the heat-
+ing rate is increased, the frequency and azimuthal number of
+the most unstable mode change smoothly, with no clear tran-
+sition from one instability to another. Modes retain character-
+istics of both instabilities and should therefore be understood
+as mixed SASI/convection/rotation modes. The dominant mixed
+modes have a relatively large angular scale with m = 1 − 3 and
+their growth rate increases with faster rotation. This regime takes
+place above a critical angular momentum such that the frequency
+of a convective mode (corotating with the maximum of Brunt-
+Väisälä frequency) is comparable to the frequency of SASI. The
+non-oscillatory nature of convection and the low frequency as-
+sociated to SASI both allow for a significant effect of moderate
+differential rotation, with the introduction of a corotation radius
+in the structure of non-axisymmetric perturbations which may
+participate in the extraction of energy and angular momentum
+from the interior region rotating faster than the region exterior
+to the corotation (Cairns 1979; Yoshida & Saijo 2017; Saijo &
+Yoshida 2006). For a PNS radius of rPNS = 50km, the mixed
+mode regime corresponds to an interval of specific angular mo-
+mentum 3 − 9 × 1015 cm2/s. Extrapolated to the radius of a cold
+neutron star 12 km, this corresponds to relatively fast rotation
+periods of ∼ 1 − 3ms.
+For high rotation rates (more than 30% than the Keplerian
+frequency at the PNS surface, i.e. j ≳ 0.3), the frequency and
+growth rate are independent or weakly dependent on the heating
+rate, when they are expressed in terms of the advection timescale
+rsh/vsh. This, together with the significant deviation of the coro-
+tation radius from the most buoyant region, suggests that the in-
+stability is dominated by rotational rather than buoyancy effects.
+The study of Walk et al. (2022) without neutrino heating pointed
+out the existence of a similar instability regime where the fre-
+quency of the dominant mode depends too little on the advection
+time to be explained by an advective-acoustic cycle. Our results
+suggest that these results obtained without heating are still valid
+when heating is taken into account. We note that the regime of
+rapid rotation appears for j ∼ 0.3 which corresponds to a small
+ratio ∼ 0.03 of the centrifugal force to the gravity at the corota-
+tion radius, even though the centrifugal displacement of the sta-
+tionary shock is not negligible. A precise understanding of this
+strong rotation regime is still missing, but it seems probable that
+the corotation radius plays an important role in the instability
+mechanism.
+A future detection of gravitational waves is expected to
+give information on the physical phenomena during stellar core-
+collapse. Our identification of three different instability regimes
+depending on the rotation rate should help clarify the still poorly
+known influence of rotation on the gravitational wave signal.
+Non-axisymmetric convective modes become oscillatory in the
+presence of rotation. It may therefore become possible to iden-
+tify their frequency in the low-frequency part of the gravitational
+wave spectrum. For low/moderate rotations, the corotation ra-
+dius of the m = 2 convective/mixed mode is close to the gain or
+Brunt-Väisälä radius. The identification of the mode frequency
+would therefore give access to the rotation frequency at this ra-
+dius. Non-axisymmetric equatorial modes with a large angular
+scale m = 1, 2 are strongly destabilised by rotation and dominate
+the dynamics in a wide region of the parameter space in particu-
+lar for moderate to strong rotation, which should be favourable to
+a strong emission of gravitational waves. In the regime of strong
+rotation, the frequency of the m = 2 mode becomes independent
+of the heating rate and depends only on the advection timescale
+rsh/vsh and the angular momentum j. This reduction of the pa-
+rameter space should help extract physical information from the
+measure of the mode frequency. These modes should be incorpo-
+rated in future asteroseismic studies similar to Torres-Forné et al.
+Article number, page 14 of 16
+
+A.-C. Buellet, T. Foglizzo, J. Guilet, E. Abdikamalov: Stellar rotation and post-shock instabilities during core-collapse
+(2018, 2019b) where non-axisymmetric perturbations would be
+taken into account.
+Our results can be compared to previous numerical simula-
+tions of core-collapse supernovae, including rotation. Fryer &
+Heger (2000) and Fryer & Warren (2004) found that convec-
+tion was quenched by rotation, especially in the equatorial plane.
+For models A and B of Fryer & Warren (2004), this may be ex-
+plained by the strong centrifugal effects for such strong rotation
+( j2 > 0.4). Convection in the slow rotating model C (j ∼ 0.03)
+does not seem to be confined to the poles, which is consistent
+with our study, where a purely convective instability can be ob-
+served in the equatorial plane up to j ∼ 0.1.
+The setup of Iwakami et al. (2014) is similar to ours in that
+it did not involve the interior of the PNS and considered a simi-
+lar range of rotation rates (corresponding to j = 0 − 0.33 in our
+units). If we focus on the structure in models D, E, and F as dis-
+played in Fig. 5 of their paper, we see that for higher rotation and
+smaller neutrino luminosity, the entropy structures are at larger
+scales and that buoyant patterns become spiral ones. This struc-
+ture change is consistent with our result that rotation favours
+larger-scale convection and transforms convective modes into
+mixed spiral modes for moderate rotation rates. The precise in-
+terpretation of their results is, however, complicated by the fact
+that both rotation and neutrino luminosity are changed such that
+it is difficult to disentangle their respective influence. In addi-
+tion, we note that they identified a pattern referred to as a spiral
+motion with buoyant-bubble (SPB), which may be related to the
+mixed SASI/convection/rotation modes we have identified for
+moderate rotation rates.
+To have a better understanding of the processes at work at the
+onset of the shock revival, a non-linear analysis of the dynamics
+of the fluid would be necessary. It would shed light on how ro-
+tation acts in the saturation mechanism (Guilet et al. 2010) and
+show how the system evolves from the linear phase to the explo-
+sion. This last part would give information on the signatures of
+linear phenomena that may remain in the explosion signal.
+One should keep in mind that our model is idealised in many
+respects. We assume a very simple equation of state of ideal gas
+with γ = 4/3, dissociation is taken into account as a fixed energy
+sink, neutrinos are parameterised through analytical cooling and
+heating terms. We also did not describe the equatorial swelling
+of the PNS due to the centrifugal force. This effect increases the
+neutrinosphere radius, leading to cooler neutrinos and a smaller
+heating rate in the equatorial plane. As discussed above, such a
+centrifugal effect is expected to be very small at low/intermediate
+rotation rates (j < 0.3) but can become significant for the fastest
+rotations considered in this analysis. Part of this complexity is
+avoided by displaying our results as a function of χ rather than
+the heating rate. Note that, when χ is kept constant and the ro-
+tation increased, the heating constant ˜Ah decreases, in a qualita-
+tively similar way to the expected impact of the neutrinosphere
+swelling.
+The centrifugal force induces a deformation with respect to
+the spherical symmetry, which is challenging to take into ac-
+count in our formalism because it would couple different spher-
+ical harmonics. To avoid such a complexity, we restrained our
+analysis to the equatorial plane while keeping the effect of con-
+vergence in spherical geometry. As a result, our conclusions are
+limited to modes with spherical harmonics indices m = ±l and
+cannot describe the dynamics outside the equatorial plane, such
+as convective motions along the polar axis. The equatorial non-
+axisymmetric modes described here are nonetheless expected to
+dominate the dynamics in most of the parameter space except at
+χ ≳ 5 − 5.5 and slow to moderate rotation, where axisymmetric
+convective modes are expected to be more unstable.
+The highest specific angular momentum considered in this
+study would be expected to be sufficient to allow for the develop-
+ment of the low-T/|W| instability inside the PNS (Takiwaki et al.
+2021; Bugli et al. 2022). Since it is restricted to the post-shock
+region without including the PNS interior, our linear analysis
+cannot describe the low-T/|W| instability. It may therefore miss
+the most unstable mode in the regime of strong rotation. A linear
+stability analysis including both the post-shock region and the
+PNS interior is therefore an important next step. Previous linear
+mode calculations focused on the prediction of GW mode fre-
+quencies and included the post-shock region in addition to the
+PNS interior, but they were restricted to axisymmetric modes,
+and they assumed a hydrostatic equilibrium neglecting advection
+(Torres-Forné et al. 2018, 2019b). To describe all unstable mode
+and their possible interaction, a linear stability analysis will have
+to face the challenge of combining the PNS in approximate hy-
+drostatic equilibrium and the advection in the post-shock region.
+Finally, the magnetic field, which was neglected in this study,
+can be expected to play an important role for strong rotation. A
+strong magnetic field can quench the development of the low-
+T/|W| instability (Bugli et al. 2022). In the absence of rotation,
+the magnetic field can have a complex influence on the post-
+shock dynamics because of the propagation of vorticity through
+Alfvén waves (Guilet & Foglizzo 2010; Guilet et al. 2011) and a
+small-scale dynamo can amplify the magnetic field (Endeve et al.
+2012; Müller & Varma 2020). It would be interesting to investi-
+gate further how the magnetic field would change the landscape
+of the post-shock instabilities in the presence of rotation.
+Acknowledgements
+JG acknowledges support from the European Research Coun-
+cil (MagBURST grant 715368). EA is supported by RK MES
+grant No. AP13067834 and NU Faculty Development Grant No.
+11022021FD2912.
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+Phonon-assisted optical absorption of SiC polytypes from first principles
+Xiao Zhang and Emmanouil Kioupakis∗
+The University of Michigan, Ann Arbor, Department of Materials Science and Engineering, Ann Arbor, 48109, USA
+Silicon carbide (SiC) is an indirect-gap semiconductor material widely used in electronic and
+optoelectronic applications. While experimental measurements of the phonon-assisted absorption
+coefficient of SiC across its indirect gap have existed for more than fifty years, theoretical investi-
+gations of phonon-assisted absorption have been hampered by their excessive computational cost.
+In this work, we calculate the phonon-assisted temperature-dependent optical absorption spectra
+of the commonly occurring SiC polytypes (3C, 2H, 4H and 6H), using first-principles approaches
+based on density functional theory and related techniques. We show that our results agree with
+experimentally determined absorption coefficients in the spectral region between the direct and in-
+direct band gaps. The temperature dependence of the spectra can be well-predicted with taking the
+temperature-dependence of the band gaps into account. Lastly, we compare the spectra obtained
+with second-order perturbation theory to those determined by the special displacement method,
+and we show that the full consideration of the electronic energy renormalization due to tempera-
+ture is important to further improve the prediction of the phonon-assisted absorption in SiC. Our
+insights can be applied to predict the optical spectra of the less common SiC polytypes and other
+indirect-gap semiconductors in general.
+I.
+INTRODUCTION
+Silicon carbide (SiC) is an indirect gap semiconduc-
+tor material that has been well established and widely
+used in many electronic and optoelectronic devices. It
+has a variety of structural polytypes, including cubic 3C;
+hexagonal 2H, 4H, 6H and several Rhombohedral struc-
+tures (9R, 15R, etc.) [1–3], that enable a broad range
+of applications. Its 6H polytype is one of the first to be
+used to create blue light emitting diodes (LEDs) that en-
+abled full color LEDs [4], though the indirect band gap
+limits the efficiency.
+Absorbing short-wavelength light
+much more efficiently, the material is also used as UV-
+sensitive diodes in flame sensors[4–6]. More recently, SiC
+is widely used as substrate material for applications such
+as extreme condition transistors [7–9], tunable photo-
+detecting devices [10], and advanced UV-detectors[11–
+13]. Recent developments in crystal growth also allowed
+novel structures such as thin films, nanoparticles, het-
+erostructures, etc., for more advanced applications[14–
+17]. Due to its early discovery as well as the wide appli-
+cations and interest, it is now one of the most well studied
+semiconductor materials from both an experimental and
+a theoretical point of view.
+Understanding the similarities and differences between
+the structural polytypes is crucial for practical applica-
+tions. The most common SiC polytypes for application
+purposes are the cubic structure (3C) and two hexagonal
+structures with different stackings (4H and 6H). It has
+been shown both by experimental and theoretical stud-
+ies [18] that at low temperatures, SiC tends to form the
+cubic 3C structure while at higher temperatures, 4H and
+6H become the more energetically favorable structures.
+In addition to 4H and 6H, the 2H structure of SiC has also
+∗ [email protected]
+been observed, although rarely due to being energetically
+less favorable. Due to its rare occurrence, the optoelec-
+tronic properties of the 2H polytype have not been stud-
+ied thoroughly from either a theoretical or an experimen-
+tal perspective, and previous studies are typically limited
+to only structural properties and basic electronic prop-
+erties such as the band gap. The rhombohedral struc-
+ture of SiC is less commonly seen but can also be grown
+on various temperatures and substrates [19–21], and has
+been shown to demonstrate great potential in MOSFET
+applications [19, 21]. Overall, the natural occurrence of
+these different polytypes make it an important task to
+study them consistently from a first-principles perspec-
+tive consistently to understand the properties of the ma-
+terial better, and potentially discover new opportunities
+for applications.
+Despite being well studied for many polytypes, the in-
+direct optical properties of SiC, which are important con-
+sidering its applications in optoelectronic devices, have
+never been thoroughly studied with first principles tech-
+niques. Experimental characterization of its indirect gap
+and optical absorption in the spectral region between the
+indirect and direct band gaps already exists ever since the
+1960s [22, 23]. There has also been a considerable num-
+ber of theoretical calculations of optical properties for
+both bulk SiC and nanostructures, however, always fo-
+cusing on direct transitions [24–30]. The commonly used
+theoretical spectroscopy approaches based on the density
+functional theory lack descriptions of electron-phonon in-
+teractions. As a result, momentum transfer is neglected
+in considering optical absorption and such approaches
+are only able to consider direct transitions. Yet indirect
+optical absorption plays an important role in the appli-
+cation of SiC as the difference between the direct gap
+and the indirect gap can range from about 1 eV up to
+more than 3 eV, depending on the polytype[2]. As an ex-
+ample, although SiC was one of the first LED materials
+for blue light mission, it can only emit visible light via
+arXiv:2301.05110v1  [cond-mat.mes-hall]  10 Jan 2023
+
+2
+phonon-assisted transitions across its indirect band gap.
+Although being considerably weaker compared to direct
+transitions, the phonon-assisted contribution enabled its
+application in optoelectronic devices. Therefore, the lack
+of theoretical studies on the indirect optical properties
+of SiC significantly limits our ability to understand the
+material and its further potential applications in opto-
+electronic devices.
+There are numerous computational challenges on eval-
+uating phonon-assisted optical properties from a theo-
+retical perspective, and tools to model them have only
+emerged in recent years. One significantly challenge is the
+necessity of dense sampling of the electronic states in the
+first Brillouin zone, due to the broad energy range of pho-
+ton frequencies, combined with the large number of pro-
+cesses needed to be considered in second-order transition.
+While the latter makes the problem more complicated
+by nature, the task of studying electron-phonon prop-
+erties with dense electronic and phonon Brillouin-zone
+sampling grids can be overcome by the approach of max-
+imally localized Wannier function interpolation.[31] The
+Wannier interpolation starts from a coarse grid of elec-
+tronic states, finds a set of maximally localized Wannier
+functions to represent the electronic wave functions in the
+real space, and utilize such set of Wannier functions to
+construct the electronic wave functions in a dense grid of
+the electronic states. It has been shown that Wannier in-
+terpolation can result in accurate interpolations for both
+the electronic structures and electron-phonon coupling
+matrix elements in a broad range of materials[32]. It thus
+becomes feasible to calculate electron-phonon properties
+on coarse grid with reasonable computational cost, then
+utilize Wannier interpolation to determine the phonon-
+assisted optical absorption spectra with adequate spec-
+tral resolution.
+In this work, we perform first-principles calculations to
+understand the phonon-assisted optical absorption spec-
+tra for common SiC polytypes (3C, 2H, 4H, 6H, and
+15R). As in experimental measurements, the absorption
+coefficients are mostly measured along the c-axis (E ⊥ c),
+thus in our work we report and compare the calculated
+absorption coefficient for the ordinary direction as well.
+By combining standard first-principles approaches (DFT,
+Many-body perturbation theory) for electronic structure
+and direct optical properties along with Wannier inter-
+polation for electron-phonon coupling, we show that our
+calculated indirect optical spectra are in good agree-
+ment with available experimental measurements. We fur-
+ther predict the phonon-assisted spectra for polytypes for
+which experimental data are lacking. Our approach well
+predicts the temperature dependence of the indirect op-
+tical spectra, which can benefit the applications of SiC in
+extreme conditions. Further, by comparing to special dis-
+placement method (SDM) [33–35], we show that to fur-
+ther improve the predictions of indirect optical properties
+s a function of temperature, both the effects of temper-
+atures on the optical spectra due to changes in phonon
+occupation factors and the effects of temperatures on the
+electronic energy renormalization are important.
+II.
+COMPUTATIONAL APPROACH
+To calculate the structural, electronic, and optical
+properties of the SiC polytypes, we performed first-
+principles calculations based on density functional the-
+ory.
+The calculation is carried out with the Quan-
+tum Espresso[36, 37] package using the Perdew-Burke-
+Ernzerhof (PBE)[38] approximation for the exchange-
+correlation functional and the SG15 Optimized Norm-
+Conserving Vanderbilt (ONCV) pseudopotentials.[39, 40]
+The wave functions are expanded into plane waves up
+to an energy cutoff of 60 Ry.
+Structural relaxation is
+done for all polytypes until all the components of the
+forces on the atoms are smaller than 10−5 Ry/Bohr.
+Converged ground-state calculations for each polytype
+were performed with Brillouin-zone (BZ) sampling grids
+of 8 × 8 × 8 for 3C, 12 × 12 × 8 for 2H, 12 × 12 × 4
+for 4H, 10 × 10 × 2 for 6H, and 8 × 8 × 8 for 15R. In all
+cases, the grid is shifted by half a grid spacing to improve
+convergence.
+To correct the underestimation of the band gap
+by PBE, we calculated quasiparticle energies with the
+one-shot GW method (G0W0) using the BerkeleyGW
+package.[41, 42] The static dielectric matrix is evaluated
+in the random phase approximation and extended to fi-
+nite frequency with the generalized plasmon-pole model
+by Hybertsen and Louie.[41] The static remainder ap-
+proach is used to reduce the number of empty orbitals
+required.[43] The GW calculations are performed with
+BZ-sampling points grid of 6 × 6 × 6 for 3C, 6 × 6 × 4 for
+2H, 6 × 6 × 3 for 4H, 6 × 6 × 2 and 6 × 6 × 6 for 15R.
+For all polytypes, the quasiparticle energies were inter-
+polated with the maximally localized Wannier function
+method[31] to obtain accurate quasiparticle band struc-
+tures as well as velocity and electron-phonon coupling
+matrix elements for fine BZ sampling grids in subsequent
+optical property calculations.
+Phonon-related properties are calculated with den-
+sity functional perturbation theory[44] as implemented in
+Quantum Espresso and interpolated to fine BZ-sampling
+grids with the maximally localized Wannier function
+method as implemented in the Electron-Phonon Wannier
+(EPW) package. [45–48] The coarse phonon BZ-sampling
+grids used for the different polytypes are 6 × 6 × 6 for
+3C, 6 × 6 × 4 for 2H, 6 × 6 × 3 for 4H, 6 × 6 × 2 for
+6H, and 6 × 6 × 6 for 15R. We note that the choices
+of the coarse BZ samplings are made to ensure both
+accurate Wannier interpolations as well as retain rea-
+sonable computational cost.
+Since SiC is a polar ma-
+terial, the F¨ohlich interaction is considered analytically
+via a long-range term when calculating the electron-
+phonon matrix elements.[47] The phonon-assisted opti-
+cal absorption spectra are determined with second-order
+time-dependent perturbation theory.
+Phonon-assisted
+optical absorption processes are second-order processes
+
+3
+in which electrons are not only excited vertically in a
+band-structure diagram through energy transfer by pho-
+ton absorption, but also horizontally through momen-
+tum transfer with the emission or absorption of phonons.
+The imaginary part of the dielectric function is derived
+from second-order time-dependent perturbation theory
+as[32, 48, 49]:
+ε2(ω) =8π2e2
+Ωω2
+1
+NkNq
+�
+νijkq
+|e · [S1,ijν(k, q) + S2,ijν(k, q)]|2
+× Pij(k, q)δ(ϵj,k+q − ϵi,k − ℏω ± ℏωνq),
+(1)
+where the upper (lower) sign represents the phonon emis-
+sion (absorption) process. Ω is the volume of the unit cell.
+k and q are the electron and phonon wave-vectors. Nk
+and Nq are the number of k and q points in the Bril-
+louin zone sampling. i, j label the band indices and ν
+labels phonon modes. ϵj,k+q and ϵi,k are the electronic
+energies of state (j, k + q) and (i, k), respectively. The
+generalized matrix elements for the two possible scatter-
+ing process is described by the terms S1,ijν(k, q) (i.e.,
+photon absorption from electronic state (i, k) to inter-
+mediate state (m, k) followed by electron-phonon scatter-
+ing from (m, k) to final state (j, k + q) and S2,ijν(k, q)
+(i.e., electron-phonon scattering from (i, k) to (m, k + q)
+followed by photon absorption from (m, k + q) to final
+state (j, k + q). Mathematically, the two terms are given
+by:[48, 49]
+S1,ijν(k, q) =
+�
+m
+vim(k)gmj,ν(k, q)
+ϵm,k − ϵi,k − ℏω + iη ,
+(2)
+and
+S2,ijν(k, q) =
+�
+m
+gim,ν(k, q)vmj(k + q)
+ϵm,k+q − ϵi,k ± ℏωνq + iη .
+(3)
+In the equations above, vij(k) represents the velocity
+matrix element for the optical transition between bands
+i and j at k, and gij,ν(k, q) represents the electron-
+phonon matrix element describing the scattering process
+from electronic state (i, k) to electronic state (j, k + q)
+through phonon mode νq. In addition, the occupation
+factor Pij(k, q) in Eq.(1) is given by combining the Bose-
+Einstein occupation factor of the phonons nνq and the
+Fermi occupation factor of the electrons fnk. Consider-
+ing energy conservation, we obtain the occupation factor
+for the phonon absorption Pa,ij(k, q) and phonon emis-
+sion Pe,ij(k, q) process as:
+Pa,ij(k, q) =nνq × fi,k × (1 − fj,k+q)
+− (nνq + 1) × (1 − fi,k) × fj,k+q,
+(4)
+and
+Pe,ij(k, q) =(nνq + 1) × fi,k × (1 − fj,k+q)
+− nνq × (1 − fi,k) × fj,k+q.
+(5)
+In addition, η in Eq.(2) and Eq.(3) is a numerical broad-
+ening parameter to prevent singularities induced by zeros
+in the denominator that occurs when direct transitions
+are allowed. In our work, we focus on the optical spectra
+between the indirect band gap and the direct band gap
+that are not affected by η.
+Eq.(1) to (5) serve as the foundation of calculating
+phonon-assisted optical properties from first principles
+utilizing second-order perturbation theory. In this work,
+we calculated ε2(ω) resulting from phonon-assisted op-
+tical absorption using the maximally localized Wannier
+function method to interpolate the quasiparticle ener-
+gies, velocity matrix elements, phonon frequencies, and
+electron-phonon-coupling matrix elements onto fine BZ-
+sampling grids needed to converge the spectra. The fine
+electronic k-point and phonon q-point sampling grids we
+employed are 32 × 32 × 32 for 3C, 24 × 24 × 16 for 2H,
+24 × 24 × 12 for 4H, 24 × 24 × 8 for 6H and 24 × 24 × 24
+for 15R. The delta function in Eq.(1) is approximated
+by a Gaussian function with a broadening of 0.05 eV to
+resolve fine features close to the absorption edge.
+The direct part of the optical spectra is calculated in-
+cluding electron-hole Coulomb interactions by solving the
+Bethe-Salpeter equation for the optical polarization func-
+tion, implemented in the BerkeleyGW package.[41, 42,
+50] The electron-hole interaction kernel is calculated on
+homogeneous electronic k-grid as in the GW calculations,
+and interpolated to the following finer grid through con-
+sidering the wavefunction overlap between the fine grids
+and coarse grids.[42] A small arbitrary shift is applied to
+all of the fine grids to ensure smooth spectra: 12×12×12
+for 3C, 8 × 8 × 6 for 2H, 8 × 8 × 3 for 4H, 9 × 9 × 3 for
+6H and 8 × 8 × 8 for 15R. For the direct part of the spec-
+tra, a Gaussian function with a broadening of 0.15 eV
+is used to model the delta function. The imaginary part
+of the dielectric function is calculated for a total number
+of bands of two times the number of valence bands for
+each polytype (corresponds to a maximum energy of at
+least 30 eV and ∼15 eV above valence band maximum),
+and the real part of the dielectric function is evaluated
+by summing the direct and phonon-assisted part of ε2(ω)
+and utilizing the Kramers-Kronig relationship. The com-
+bined real and imaginary parts of the dielectric function
+are then used to evaluate the refractive indices (nr) and
+absorption coefficients (See appendix section V A for the
+comparison between the calculated nr and experimental
+measurements). We mention that the imaginary part of
+the dielectric function resulting from the phonon-assisted
+process is typically a few orders of magnitude smaller
+than that resulting from direct transitions.
+As a re-
+sult, the contribution of the phonon-assisted process to
+the overall integral in the Kramers-Kronig relationship
+is usually negligible, i.e., the difference between the re-
+fractive index obtained with and without considering the
+phonon-assisted contribution is small.
+In addition to the approach derived above from second-
+order perturbation theory, phonon-assisted optical prop-
+erties can be obtained by properly displacing the atoms
+
+4
+TABLE I. Lattice constants (in ˚A) of the investigated SiC polytypes as calculated in the present study and compared to
+previous theoretical and experimental studies. For the 3C structure, the lattice parameters are also converted to the equivalent
+3H structure and shown in parentheses.
+Our calculated values are in excellent agreement with experimental data for all
+polytypes with a maximum difference of 0.8%.
+This work
+Previous theory
+Experiment
+Polymorph
+Space group
+a
+c
+a
+c
+a
+c
+3C (3H)
+F¯43m (206)
+4.382 (3.099)
+-
+4.372 (3.091)[51]
+-
+4.358 (3.082)[2]
+-
+2H
+P63mc (186)
+3.094
+5.077
+3.086[51]
+5.065[51]
+3.076[2]
+5.048[2]
+4H
+P63mc (186)
+3.096
+10.135
+3.094[51]
+10.129[51]
+3.073[2]
+10.053[2]
+6H
+P63mc (186)
+3.097
+15.194
+3.094[51]
+15.185[51]
+3.081[2]
+15.117[2]
+15R
+R3m (160)
+3.090
+37.910
+3.082[52]
+37.796[52]
+3.07-3.08[53]
+37.30-37.80[53]
+in supercells according to the eigenvalues and eigenmodes
+of the dynamical matrix and calculating direct transi-
+tions in the resulting distorted supercells, namely the
+special displacement method (SDM).[33–35] The benefit
+of the SDM approach is that the phonon-induced band
+gap renormalization, as well as the temperature depen-
+dence of the band structure, are taken into consideration
+by nature with the construction of the specific supercell.
+The SDM approach uses the vibrational eigenmodes and
+eigenfrequencies obtained from density functional pertur-
+bation theory to generate the optimal supercell.[33] In the
+SDM, phonon-related effects are directly captured via the
+construction of the supercell using the specific atomic
+displacements, the indirect part of the optical spectra
+can be directly calculated out using standard approaches
+utilizing first-order Fermi’s Golden rule. In our work, we
+adopted the SDM approach to construct 3×3×3, 4×4×4
+and 6 × 6 × 6 supercells for the 3C polytype to compare
+the SDM approach with the standard approach. We note
+that in the SDM approach, due to the requirement of
+calculations for large supercells, evaluating electron-hole
+interactions via the BSE formalism becomes computa-
+tionally expensive, thus our optical calculation is limited
+to the independent-particle picture.
+III.
+RESULT AND DISCUSSION
+A.
+Structural, electronic and phonon properties
+The calculated structural parameters for all polytypes
+considered in this study are listed in Table I, and are
+in good agreement with previously reported computa-
+tional results and experimental measurements. Our re-
+sults for the lattice constants of all polytypes agree well
+with literature values using the same type of exchange-
+correlation functional (PBE), with the maximum differ-
+ence not exceeding 0.3%. For better comparison to the
+hexagonal polytypes, we convert the cubic lattice param-
+eter of the 3C structure to the equivalent hexagonal 3H
+structure[2], i.e., a3H =
+1
+√
+2a3C, which is shown in the
+parentheses in I. For hexagonal structures, our calcula-
+tions show a minor expansion of the a lattice constant
+and a contraction of the c lattice constant as the number
+of hexagonal layers increases. Their variation, however,
+is on the order of the variation of the experimental lattice
+constant reported by different authors.[2, 54–56] Overall,
+the theoretical lattice constants are consistently overes-
+timated in comparison to experiment, which is expected
+for GGA-type of exchange-correlation functionals due to
+its known underbinding of chemical bonds. However, the
+calculated trend of the variation of lattice parameters
+across different polytypes agrees well with experimen-
+tal measurements. The maximum overestimation of the
+calculated lattice constant compares to the experimen-
+tal value is only 0.8%. The electronic band structures of
+all investigated polytypes of SiC are shown in Figure 1.
+From the figure it can be clearly seen that all polytypes
+exhibit indirect band gaps that are much smaller than
+their direct band gaps. Overall, we see that the band
+structure of the 3C structures shows clearly the smallest
+indirect band gap and the largest direct gap compared
+to other polytypes. It is interesting to notice that the
+2H structure exhibits a different feature that the other
+hexagonal structures (4H and 6H): whereas the conduc-
+tion band minimum (CBM) of 4H and 6H SiC occurs at
+the M point of the BZ, the CBM of 2H SiC occurs at
+the K point, while the local minimum at M lies about
+0.4 eV higher in energy. Later, we show that this differ-
+ence between the 2H structure and the other hexagonal
+polytypes induce a unique double bump-like feature in its
+indirect optical absorption as described in section III C.
+We next show that our calculated electronic band gaps
+agree well both with both experiment and with previ-
+ous calculations for all SiC polytypes.
+The calculated
+band gaps of the different SiC polytypes are listed in
+II. The PBE results exhibit the well-known underesti-
+mation of the band gap for all polytypes. However, in-
+cluding quasiparticle corrections with the GW approxi-
+mation, the calculated band gap is in much better agree-
+ment with experimental measurements, with the differ-
+ences being within 0.15 eV. Our results exhibit a consis-
+tent trend as other theoretical and experimental studies:
+the indirect gap of the material decreases as the number
+of hexagonal layers increase. Meanwhile, the 3C struc-
+ture shows a band gap about 1 eV narrower than all
+other polytypes, while the 15R structures show slightly
+smaller band gap compared to the 6H structure. It can
+be seen, however, that our calculated GW band gap does
+
+5
+G
+X
+W
+K
+G
+L
+U
+W
+L
+K|U X
+-5
+0
+5
+Energy (eV)
+G
+M
+K
+G
+A
+L
+H
+A|L
+M|K
+H
+-5
+0
+5
+Energy (eV)
+G
+M
+K
+G A
+L
+H
+A|L  M|K  H
+-5
+0
+5
+Energy (eV)
+G
+M
+K
+G A
+L
+H
+A|L
+     M|K
+       H
+-5
+0
+5
+Energy (eV)
+G
+L
+B1|B
+Z G
+X|Q
+F   P1
+Z|L   P
+-5
+0
+5
+Energy (eV)
+(a)
+(b)
+(c)
+(d)
+(e)
+FIG. 1.
+Quasiparticle band structures of the five investigated SiC polytypes: (a) 3C, (b) 2H, (c) 4H, (d) 6H and (e) 15R
+calculated with the GW method and interpolated with the Maximally Localized Wannier Function method. All polytypes are
+indirect-gap semiconductors (the indirect gap is marked with red arrows).
+TABLE II. Calculated band gaps (in eV) of the five investigated SiC polytypes both within PBE and with the GW approxi-
+mation, in comparison to previous theoretical and experimental results. Our data are in good agreement with previous work
+with a maximum difference of 0.15 eV.
+Polytype
+This work, PBE
+Previous theory, GGA
+This work, GW
+Previous theory, GW
+Experiment
+3C
+1.373
+1.391[51],1.410[57]
+2.514
+2.38[58], 2.24[59], 2.59[60]
+2.360[61]
+2H
+2.335
+2.350[57]
+3.311
+3.33[58]
+3.330[62]
+4H
+2.244
+2.238[57]
+3.205
+3.26[58], 3.11[59]
+3.230[61]
+6H
+2.050
+2.034[51],2.031[57]
+3.127
+3.05[58]
+3.000[61]
+15R
+1.977
+2.16[63]
+3.062
+-
+2.986[64]
+not consistently overestimate or underestimate the exper-
+imental gap. The difference between our GW result in
+comparison to other literature values may result from the
+plasmon-pole model, the different starting point due to
+the exchange-correlation functional, etc. The difference
+between the calculated value from the GW approxima-
+tion and the experimentally measured value is attributed
+in part to the lack of temperature-related renormalization
+(∼ 0.03-0.05 eV[22]) of the band gap and the zero-point
+motion. Nevertheless, the differences between our calcu-
+lated GW band gap and experimental measurements are
+in general within 0.15 eV, with the 3C structure being the
+largest but not exceeding 7%. We note that fine tuning
+the electronic-structure methodology to obtain a better
+match of the gap to experiment is not the focus of this
+work, thus we restrict our band-structure calculation to
+the one-shot GW approximation.
+We further show the calculated phonon band struc-
+tures, and we show that the phonon frequencies agree
+well with both experimental measurements as well as
+theoretical studies. The calculated phonon band struc-
+tures for all different SiC polytypes are shown in Fig-
+ure 2.
+The maximum phonon frequency at Γ is very
+similar across all different polytypes, with a variation
+within the range from 117.0 meV to 117.2 meV. Com-
+pared to available experimental data (See marks for the
+data and references in Figure 2), the differences are less
+than 3%. We also compared the calculated phonon fre-
+quencies to first-principles results[69] using ABINIT[70],
+and the differences of the calculated phonon frequencies
+are no larger than 2%. The calculated electron phonon
+matrix elements are interpolated onto a fine q and k-
+grid utilizing Wannier interpolation to obtain converged
+phonon-assisted optical spectra that we show later in sec-
+tion III C.
+
+6
+G
+X
+W
+K
+G
+L
+U
+W
+L
+K|U X
+0
+20
+40
+60
+80
+100
+120
+Energy (meV)
+G
+M
+K
+G
+A
+L
+H
+A|L M|K K|H
+0
+20
+40
+60
+80
+100
+120
+Energy (meV)
+G
+M
+K
+G A
+L
+H
+0
+20
+40
+60
+80
+100
+120
+Energy (meV)
+A|L M|K H
+G
+M
+K
+G A
+L
+H
+0
+20
+40
+60
+80
+100
+120
+Energy (meV)
+A|L M|K H
+G
+L
+B1|B
+Z G
+X|Q
+0
+20
+40
+60
+80
+100
+120
+Energy (meV)
+F P1
+Z|L P
+(a)
+(b)
+(c)
+(d)
+(e)
+FIG. 2. Calculated phonon dispersions of the investigated SiC polytypes (a) 3C, (b) 2H, (c) 4H, (d) 6H, and (e) 15R. Our
+calculated phonon frequencies underestimate available experimental measurements by less than 3%. Experimental data are
+plotted from: Ref. [65] (3C, purple X symbols), Ref. [66] (4H and 15R, red crosses), Ref. [67] (4H, orange circles) and Ref.[68]
+(6H, blue upper triangles).
+Compared to previous theoretical studies[69], we find very good agreement of the dispersion
+compared to experiment, with maximum differences of the phonon frequencies on the order of 2%.
+B.
+Direct optical properties
+We first report the calculated direct part of the absorp-
+tion spectra and the dielectric constants, with excitonic
+effects included.
+The optical spectra shown in Figure
+3 (ordinary direction for the hexagonal structures) are
+calculated with quasiparticle energies from the GW ap-
+proximation and with electron-hole interaction from the
+BSE approach. Our calculated optical spectra agree well
+with literature[71] for the 3C, 2H and 6H structure, espe-
+cially considering the relative height and position of the
+main peak. Similar to Ref.[71] we observe a significant
+shoulder close to the absorption onset (∼6 eV) that is
+clearest for 3C SiC, while being less significant for 2H
+and 6H. It is worthwhile to note that our spectra are no-
+ticeably higher than in Ref.[71], which is due to the differ-
+ent choice of broadening parameters used to approximate
+the delta function (0.15 eV in this work versus 0.25 eV in
+the reference). We report the calculated dielectric con-
+stants in Table III. Our calculated dielectric constants are
+higher than Ref.[71] and this can be affected by many fac-
+tors, including possible different BSE energy cutoffs and
+different quasiparticle band gaps, etc. Overall, the dif-
+ferences between our calculated dielectric constant and
+experimental values are within 5%. Since in the indirect
+part of the spectra, the imaginary part of the dielectric
+functions are orders of magnitudes smaller than the di-
+rect part, it is reasonable to assume that the refractive
+index is approximately proportional to the square root of
+the real part of the dielectric function. As a result, in this
+TABLE III. Dielectric constant for the ordinary polarization
+(ε⊥
+∞) calculated with the BSE approach for the five SiC poly-
+types.
+Our calculated dielectric constants overestimate ex-
+perimental values by less than 5%, resulting in less than 3%
+overestimation for the refractive index.
+Polytype
+This work,
+BSE
+Theory,
+BSE (Ref.[71])
+Experiment
+(Ref.[72])
+3C
+6.93
+6.36
+6.52
+2H
+6.89
+6.27
+6.51
+4H
+6.84
+-
+-
+6H
+6.79
+6.31
+6.52
+15R
+6.73
+-
+-
+case the difference in the calculated direct spectra from
+theory to experiment does not affect the calculated indi-
+rect part of the spectra by more than 5%. Later in this
+section, we show that this is a minor effect compared to
+other factors, for example, the finite-temperature band-
+gap renormalization, or the mere difference of the band
+gap from GW approximation to experiment.
+C.
+Phonon-assisted optical properties from
+second-order perturbation theory
+Following the calculation of the direct spectra, we de-
+termine the phonon-assisted indirect optical spectra of
+the SiC polytypes, evaluated with the second-order time-
+dependent perturbation theory. We find that our results
+
+7
+0
+10
+20
+30
+40
+0
+10
+20
+30
+40
+0
+10
+20
+30
+40
+Im e(w)
+0
+10
+20
+30
+40
+2
+4
+6
+8
+10
+12
+14
+Energy (eV)
+0
+10
+20
+30
+40
+3C
+2H
+4H
+6H
+15R
+3C
+2H
+4H
+6H
+15R
+FIG. 3. Solid: Calculated direct part of the optical absorption
+spectra (imaginary part of the complex dielectric function) for
+the ordinary light polarization for the five different SiC poly-
+types, including quasiparticle effects with the GW approxi-
+mation and electron-hole interactions via the BSE equation.
+Dashed: Theoretical BSE data from Ref.[71] for 3C, 2H and
+6H polytypes. We find very good agreement in terms of the
+peak positions and shapes, with the height of the peak poten-
+tially affected by the choice of the broadening parameters.
+agree well with experiments for the most common poly-
+types (3C, 4H and 6H), and we can predict the indirect
+spectra for 2H and 15R polytypes. Figure 4 shows the
+calculated absorption coefficient in the region of indirect
+absorption for all polytypes. Additional rigid shifts ∆ are
+applied to the quasiparticle energies from the GW ap-
+proximation to match the indirect gap from experiment:
+∆ = −0.154 eV for 3C, ∆ = 0.019 eV for 2H, ∆ = 0.025
+eV for 4H, ∆ = −0.126 eV for 6H and ∆ = −0.076 eV
+for 15R. For the indirect part of the spectra, it is reason-
+able to assume that the correction on the band gap can
+be well represented with a rigid shift of the absorption
+coefficient (i.e., no scaling of the spectra is needed).[33]
+We compared the calculated absorption coefficient in the
+indirect region of the 3C, 4H and 6H polytypes to ex-
+perimental measurements, and we find a good agreement
+with the experimental results.
+For the 15R structure,
+we observe that the absorption coefficient in the indi-
+rect region is similar to the 6H structure. Such similar-
+ity is expected due to the similarities in the electronic
+band structure and phonon band structures.
+Interest-
+ingly, for the 2H structure, we observe two bump-like
+features around photon energies of 3.45 eV and 3.75 eV.
+We note that these two energies correspond to the en-
+ergy difference between the valence band maximum at Γ
+and the conduction band minimum at K, as well as the
+second minimum at M, respectively. This is a unique
+feature that we only observe in the 2H structure, as for
+all other hexagonal structures, the CBM is located at M,
+rather than K.
+2
+2.5
+3
+3.5
+4
+Energy (eV)
+0.01
+0.1
+1
+10
+100
+a(w)
+1/2 (cm
+-1/2)
+3C
+6H
+15R
+2H
+4H
+FIG. 4. Calculated optical absorption coefficient for the ordi-
+nary direction (E⊥ c) at 300 K for the five investigated SiC
+polytypes in the phonon-assisted spectral region between the
+indirect and direct band gaps. The experimental results from
+the literature are taken from Ref.[73] (3C, 4H and 6H, cross
+mark), Ref.[74] (4H and 6H, dot marks) and Ref.[75] (4H and
+6H, upper triangle mark). A rigid shift has been applied to
+each theoretical curve to correct the difference between the
+GW-calculated gaps and the experimental values at 300 K
+listed in Table II for each polytype. The calculated absorp-
+tion spectra are in overall good agreement with experiment.
+Stepping forward, our first-principles tools allow us to
+investigate the temperature dependency of the indirect
+optical properties to examine the factors that affect the
+spectra as temperature changes. As a thermally stable
+material, many of the novel optoelectronic applications
+of SiC involve extreme conditions.[76–78] Therefore, be-
+ing able to predict temperature-dependent optical prop-
+erties from a first-principles perspective can provide cru-
+cial information for novel applications.
+We calculated
+the temperature-dependent indirect optical spectra for
+the 4H and 6H polytypes at three different temperatures
+(300 K, 473 K and 573 K). The calculated temperature-
+dependent absorption coefficients in the indirect region
+are shown in Figure 5.
+Additional rigid shifts are ap-
+plied to match the calculated electronic band gap with
+experimentally determined values.[61] It can be seen from
+the figure that our calculation reproduces the relative
+change in the indirect optical absorption due to temper-
+ature changes very well after taking the band gap renor-
+malization into account (See appendix section V B for
+
+8
+3
+3.5
+4
+Energy (eV)
+0.1
+1
+10
+100
+a(w)
+1/2 (cm
+-1/2)
+D573=-0.054 eV
+D473=-0.023 eV
+D300=0.025 eV
+3
+3.5
+4
+Energy (eV)
+0.1
+1
+10
+100
+a(w)
+1/2 (cm
+-1/2)
+D573=-0.207 eV
+D473=-0.174 eV
+D300=-0.127 eV
+(a)
+(b)
+FIG. 5.
+Calculated absorption coefficient for (a) 4H SiC and
+(b) 6H SiC as a function of photon energy in the phonon-
+assisted region (lines) for a temperature of 300 K (solid), 473
+K (dashed), and 573 K (dotted), and compared to experimen-
+tal measurements (data points) from Ref.[75] at 300 K (upper
+triangle), 473 K (square) and 573 K (plus). The theoretical
+curves have been rigidly shifted to match the temperature de-
+pendence of the experimental band gap according to Ref.[61].
+The change in the spectra with respect to temperature change
+is well-predicted with the changes in band gaps included.
+spectra without additional rigid shift). We conclude that
+the effects of temperature on both the phonon occupa-
+tion numbers as well as on the renormalization of the
+band gap itself are important to obtain reliable predic-
+tion of the indirect absorption spectra. However, for the
+6H polytype, we find that the simple rigid shift of the gap
+does not accurately provide the correct positions of the
+shoulder of the absorption curve. In the next section, we
+examine an alternative route to calculate phonon-assisted
+absorption that avoids the rigid-shift approach and con-
+siders implicitly the temperature renormalization of the
+band structure.
+2
+2.5
+3
+3.5
+4
+Energy (eV)
+0.01
+0.1
+1
+10
+100
+a
+1/2(cm
+-1/2)
+SDM
+Second-order perturbation
+FIG. 6. Calculated absorption coefficient as a function of en-
+ergy for the 3C polytype with 6 × 6 × 6 supercell with the
+special displacement (SDM) approach[35] using a 4 × 4 × 4
+k-points grid (violet) and the perturbation approach (red
+solid) (32×32×32 k/q-points grid). Experimental data from
+Ref.[73] (black dots) and Ref.[74] (black cross) are shown for
+comparison. The same rigid shift is applied as in Figure 4 for
+the perturbation approach, while no rigid shift is applied for
+the SDM approach. The renormalization of the band gap due
+to temperature is physically included with the SDM approach.
+D.
+Phonon-assisted optical properties from the
+special displacement method (SDM)
+Next, we show that the effect of temperature-related
+renormalizations of the band gap on the optical absorp-
+tion spectra of SiC can be overcome by applying the spe-
+cial displacement method. Although the SDM approach
+only requires a single snapshot of the atomic displace-
+ment configuration,[33–35] the need to perform calcu-
+lations on relatively large supercells (approaching thou-
+sands of atoms) results in a drastic increase in the com-
+putational cost. As a result, we take quasiparticle cor-
+rections into account using a rigid increase of the gap
+(1.141 eV) to account for the difference between the PBE
+band gap and the GW band gap from unit cell calcula-
+tions. In Figure 6, we show the best converged results
+using the SDM approach compared to the perturbation
+approach for 3C SiC. The same rigid shift is applied for
+the perturbation approach as indicated in the previous
+sections. In the SDM approach, no artificial shifts are
+applied except for the indirect-band-gap correction from
+the PBE to the GW value. At energies beyond 3 eV, the
+SDM approach shows better agreement with experiment
+quantitatively compared to experimental data. This is
+because, by taking the renormalization of the band gap
+with respect to temperature into account, the SDM ap-
+proach is more physical in terms of predicting the correct
+position of the absorption peaks. However, below a pho-
+ton energy of 3 eV, the perturbation approach predicts
+the shape of the curve better. There are a few possible
+factors: First, it is straightforward to use the perturba-
+
+9
+tion approach utilizing Wannier interpolation to obtain
+matrix elements on the fine grid, which is important in
+order to obtain converged spectra near the absorption
+onset that require a fine sampling of the Brillouin zone.
+For the SDM approach, however, convergence with re-
+spect to supercell size is required. In our work, we find
+that under-converged supercell can incur significant error
+for the absorption onset, or incorrect absorption shoul-
+ders, and a 6 × 6 × 6 supercell is needed for converged
+phonon-assisted spectra (See appendix section V C). Sec-
+ond, the nature of the approach corresponds to neglecting
+the explicit phonon frequencies in the delta-function of
+calculating ε2(ω) in Eq.(1), thus, the absorption onset
+can be affected.[33–35]
+Overall, we show that although both the SDM ap-
+proach and the perturbation approach provide satisfy-
+ing results for predicting the phonon-assisted optical
+absorption properties for 3C SiC, the SDM approach
+clearly illustrates the importance of including the cor-
+rect temperature-dependent renormalization of the band
+gap to obtain higher quantitative accuracy. The special
+displacement approach also shows the potential of fur-
+ther investigations, e.g., excitonic effects on the indirect
+part of the optical spectra by performing BSE calcula-
+tions on properly converged supercells. However, fully
+converged BSE calculations on such large supercells is
+computationally challenging and is beyond the scope of
+this paper. Meanwhile, it is also clear that, to resolve the
+finer structures of the optical absorption around the ab-
+sorption edge, it is important to fully consider the effects
+of phonons including the phonon energies.
+IV.
+CONCLUSION
+In this work, we investigate the phonon-assisted opti-
+cal properties of five different SiC polytypes with first-
+principles calculations. We first calculate the structural
+properties of all polytypes with density functional theory,
+and electronic properties with the GW approximation,
+and we show that the calculated results agree well with
+both other theoretical studies and experimental measure-
+ments.
+We then utilize Wannier interpolation to ob-
+tain the electron-phonon coupling in dense electronic and
+phonon grids, and we use second-order time-dependent
+perturbation theory to obtain phonon-assisted optical
+spectra in the indirect gap region of all polytypes. We
+show that the simulated absorption coefficient in the indi-
+rect region agrees well with experimental measurements
+for the most common 3C, 4H and 6H polytypes, and pro-
+vides prediction for 15R and 2H polytypes. Further anal-
+ysis on the temperature-dependent indirect optical prop-
+erties shows that our theoretical simulation predicts the
+trend of the variation in optical properties with respect
+to temperature well, and we show that our simulations
+can be particularly useful in understanding the practical
+application of the material at various finite temperatures.
+However, our results shows that the renormalization of
+the band gaps due to temperature is important in pre-
+dicting the temperature-dependent spectra. Therefore,
+we further compare the conventional approach utilizing
+second-order perturbation theory to the method employ-
+ing special displacement of specific atoms in supercells,
+and we show that improvement of the predictions of
+the temperature dependency of the indirect optical spec-
+tra can be obtained by take the renormalization of the
+electronic energies implicitly due to temperature change
+into account. Our work shows that these methods en-
+able quantitively predictions of temperature-dependent
+phonon-assisted absorption spectra in indirect-gap semi-
+conductor materials.
+ACKNOWLEDGEMENT
+We thank Dr. Marios Zacharias for the fruitful dis-
+cussion about the SDM approach.
+The work is sup-
+ported as part of the Computational Materials Sciences
+Program funded by the U.S. Department of Energy, Of-
+fice of Science, Basic Energy Sciences, under Award No.
+DE-SC0020129. Computational resources were provided
+by the National Energy Research Scientific Computing
+Center, which is supported by the Office of Science of
+the U.S. Department of Energy under Contract No. DE-
+AC02-05CH11231.
+V.
+APPENDIX
+A.
+Refractive index
+1 1.5
+2
+2.5
+3
+2.4
+2.5
+2.6
+2.7
+2.8
+2.9
+Refractive index nr
+1 1.5
+2
+2.5
+3
+Photon energy (eV)
+1 1.5
+2
+2.5
+3
+FIG. 7. Calculated refractive index in the region close to the
+indirect band gap for 4H SiC (black solid), 3C SiC (red solid)
+and 6H SiC (green solid), as well as comparison to experi-
+mental curve (dashed) from Ref.[79] (4H and 6H) and Ref.[80]
+(3C).
+In this section we show the calculated refractive in-
+dex compared to experimental measurements. Figure 7
+shows that our calculated refractive index of 3C, 4H and
+
+10
+6H polytypes agrees very well with experimental mea-
+surements, with the difference being an overestimation
+by less than 3%. These differences in the refractive in-
+dex from our calculation compared to experiments are
+a negligible source of error even compared to the mere
+difference of the band gaps themselves.
+B.
+Temperature dependent indirect spectra
+3
+3.5
+4
+Energy (eV)
+0.1
+1
+10
+100
+a(w)
+1/2 (cm
+-1/2)
+(a)
+3
+3.5
+4
+Energy (eV)
+0.01
+0.1
+1
+10
+100
+a(w)
+1/2 (cm
+-1/2)
+(b)
+FIG. 8.
+Calculated absorption coefficient for 4H SiC (a)
+and 6H SiC (b) as a function of photon energy in the indi-
+rect region (lines) for a temperature of 300 K (solid), 473 K
+(dashed), and 573 K (dotted), and compared to experimental
+measurements (data points) from Ref.[75] at 300 K (upper tri-
+angle), 473 K (square) and 573 K (plus). The difference with
+Figure 5 is the omission of the correction to the band-gap
+value to match experiment.
+In this section, we demonstrate the temperature-
+dependent indirect spectra without including the ad-
+ditional rigid shifts to account for the temperature-
+dependent band-gap renormalization.
+In Figure 8, we
+show the calculated optical spectra without considering
+the difference between calculated band gap and exper-
+imentally measured band gap.
+It can be seen clearly
+that first, the effects of temperature on the indirect op-
+tical spectra itself is still clear, and this is mainly due
+to the change in Bose-Einstein occupation factor of the
+phonons. However, without considering the difference in
+band gap from theory and from experiment, especially in
+the 6H case, the onset of absorption is severely overes-
+timated, and more importantly in this context, the dif-
+ferences between the curves at different temperatures is
+underestimated.
+This clearly shows the importance of
+considering both the change in occupations due to tem-
+perature and the changes in band gap itself due to tem-
+perature change, as the change in spectra is clearly a
+combined effect of both.
+C.
+Convergence of the supercell approach
+2
+2.5
+3
+3.5
+4
+Energy (eV)
+0.01
+0.1
+1
+10
+100
+a
+1/2(cm
+-1/2)
+3´3´3 supercell
+4´4´4 supercell
+6´6´6 supercell
+FIG. 9.
+Calculated absorption coefficient using the special
+displacement approach[35] with three different supercell sizes:
+3 × 3 × 3 (dotted), 4 × 4 × 4 (dashed) and 6 × 6 × 6 (solid).
+The k-points sampling are 8 × 8 × 8, 6 × 6 × 6 and 4 × 4 ×
+4, respectively.
+All k-point grids are randomly shifted for
+better convergence. Convergence beyond 6×6×6 cell require
+supercells with more than 1000 atoms and is not tested due
+to the large computational cost.
+To test the convergence of the supercell approach ver-
+sus the supercell size, we calculated the indirect optical
+spectra with 3 × 3 × 3, 4 × 4 × 4 and 6 × 6 × 6 supercells.
+In this section we report the convergence behavior of the
+calculated spectra near the absorption onset in Figure
+9. It can be seen from the figure that the two smaller
+supercells are not sufficient for converged optical spectra
+in the indirect region. The best converged results of the
+6 × 6 × 6 supercell is reported in the main text.
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+Mask-then-Fill: A Flexible and Effective Data Augmentation Framework
+for Event Extraction
+Jun Gao1,3∗ Changlong Yu4 Wei Wang5 Huan Zhao6 Ruifeng Xu1,2,3†
+1Harbin Institute of Technology (Shenzhen)
+2Peng Cheng Laboratory
+3Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies
+[email protected]
+[email protected]
+4HKUST, Hong Kong, China
+5Tsinghua University
+64Paradigm. Inc.
+[email protected]
+[email protected]
+[email protected]
+Abstract
+We present Mask-then-Fill, a flexible and ef-
+fective data augmentation framework for event
+extraction. Our approach allows for more flex-
+ible manipulation of text and thus can gener-
+ate more diverse data while keeping the origi-
+nal event structure unchanged as much as pos-
+sible.
+Specifically, it first randomly masks
+out an adjunct sentence fragment and then in-
+fills a variable-length text span with a fine-
+tuned infilling model. The main advantage lies
+in that it can replace a fragment of arbitrary
+length in the text with another fragment of vari-
+able length, compared to the existing methods
+which can only replace a single word or a fixed-
+length fragment. On trigger and argument ex-
+traction tasks, the proposed framework is more
+effective than baseline methods and it demon-
+strates particularly strong results in the low-
+resource setting. Our further analysis shows
+that it achieves a good balance between diver-
+sity and distributional similarity.
+1
+Introduction
+Event Extraction (EE), which aims to extract trig-
+gers with specific types and their arguments from
+unstructured texts, is an important yet challenging
+task in natural language processing. In recent years,
+deep learning methods have emerged as one of the
+most prominent approaches for this task (Nguyen
+and Nguyen, 2019; Lin et al., 2020; Du and Cardie,
+2020; Paolini et al., 2021; Lu et al., 2021; Lou et al.,
+2022). However, they are notorious for requiring
+large labelled data, which limits the scalability of
+EE models. Annotating data for EE is usually
+costly and time-consuming, as it requires expert
+knowledge. One possible solution is to leverage
+data augmentation (DA) (Simard et al., 1998).
+Existing DA methods for NLP can be broadly
+classified into two types: (1) the first is to augment
+∗Work done when Jun Gao was interning at 4Paradigm
+† Corresponding author
+The police said Mike left this town yesterday.
+Trigger
+Event Type: Transport
+Arg1
+Role: Artifact
+Arg2
+Role: Origin
+Arg3
+Role: Time-Within
+The cop said Mike left this town yesterday.
+The police said Mike went away from this town yesterday.
+The cop saw Mike left this town yesterday.
+(1) BackTranslation
+(2) Synonym Replacement
+(3) BERT
+Figure 1: Visualization of three different data aug-
+mentation methods applied to a sentence containing a
+“Transport” event. Spans marked with different colors
+are event triggers and arguments. The parts of the aug-
+mented sample that differ from the original are colored
+in gray. Backtranslation (Xie et al., 2020) translates
+the input sentence into another language and back to
+the original. Synonym Replacement (Dai and Adel,
+2020) and BERT (Yang et al., 2019) replace words in
+the sentence.
+training data by modifying existing examples (Sen-
+nrich et al., 2016; ¸Sahin and Steedman, 2018; Dai
+and Adel, 2020; Wei and Zou, 2019), and (2) the
+second is to generate new data by estimating a gen-
+erative process and sample from it (Anaby-Tavor
+et al., 2020; Quteineh et al., 2020; Yang et al., 2020;
+Ye et al., 2022). Since the EE task requires DA
+methods to generate augmented samples and token-
+level labels jointly, the second type of DA method
+is inapplicable here. In this study, we mainly focus
+on the first type of method.
+Applying existing DA methods to the EE task
+is more challenging than to translation or classi-
+fication tasks, because we need to augment train-
+ing data while keeping the event structure (trigger
+and arguments) unchanged. Figure 1 presents ex-
+amples of three different DA methods applied to
+a sentence containing a “Transport” event. The
+event is triggered by word “left” and it has three
+arguments with different roles (“Mike”, “this town”
+arXiv:2301.02427v1  [cs.CL]  6 Jan 2023
+
+and “yesterday”). As shown in Figure 1, it is in-
+feasible to apply sentence-level DA methods such
+as BackTranslation (Xie et al., 2020), because it
+may change the event structure (change “left” to
+“went away from”). Previous attempts on DA for
+such tasks typically use heuristic rules such as
+synonym replacement (Dai and Adel, 2020; Cai
+et al., 2020) or context-based words substitution
+with BERT (Yang et al., 2019). Their idea is to
+replace adjunct tokens (the tokens in sentences ex-
+cept triggers and arguments) with other tokens, and
+thus can ensure the event structure is unchanged as
+much as possible. However, recent studies (Ding
+et al., 2020; Yang et al., 2020; Ye et al., 2022) find
+that such methods provide limited data diversity.
+In NLP, the diversity of data is mainly reflected
+in two aspects: expression diversity and seman-
+tic diversity (Zhao et al., 2019). The Synonym
+Replacement and BackTranslation methods lack
+semantic diversity, because they can only produce
+samples with similar semantics. The BERT-based
+method can only replace words and cannot change
+the syntax, so it cannot generate samples with a
+wide variety of expressions. The lack of sufficient
+diversity may lead to greater overfitting or poor
+performance through training on examples that are
+not representative.
+To this end, we present Mask-then-Fill, a flexi-
+ble and effective data augmentation framework for
+event extraction. Our approach allows for more
+flexible manipulation of text and thus can generate
+more diverse data while keeping the original event
+structure unchanged as much as possible. Specif-
+ically, we first define two types of text fragments
+in a sentence: event-related fragments (trigger and
+arguments) and adjunct fragments (e.g. “The po-
+lice said”). Then, we model DA for the EE task
+as a Mask-then-Fill process: we first randomly
+masks out an adjunct sentence fragment and then
+infills a variable-length text span with a fine-tuned
+infilling model (T5) (Raffel et al., 2020). The main
+advantage lies in that it can replace a fragment of
+arbitrary length in the text with another fragment of
+variable length, compared to the existing methods
+which can only replace a single word or a fixed-
+length fragment.
+To the best of our knowledge, we are the first
+to augment training data for event extraction via
+text infilling. We empirically show that the Mask-
+then-Fill framework improves performance for
+both classification-based (EEQA) and generation-
+The police said Mike left this town yesterday.
+[MASK] Mike left this town yesterday.
+Output1: The next door neighbor said
+Output2: Because of a fight with his girlfriend,
+Output3: Since he found a new job,
+…
+Infilling Model (T5)
+② Decoding by Sampling
+[MASK] Mike left this town yesterday.
+① Mask out an adjunct fragment
+③ Fill in the blank
+Figure 2: Overview of the proposed Mask-then-Fill
+framework.
+based (Text2Event) event extraction models on a
+well-known EE benchmark dataset (ACE2005). Es-
+pecially, it demonstrates strong results in the low
+resource setting. We further investigate reasons for
+its effectiveness by introducing two metrics, Affin-
+ity and Diversity, and find that the data augmented
+by our approach have better diversity with less dis-
+tribution shifts, achieving a good balance between
+diversity and distributional similarity.
+2
+The Mask-then-Fill Framework
+Figure 2 presents an overview of Mask-then-Fill
+framework. The input sentence contains two types
+of text fragments: event-related fragments (words
+with colors) and adjunct fragments (underlined).
+Our idea is to rewrite the whole adjunct fragment
+instead of replacing some words, and the rewritten
+sentence fragment should fit the context and should
+not introduce new events. To this end, we model
+DA for EE as a Mask-then-Fill process: we first
+randomly mask out an adjunct sentence fragment
+and then infills a variable-length text span with a
+fine-tuned infilling model. In the following, we
+describe in detail the Mask-then-Fill framework.
+Mask out an adjunct fragment.
+Given a pro-
+totype sentence X = {x1, · · · , xL} of length L
+from the training set, we first define an adjunct frag-
+ment as a set of non-overlapping spans of x that
+do not contain the event triggers and arguments.
+We then replace one of the adjunct fragments with
+a [MASK] symbol. The incomplete sentence ˆx
+
+is a version of x with a fragment replaced with a
+[MASK] symbol.
+Blank Infilling Model.
+We formulate our blank
+infilling process as the task of predicting the miss-
+ing span of text which is consistent with the preced-
+ing and subsequent text. Figure 2 gives an example
+with an incomplete input sentence �x, where the
+[MASK] is a placeholder for a blank, which has
+masked out multiple tokens. Our goal is to pre-
+dict only the missing span y which will replace the
+[MASK] token in �x. Therefore, the infilling task
+can be cast as learning p(y|�x).
+To train our infilling model, we fine-tune a pre-
+trained sequence-to-sequence model T5 (Raffel
+et al., 2020) on the Gigaword corpus (Graff et al.,
+2003), which is from similar domains as the event
+extraction dataset ACE2005 adopted by our work.
+Given a corpus consisting of plain sentences, we
+first produce large pools of infilling examples and
+then train the T5 model on these examples. For a
+given complete sentence x from the training cor-
+pus, we generate an infilling example �x with the
+following procedure: (1) randomly sample a span
+length l from the range of [1, min(10, l)]; (2) split
+the sentence into l spans; (3) randomly select a
+span to be replaced with a [MASK] symbol. The
+replaced span is used as the target y. We then
+fine-tune the T5 model on these infilling examples,
+yielding the model of the form pθ(y|�x).
+Fill in the blank.
+Once trained, the infilling
+model can be used to take the incomplete sentence
+�x, containing one missing span, and return a pre-
+dicted span y. We then replace the [MASK] token
+in �x with the predicted span y to generate an aug-
+mented example. Note that we can produce large
+pools of augmented samples using top-k sampling.
+3
+Experimental Setup
+Dataset.
+We empirically evaluate our proposed
+data augmentation method for event extraction
+on the ACE2005 corpus1 with the same train-
+dev-test split and preprocessing step as previous
+works (Zhang et al., 2019; Wadden et al., 2019).
+We simulate a low-resource setting by randomly
+sampling 1,000, 4,000 and 8,000 examples from
+the training set to create the small, medium, and
+large training sets (denoted as S, M, L in Table 1,
+whereas the complete training set is denoted as F).
+1https://catalog.ldc.upenn.edu/LDC2006T06
+We only augment the training data and keep the
+dev set and test sets unchanged.
+Evaluation Metrics.
+Following the previous
+works (Du and Cardie, 2020; Lu et al., 2021) on
+event extraction, we adopt the same evaluation cri-
+teria defined in Li et al. (2013): (i) An event trigger
+is correctly identified and classified (Trig-ID+C) if
+its offsets match a gold trigger and its event type is
+also correct. (ii) An argument is correctly identified
+and classified (Arg-ID+C) if its offsets and event
+type match a gold argument and its event role is
+also correct.
+Event Extraction Models.
+In our study, we con-
+sider two representative models for event extrac-
+tion:
+• Text2Event (Lu et al., 2021) is a framework to
+solve the event extraction task by casting it as
+a SEQ2SEQ generation task. All triggers, argu-
+ments, and their labels are generated as natural
+language words.
+• EEQA (Du and Cardie, 2020) formulates the
+event extraction task as a question answering
+task. They develop two BERT-based QA models
+– one for event trigger detection and the other for
+argument extraction.
+Comparison Methods.
+We compare our pro-
+posed data augmentation method Ours (t5-small)
+with three baselines: (1) Synonym Replacement
+replaces adjunct tokens with one of their synonyms
+retrieved from WordNet (Miller, 1992) at random;
+(2) BERT replaces adjunct tokens with others ran-
+domly drawn according to the pretrained BERT’s
+distribution; (3) Span-BackTranslation: Inspired
+by Yaseen and Langer (2021), we only “back trans-
+late” randomly selected adjunct spans to prevent
+the model from changing the event structure.
+Hyperparameters.
+For all data augmentation
+methods, we tune the number of augmentation sam-
+ples per training sample from a list of numbers:
+{1, 3, 6, 10}.
+4
+Results and Analysis
+Main Results.
+The main results are presented
+in Table 1,
+where we use two EE mod-
+els (Text2EVent and EEQA) to test the perfor-
+mance of different DA methods in both low-
+resource (S, M and L) and normal (F) settings. As
+shown in the table, we observe that Ours (t5-small)
+achieves the best overall performance among all
+
+EE Model
+DA Method
+Trig-ID+C (F1)
+Arg-ID+C (F1)
+S
+M
+L
+F
+S
+M
+L
+F
+Text2Event
+No Augmentation
+45.44
+59.75
+63.55
+67.06
+22.05
+36.04
+40.35
+49.30
+Synonym Replacement
+49.14
+61.96
+63.73
+69.09
+27.71
+39.64
+43.63
+48.95
+BERT
+48.66
+60.75
+63.81
+68.33
+26.71
+38.75
+41.44
+48.41
+Span-BackTranslation
+47.91
+61.54
+64.59
+67.58
+26.68
+38.18
+43.39
+47.93
+Ours (t5-small)
+52.32
+63.38
+67.25
+69.03
+28.68
+39.79
+44.73
+50.29
+EEQA
+No Augmentation
+48.05
+64.20
+64.06
+67.13
+39.81
+56.30
+59.27
+61.93
+Synonym Replacement
+54.86
+64.03
+65.71
+68.05
+42.99
+56.62
+56.40
+61.50
+BERT
+53.61
+63.23
+65.90
+68.35
+38.80
+52.82
+59.49
+61.62
+Span-BackTranslation
+53.26
+62.64
+65.46
+68.40
+42.47
+56.64
+55.87
+61.55
+Ours (t5-small)
+54.80
+64.37
+67.33
+69.62
+47.67
+56.70
+58.52
+61.62
+Table 1: Results on trigger extraction and argument extraction using different subsets of the training data. The best
+results are marked in bold, and the second best is underlined.
+DA methods on both trigger extraction (F1) and
+argument extraction (F1). Using our DA method
+gives the best results for the Text2event model on
+7 out of 8 datasets. For the EEQA model, our
+method achieves the best results on 6 out of 8
+datasets, where the difference between our method
+and the best results on Trig-S and Arg-L is very
+small, with only 0.06 and 0.97 points difference be-
+tween them, respectively. Particularly, our methods
+demonstrates strong results in the low-resource set-
+ting. Using our DA gives the Text2Event model a
+performance improvement of 15.14% and 30.07%
+on Trig-S and Arg-S, respectively.
+We also notice that as the amount of data in-
+creases, the improvement from all DA method de-
+creases, and in some cases (EEQA model on Arg-L
+and Arg-F), there is even a slight decrease in per-
+formance. In the case of more data, the model may
+overfit if the augmented data are just some similar
+samples rather than data with large variations.
+DA Method
+Affinity
+Dist-1
+Dist-2
+Synonym Replacement
+-0.118
+0.400
+0.523
+BERT
+-0.082
+0.374
+0.496
+Span-BackTranslation
+-0.155
+0.407
+0.513
+Ours (t5-small)
+-0.086
+0.488
+0.612
+Table 2: Results on Affinity and Diversity. The best re-
+sults are marked in bold. The second best is underlined.
+Affinity and Diversity.
+Inspired by Gontijo-
+Lopes et al. (2020), we further investigate reasons
+for its effectiveness by introducing two metrics,
+Affinity and Diversity, where Affinity quantifies how
+augmentation shifts data distribution and Diversity
+measures the complexity of the augmented data.
+We measure Affinity by computing the difference
+between the loss of a model trained on the original
+training set and tested on the original example, and
+the loss of the same model tested on an augmented
+example. We use the Dist-1/2 metric (Celikyilmaz
+et al., 2020), commonly used in text generation, to
+Furthermore , the United States supported him in the war against Iran.
+In addition, the United States supported him in the war against Iran.
+Furthermore, the United States supported him in the war against Iran.
+Later, the united States supported him in the war against Iran .
+Furthermore, the United States supported iraq in the war against iraq .
+Moreover, the combine States supported him in the war against Iran.
+What is more, the unify DoS supported him in the war against Iran.
+He also called for an end to the war against Iran.
+The U.S. military has been fighting a war against Iran.
+Event Type: Attack | Trigger: war
+(1) Synonym Replacement
+(2) BERT
+(3) Span-BackTranslation
+(4) Ours (t5-small)
+Figure 3: Augmented examples of four different DA
+methods.
+Given a sentence containing an “Attack”
+event triggered by the word "war", we generate two
+new samples for each DA method. The parts of the
+new sample that differ from the original are colored in
+gray.
+assess the Diversity of the augmented data. For im-
+plementation details of two metrics, see Appendix.
+We first construct a new test set by generating a
+new sample for each data in the test set. We then
+calculate the Affinity and Dist-1/2 scores between
+the new data set and the original data set, respec-
+tively. As shown in Table 2, it is clear that the data
+augmented by our DA method have better diversity
+with less distribution shifts, obtaining a balance
+between diversity and distributional similarity.
+Case Study.
+Figure 3 presents examples gener-
+ated by different DA methods. Given a sentence
+containing an “Attack” event triggered by the word
+"war", we generated two new samples for each DA
+method, and the parts of the new sample that differ
+from the original are colored in gray. Obviously,
+The synonym replacement based on WordNet can-
+not avoid introducing some words that do not fit the
+context (e.g “unify” and “DoS”), while the BERT-
+based word replacement can consider the context
+
+better. However, they both provide limited diver-
+sity. BackTranslation method performs even worse
+in terms of data diversity. Its generated data differs
+very little from the original sentence. Finally, com-
+pared with the original sentences, the new samples
+generated by our method are more fluent and more
+different in expression and semantics. Therefore, it
+not only generates data that fits the context better,
+but also provides better diversity.
+5
+Conclusion
+In this paper, we present Mask-then-Fill, a flexi-
+ble and effective data augmentation framework for
+event extraction. Our approach allows for more
+flexible manipulation of text and thus can gen-
+erate more diverse data while keeping the origi-
+nal event structure unchanged. The main advan-
+tage lies in that it can replace a fragment of ar-
+bitrary length in the text with another fragment
+of variable length. We empirically show that the
+Mask-then-Fill framework improves performance
+for both EEQA and Text2Event EE models on
+the ACE2005 dataset. It demonstrates particularly
+strong results in the low-resource setting. Our fur-
+ther analysis shows that it achieves a good balance
+between diversity and distributional similarity.
+Limitations
+This paper presents a flexible and effective data
+augmentation framework for event extraction tasks.
+Here, we note some of Mask-then-Fill frame-
+work’s limitations. First, performance gains can
+be marginal when data is sufficient. We believe
+this approach has much room for improvement in
+generating more diverse data. In this work, we
+select only one adjunct fragment at a time for modi-
+fication, and modifying multiple adjunct fragments
+in an event mention can further enhance the diver-
+sity of the generated data. Second, currently this
+method can only replace one fragment at a time.
+This makes it easier to control the properties of
+the generated fragments, such as length or style.
+It is possible to modify multiple fragments at the
+same time using some existing techniques (Don-
+ahue et al., 2020; Du et al., 2022; Chen et al., 2022).
+This approach is more efficient, but it is prone to
+generate incoherent augmented samples and thus
+introduce more noise. A possible approach to solve
+this problem is to design some sample selection
+strategies.
+Acknowledgements
+This work was partially supported by the Na-
+tional Natural Science Foundation of China
+62006062 and 62176076, Shenzhen Foundational
+Research
+Funding
+JCYJ20200109113441941,
+JCYJ20210324115614039,
+The
+Major
+Key
+Project of PCL2021A06, Guangdong Provincial
+Key Laboratory of Novel Security Intelligence
+Technologies 2022B1212010005.
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+understanding. In Proceedings of the 2019 Confer-
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+sociation for Computational Linguistics.
+A
+Affinity and Diversity
+Inspired by Gontijo-Lopes et al. (2020) and Arora
+et al. (2021), we proposed to use a calibration
+method to quantify how augmentation shifts data.
+They all note that a trained model is often sensitive
+to the distribution of the training data.
+Given the original example x and one of its aug-
+mented example x+, we measure distribution shifts
+by computing the difference between the loss of a
+model trained on the original training set and tested
+on the original example, and the loss of the same
+model tested on an augmented example:
+τα = ℓ(M, x) − ℓ(M, x+),
+(1)
+where M is an EE model trained on the original
+training set and ℓ(M, x+) denotes the model’s val-
+idation loss when evaluated on the augmented ex-
+ample y.
+We use the Dist-1/2 metric (Celikyilmaz et al.,
+2020), commonly used in text generation, to assess
+the Diversity of the augmented data.
+B
+Implementation Details
+Parameter
+Value
+Training Epochs
+3
+Optimizer
+AdamW
+Batch Size
+64
+Learning rate
+1e-5
+Seed
+1024
+Top-k
+100
+Top-p
+0.7
+Beam Size
+5
+Table 3:
+Implementation details of our infilling
+model (t5-small).
+Parameter
+Value
+Training Epochs
+30
+Optimizer
+AdamW
+Batch Size
+64
+Learning rate
+5e-5
+Seed
+1024
+Table 4: Implementation details of Text2Event.
+
+Parameter
+Value
+Training Epochs
+30
+Optimizer
+AdamW
+Batch Size
+64
+Learning rate
+4e-5
+Seed
+1024
+nth query
+5
+Table 5: Implementation details of EEQA.
+

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+page_content='Mask-then-Fill: A Flexible and Effective Data Augmentation Framework  for Event Extraction  Jun Gao1,3∗ Changlong Yu4 Wei Wang5 Huan Zhao6 Ruifeng Xu1,2,3†  1Harbin Institute of Technology (Shenzhen)  2Peng Cheng Laboratory  3Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies  imgaojun@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
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+page_content='cn  4HKUST, Hong Kong, China  5Tsinghua University  64Paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  cyuaq@cse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
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+page_content='hk  weiwangorg@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='com  zhaohuan@4paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='com  Abstract  We present Mask-then-Fill, a flexible and ef-  fective data augmentation framework for event  extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Our approach allows for more flex-  ible manipulation of text and thus can gener-  ate more diverse data while keeping the origi-  nal event structure unchanged as much as pos-  sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Specifically, it first randomly masks  out an adjunct sentence fragment and then in-  fills a variable-length text span with a fine-  tuned infilling model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The main advantage lies  in that it can replace a fragment of arbitrary  length in the text with another fragment of vari-  able length, compared to the existing methods  which can only replace a single word or a fixed-  length fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' On trigger and argument ex-  traction tasks, the proposed framework is more  effective than baseline methods and it demon-  strates particularly strong results in the low-  resource setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Our further analysis shows  that it achieves a good balance between diver-  sity and distributional similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  1  Introduction  Event Extraction (EE), which aims to extract trig-  gers with specific types and their arguments from  unstructured texts, is an important yet challenging  task in natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' In recent years,  deep learning methods have emerged as one of the  most prominent approaches for this task (Nguyen  and Nguyen, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Du and Cardie,  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Paolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Lou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' However, they are notorious for requiring  large labelled data, which limits the scalability of  EE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Annotating data for EE is usually  costly and time-consuming, as it requires expert  knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' One possible solution is to leverage  data augmentation (DA) (Simard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Existing DA methods for NLP can be broadly  classified into two types: (1) the first is to augment  ∗Work done when Jun Gao was interning at 4Paradigm  † Corresponding author  The police said Mike left this town yesterday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Trigger  Event Type: Transport  Arg1  Role: Artifact  Arg2  Role: Origin  Arg3  Role: Time-Within  The cop said Mike left this town yesterday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  The police said Mike went away from this town yesterday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  The cop saw Mike left this town yesterday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  (1) BackTranslation  (2) Synonym Replacement  (3) BERT  Figure 1: Visualization of three different data aug-  mentation methods applied to a sentence containing a  “Transport” event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Spans marked with different colors  are event triggers and arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The parts of the aug-  mented sample that differ from the original are colored  in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Backtranslation (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020) translates  the input sentence into another language and back to  the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Synonym Replacement (Dai and Adel,  2020) and BERT (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2019) replace words in  the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  training data by modifying existing examples (Sen-  nrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' ¸Sahin and Steedman, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Dai  and Adel, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Wei and Zou, 2019), and (2) the  second is to generate new data by estimating a gen-  erative process and sample from it (Anaby-Tavor  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Quteineh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Since the EE task requires DA  methods to generate augmented samples and token-  level labels jointly, the second type of DA method  is inapplicable here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' In this study, we mainly focus  on the first type of method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Applying existing DA methods to the EE task  is more challenging than to translation or classi-  fication tasks, because we need to augment train-  ing data while keeping the event structure (trigger  and arguments) unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Figure 1 presents ex-  amples of three different DA methods applied to  a sentence containing a “Transport” event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The  event is triggered by word “left” and it has three  arguments with different roles (“Mike”, “this town”  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='02427v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='CL]  6 Jan 2023  and “yesterday”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' As shown in Figure 1, it is in-  feasible to apply sentence-level DA methods such  as BackTranslation (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020), because it  may change the event structure (change “left” to  “went away from”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Previous attempts on DA for  such tasks typically use heuristic rules such as  synonym replacement (Dai and Adel, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Cai  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020) or context-based words substitution  with BERT (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Their idea is to  replace adjunct tokens (the tokens in sentences ex-  cept triggers and arguments) with other tokens, and  thus can ensure the event structure is unchanged as  much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' However, recent studies (Ding  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2022) find  that such methods provide limited data diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  In NLP, the diversity of data is mainly reflected  in two aspects: expression diversity and seman-  tic diversity (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The Synonym  Replacement and BackTranslation methods lack  semantic diversity, because they can only produce  samples with similar semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The BERT-based  method can only replace words and cannot change  the syntax, so it cannot generate samples with a  wide variety of expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The lack of sufficient  diversity may lead to greater overfitting or poor  performance through training on examples that are  not representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  To this end, we present Mask-then-Fill, a flexi-  ble and effective data augmentation framework for  event extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Our approach allows for more  flexible manipulation of text and thus can generate  more diverse data while keeping the original event  structure unchanged as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Specif-  ically, we first define two types of text fragments  in a sentence: event-related fragments (trigger and  arguments) and adjunct fragments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' “The po-  lice said”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Then, we model DA for the EE task  as a Mask-then-Fill process: we first randomly  masks out an adjunct sentence fragment and then  infills a variable-length text span with a fine-tuned  infilling model (T5) (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The main  advantage lies in that it can replace a fragment of  arbitrary length in the text with another fragment of  variable length, compared to the existing methods  which can only replace a single word or a fixed-  length fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  To the best of our knowledge, we are the first  to augment training data for event extraction via  text infilling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' We empirically show that the Mask-  then-Fill framework improves performance for  both classification-based (EEQA) and generation-  The police said Mike left this town yesterday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  [MASK] Mike left this town yesterday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Output1: The next door neighbor said  Output2: Because of a fight with his girlfriend,  Output3: Since he found a new job,  …  Infilling Model (T5)  ② Decoding by Sampling  [MASK] Mike left this town yesterday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  ① Mask out an adjunct fragment  ③ Fill in the blank  Figure 2: Overview of the proposed Mask-then-Fill  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  based (Text2Event) event extraction models on a  well-known EE benchmark dataset (ACE2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Es-  pecially, it demonstrates strong results in the low  resource setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' We further investigate reasons for  its effectiveness by introducing two metrics, Affin-  ity and Diversity, and find that the data augmented  by our approach have better diversity with less dis-  tribution shifts, achieving a good balance between  diversity and distributional similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  2  The Mask-then-Fill Framework  Figure 2 presents an overview of Mask-then-Fill  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The input sentence contains two types  of text fragments: event-related fragments (words  with colors) and adjunct fragments (underlined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Our idea is to rewrite the whole adjunct fragment  instead of replacing some words, and the rewritten  sentence fragment should fit the context and should  not introduce new events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' To this end, we model  DA for EE as a Mask-then-Fill process: we first  randomly mask out an adjunct sentence fragment  and then infills a variable-length text span with a  fine-tuned infilling model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' In the following, we  describe in detail the Mask-then-Fill framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Mask out an adjunct fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Given a pro-  totype sentence X = {x1, · · · , xL} of length L  from the training set, we first define an adjunct frag-  ment as a set of non-overlapping spans of x that  do not contain the event triggers and arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  We then replace one of the adjunct fragments with  a [MASK] symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The incomplete sentence ˆx  is a version of x with a fragment replaced with a  [MASK] symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Blank Infilling Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  We formulate our blank  infilling process as the task of predicting the miss-  ing span of text which is consistent with the preced-  ing and subsequent text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Figure 2 gives an example  with an incomplete input sentence �x, where the  [MASK] is a placeholder for a blank, which has  masked out multiple tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Our goal is to pre-  dict only the missing span y which will replace the  [MASK] token in �x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Therefore, the infilling task  can be cast as learning p(y|�x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  To train our infilling model, we fine-tune a pre-  trained sequence-to-sequence model T5 (Raffel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020) on the Gigaword corpus (Graff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=',  2003), which is from similar domains as the event  extraction dataset ACE2005 adopted by our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Given a corpus consisting of plain sentences, we  first produce large pools of infilling examples and  then train the T5 model on these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' For a  given complete sentence x from the training cor-  pus, we generate an infilling example �x with the  following procedure: (1) randomly sample a span  length l from the range of [1, min(10, l)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' (2) split  the sentence into l spans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' (3) randomly select a  span to be replaced with a [MASK] symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The  replaced span is used as the target y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' We then  fine-tune the T5 model on these infilling examples,  yielding the model of the form pθ(y|�x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Fill in the blank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Once trained, the infilling  model can be used to take the incomplete sentence  �x, containing one missing span, and return a pre-  dicted span y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' We then replace the [MASK] token  in �x with the predicted span y to generate an aug-  mented example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Note that we can produce large  pools of augmented samples using top-k sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  3  Experimental Setup  Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  We empirically evaluate our proposed  data augmentation method for event extraction  on the ACE2005 corpus1 with the same train-  dev-test split and preprocessing step as previous  works (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Wadden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  We simulate a low-resource setting by randomly  sampling 1,000, 4,000 and 8,000 examples from  the training set to create the small, medium, and  large training sets (denoted as S, M, L in Table 1,  whereas the complete training set is denoted as F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  1https://catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='ldc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='edu/LDC2006T06  We only augment the training data and keep the  dev set and test sets unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Following the previous  works (Du and Cardie, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2021) on  event extraction, we adopt the same evaluation cri-  teria defined in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' (2013): (i) An event trigger  is correctly identified and classified (Trig-ID+C) if  its offsets match a gold trigger and its event type is  also correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' (ii) An argument is correctly identified  and classified (Arg-ID+C) if its offsets and event  type match a gold argument and its event role is  also correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Event Extraction Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  In our study, we con-  sider two representative models for event extrac-  tion:  Text2Event (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2021) is a framework to  solve the event extraction task by casting it as  a SEQ2SEQ generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' All triggers, argu-  ments, and their labels are generated as natural  language words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  EEQA (Du and Cardie, 2020) formulates the  event extraction task as a question answering  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' They develop two BERT-based QA models  – one for event trigger detection and the other for  argument extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Comparison Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  We compare our pro-  posed data augmentation method Ours (t5-small)  with three baselines: (1) Synonym Replacement  replaces adjunct tokens with one of their synonyms  retrieved from WordNet (Miller, 1992) at random;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  (2) BERT replaces adjunct tokens with others ran-  domly drawn according to the pretrained BERT’s  distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' (3) Span-BackTranslation: Inspired  by Yaseen and Langer (2021), we only “back trans-  late” randomly selected adjunct spans to prevent  the model from changing the event structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  For all data augmentation  methods, we tune the number of augmentation sam-  ples per training sample from a list of numbers:  {1, 3, 6, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  4  Results and Analysis  Main Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  The main results are presented  in Table 1,  where we use two EE mod-  els (Text2EVent and EEQA) to test the perfor-  mance of different DA methods in both low-  resource (S, M and L) and normal (F) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' As  shown in the table, we observe that Ours (t5-small)  achieves the best overall performance among all  EE Model  DA Method  Trig-ID+C (F1)  Arg-ID+C (F1)  S  M  L  F  S  M  L  F  Text2Event  No Augmentation  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='44  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='75  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='55  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='06  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='05  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='04  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='35  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='30  Synonym Replacement  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='14  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='96  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='73  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='09  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='71  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='64  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='63  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='95  BERT  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='66  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='75  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='81  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='33  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='71  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='75  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='44  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='41  Span-BackTranslation  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='91  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='54  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='59  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='58  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='68  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='18  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='39  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='93  Ours (t5-small)  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='32  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='38  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='25  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='03  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='68  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='79  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='73  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='29  EEQA  No Augmentation  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='05  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='20  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='06  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='13  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='81  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='30  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='27  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='93  Synonym Replacement  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='86  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='03  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='71  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='05  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='99  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='62  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='40  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='50  BERT  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='61  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='23  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='90  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='35  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='80  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='82  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='49  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='62  Span-BackTranslation  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='26  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='64  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='46  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='40  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='47  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='64  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='87  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='55  Ours (t5-small)  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='80  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='37  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='33  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='62  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='67  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='70  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='52  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='62  Table 1: Results on trigger extraction and argument extraction using different subsets of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The best  results are marked in bold, and the second best is underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  DA methods on both trigger extraction (F1) and  argument extraction (F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Using our DA method  gives the best results for the Text2event model on  7 out of 8 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' For the EEQA model, our  method achieves the best results on 6 out of 8  datasets, where the difference between our method  and the best results on Trig-S and Arg-L is very  small, with only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='06 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='97 points difference be-  tween them, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Particularly, our methods  demonstrates strong results in the low-resource set-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Using our DA gives the Text2Event model a  performance improvement of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='14% and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='07%  on Trig-S and Arg-S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  We also notice that as the amount of data in-  creases, the improvement from all DA method de-  creases, and in some cases (EEQA model on Arg-L  and Arg-F), there is even a slight decrease in per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' In the case of more data, the model may  overfit if the augmented data are just some similar  samples rather than data with large variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  DA Method  Affinity  Dist-1  Dist-2  Synonym Replacement  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='118  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='523  BERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='082  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='374  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='496  Span-BackTranslation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='155  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='407  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='513  Ours (t5-small)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='086  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='488  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='612  Table 2: Results on Affinity and Diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The best re-  sults are marked in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The second best is underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Affinity and Diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Inspired by Gontijo-  Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' (2020), we further investigate reasons  for its effectiveness by introducing two metrics,  Affinity and Diversity, where Affinity quantifies how  augmentation shifts data distribution and Diversity  measures the complexity of the augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  We measure Affinity by computing the difference  between the loss of a model trained on the original  training set and tested on the original example, and  the loss of the same model tested on an augmented  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' We use the Dist-1/2 metric (Celikyilmaz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020), commonly used in text generation, to  Furthermore , the United States supported him in the war against Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  In addition, the United States supported him in the war against Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Furthermore, the United States supported him in the war against Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Later, the united States supported him in the war against Iran .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Furthermore, the United States supported iraq in the war against iraq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Moreover, the combine States supported him in the war against Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  What is more, the unify DoS supported him in the war against Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  He also called for an end to the war against Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' military has been fighting a war against Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Event Type: Attack | Trigger: war  (1) Synonym Replacement  (2) BERT  (3) Span-BackTranslation  (4) Ours (t5-small)  Figure 3: Augmented examples of four different DA  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Given a sentence containing an “Attack”  event triggered by the word "war", we generate two  new samples for each DA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The parts of the  new sample that differ from the original are colored in  gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  assess the Diversity of the augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' For im-  plementation details of two metrics, see Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  We first construct a new test set by generating a  new sample for each data in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' We then  calculate the Affinity and Dist-1/2 scores between  the new data set and the original data set, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' As shown in Table 2, it is clear that the data  augmented by our DA method have better diversity  with less distribution shifts, obtaining a balance  between diversity and distributional similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Case Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Figure 3 presents examples gener-  ated by different DA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Given a sentence  containing an “Attack” event triggered by the word  "war", we generated two new samples for each DA  method, and the parts of the new sample that differ  from the original are colored in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Obviously,  The synonym replacement based on WordNet can-  not avoid introducing some words that do not fit the  context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='g “unify” and “DoS”), while the BERT-  based word replacement can consider the context  better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' However, they both provide limited diver-  sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' BackTranslation method performs even worse  in terms of data diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Its generated data differs  very little from the original sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Finally, com-  pared with the original sentences, the new samples  generated by our method are more fluent and more  different in expression and semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Therefore, it  not only generates data that fits the context better,  but also provides better diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  5  Conclusion  In this paper, we present Mask-then-Fill, a flexi-  ble and effective data augmentation framework for  event extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Our approach allows for more  flexible manipulation of text and thus can gen-  erate more diverse data while keeping the origi-  nal event structure unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' The main advan-  tage lies in that it can replace a fragment of ar-  bitrary length in the text with another fragment  of variable length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' We empirically show that the  Mask-then-Fill framework improves performance  for both EEQA and Text2Event EE models on  the ACE2005 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' It demonstrates particularly  strong results in the low-resource setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Our fur-  ther analysis shows that it achieves a good balance  between diversity and distributional similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Limitations  This paper presents a flexible and effective data  augmentation framework for event extraction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Here, we note some of Mask-then-Fill frame-  work’s limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' First, performance gains can  be marginal when data is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' We believe  this approach has much room for improvement in  generating more diverse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' In this work, we  select only one adjunct fragment at a time for modi-  fication, and modifying multiple adjunct fragments  in an event mention can further enhance the diver-  sity of the generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Second, currently this  method can only replace one fragment at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  This makes it easier to control the properties of  the generated fragments, such as length or style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  It is possible to modify multiple fragments at the  same time using some existing techniques (Don-  ahue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  This approach is more efficient, but it is prone to  generate incoherent augmented samples and thus  introduce more noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' A possible approach to solve  this problem is to design some sample selection  strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Acknowledgements  This work was partially supported by the Na-  tional Natural Science Foundation of China  62006062 and 62176076, Shenzhen Foundational  Research  Funding  JCYJ20200109113441941,  JCYJ20210324115614039,  The  Major  Key  Project of PCL2021A06, Guangdong Provincial  Key Laboratory of Novel Security Intelligence  Technologies 2022B1212010005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
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+page_content='  Zerogen: Efficient zero-shot learning via  dataset generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' ArXiv, abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='07922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Tongtao Zhang, Heng Ji, and Avirup Sil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Joint  entity and event extraction with generative adversar-  ial imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Data Intelligence, 1(2):99–  120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Zijian Zhao, Su Zhu, and Kai Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' Data augmen-  tation with atomic templates for spoken language  understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' In Proceedings of the 2019 Confer-  ence on Empirical Methods in Natural Language  Processing and the 9th International Joint Confer-  ence on Natural Language Processing (EMNLP-  IJCNLP), pages 3637–3643, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' As-  sociation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  A  Affinity and Diversity  Inspired by Gontijo-Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' (2020) and Arora  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=' (2021), we proposed to use a calibration  method to quantify how augmentation shifts data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  They all note that a trained model is often sensitive  to the distribution of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Given the original example x and one of its aug-  mented example x+, we measure distribution shifts  by computing the difference between the loss of a  model trained on the original training set and tested  on the original example, and the loss of the same  model tested on an augmented example:  τα = ℓ(M, x) − ℓ(M, x+),  (1)  where M is an EE model trained on the original  training set and ℓ(M, x+) denotes the model’s val-  idation loss when evaluated on the augmented ex-  ample y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  We use the Dist-1/2 metric (Celikyilmaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content=',  2020), commonly used in text generation, to assess  the Diversity of the augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  B  Implementation Details  Parameter  Value  Training Epochs  3  Optimizer  AdamW  Batch Size  64  Learning rate  1e-5  Seed  1024  Top-k  100  Top-p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='7  Beam Size  5  Table 3:  Implementation details of our infilling  model (t5-small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Parameter  Value  Training Epochs  30  Optimizer  AdamW  Batch Size  64  Learning rate  5e-5  Seed  1024  Table 4: Implementation details of Text2Event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}
+page_content='  Parameter  Value  Training Epochs  30  Optimizer  AdamW  Batch Size  64  Learning rate  4e-5  Seed  1024  nth query  5  Table 5: Implementation details of EEQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE0T4oBgHgl3EQfhAHl/content/2301.02427v1.pdf'}

G9AyT4oBgHgl3EQffPiq/content/tmp_files/2301.00337v1.pdf.txt

@@ -0,0 +1,2065 @@
+Design, Modeling, and Evaluation of Separable
+Tendon-Driven Robotic Manipulator with Long,
+Passive, Flexible Proximal Section
+Christian DeBuys
+Texas A&M University
+Mechanical Engineering
+College Station, TX, USA
+Email: [email protected]
+Florin C. Ghesu
+Siemens Healthineers
+Digital Technology & Innovation
+Princeton, NJ, USA
+Email: fl[email protected]
+Jagadeesan Jayender
+Surgical Planning Laboratory
+Brigham and Women’s Hospital
+Harvard Medical School, Boston, USA
+Email: [email protected]
+Reza Langari
+Texas A&M University
+Mechanical Engineering
+College Station, TX, USA
+Email: [email protected]
+Young-Ho Kim
+Siemens Healthineers
+Digital Technology & Innovation
+Princeton, NJ, USA
+Email: [email protected]
+The purpose of this work was to tackle practical issues
+which arise when using a tendon-driven robotic manip-
+ulator with a long, passive, flexible proximal section in
+medical applications. A separable robot which overcomes
+difficulties in actuation and sterilization is introduced, in
+which the body containing the electronics is reusable and
+the remainder is disposable. A control input which re-
+solves the redundancy in the kinematics and a physical
+interpretation of this redundancy are provided. The effect
+of a static change in the proximal section angle on bend-
+ing angle error was explored under four testing conditions
+for a sinusoidal input. Bending angle error increased for
+increasing proximal section angle for all testing condi-
+tions with an average error reduction of 41.48% for re-
+tension, 4.28% for hysteresis, and 52.35% for re-tension
++ hysteresis compensation relative to the baseline case.
+Two major sources of error in tracking the bending angle
+were identified: time delay from hysteresis and DC off-
+set from the proximal section angle. Examination of these
+error sources revealed that the simple hysteresis compen-
+sation was most effective for removing time delay and
+re-tension compensation for removing DC offset, which
+was the primary source of increasing error. The re-tension
+compensation was also tested for dynamic changes in the
+proximal section and reduced error in the final configura-
+tion of the tip by 89.14% relative to the baseline case.
+1
+INTRODUCTION
+Tendon-driven robotic manipulators (TDRM) have
+been used in many fields for various applications, such
+as remote inspection and maintenance in aerospace, min-
+imally invasive surgery in medicine, and general search
+and rescue. They are preferred in these areas for their abil-
+ity to maneuver in tight spaces in a compliant and safe
+manner. Many devices have been developed in the medi-
+cal domain for cardiac catheterization [1, 2, 3] and bron-
+choscopy [4, 5, 6]. Some of these devices are “flexible-
+arXiv:2301.00337v1  [cs.RO]  1 Jan 2023
+
+steerable” as classified by Dupont at al. [7], meaning they
+are comprised of a steerable articulation section and a
+long, passive proximal section between the articulation
+section and the actuators. Such devices are particularly
+difficult to model and control accurately, and the prox-
+imal section is not included in the state-of-the-art kine-
+matic and dynamic models. Devices with sensors at the
+distal tip using Electromagnetic (EM) sensor (e.g., [6]) or
+Fiber bragg grating (FBG) based shape sensors(e.g., [5])
+can circumvent the unmodeled behavior of the proximal
+section via closed-loop control. However, some devices,
+such as heart catheters, are one-time or limited-time use,
+meaning that it would be too costly to implement such
+sensors and that the unmodeled behavior cannot be com-
+pensated as with the aforementioned devices.
+We introduce a separable TDRM for a practical set-
+ting. The separable design tackles the issue of reusability
+that is common among medical devices, where the part
+which interacts with the anatomy is disposable and the
+part containing the actuators is reusable. In addition, we
+utilize a practical model and calibration method for our
+proposed mechanism so that the four tendons are actu-
+ated simultaneously, allowing for precise tip control and
+mitigating issues with conventional devices, such as dead-
+zone and hysteresis, with simple linear compensation. We
+consider an open-loop controller since many available de-
+vices [2, 8] are used without position-tracking sensors at
+the tip due to costs and single use. We analyze the effect
+of the shape of the passive proximal section for different
+compensation types and offer insight on how this behav-
+ior might be accounted for in open and closed-loop sys-
+tems.
+2
+RELATED WORKS
+The tendon-sheath mechanism facilitates maneuver-
+ability and compliance in highly constrained anatomies
+by allowing for a long, thin, flexible, proximal tail-like
+structure. Thus, the TDRM is an actuation method that
+has been used in many therapeutic [9, 10, 11, 12, 13] and
+real-time diagnostic (e.g., endoscope [14, 15, 16, 17, 18],
+and Intracardiac echocardiography (ICE) [19, 20, 21])
+manipulators. The increased compliance of tendon-driven
+manipulators is not without drawbacks. Much work has
+gone into accurately modeling the kinematics; accounting
+for phenomena such as hysteresis, deadzone, and slack;
+and overcoming the redundancy in the control input.
+With respect to forward kinematics, many publica-
+tions have presented a lumped-parameter approach with
+the constant curvature model (CCM) [20, 22, 23]. Re-
+views for modeling tendon-driven continuum manipula-
+tors have been given [24, 25], but the shape of the passive
+proximal section–which can be very long for devices such
+as catheters–is not included in these kinematic models.
+Shape sensing is possible and has been reviewed [26, 27],
+but the sensors necessary (e.g. FBG or EM) would be pro-
+hibitively expensive for a disposable device or disposable
+portion of a device. We take first steps to characterize the
+effect of the proximal section on bending angle error of
+the articulation/bending section.
+Hysteresis has been addressed for TDRM via com-
+pensation methods [28, 29, 30, 31] and modeling ap-
+proaches [28, 32, 33]. Since the modeling approaches in-
+volve many hyper-parameters, which themselves require
+a complicated identification process, Lee et al. [8] pro-
+vided a simplified hysteresis model which included dead-
+zone and backlash. Kim et al. [34] proposed a practical
+shape-adaptive hysteresis compensation based on dead-
+zone detection using motor current, where compensation
+is adjusted based on arbitrary shape change of the proxi-
+mal shaft. We implement a simple hysteresis compensa-
+tion which does not require hyper-parameters and a re-
+dundant control input which removes deadzone.
+Redundant control strategies for tendon-driven de-
+vices have been implemented for cable driven parallel
+robots [35], dexterous robot hands [36], and continuum
+manipulators [37]. Fang et al. [35] handled platform and
+cable dynamics, provided a redundant control input with
+tension and actuator constraints to prevent slack, and
+solved an optimization real-time to determine the control
+input. However, due to their different application, their
+tendons were free floating and thus did not have tendon-
+sheath friction. Abdallah et al. [36] handled multi-joint
+finger and cable dynamics with an optimization and con-
+straints similar to [35], but with the addition of tendon-
+sheath friction. Camarillo et al [37] gave a redundant con-
+trol scheme for quasi-static motion of a catheter with two
+articulation sections that decoupled the inverse kinemat-
+ics. Their constraints and optimization were similar to
+[35, 36] except the solution allowed for slack tendons
+to be present (as long as they contribute no force). We
+propose a redundant control input which prevents slack
+tendons as in [35, 36] but does not require an optimiza-
+tion and which yields a physical interpretation of the re-
+dundancy. Unlike [35, 36], the state transition matrix (C
+in this paper) is not dependent on the configuration of
+the manipulator, resulting in a single solution to the in-
+verse kinematics for a given desired configuration. We
+also present simplified kinematics for an incompressible
+articulation section.
+
+Fig. 1: An overview of the catheter robot.
+3
+MATERIALS & METHODS
+The following subsections will cover the design of
+the robot (Section 3.1), the derivation of the redundant
+control scheme and determination of its parameters (Sec-
+tion 3.2), and the simple hysteresis and re-tension com-
+pensations (Section 3.3).
+3.1
+Design of Separable Tendon-Driven Robotic Ma-
+nipulator
+The proposed robotic system shown in Fig. 1 consists
+of two parts: the back portion or reusable portion contains
+four Faulhaber linear motors (LM 1483-080-11-C) with
+motor drivers mounted to a plastic core. The plastic core
+contains a channel along its central axis for an ultrasound
+(US) cable or for any other sensor cables, which is shown
+in blue in Fig. 1(a). The front portion or disposable por-
+tion contains all of the tendons and includes the catheter’s
+passive, proximal section and articulation section. The in-
+terface from one tendon in the disposable portion shown
+in Fig. 2 to its respective linear motor is as follows: 1)
+the tendon (green) from the catheter is bent 90 degrees
+around a low-friction roller and is wrapped around and
+fastened to the small radius of the spool, 2) a separate
+tendon (red) is wrapped around and fastened to the large
+radius of the spool and then attached to the tendon anchor,
+and 3) the tendon anchor is held in place by the fan lock
+until the catheter is clipped together and the motor rod
+Fig. 2: Disposable portion of the robot.
+has connected to the tendon anchor via magnetic force.
+The spools serve to increase the pulling force of the mo-
+tors with a pulley ratio of R:r, where R is the larger radius
+and r is the smaller radius; and in our case the ratio is
+3:1. The wave-disc spring serves to push the front portion
+against a locking mechanism consisting of plastic hooks
+in the back portion after the system has been clamped.
+
+The back portion (i.e., reusable)
+The front portion
+(i.e., disposable)
+motor driver
+linear motor
+O
+ww 99
+channel for sensor cables
+motor rod
+(a)
+298 mm
+bending section
+Existing device with passive,
+flexible, proximal section (ACUSON
+Conjoined front and back
+AcuNavTM, Siemens Healthineers)
+portions of the real robot
+(b)
+(c)fan lock
+spool
+tendons
+slider
+wave disc spring
+tendon anchor
+magnetFig. 3: Reusable portion of the robot.
+The proposed design has several advantages:
+1) Direct actuation of the tendons via linear motors fa-
+cilitates a more transparent control input and gives a
+reliable measure of tendon tension via motor current.
+This prevents the need for additional tension sensors
+and reduces the complexity of hysteresis phenomena
+compared to [8], simplifying the control problem.
+2) The reusable portion contains no tendons, meaning
+that the issue of tendon wear present in many devices
+is avoided. The disposable portion (Fig. 2) can be un-
+clipped and reclipped indefinitely, since the fan lock
+mechanism can clamp the tendons in place when the
+disposable portion is not attached and since the inter-
+face between the motor rods and the tendon anchors is
+magnetic (it does not involve a single use fastener).
+3) The clipping system has a closing mechanism simi-
+lar to that of a child-proof medicine bottle: the front
+portion is pushed into the body until it makes contact,
+pushed slightly farther and twisted, then released. Af-
+ter releasing, the front is locked, the motors move their
+rods toward the front until their magnets make con-
+tact with the magnets on the tendon anchors, the slid-
+ers are pushed away from each other to manually open
+the fan-lock, and the motors are free to pull on the ten-
+dons.
+3.2
+Derivation of redundant control
+3.2.1
+Moment balance equation
+We borrow nomenclature from [37] and define the
+robot’s configuration for a compressible bending section
+as q = [κx,κz,εa]T, where curvature about the x-axis κx
+[1/mm] corresponds to bending in the yz-plane, curva-
+ture about the z-axis κz [1/mm] corresponds to bending
+in the xy-plane, and axial strain εa [unitless] corresponds
+to compression along the y-axis. Just as in [37], curvature
+is chosen rather than bending angle for the configuration
+variables to obtain linear kinematic and static equations.
+First, we write the moment balance equation for a com-
+pressible bending section and n tendons:
+�
+�
+Kb
+0
+0
+0 Kb 0
+0
+0 Ka
+�
+�
+�
+�
+κx
+κz
+εa
+�
+� =
+�
+�
+dz1
+dz2 ···
+dzn
+−dx1 −dx2 ···−dxn
+1
+1 ···
+1
+�
+�
+�
+����
+T1
+T2
+...
+Tn
+�
+����,
+(1)
+where the matrix K is a stiffness matrix containing Kb
+[N·mm2], which is bending stiffness with respect to cur-
+vature κ, and Ka [N], which is axial stiffness with respect
+to strain εa. The matrix D results from the cross prod-
+uct of the tendon locations (dxi,dzi) [mm] with the tendon
+tensions Ti [N] where i ∈ 1,2,...,n. A conceptual side-
+view of the xy-plane and cross-section of the xz-plane are
+shown in Fig. 4.
+If the bending section undergoes negligible compres-
+sion, we can deviate from Eq. (1) and reduce the robot’s
+configuration to q = [κx,κz]T. The corresponding moment
+
+sensorcable
+motor rod
+linear motor
+motor
+driverFig. 4: Conceptual side-view and cross-section. θz, κz,
+and rz are the bending angle, curvature, and radius of cur-
+vature about the z-axis (in the xy-plane). (dxi,dzi) are the
+coordinates of tendon i relative to the central axis (y-axis)
+of the robot.
+balance for an incompressible bending section is:
+�
+Kb 0
+0 Kb
+��
+κx
+κz
+�
+=
+�
+dz1
+dz2 ···
+dzn
+−dx1 −dx2 ···−dxn
+�
+�
+����
+T1
+T2
+...
+Tn
+�
+����.
+(2)
+We can write Eq. (1) or Eq. (2) in matrix form:
+Kq = Dτ.
+(3)
+The relation in Eq. (3) describes the static equilibrium,
+where the tendon tensions must balance against each
+other and the spring-like forces inherent to the bending
+of the catheter.
+Let dim(τ) = m and dim(q) = n. When m > n, the
+system is redundant since there are more inputs m than
+outputs n. This means, for a given desired configuration
+qd, there are infinitely many solutions τ to Eq. (3). It is
+possible to solve this redundancy by minimizing the con-
+trol effort via real-time optimization, but this will not
+solve the issue of slack tendons. We could include the
+constraint τ ≥ 0 in the optimization, but we can develop a
+control scheme which resolves the redundancy and avoids
+real-time optimization. Derivation of the control scheme
+is handled for both the compressible and incompressible
+cases.
+3.2.2
+Compressible bending section (n=3, m=4)
+Suppose we have four tendons located at 90 degree
+increments around the central axis of the beam. If the
+bending section is compressible, then we are using Eq. (1)
+and have three outputs. With one more input than output,
+we have one dimension of redundancy. Thus, the family
+of solutions for a given desired output can be parameter-
+ized along one vector, namely the vector which spans the
+nullspace of the map from the input to the output. We re-
+organize Eq. (1) and define C = K−1D, which is the map
+from the input τ to the output q:
+q = Cτ.
+(4)
+For a fully controllable system, rank(C) = n, and the lo-
+cations of the tendons in this system render it fully con-
+trollable (even if tendons can only pull, i.e., even if τ ≥ 0).
+If we label the last tendon as redundant, we can partition
+C into a full rank portion B, which is square, and a re-
+dundant portion h, which is a column vector, as in [35]:
+q = [B h]τ.
+(5)
+Given a desired output qd, the control input τ is:
+τ =
+�
+−B−1h
+1
+�
+µ +
+�
+B−1qd
+0
+�
+,
+(6)
+where the desired output with a scalable parameter µ has
+no affect on the output. In other words, our family of so-
+lutions is parameterized by µ along the direction of the
+null space vector n = [−B−1h 1]T. The variable µ does
+not affect the output because n is in the null space of C:
+Cn = [B h]
+�
+−B−1h
+1
+�
+= 0.
+(7)
+The problems of redundancy and preventing tendon slack
+are solved with Eq. (6) and an appropriate choice of µ. An
+obvious choice for µ is the one that minimizes the control
+effort [38, 39] as follows:
+min
+µ
+1
+2
+n
+∑
+i=1
+T 2
+i ,
+s.t.
+Ti ≥ 0,
+(8)
+where the constraints on Ti (no slack) and the bottom row
+of Eq. (6) imply that µ ≥ 0. The choice of control input
+in Eq. (6) allows us to directly calculate µmin instead of
+
+y
+Z
+4
+1
+Bending
+1
+Section
+d
+Kz
+Z,4
+d
+z,1
+0z
+x
+d
+dz,2
+Z,3
+Passive
+Proximal
+3
+2
+Section
+x,2solving this optimization computationally. In the uncon-
+strained case, the minimum is found as follows:
+n
+∑
+i=1
+Ti
+∂Ti
+∂µ = 0.
+(9)
+If the solution to Eq. (9) does not violate the constraints
+in Eq. (8), then that minimum is sufficient. If it does vi-
+olate those constraints, then we simply pick the smallest
+µ which satisfies all of the constraints. To highlight, let
+the rows of B−1 be denoted by βi where i ∈ 1,2,3 such
+that B−1 = [β1 β2 β3]T. Then we can write the constraints
+from Eq. (8) for tension in each tendon:
+Ti = −βihµ +βiqd ≥ 0 for i ∈ 1,2,3 ,
+µ ≥ 0,
+(10)
+where we can see that every term in the equations except
+for µ is determined by the structure of the robot (βi and
+h) and the desired configuration (qd). The strength of this
+controller is that, as long as the structure of the robot in C
+does not vary explicitly with configuration or time, we can
+choose a constant value for µ and achieve a control input
+for any desired state qd without real-time optimization.
+We elaborate on the previous statement and note the
+limitations of the analysis so far. In our case, the choice
+the redundant tendon and thus the redundant column of
+C was arbitrary; we could choose any tendon and get the
+same result. This is because the locations of the tendons
+relative to the central axis do not depend on the system
+configuration, which means that B is independent from
+the input. In other robots [35], it is possible for B to
+become ill-conditioned with changing configuration, in
+which case an algorithm is required to ensure that an ap-
+propriate tendon is chosen.
+To summarize, a redundant control scheme is used
+with a constant curvature assumption to resolve and take
+advantage of this redundancy, in which the control ef-
+fort is minimized while respecting feasibility constraints
+(such as requiring a minimum tension for all tendons) as
+in [35] but with two key differences. The first difference
+is that the minimum is easily found analytically since µ
+is the only unknown, and so there is no need to optimize
+in real-time. The second difference is that the axial po-
+sitions of the tendons do not change appreciably during
+actuation, meaning that we will not encounter a singular
+configuration and will not need to reconstruct B as in [35].
+3.2.3
+Incompressible bending section (n=2, m=4)
+In this case, the compression of the bending section is
+negligible compared to the magnitude of bending. There-
+fore, we can use Eq. (2) and will have two outputs. This
+results in two dimensions of redundancy and a family of
+solutions which can be parameterized by two scalars, µ1
+and µ2, along the directions of two vectors, n1 and n2,
+which span the null space of C. We arbitrarily label the
+last two tendons as redundant and partition C into a full
+rank portion B, which is square, and two redundant por-
+tions h1 and h2, which are both column vectors. For a
+given desired output qd, the control input τ as follows:
+τ =
+�
+�
+−B−1h1
+1
+0
+�
+�µ1 +
+�
+�
+−B−1h2
+0
+1
+�
+�µ2 +
+�
+�
+B−1qd
+0
+0
+�
+�, (11)
+where µ1 and µ2 are the desired output with two scal-
+able parameters. As before, adjusting µ1 or µ2 has no
+affect on the output since n1 = [−B−1h1
+1
+0]T and
+n2 = [−B−1h2 0 1]T lie in the null space of C:
+Cn1 = [B h1 h2]
+�
+�
+−B−1h1
+1
+0
+�
+� = 0,
+Cn2 = [B h1 h2]
+�
+�
+−B−1h2
+0
+1
+�
+� = 0.
+(12)
+If the tendons are located equidistant from each other
+about the central axis, the assumption of incompressibil-
+ity decouples the bending caused by one pair of tendons
+(1 and 3) from the bending caused by the other pair of
+tendons (2 and 4). In other words, n1 = [1 0 1 0]T and
+n2 = [0 1 0 1]T, which means µ1 and µ2 do not appear in
+the same tension equation. Thus, the minimum µ1 and µ2
+can be determined independently, in a procedure similar
+to the compressible case via Eq. (8) and Eq. (9). If we de-
+note the rows of B by βi such that B = [β1 β2]T, we can
+write as follows:
+Ti = −βihµi +βiqd ≥ 0 for i ∈ 1,2 ,
+µ1 ≥ 0,
+µ2 ≥ 0,
+(13)
+where we can see that µ1 and µ2 are the only unknowns
+and can be chosen independently. Again we see that we
+need only pick µ1 and µ2 to obtain feasible tensions with-
+out real-time optimization. For a minimum allowable ten-
+sion of 0 (i.e., no slack condition), µ1 = µ2 = 0, but the
+constraint τ ≥ 0 is generally not strict enough to ensure
+that no slack occurs. Therefore, small constants should be
+
+found experimentally for µ1 and µ2 to ensure a low pre-
+tension which prevents slack tendons. Why this is referred
+to as ”pre-tension” will be addressed in the discussion.
+With µ1 and µ2 discussed, we move on to the kinematics
+and summarize the procedure for determining the control
+input.
+3.2.4
+Kinematics
+The preceding discussion dealt with determining the
+force input (tendon tensions) for our redundant system.
+However, it is often preferred and more stable to com-
+mand actuator positions (tendon displacements). To con-
+vert our control input from actuator forces τ to actuator
+displacements y, we consider the conservation of strain
+equation [37]:
+y = DTL0q+LtK−1
+t
+τ,
+(14)
+where L0, Lt, and Kt are diagonal matrices containing the
+undeformed bending section length L0, the unstretched
+tendon lengths lt,i, and the length-normalized tendon stiff-
+nesses kt,i, respectively. For instance, tendon 1 would
+have unstretched length lt,1 located in row 1 and column 1
+of Lt. The moment balance in Eq. (3) along with Eq. (14)
+define a planar spring model in which the moments and
+forces exerted by the tendons balance against the inherent
+bending and axial stiffness of the articulation section. An
+example for a two-tendon planar spring model is given in
+Fig. 5. Using Eq. (3) we can finally obtain a direct relation
+between y and τ:
+y = (DTL0K−1D+LtK−1
+t
+)τ.
+(15)
+If the actuators pulled the tendons directly, Eq. (15)
+would be sufficient. Since our system uses pulleys to in-
+crease the effective pulling force of the actuators, we have
+additional steps. The force felt by the motors τm is given
+by:
+τm = R−1τ,
+(16)
+where R is a diagonal matrix containing the pulley ratio.
+Then, the displacement of the motors ym, which de-
+pends on the displacement of the catheter tendons y and
+the deflection of the additional tendons due to the motor
+force τm, is as follows:
+ym = Ry+LmK−1
+t
+τm,
+(17)
+Fig. 5: Planar spring model for two tendons: L0, Ka, and
+Kb are the length, axial stiffness, and bending stiffness of
+the articulation section; lt,i, kt,i, di, and yi are the unde-
+formed length, stiffness, distance from the central axis,
+and displacement of tendon i.
+where Lm is a diagonal matrix containing the undeformed
+lengths of the additional tendons lm,i, which are shown in
+red in Fig. 2. Substituting Eq. (15) into Eq. (17) yields:
+ym = R(DTL0K−1D+LtK−1
+t
+)τ +LmK−1
+t
+τm,
+(18)
+and replacing τ with τm using Eq. (16), we can rewrite
+Eq. (17) in terms of the displacement of the actuators ym
+and forces felt by the actuators τm:
+ym = (R2(DTL0K−1D+LtK−1
+t
+)+LmK−1
+t
+)τm.
+(19)
+Now that we have Eq. (19), the process of determining the
+control input can be summarized.
+3.2.5
+Redundant control summary
+The scalars µ1 and µ2 are found experimentally and
+are chosen such that tendons remain in tension. Since
+these values can effect parameter estimates, it is best to
+leave them constant once they are chosen. Given µ1 and
+µ2, the tendon tensions in τ which correspond to the
+desired configuration qd are computed using Eq. (6) or
+Eq. (11), depending on whether the manipulator is com-
+pressible. The tensions in τ are converted into tendon dis-
+
+Bending
+Section
+Passive
+Proximal
+Sectionplacements in y using Eq. (15). This method gives a sin-
+gle vector y for any desired configuration qd, resolving
+the redundancy while preventing slack tendons. The re-
+dundant control input is considered the baseline input for
+the remainder of this work.
+Prior works have included τ ≥ 0 as a constraint in an
+optimization which solves for minimum τ in cable-driven
+platform [35] and dexterous hand applications [36]. Ca-
+marillo et. al. [37] developed an algorithm which opti-
+mizes τ while allowing slack tendons (as long as they
+contribute no force) for a continuum manipulator. We dif-
+fer from these works in that we leverage the structure
+of the constant curvature kinematics to get an analytical
+solution and avoid real-time optimization altogether. We
+also differ from [37] since our method does not permit
+slack tendons. With the baseline control input established,
+we move on to the compensation methods.
+3.3
+Practical Compensation methods
+3.3.1
+Preliminaries: Time-shift and DC Offset
+Before introducing the compensation methods, we
+must mention the two phenomena which they will com-
+pensate. In the presence of unmodeled hysteresis, an out-
+put will tend to lag an input in an open-loop system each
+time the direction of the input changes, due to backlash
+from the hysteresis. For a periodic input, this lag mani-
+fests as an obvious phase lag or time-shift as idealized in
+Fig. 6a. We hypothesize that hysteresis compensation can
+reduce this phenomena and thereby reduce error.
+Bending of the passive proximal section will natu-
+rally occur during operation of such flexible-steerable de-
+vices, for instance in traversing blood vessels to reach the
+heart in catheter applications, and this behavior is typi-
+cally unmodeled. We found that introducing an angle in
+the proximal section in one direction causes the articula-
+tion section to bend in the other direction, as visible when
+comparing Fig. 8a with 8b. We guess that this is caused
+by the increased stretching and thus increased tension in
+the outer tendon(s) and decreased tension in the inner ten-
+don(s). This change in tensions causes an offset of the
+unbent configuration of the articulation section, which re-
+sults in an offset of the output. For a sinusoidal input in
+the same plane as the proximal section deformation, the
+offset in the output would manifest as a DC offset of the
+sinusoid as shown in Fig. 6b.
+We further predict that these two sources of error are,
+for the most part, independent. For a sinusoidal input xin
+with amplitude A and frequency ω, we would expect the
+output xout to be shifted in time due to hysteresis by some
+amount ts and offset due to the passive proximal section
+0
+0.2
+0.4
+0.6
+0.8
+1
+time
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+amplitude
+baseline
+shifted
+(a) Idealized time-shift due to hysteresis
+0
+0.2
+0.4
+0.6
+0.8
+1
+time
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+amplitude
+baseline
+offset
+(b) Idealized DC offset due to proximal section angle
+Fig. 6: Phenomena in need of compensation.
+angle by some amount As, as follows:
+xin = Asin(ωt),
+xout = Asin(ω(t −ts))+As.
+(20)
+3.3.2
+Simple hysteresis compensation for time-shift
+The design of the robot and the control scheme result
+in a hysteresis curve, shown in Fig. 7, which is very lin-
+ear and has approximately no deadzone. The simplicity
+of this hysteresis curve allows us to implement a simpli-
+fied hysteresis compensation on top of the control scheme
+which does not rely on extra sensors or tuning a large
+number of parameters; however, those methods are com-
+patible with the control scheme and could still be imple-
+mented in future works.
+The simple hysteresis compensation consists of a
+constant value added to the desired input depending on
+
+Fig. 7: Hysteresis curve with width w.
+the direction of motion. This constant value is based on
+the width w of the hysteresis curve, also shown in Fig. 7,
+which can be determined from the input-output map of a
+continuum manipulator and is device dependent. The in-
+put and output are given as angles in degrees since bend-
+ing angle is easier to conceptualize than curvature, and
+the compensation equation is also given as an angle:
+θd,h = θd + w
+2 .
+(21)
+This desired angle with hysteresis compensation θd,h
+is converted into a desired curvature using the usual equa-
+tion for constant curvature in the incompressible case:
+κd,h = θd,h
+L0
+.
+(22)
+This additional compensation involves an adjustment
+of the input configuration in order to achieve the configu-
+ration which is truly desired, and thus it completely com-
+patible with the previously described control scheme.
+3.3.3
+Re-tension compensation for DC offset
+The re-tension compensation aims to remove error
+introduced by the passive proximal section angle, which
+manifests as a DC offset for a sinusoidal input. This angle
+can come about naturally in medical applications, such as
+in the tortuous path of a catheter from the incision point to
+the heart. In the baseline case, the robot is initialized, and
+the passive proximal section angle is introduced with no
+movement of the actuators. That is, the motors are pow-
+ered and holding their positions under the assumption that
+(a) Straight passive proximal section
+(b) passive proximal section bent to 45◦ angle
+Fig. 8: Experimental setup: testbed of robot has outer
+cover removed; device is clipped a few centimeters proxi-
+mal to the articulation section; and black tape marks prox-
+imal section angles.
+the proximal section is straight. This is not unreasonable,
+as current kinematic and dynamic representations of con-
+tinuum robots with bending sections make the assumption
+that the passive proximal section is straight. In the case
+with re-tension compensation, the robot is given approx-
+imately 3-5 seconds to re-tension the tendons to the de-
+sired pre-tension (0.25 N) after the passive proximal sec-
+tion angle has been introduced but before data for the si-
+nusoidal input is collected. Motor current is filtered using
+a 3rd order Butterworth filter to remove high frequency
+noise, and this filtered motor current is used along with
+the force constant from the manufacturer to get a measure
+of motor force and thus of tendon tension. It is anticipated
+that re-tensioning the tendons will remove some of the er-
+ror from the passive proximal section angle.
+4
+EXPERIMENTS AND RESULTS
+4.1
+Equipment and setup
+The experimental setup is shown in Fig. 8. The pro-
+totype was laid on a table, and the proximal section was
+lightly clamped just proximal to the bending section using
+a vise. A protractor was used to measure the angles for the
+
+PNLMGREN
+USB-to-CAN60
+40
+[6ap] a]bue ndino
+20
+0
+W
+20
+-40
+-60
+-50
+0
+50
+input angle [deg]PNLMGRENcatheter body, and tape was used to mark the locations on
+the table. Special care was taken to keep the curve of the
+proximal section tangent to the axis of the catheter body
+and of the vise at all angles. Bending angle was measured
+using an EM sensor (3D Guidance trakSTAR 2) for vali-
+dation purposes only, meaning that the EM measurement
+was not used in the control loop and that the control was
+performed entirely using the kinematics.
+4.2
+Parameter identification
+Since compression of our articulation section is neg-
+ligible, we use the equations and associated parameters
+for an incompressible device. All parameters in the equa-
+tions must be determined empirically. L0, Lt, Lm, R, and
+Kt can be measured directly. µ1 and µ2 are determined
+heuristically; they are chosen as the smallest constants
+which result in no slack tendons–as measured from motor
+current–for a few test inputs. K and D can be approx-
+imated by the following parameter identification proce-
+dure:
+1. Make initial guess for parameters.
+2. Input motor position trajectories which should
+achieve desired bending angle as calculated from in-
+verse kinematics and redundant control input.
+3. Update parameter guesses based on difference be-
+tween desired and measured bending angle.
+4. Repeat steps 2 and 3 until parameter values converge.
+This procedure is also rather heuristic, but it achieved
+consistent values for K and D with a bisection search.
+Parameter values are listed in Table 1. The tendon loca-
+tions di, as seen in Fig. 4, are given as coordinates (x,z).
+For simplicity, the tendon locations were assumed to be at
+90◦ increments around the central axis, but this assump-
+tion is not necessary to complete the parameter identifi-
+cation. For values which were not measured (i.e., Kb and
+di), resolution refers to the smallest step of the bisection
+used to identify the value.
+4.3
+Experiments
+The robot was tested under three scenarios of valida-
+tion which are listed here in brief and explained in detail
+in the subsections to follow.
+1. Straight Condition – sinusoidal input; no passive
+proximal section angle; baseline and hysteresis com-
+pensation
+2. Curved Condition – sinusoidal input; various passive
+proximal section angles; baseline, re-tension, hys-
+teresis, and re-tension + hysteresis (both) compensa-
+tion
+Table 1: Kinematic Parameters
+Parameter
+Value
+Resolution
+Units
+d1
+(0.492, -0.0868)
+0.00625
+mm
+d2
+(0.0868, 0.492)
+0.00625
+mm
+d3
+(-0.492, 0.0868)
+0.00625
+mm
+d4
+(-0.0868, -0.492)
+0.00625
+mm
+Kb
+360.0
+3.125
+N·mm2
+kt
+1080
+5.0
+N
+L0
+60
+0.5
+mm
+lm,1
+20
+0.5
+mm
+lm,2
+25
+0.5
+mm
+lm,3
+20
+0.5
+mm
+lm,4
+30
+0.5
+mm
+lt,1
+775
+0.5
+mm
+lt,2
+785
+0.5
+mm
+lt,3
+775
+0.5
+mm
+lt,4
+785
+0.5
+mm
+R
+3.0
+0.1
+unitless
+µ1
+0.25
+N/A
+unitless
+µ2
+0.25
+N/A
+unitless
+3. Dynamic Condition – re-tension compensation is
+tested while passive proximal section angle is in-
+creased dynamically
+4.3.1
+First Scenario: Straight Condition
+The first scenario tests the robot and the redundant
+controller under the default condition of a straight pas-
+sive proximal section. The proposed controller was val-
+idated for the straight condition with three trials on a
+test bed by following a desired sinusoidal input for bend-
+ing angle in one dimension. For each trial, the robot fol-
+lowed two cycles of the sinusoid, and the ground truth
+output angle was measured with an EM sensor, which was
+only used to evaluate performance.Tip position and bend-
+ing angle errors were described as mean absolute error
+(MAE) and standard deviation (StD). Performance was
+evaluated without and with simple hysteresis compensa-
+tion. The simple hysteresis compensation involved adjust-
+ing the input angle by a constant ± 10 degrees depending
+on the direction of motion as described in Eq. (21). The
+backlash width or width of the hysteresis curve (20◦) was
+
+obtained based on the unique physical properties of the
+catheter [8].
+4.3.2
+Second Scenario: Curved Condition
+We hypothesized that the shape of the passive prox-
+imal section affects the accuracy of the kinematics by al-
+tering the tensions in the tendons and that any increase
+in error would be caused by these changes tension. We
+implemented two types of compensation–one targeted at
+hysteresis and the other targeted at pre-tension errors
+caused by the passive proximal section angle–and hypoth-
+esized that these are two separate sources of error that
+would both require compensation. Data were collected
+using a test-bed of the prototype under various conditions
+for two periods of a sinusoidal input with an amplitude of
+45◦ and a frequency of 0.1 Hz. Note that the input refers
+to bending angle, which is then converted to curvature for
+use in the kinematics. For baseline and re-tension com-
+pensation, trials were gathered for six passive proximal
+section angles (15, 30, 45, 60, 75, and 90 degrees) and
+two bending planes (yz and xy). For hysteresis compensa-
+tion and re-tension + hysteresis compensation, trials were
+gathered for three passive proximal section angles (30, 60,
+and 90 degrees) and two bending planes (yz and xy). Two
+bending planes are tested to confirm that error introduced
+by the passive proximal section angle is independent of
+the input plane.
+4.3.3
+Third Scenario: Dynamic Condition
+Where the previous scenario examines the effect of
+the re-tension compensation on the output under static
+conditions, this scenario examines the re-tension compen-
+sation itself under dynamic conditions. In the previous
+cases, the re-tension compensation involved giving the
+robot a few seconds to recover the desired minimum ten-
+sions, which was followed by a desired input trajectory; in
+this case, the controller attempts to maintain the desired
+minimum tensions for all 30 seconds of each trial, and
+there is no trajectory input (only the re-tension compen-
+sation). The robot is moved manually for approximately
+20 seconds to change the passive proximal section angle
+from 0 to 60 degrees and then allowed to settle for an ad-
+ditional 10 seconds. Data were collected for three trials,
+both without and with re-tension compensation. It was ex-
+pected that a large mismatch between the measured angle
+and the desired angle of 0 degrees would develop without
+re-tension compensation.
+Table 2: Bending angle error
+Angle
+Compensation
+Error ± StDev
+% Reduction
+0◦
+baseline
+6.11◦ ± 4.03◦
+N/A
+hysteresis
+3.26◦ ± 2.90◦
+46.6
+Fig. 9: One result for yz-plane bending: (a) Time ver-
+sus output angle without compensation (b) Time versus
+output angle with hysteresis compensation (c)(d) The in-
+put angle versus the output angle without/with hysteresis
+compensation.
+4.4
+Results
+4.4.1
+First Scenario: Straight Condition Results
+Fig. 9 shows bending angle without and with the sim-
+ple hysteresis compensation. The control of all tendons
+simultaneously allowed the removal of tendon slack and
+thereby deadzone present in conventional catheters. The
+overall performance evaluation is described in Table 2.
+The tip pose error (MAE) is reported as 6.1◦. With hys-
+teresis compensation, the position and orientation errors
+are improved 31% and 47%, respectively. Bending angle
+error is reduced by including the simple hysteresis com-
+pensation. We also see from Fig. 9c and 9d that hysteresis
+is reduced noticeably even with simple compensation.
+4.4.2
+Second Scenario: Curved Condition Results
+For the xy-plane input, bending occurs in the same
+plane as the passive proximal section angle, since the pas-
+sive proximal section angle is created by moving the body
+to the right. Error bars denote the standard deviation over
+the three trials at that data point. Fig. 10 shows bending
+angle error for the six proximal section angles and four
+compensation types in the xy-plane and yz-plane, respec-
+
+80
+Input angle
+80
+Inputangle
+60
+Outputangle
+60
+Outputangle
+[deg]
+40
+40
+e
+20
+20
+angl
+bending
+-20
+20
+-40
+-40
+-60
+(a)
+60
+(b)
+-80
+20
+40
+60
+80
+100
+120
+-80
+0
+time [s]
+20
+40
+time [s]
+60
+80
+100
+120
+80
+80
+60
+60
+[deg]
+40
+ output angle [deg]
+40
+20
+output angle
+0
+-20
+-20
+-40
+-40
+-60
+(c)
+60
+(d)
+-80
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+-60
+-40
+-20
+0
+20
+40
+60
+input angle[deg]
+input angle [deg]tively. From Fig. 10a it is clear that bending angle error in
+15
+30
+45
+60
+75
+90
+tail angle [deg]
+-5
+0
+5
+10
+15
+20
+25
+30
+35
+40
+bending angle error [deg]
+baseline
+re-tension
+hysteresis
+re-tension + hysteresis
+(a) xy-plane bending angle error
+15
+30
+45
+60
+75
+90
+tail angle [deg]
+-5
+0
+5
+10
+15
+20
+25
+30
+35
+40
+bending angle error [deg]
+baseline
+re-tension
+hysteresis
+re-tension + hysteresis
+(b) yz-plane bending angle error
+Fig. 10: Bending angle error for different compensation
+types and proximal section angles.
+the xy-plane increases with increasing proximal section
+angle for all compensation types; however, both compen-
+sations with re-tension were less susceptible to this error
+increase. Error in the yz-plane is unaffected because the
+passive proximal section is bent in the xy-plane. The er-
+ror values (RMSE), standard deviations, and percent re-
+duction in error relative to the baseline case are given in
+Table 3, and the averages for the proximal section angles
+with all compensation types are in Table 4.
+To give a closer look at how error increases with
+proximal section angle, bending angle for 30, 60, and 90
+degrees is shown in Fig. 11, where shaded regions denote
+the standard deviation of the trajectory for its correspond-
+ing color.
+Table 3: xy-plane bending angle error
+Angle
+Compensation
+Error ± StDev
+% Reduction
+15◦
+baseline
+8.52◦ ± 3.62◦
+N/A
+re-tension
+8.66◦ ± 2.43◦
+-1.62
+hysteresis
+–
+–
+both
+–
+–
+30◦
+baseline
+13.54◦ ± 3.52◦
+N/A
+re-tension
+9.26◦ ± 2.64◦
+31.64
+hysteresis
+8.17◦ ± 1.23◦
+39.66
+both
+4.66◦ ± 1.65◦
+65.55
+45◦
+baseline
+16.98◦ ± 2.90◦
+N/A
+re-tension
+9.12◦ ± 1.79◦
+46.28
+hysteresis
+–
+–
+both
+–
+–
+60◦
+baseline
+19.37◦ ± 3.38◦
+N/A
+re-tension
+11.50◦ ± 4.14◦
+40.62
+hysteresis
+24.76◦ ± 1.43◦
+-27.86
+both
+11.17◦ ± 4.25◦
+42.31
+75◦
+baseline
+27.53◦ ± 5.98◦
+N/A
+re-tension
+12.39◦ ± 4.45◦
+55.00
+hysteresis
+–
+–
+both
+–
+–
+90◦
+baseline
+32.79◦ ± 6.17◦
+N/A
+re-tension
+15.68◦ ± 5.18◦
+52.19
+hysteresis
+32.45◦ ± 1.91◦
+1.03
+both
+16.66◦ ± 1.54◦
+49.19
+Table 4: Average xy-plane bending angle error, average
+standard deviation, and average % error reduction for the
+30◦, 60◦, and 90◦ trials
+Compensation
+Error ± StDev
+% Reduction
+baseline
+21.90◦ ± 4.36◦
+N/A
+re-tension
+12.15◦ ± 3.99◦
+41.48
+hysteresis
+21.79◦ ± 2.48◦
+4.28
+both
+10.83◦ ± 1.54◦
+52.35
+
+0
+5
+10
+15
+20
+time [s]
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+80
+bending angle [deg]
+baseline
+re-tension
+hysteresis
+both
+desired
+(a) xy-plane angle at 30◦ proximal section angle
+0
+5
+10
+15
+20
+time [s]
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+80
+bending angle [deg]
+baseline
+re-tension
+hysteresis
+both
+desired
+(b) xy-plane angle at 60◦ proximal section angle
+0
+5
+10
+15
+20
+time [s]
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+80
+bending angle [deg]
+baseline
+re-tension
+hysteresis
+both
+desired
+(c) xy-plane angle at 90◦ proximal section angle
+Fig. 11: Bending angle for different compensation types
+at three proximal section angles.
+15
+30
+45
+60
+75
+90
+tail angle [deg]
+-40
+-35
+-30
+-25
+-20
+-15
+-10
+-5
+0
+5
+DC offset [deg]
+baseline
+re-tension
+hysteresis
+both
+Fig. 12: Bending angle offset for different compensation
+types and proximal section angles.
+Going from Fig. 11a to 11b and from Fig. 11b to 11c,
+the source of the increasing error becomes more obvious:
+increases in passive proximal section angle caused an in-
+crease in the offset or bias error of the catheter tip. This
+phenomenon was visible in the bending section during ex-
+perimentation, and can be seen when comparing the tip in
+Fig. 8a to the tip in Fig. 8b. Also of note, the quality of the
+output angle degrades as the offset increases; this can be
+seen by examining the baseline or hysteresis compensated
+angle as the offset gets large and may be due to increased
+friction in the tendon sheath mechanism.
+To better show how the re-tension and hysteresis
+compensation reduce error from different sources, we ex-
+amine the DC offset and the time delay of the bending
+angle in Fig. 12 and Fig. 13, respectively. Time delay was
+computed by filtering the output (low pass, cutoff 2 Hz)
+and finding the cross-correlation between each output and
+the input. DC offset is the distance of the mean of each fil-
+tered output from zero.
+The increasing DC offset for increasing proximal
+section angle in Fig. 13 reflects the increasing error seen
+in Fig. 10a. It also mirrors Fig. 10a in that the re-tension
+and re-tension + hysteresis compensation trials are less af-
+fected by the offset, just as these compensations both had
+lower error for increasing proximal section angle. This
+strongly suggests that this offset is the primary source
+of error, and it is caused by the passive proximal section
+angle. From Fig. 13, it is clear that the simple hysteresis
+compensation reduces the time delay from hysteresis by
+half. However, Fig. 10a shows that the reduction in hys-
+teresis does not reduce the error substantially for increas-
+ing proximal section angle, relative to the error reduction
+from the re-tension compensation. This furthers the idea
+
+15
+30
+45
+60
+75
+90
+tail angle [deg]
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+time delay [seconds]
+baseline
+re-tension
+hysteresis
+both
+Fig. 13: Bending angle time delay for different compen-
+sation types and proximal section angles.
+that the DC offset caused by the passive proximal sec-
+tion angle is the larger source of error, especially at higher
+proximal section angles, and that the effect of the proxi-
+mal section angle on the bending section should not be
+ignored.
+For the yz-plane input, bending occurs perpendicular
+to the plane in which the passive proximal section is bent.
+Fig. 14 shows that the error in the xy-plane still increases
+with proximal section angle, even though the input is in
+the yz-plane. The trials with re-tension compensation still
+exhibit less error for increasing proximal section angle as
+well. There is no data for trials with hysteresis compensa-
+tion at 60 degrees.
+We again see the trend on increasing error in the xy-
+plane reflected in the DC offset of the output in the xy-
+plane in Fig. 15, although it is less pronounced than in
+the xy-input case. The trials with re-tension compensa-
+tion have lower offset at 90 degrees compared to those
+without, but it is difficult to say for certain whether the
+overall trend from the xy-input case is preserved with the
+lower offset numbers and missing 60 degree hysteresis
+data. Fig. 16 shows that the hysteresis compensation re-
+duced the time delay by half as expected from the xy-
+input case.
+4.4.3
+Third Scenario: Dynamic Condition Results
+The effect of introducing the passive proximal sec-
+tion angle in a dynamic scenario is shown in Fig. 17. If
+the passive proximal section angle had no effect on the
+kinematics, we would expect to see θx,EM and θz,EM both
+remain close to 0 for the duration of the trial in Fig. 17a.
+However, in Fig. 17a we instead see the consequence of
+increasing the passive proximal section angle from 0 to
+15
+30
+45
+60
+75
+90
+tail angle [deg]
+-5
+0
+5
+10
+15
+20
+25
+30
+35
+40
+bending angle error [deg]
+baseline
+re-tension
+hysteresis
+re-tension + hysteresis
+(a) xy-plane bending angle error
+15
+30
+45
+60
+75
+90
+tail angle [deg]
+-5
+0
+5
+10
+15
+20
+25
+30
+35
+40
+bending angle error [deg]
+baseline
+re-tension
+hysteresis
+re-tension + hysteresis
+(b) yz-plane bending angle error
+Fig. 14: Bending angle error for different compensation
+types and proximal section angles (yz-plane input).
+15
+30
+45
+60
+75
+90
+tail angle [deg]
+-40
+-30
+-20
+-10
+0
+10
+20
+30
+40
+DC offset [deg]
+baseline
+re-tension
+hysteresis
+both
+Fig. 15: Bending angle offset for different compensation
+types and proximal section angles (yz-plane input).
+
+15
+30
+45
+60
+75
+90
+tail angle [deg]
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+time delay [seconds]
+baseline
+re-tension
+hysteresis
+both
+Fig. 16: Bending angle time delay for different compen-
+sation types and proximal section angles (yz-plane input).
+0
+5
+10
+15
+20
+25
+30
+time [s]
+-5
+0
+5
+10
+15
+20
+25
+angle [deg]
+x
+z
+x,EM
+z,EM
+(a) baseline (no re-tension compensation)
+0
+5
+10
+15
+20
+25
+30
+time [s]
+-40
+-30
+-20
+-10
+0
+10
+20
+angle [deg]
+x
+z
+x,EM
+z,EM
+(b) re-tension compensation
+Fig. 17: Dynamic tests with re-tension compensation:
+dashed lines are angles according to the kinematics, and
+solid lines are measured by the EM sensor.
+Table 5: Configuration error after introduction of proxi-
+mal section angle
+Angle
+Compensation
+Error ± StDev
+% Reduction
+θx,EM
+baseline
+-1.69◦ ± 0.12◦
+N/A
+re-tension
+-0.47◦ ± 0.82◦
+72.29
+θz,EM
+baseline
+20.94◦ ± 1.09◦
+N/A
+re-tension
+-2.27◦ ± 0.89◦
+89.14
+60 degrees in the xy-plane without compensation is that
+the measured angle θz,EM accumulates over 20 degrees of
+error (over a 20 degree offset from 0). This agrees closely
+with the mean error for the baseline trial in the static case
+in Fig. 10a, further supporting that most of the error in the
+static case comes from this offset error. We also see that
+this behavior is not captured well by the kinematics, since
+θx and θz only deviate by a few degrees.
+In Fig. 17b, the re-tension compensation succeeds in
+bringing θx,EM and θz,EM back to 0 by the end of the trial.
+There is a trade-off visible in the trajectory for θz,EM in
+Fig. 17b. If the sensitivity of the controller is increased,
+the trajectory will be held closer to 0, but the oscillations
+in the trajectory will increase. Table 5 gives the measured
+error in the configuration–in this case deviation from the
+starting configuration (0,0) by the end of the trial–along
+with the reduction in error due to the re-tension compen-
+sation.
+5
+DISCUSSION
+The design of the robot allowed for the easy imple-
+mentation of the redundant controller and additional com-
+pensation methods. The current actuators are not strong
+enough to exceed the breaking force of the magnets, but
+if more powerful actuators were used, this could become
+an issue. A good addition to the design to overcome this
+would be a rail structure which contains the tendon an-
+chors without obstructing the tendons; this rail structure
+would allow the motors to reestablish contact with the
+tendon anchors after a disconnect.
+During the procedure for determining µ1 and µ2 in
+the methods section, we referred to these values as “pre-
+tension”. This is because mathematical redundancy in real
+systems often points to a physical phenomenon, in this
+case the magnitude of pre-tension or co-contraction of
+opposing tendons. If we assume the four tendons are at
+90 degree increments and incompressibility (as were both
+the case with our experiments), the redundancy represents
+the co-contraction or simultaneous pulling of opposing
+
+tendons, such as tendons 1 and 3 or tendons 2 and 4,
+just as opposing tendons are pulled in muscle contrac-
+tions. Furthermore, the magnitude of this co-contraction
+is determined by µ1 and µ2. In theory, µ1 and µ2 could
+be any positive value without affecting the configuration
+of the manipulator. In practice, large values of µ1 or µ2
+could cause compression of the “incompressible” bend-
+ing section, buckling of the passive proximal section, and
+changes in the stiffness estimates. Thus, it is best to use
+small values of µ1 and µ2 as a method for preventing slack
+tendons in that any µ1 or µ2 value above zero effectively
+enforces a pre-tension in the corresponding tendons equal
+to the magnitude of µ1 or µ2. We also suggested keeping
+the scalable redundant parameters µ1 and µ2 constant to
+avoid affecting other parameter estimates such as bend-
+ing stiffness. There is room for future experimentation in
+which these parameters are intentionally varied in order
+to obtain variable stiffness behavior, akin to impedance
+control.
+The deflection of the passive proximal section to the
+right caused an offset of the bending section to the left,
+irrespective of the plane of the input. This offset was the
+primary source of bending angle error and increased with
+increasing passive proximal section deflection. Simple
+hysteresis compensation alone did not substantially re-
+duce bending angle error, although it did reduce the phase
+lag of the output. Re-tension compensation reduced DC
+offset and bending angle error, both alone and with hys-
+teresis compensation. Compensation methods were kept
+simple so that the sources of error were not obscured and
+so that they can be implemented in the absence of a tip
+sensor; however, there is room for future work involv-
+ing more complex re-tension or hysteresis compensations,
+perhaps relying on additional sensors.
+The trials with re-tension compensation had reduced
+error at all angles in the passive proximal section relative
+to trials without. By adjusting the tendon tensions to their
+minimum values, the compensation can be seen as get-
+ting much closer to a neutral, unstretched state which the
+proximal section is assumed to have by kinematics mod-
+els in the state of the art. However, even for trials with
+re-tension compensation, error still increases for increas-
+ing proximal section angle. That is likely because, even
+with the tendon tensions adjusted, the proximal section
+is still deformed, deforming the tendon sheaths and the
+outer casing of the proximal section and articulation sec-
+tion. These deformations are not fully compensated by the
+re-tension compensation and should be investigated in fu-
+ture works.
+The amount of time the re-tension compensation was
+given to reach the desired tension was arbitrary. Since the
+re-tension occurs before the trial in the second scenario,
+the time spent re-tensioning does not affect the controller
+during the trial. This compensation is useful for any appli-
+cation where the proximal shape will remain unchanged
+for a long period of time, such as after insertion of a heart
+catheter. For the dynamic case in the third scenario, the re-
+tension occurs while the proximal section shape is chang-
+ing. This type of compensation would be useful for ap-
+plications where the shape is changing over time, such as
+during insertion of a catheter.
+The trials with hysteresis compensation tended to
+have lower error than those without for the lower prox-
+imal section angles, and the same or slightly higher er-
+ror for the higher angles. That hysteresis compensation is
+less helpful at higher angles is unsurprising, as we would
+expect errors from the proximal section angle to be the
+dominant source of error at higher angles. It is a little sur-
+prising that hysteresis compensation would cause a slight
+increase in error at some of the higher angles, and this
+would suggest that the two sources of error, proximal sec-
+tion angle and hysteresis, are not entirely decoupled. That
+said, the data presented regarding DC offset and time-
+shift and the fact the error increase is small both suggest
+that the phenomena are mostly independent.
+The amount of hysteresis compensation was based
+on the width of the hysteresis curve. This curve was rel-
+atively constant for different input amplitudes but not ex-
+actly constant. There is room for future work in augment-
+ing this simple hysteresis compensation by varying the
+magnitude slightly based on the input amplitude or some
+other factor.
+The passive proximal section angle affects the accu-
+racy of the kinematics, and the magnitude of this effect
+should not be ignored. How it is included depends on
+whether a sensor is present at the tip of the continuum
+manipulator. With a tip sensor for closed-loop control, the
+deviation between the desired output and the measured
+output can be used to update the kinematics or dynamics
+to account for the proximal section angle, whether it up-
+dates the system parameters directly or simply adds an ad-
+ditional term to the equations. Without a tip sensor there
+is no way to update the kinematics or dynamics real-time
+without first measuring the behavior for different passive
+proximal section angles off-line, and so they should be
+adjusted off-line based on the behavior of the system at
+different expected configurations of the passive proximal
+section.
+A strength of this study is that a one-dimensional an-
+gle in the passive proximal section causes a clear one-
+dimensional change in the bending section, but this is also
+its limitation. The effect of multiple bends in different di-
+rections, such as two bends in an S-shaped curve, of the
+passive proximal section should be investigated in future
+
+work, though the relationship between the passive proxi-
+mal section shape and the bending section may not be as
+simple to characterize.
+6
+CONCLUSION
+We introduced a separable tendon-driven robot ma-
+nipulator which addresses practical issues in actuation
+and sterilization. The separable design allows for reuse of
+all electromechanical components and does not use ten-
+dons in the reusable portion, avoiding mechanical wear.
+The interface between the actuators and the tendons is
+very transparent and all tendons are actuated, allowing
+for easier mitigation of backlash and deadzone. The fan-
+lock and magnetic interface allow for un-clipping and re-
+clipping.
+We presented a control scheme which utilizes the
+simplicity of constant curvature to yield a single solution
+to the inverse kinematics without the need for real-time
+optimization. The control scheme can be used without a
+tip sensor and does not require high fidelity knowledge
+of system parameters a priori, which is frequently lacking
+for mass-produced medical devices.
+The control scheme was validated along with addi-
+tional re-tension compensation for proximal section an-
+gle and hysteresis compensation. On average, error was
+reduced by 41.48% for re-tension, 4.28% for hysteresis,
+and 52.35% for re-tension + hysteresis compensation rel-
+ative to the baseline case. The re-tension compensation
+was tested for dynamic changes in the proximal section.
+The error in the final configuration of the tip was reduced
+by 89.14% relative to the baseline case
+7
+DISCLAIMER
+The concepts and information presented in this paper
+are based on research results that are not commercially
+available. Future availability cannot be guaranteed.
+8
+ACKNOWLEDGEMENT
+Research reported in this publication was supported
+in part by the National Institute of Biomedical Imaging
+and Bioengineering of the National Institutes of Health
+under award number R01EB028278. The content is solely
+the responsibility of the authors and does not necessarily
+represent the official views of the National Institutes of
+Health.
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+[39] Liberzon, D., 2012, Calculus of Variations and Optimal
+Control Theory: A Concise Introduction Princeton Uni-
+versity Press.
+

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+arXiv:2301.04905v1  [math.DS]  12 Jan 2023
+GENERICITY OF TRIVIAL LYAPUNOV SPECTRUM FOR Lp-COCYCLES DERIVED
+FROM SECOND ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS
+DINIS AMARO, MÁRIO BESSA†, AND HELDER VILARINHO
+CENTRO DE MATEMÁTICA E APLICAÇÕES (CMA-UBI)
+UNIVERSIDADE DA BEIRA INTERIOR
+RUA MARQUÊS D’ÁVILA E BOLAMA, 6201-001, COVILHÃ, PORTUGAL.
+A�������. Given an ergodic flow ϕt : M → M defined on a probability space M we study a family of
+continuous-time kinetic linear cocycles associated to the solutions of the second order linear homo-
+geneous differential equations ¨x + α(ϕt(ω))˙x + β(ϕt(ω))x = 0, where the parameters α, β evolve along
+the ϕt-orbit of ω ∈ M. Our main result states that for a generic subset of kinetic continuous-time
+linear cocycles, where generic means a Baire second category with respect to an Lp-like topology on
+the infinitesimal generator, the Lyapunov spectrum is trivial.
+1. I�����������
+We know, since the first half of nineteenth-century and by Liouville’s theorem, that serious
+restrictions are present when we try to apply analytic methods to integrate most functions. This
+result can be seen as a kind of differential Galois theory and represent a deep obstacle in solving
+differential equations explicitly. The way we bypass this inevitable fact is twofold: in one hand
+powerful numerical methods were developed to approximate the solutions and on the other hand
+a qualitative theory of differential equations emerged from the pioneering works of Poincaré and
+Lyapunov. We will be interested in following the last mentioned approach.
+With the study that we carry out we intend to understand the asymptotic behavior of the so-
+lutions of the second order homogeneous linear differential equations with coefficients displaying
+Lp regularity, varying in time and allowing an Lp-small perturbation. Namely, to describe its Lya-
+punov spectrum under Lp-generic conditions of its coefficients. Families of this type of equations
+will be considered indexed in a flow which keep invariant the probability measure in a measure
+space M.
+We have as motivation in the first instance a family of equations that describe the
+motion of the simple damped pendulum free from external forces, of type
+¨x(t) + α(ϕt(ω))˙x(t) + β(ϕt(ω))x(t) = 0,
+(1)
+where α and β are functions depending on ω ∈ M evolving along a flow ϕt : M → M for t ∈ R.
+Clearly, if α and β are first integrals related with ϕt (i.e. constant along the ϕt-orbits) the equation
+(1) is easily solved by elementary methods of a first course on differential equations. When the
+parameters vary in time, explicit solutions are hard to get. This is the case when the frictional
+force α and the frequency of the oscillator β change over time.
+In [8] the second author deal with a similar case but with periodic coefficients along periodic
+closed orbits and proved that small C0-perturbations on the parameters allows us to obtain that
+asymptotic unstable solutions are precisely the uniformly hyperbolic saddle-type ones.
+In the
+Date: January 13, 2023.
+Key words and phrases. Kinetic cocycles; Linear cocycles; Linear differential systems; Multiplicative ergodic theorem;
+Lyapunov exponents; Random dynamical systems.
+†Corresponding author: [email protected].
+2020 Mathematics Subject Classification: Primary: 34D08, 37H15, Secondary: 34A30, 37A20.
+1
+
+present paper we intend to focus on a perturbative theory with a coarser topology, namely allowing
+perturbations in an Lp-type topology and with random (non-periodic) base dynamics.
+Differential equations like (1) are ubiquitous in physics, engineering, biology and numerous
+applications of mathematics like, e.g., solid state physics, structural stability, wave propagation in
+one-dimensions, stability of synchronous electrical machines, etc. Fixing position and momentum
+(x(0), ˙x(0)) we intend to study the asymptotic behavior when t → ∞ of the pair (x(t), ˙x(t)), namely
+the asymptotic exponential growth rate given by the Lyapunov exponent.
+The literature on the
+subject with more or less similar nuances is substantial (see [3, 4, 8, 15, 19] and the references
+therein). The literature on the broader matter of Lyapunov exponents had a substantial grown in
+the last decade as it can be seen by several books published lately [6, 12, 16, 26, 28].
+The Lp-generic point of view on quite general linear differential systems was studied in [9]
+by two of the authors and after Arbieto-Bochi [1] (see also the references therein). In [9] was
+proved that the class of accesible (aka twisting) linear differential systems, a wider class that
+includes cocycles that evolve in GL(d, R), SL(d, R) and Sp(d, R), have a trivial Lyapunov spectrum
+Lp-genericaly. When considering the sharper C0-norm it is known since Millionshchikov’s work in
+the late sixties that the generic behaviour changes (see [22]). A complete treatment on the C0-case
+was done in [7, 8] after the discrete approach done in [10, 11]. The question of knowing the C0-
+generic asymptotic behaviour of linear differential systems arising from equations like (1) is a work
+in progress. There as been recently a growing interest in understanding the discrete ‘dynamical
+cocycle’ from an Lp-perturbative viewpoint and Lp-generic properties [5, 13, 14]. Since perturbing
+the cocycle given by the derivative depends on a perturbation of the map, these dynamical cocycles
+tend to drag several other difficulties.
+When we intend to change all the Lyapunov exponents in order them to become equal the
+naive idea is to distribute expansion/contraction rates equally for all directions.
+This can be
+made by rotating directions in a convenient way to have, at the end of the day, those tax rates
+identically scattered. Indeed, rotating in a systematic way, is a crucial idea which was developed
+in certain contexts in the Sovietic literature of the 1970s (see [23]) and in the 1980s by Mañé
+[20, 21]. Issues related with the continuity of Lyapunov exponents are much more complicated
+within feeble topologies like the Lp one. Here, we dedicate a substantial effort understanding the
+Lp-continuous dependence on the Lyapunov exponents as our arguments are supported on this
+assumption.
+Moreover, and also related with continuity, we know that typically squeezing the
+distance between vector fields implies in a decrease of the distance between its flows trajectories
+on compact times. Here, and once again, the Lp topology creates additional difficulties.
+In overall, the main result in the present paper (Theorem 1) can be summarized in the following
+way:
+For an Lp-generic choice of a kinetic linear differential system (as in (1)) and for almost every driving
+realization, no matter what position and momentum (x(0), ˙x(0)) we chose as initial conditions, the asymptotic
+exponential behaviour of the solutions will be the same.
+This paper is organized as follows: in §2 we will present the basic definitions and state our
+main result (Theorem 1); section §3 is devoted to the perturbation framework; in section §4 we
+will deal with continuity issues of the Lyapunov exponents with respecto to an Lp distance and,
+finally, in §5 we prove Theorem 1.
+2. D���������� ��� ���� ������
+2.1. Linear cocycles. In this section we present some definitions that will be useful for the
+development of this work. Let (M, M, µ) be a probability space and let ϕ: R × M → M be a metric
+dynamical system (or flow) in the sense that is a measurable map and
+(1) ϕt : M → M given by ϕt(ω) = ϕ(t, ω) preserves the measure µ for all t ∈ R;
+(2) ϕ0 = IdM and ϕt+s = ϕt ◦ ϕs for all t, s ∈ R.
+2
+
+Unless stated otherwise we will consider in the sequel that the flow is ergodic in the usual sense
+that there exist no invariant sets except null sets and their complements.
+Let B(X) be the Borel σ-algebra of a topological space X. A (continuous-time) linear random
+dynamical system (RDS) on (R2, B(R2)), or a (continuous-time) linear cocycle, over ϕ is a (B(R) ×
+M/B(GL(2, R))-measurable map
+Φ : R × M → GL(2, R)
+such that the mappings Φ(t, ω) forms a cocycle over ϕ, i.e.,
+(1) Φ(0, ω) = Id for all ω ∈ M;
+(2) Φ(t + s, ω) = Φ(t, ϕs(ω)) ◦ Φ(s, ω), for all s, t ∈ R and ω ∈ M,
+and t �→ Φ(t, ω) is continuous for all ω ∈ M. We recall that having ω �→ Φ(t, ω) measurable for
+each t ∈ R and t �→ Φ(t, ω) continuous for all ω ∈ M implies that Φ is measurable in the product
+measure space. These objects are also called linear differential systems in the literature (see §2.2).
+2.2. Kinetic Linear Differential Systems. We begin by considering as motivation the non-
+autonomous linear differential equation, which describes a motion of the damped harmonic oscil-
+lator as the ‘simple pendulum’ along the path (ϕt(ω))t∈R, with ω ∈ M described by the flow ϕ. Let
+K ⊂ R2×2 be the set of matrices 2 × 2 of type
+A =
+�
+0
+1
+b
+a
+�
+for real numbers a, b, and denote by G the set of measurable applications A : M → R2×2 and
+by K ⊂ G the set of kinetic measurable applications A : M → K.
+As usual we identify two
+applications on G that coincide on a µ full measure subset of M.
+Consider measurable maps
+α: M → R and β: M → R. Take the random differential equation given by (1) and defined by:
+¨x(t) + α(ϕt(ω))˙x(t) + β(ϕt(ω))x(t) = 0.
+Considering the change of variables y(t) = ˙x(t) we may rewrite (1) as the following vectorial linear
+system
+˙X = A(ϕt(ω)) · X,
+(2)
+where X = X(t) = (x(t), y(t))T = (x(t), ˙x(t))T and A ∈ K is given by
+A(ω) =
+�
+0
+1
+−β(ω)
+−α(ω)
+�
+.
+It follows from [2, Thm. 2.2.2] (see also Lemma 2.2.5 and Example 2.2.8 in this reference) that if
+A ∈ G1 =: G ∩ L1(µ), i.e.
+�
+M ∥A∥ dµ < ∞, generates a unique (up to indistinguishability) linear RDS
+ΦA satisfying
+ΦA(t, ω) = Id +
+� t
+0
+A(ϕs(ω))ΦA(s, ω) ds.
+(3)
+The solution ΦA(t, ω) defined in (3) is called the Carathéodory solution or weak solution. Given
+an initial condition X(0) = v ∈ R2, we say that t �→ ΦA(t, ω)v solves or is a solution of the Random
+Differential Equation (RDE) (2), or that the RDE (2) generates ΦA(t, ω). Note that ΦA(0, ω)v = v
+for all ω ∈ M and v ∈ R2. If the solution (3) is differentiable in time (i.e. with respect to t) and
+satisfies for all t
+d
+dtΦA(t, ω)v = A(ϕt(ω))ΦA(t, ω)v
+and
+ΦA(0, ω)v = v,
+(4)
+then it is called a classical solution of RDE (2).
+Of course that t �→ ΦA(t, ω)v is continuous for all ω and v. Due to (4) we call A : M → K
+a (kinetic) ‘infinitesimal generator’ of ΦA. Sometimes, due to the relation between A and ΦA, we
+refer to both A and ΦA as a kinetic linear cocyle/RDS/differential system.
+If the RDE (2) has
+initial condition X(0) = v then ΦA(0, ω)v = v and X(t) = ΦA(t, ω)v.
+3
+
+2.3. The Lp topology. We begin by defining an Lp-like topology generated by a metric that com-
+pares the infinitesimal generators on G.By a standard use of Grönwall’s inequality arguments it
+is usual to obtain the approximation of flows (on compact times) by assuming the approximation
+of the corresponding infinitesimal generators. However, Lp-estimates, 1 ≤ p < ∞, are more de-
+manding and it is not clear if such Lp-continuous dependence holds (cf. Lemma 4.3). Related to
+this problem see e.g. [25] where these issues are treated for L1-approximation on initial conditions
+in quasilinear elliptic-parabolic equations. With this in mind we define now the distance we are
+going to consider along this work. Given 1 ≤ p < ∞ and A, B ∈ G we set
+ˆσp(A, B) :=
+
+��
+M
+∥A(ω) − B(ω)∥p dµ(ω)
+� 1
+p
+,
+∞ if the above integral does not exists,
+and define
+σp(A, B) :=
+
+ˆσp(ΦA,ΦB)
+1+ ˆσp(ΦA,ΦB),
+if ˆσp(A, B) < ∞
+1,
+if ˆσp(A, B) = ∞ .
+Clearly, σp is a distance in G.
+Proposition 2.1. Consider 1 ≤ p < ∞. Then:
+(i) σp(A, B) ≤ σq(A, B) for all q ≥ p and all A, B ∈ G.
+(ii) If A ∈ G1 then for any B ∈ G satisfying σp(A, B) < 1 we have B ∈ G1.
+Therefore,
+sup
+0≤t≤1
+log+ ∥ΦB(t, ω)±1∥ ∈ L1(µ).
+Proof. (i) For all 1 ≤ p ≤ q < ∞ it is a standard result on Lebesgue spaces that Lq-norms are
+thinner than Lp ones. So, we have ˆσp(A, B) ≤ ˆσq(A, B). Hence, σp(A, B) ≤ σq(A, B).
+(ii) Pick any 1 ≤ p < ∞ and let A ∈ G1 and B ∈ G satisfying σp(A, B) < 1.
+By (i) we have
+σ1(A, B) < 1, thus ˆσ1(A, B) < ∞, and we also have that ∥B(ω)∥ ≤ ∥A(ω)∥ + ∥B(ω) − A(ω)∥ for all
+ω ∈ M. Therefore, B ∈ L1(µ). The last statement follows from (5).
+□
+The following result will be used to prove that the kinetic subspace of linear cocycles is a Baire
+space.
+Lemma 2.2. (K1, σp) is closed, for all 1 ≤ p < ∞.
+Proof. Given any sequence (An)n in K1, with An
+σp
+−→ A∗, for some A∗ ∈ G we must prove that
+A∗ ∈ K1. As (An)n ∈ K1 we get
+An(ω) =
+�
+0
+1
+−βn(ω)
+−αn(ω)
+�
+and
+A∗ =
+�
+a(ω)
+b(ω)
+−β(ω)
+−α(ω)
+�
+for maps αn, βn ∈ L1(µ) and measurable maps a, b, α and β. From (i) in Proposition 2.1 we have
+An
+σ1
+−→ A∗. From (ii) in Proposition 2.1 we have that a, b, α and β are also L1 maps, i.e. A∗ ∈ G1.
+Hence,
+�
+M |a(ω)| dµ(ω) = 0 and
+�
+M |b(ω) − 1| dµ(ω) = 0, which implies a(ω) = 0 and b(ω) = 1 for µ
+almost every ω. Therefore, A∗ ∈ K. In conclusion, A∗ ∈ G1 ∩ K = K1.
+□
+Since for all 1 ≤ p < ∞, the metric space (G1, σp) is complete and K1 ⊂ G1 is σp-closed, we
+conclude the following.
+Corollary 2.3. For all 1 ≤ p < ∞, (K1, σp) is a complete metric space and, therefore, a Baire space.
+2.4. Lyapunov exponents. Let K1 = K ∩ L1(µ) ⊂ G1. Observe that if A ∈ K1 then the cocycle ΦA
+satisfies the following integrability condition
+sup
+0≤t≤1
+log+ ∥ΦA(t, ω)±1∥ ∈ L1(µ),
+(5)
+4
+
+where f + = max{0, f}. Indeed, consider ω in the full measure ϕt-invariant subset of M where
+t �→ A(ϕt(ω)) is locally integrable. By (3) we get
+∥ΦA(t, ω)±1∥ ≤ 1 +
+� t
+0
+���A(ϕs(ω))
+���
+���ΦA(s, ω)±1��� ds.
+By Grönwall’s inequality (see [2]) we have
+ΦA(t, ω)±1∥ ≤ exp
+�� t
+0
+∥A(ϕs(ω))∥ ds
+�
+(6)
+that is, for all t ≥ 0 we have log+ ∥ΦA(t, ω)±1∥ ≤
+� t
+0 ∥A(ϕs(ω))∥ ds. Therefore
+sup
+0≤t≤T
+log+ ∥ΦA(t, ω)±1∥ ≤
+� T
+0
+∥A(ϕs(ω))∥ ds =: ψ(ω, T).
+(7)
+By [2, Lemma 2.2.5] we have ψ(·, T) ∈ L1(µ). Thus sup
+0≤t≤1
+log+ ∥ΦA(t, ω)±1∥ ∈ L1(µ), getting the claim.
+Moreover, Fubini’s theorem allow us to obtain
+�
+M
+log+ ∥ΦA(t, ω)±1∥ dµ(ω) ≤
+�
+M
+� t
+0
+∥A(ϕs(ω))∥ ds dµ(ω)
+=
+� t
+0
+�
+M
+∥A(ϕs(ω))∥ dµ(ω) ds = t∥A∥1.
+If A ∈ G1 then the integrability condition (5) holds and Oseledets theorem (see e.g.
+[24, 2])
+guarantees that for µ almost every ω ∈ M, there exists a ��A-invariant splitting called Oseledets
+splitting of the fiber R2
+ω = E1
+ω⊕E2
+ω and real numbers called Lyapunov exponents λ1(A, ω) ≥ λ2(A, ω),
+such that:
+lim
+t→±∞
+1
+t log ∥ΦA(t, ω)vi∥ = λi(A, ω),
+for any vi ∈ Ei
+ω \ {⃗0} and i = 1, 2. Moreover, given any of these subspaces E1
+ω and E2
+ω, the angle
+between them along the orbit has subexponential growth, meaning that
+lim
+t→±∞
+1
+t log sin
+�
+∡(E1
+ϕt(ω), E2
+ϕt(ω))
+�
+= 0.
+(8)
+If the flow ϕt is ergodic, then the Lyapunov exponents and the dimensions of the associated
+subbundles are constant µ almost everywhere and we refer to them as λ1(A) and as λ2(A), with
+λ1(A) ≥ λ2(A). We say that A has one-point Lyapunov spectrum or trivial Lyapunov spectrum if for
+µ a.e. ω ∈ M, λ1(A, ω) = λ2(A, ω). Otherwise we say A has simple Lyapunov spectrum. For details
+on these results on linear differential systems see [2] (in particular, Example 3.4.15). See also [17].
+2.5. Statement of the main result. We are now in position to state our main contribution. In
+the present paper we establish the existence of a σp-residual of K1 displaying one-point spectrum.
+More precisely we prove the following:
+Theorem 1. For all 1 ≤ p < ∞ there exists a σp-residual subset R ∈ K1 such that any A ∈ R has
+one-point spectrum.
+3. I������������ O��������’ ����������
+In this section we build up the fundamental perturbation tool which allows us to interchange
+Oseledets directions. First, we will discuss some topics about the perturbations that we will be
+tailor-made to our purpose.
+5
+
+3.1. Perturbations supported in compact sets. Take A ∈ G and a non-periodic orbit ω ∈ M.
+We will consider a perturbation Bω of A only along a segment of the orbit of ω with extremes ω
+and ϕ1(ω). Let P ∈ G be given and define B: M → R2×2 such that B( ˆω) = A( ˆω) for all ˆω outside
+ϕ[0,1](ω) = {ϕs(ω) : s ∈ [0, 1]} and B( ˆω) = P( ˆω) otherwise. The map B is called a perturbation of A
+by P supported on ϕ[0,1](ω).
+Lemma 3.1. Given ω ∈ M, u, v ∈ R2 \ {0}, A ∈ K1, there is γ � 0, and a perturbation Bω ∈ K1 of A
+supported on ϕ[0,1](ω) such that:
+(i) ∥Bω( ˆω)∥ ≤ 4π2 for all ˆω on ϕ[0,1](ω) and
+(ii) ΦBω(1, ω)u = γ v.
+Proof. Let θ = ∡(Ru, Rv) ∈ [π, 2π] measured clockwise.
+Set a constant infinitesimal generator
+P: M → R2×2 given by
+P =
+�
+0
+1
+−θ2
+0
+�
+.
+(9)
+We consider the perturbation Bω ∈ K1 of A by P supported on ϕ[0,1](ω). The infinitesimal generator
+in (9) generates a linear differential system with fundamental classical solution (4) given, for all
+ˆω ∈ M and t ∈ R by the ‘clockwise elliptical rotation’ defined by:
+ΦP(t, ω) =
+�
+cos(θt)
+θ−1 sin(θt)
+−θ sin(θt)
+cos(θt)
+�
+,
+and such that ΦBω(1, ω)u = ΦP(1, ω)u = γv, for some γ � 0 fulfilling (ii).
+□
+Remark 3.1. In the application of Lemma 3.1 further ahead we will consider u and v such that
+Ru = E1
+ω and Rv = E2
+ϕ1(ω) where Ei are the Oseledets directions associated to A. In particular,
+ΦBω(1, ω)E1
+ω = E2
+ϕ1(ω). Morever, as the canonic norm and the maximum norm are equivalent item (i)
+of Lemma 3.1 and (6) impies ∥ΦBω(1, ω)∥max ≤ C∥ΦBω(1, ω)∥ ≤ Ce4π2.
+3.2. Special flows. Consider a measure space Σ, a map T : Σ → Σ, a T -invariant probability
+measure ˜µ defined in Σ and a roof function h: Σ → R+ satisfying h(ω) ≥ H > 0, for some H > 0
+and all ω ∈ Σ, and
+�
+Σ h(ω)d˜µ(ω) < ∞.
+Define the space Mh ⊆ Σ × R+ by
+Mh = {(ω, t) ∈ Σ × R+ : 0 ≤ t ≤ h(ω)}
+with the identification between the pairs (ω, h(ω)) and (T (ω), 0). The semiflow defined on Mh by
+S s(ω, r) = (T n(ω), r + s − �n−1
+i=0 h(T i(ω))), where n ∈ N is uniquely defined by
+n−1
+�
+i=0
+h(T i(ω)) ≤ r + s <
+n
+�
+i=0
+h(T i(ω))
+is called a suspension semiflow. If T is invertible then (S t)t is a flow. Furthermore, if ℓ denotes
+the one dimensional Lebesgue measure the measure µ = (˜µ × ℓ)/
+�
+h d˜µ defined on Mh by
+�
+g dµ =
+1
+�
+h d˜µ
+� �� h(ω)
+0
+g(ω, t)dt
+�
+d˜µ(ω),
+∀g ∈ C0(Mh)
+is a probability measure and it is invariant by the suspension semiflow (S t)t. Flows with such
+representation are called special flows.
+Next result, despite simple, will be the key step to prove second part of Proposition 3.3. We fix
+a special flow described by the quadruple (ϕt, Σ, T , h) endowed with product measure ˜µ × ℓ, Σ0 ⊆ Σ
+with ˜µ(Σ0) > 0. Moreover, for b > a > 0 we define the set
+ϕ[a,b](Σ0) = {ϕt(ω): ω ∈ Σ0, t ∈ [a, b]}.
+6
+
+Given A ∈ G1, P ∈ G, Σ0 ⊆ Σ and a > 0, we may extend the perturbation of A by P to be supported
+on the flowbox ϕ[a,a+1](Σ0) in the following way: for ω ∈ ϕ[a,a+1](Σ0) we project ω in ˜ω ∈ Σ0 and
+let B ˜ω be perturbation of A by P supported on ϕ[0,1](ϕa(ω)), and define
+B(ω) :=
+� A(ω),
+if ω � ϕ[a,a+1](Σ0)
+B ˜ω(ω),
+if ω ∈ ϕ[a,a+1](Σ0) .
+(10)
+To distinguish the situations we refer for B(ω) as a global perturbation of A by P supported in
+ϕ[a,a+1](Σ0).
+Lemma 3.2. For all A ∈ G1, a > 0 and ε ∈]0, 1[, there exists a measurable set Σ0 ⊂ Σ with
+˜µ(Σ0) > 0 such that for any global perturbation B ∈ G1 of A supported in the flowbox ϕ[a,b](Σ0) with
+∥B(ϕt(ω))∥ ≤ 4π2 for all ω ∈ Σ0 and t ∈ [a, b], we have that σ1(A, B) < ε.
+Proof. Let A ∈ G1, b > a > 0 and ε ∈]0, 1[ be fixed.
+Since A ∈ G1 we let L > 0 be such that
+�
+M ∥A(ω)∥ dµ(ω) < L. For any Σ0 ⊂ Σ we have µ(ϕ[a,b](Σ0)) = (b − a)˜µ(Σ0). Let ε′ =
+ε
+1−ε and choose
+a measurable set Σ0 ⊂ Σ satisfying
+˜µ(Σ0) ∈
+�
+0,
+ε′
+(b − a)(L + 4π2)
+�
+.
+(11)
+Finally, we will check that σ1(A, B) < ε. Indeed, from (11) we get:
+ˆσ1(A, B)
+=
+�
+M
+∥A(ω) − B(ω)∥ dµ(ω) =
+�
+ϕ[a,b](Σ0)
+∥A(ω) − B(ω)∥ dµ(ω)
+≤
+�
+ϕ[a,b](Σ0)
+∥A(ω)∥ + ∥B(ω)∥ dµ(ω) ≤ µ(ϕ[a,b](Σ0))(L + 4π2)
+=
+(b − a)(L + 4π2)˜µ(Σ0) < ε′,
+which implies σ1(A, B) < ε.
+□
+3.3. Lowering the maximal Lyapunov exponent. The next result asserts that is possible to
+lower the maximal Lyapunov exponent of a kinetic cocycle with simple spectrum by a σp-small
+perturbation.
+Proposition 3.3. Given A ∈ K1 and ε, δ > 0, there exists B ∈ K1 such that σ1(A, B) < ε and
+λ1(B) ≤ λ1(A) + λ2(A)
+2
++ δ.
+(12)
+Proof. Let A ∈ K1 and ε, δ > 0 be given.
+We assume that A has simple spectrum, otherwise
+the conclusion is trivial. We are going to perform perturbations supported on a segment of size
+1 (cf.
+Lemma 3.1) in points ω of a subset Σ0 of positive measure, with the perturbation to be
+taken approximately at ϕN0/2(ω), where N0 is large enough to fulfill the asymptotic properties of
+Oseletets’ theorem, up to some given accuracy η, both for ω and ϕ
+N0
+2 +1(ω). To avoid unpleasant
+overlapping situations, we codify ϕ by a special flow along a Kakutani castle with a finite roof
+function.
+For a better understanding and since the flow ϕ is ergodic a simple application of
+Rudolph’s two symbol code representation (see [27]) allows us to consider a special flow S t with
+basis Σ ⊂ M and a roof function h: Σ → {N0 +1, N0 +q}, for some large N0 and some 1 < q ∈ R\Q.
+We recall that Rudolph’s theorem gives a step function of irrationally independent high levels, and
+we need time enough to see the Lyapunov exponent in time N0/2 (with accuracy η), to perform
+the perturbation in time less than 1, and so see again the Lyapunov exponent in time N0/2 before
+the return to the basis. In this way we prevent overlaps of the perturbations. Moreover, we have
+a decomposition of µ = ˜µ × ℓ, where ℓ is the time length and ˜µ is a measure on Σ invariat by the
+return map. By abuse of notation in the following we still denote the special flow by ϕt. Up to
+7
+
+some rearrangement of the castle, given η > 0 there is N ∈ N (N ≥ N0) such that by the Oseledets
+theorem there is a subset ˜Σ ⊆ Σ, with ˜µ(˜Σ) > 0, such that for every ω ∈ ˜Σ we have:
+�����λi − 1
+t log
+���ΦA(t, ω)|Eiω
+���
+����� < η/8 < η
+(13)
+for i = 1, 2 and t ≥ N/2. Notice that from the cocycle property, by taking N larger if necessary, for
+ω ∈ ˜Σ and setting ω′ = ϕN/2+1(ω), we also have
+�����λi −
+1
+N/2 − 1 log
+����ΦA(N/2 − 1, ω′)|Ei
+ω′
+����
+����� < η.
+(14)
+Indeed, to obtain (14) we notice that
+�����λi −
+1
+N/2 − 1 log
+����ΦA(N/2 − 1, ω′)|Ei
+ω′
+����
+����� =
+�����λi −
+1
+N/2 − 1 log
+����ΦA(N, ω)|EiωΦA(−(N/2 + 1), ω′)|Ei
+ω′
+����
+�����
+≤
+�����2λi −
+N
+N/2 − 1
+1
+N log
+���ΦA(N, ω)|Eiω
+���
+�����
++
+�����−λi − N/2 + 1
+N/2 − 1
+1
+N/2 + 1 log
+����ΦA(N/2 + 1, ω)−1|Ei
+ω′
+����
+����� .
+The first term is less or equal than
+�����2 −
+N
+N/2 − 1
+����� |λi| +
+�����λi − 1
+N log ∥ΦA(N, ω)|Eiω
+�����
+�����
+N
+N/2 − 1
+����� ,
+which can be made smaller than η/2 for sufficiently large N, as well the second term because is
+less or equal than
+�����1 − N/2 + 1
+N/2 − 1
+����� |λi| +
+�����λi −
+1
+N/2 + 1 log ∥ΦA(N/2 + 1, ω)|Eiω
+�����
+�����
+N/2 + 1
+N/2 − 1
+����� .
+since
+�����−λi −
+1
+N/2 − 1 log ∥ΦA(N/2 + 1, ω)−1|Ei
+ω′ ∥
+����� =
+�����−λi +
+1
+N/2 − 1 log
+����ΦA(N/2 + 1, ω)|Ei
+ω′
+����
+����� .
+For every ˜ω ∈ ˜Σ let B ˜ω be the perturbation of A by P, depending on Ru = E1
+ϕN/2( ˜ω) and
+Rv = E2
+ϕ1+N/2( ˜ω), as in (9) given by Lemma 3.1 supported on ϕ[N/2,N/2+1]( ˜ω) such that we have
+ΦB ˜ω(1, ϕN/2( ˜ω))E1
+ϕN/2( ˜ω) = E2
+ϕ1+N/2( ˜ω).
+We may consider now a global perturbation B of A supported on the flowbox ϕ[N/2,N/2+1](˜Σ) as
+before: for ω ∈ ϕ[N/2,N/2+1](˜Σ) we project ω in ˜ω ∈ ˜Σ and define
+B(ω) :=
+�
+A(ω),
+if ω � ϕ[N/2,N/2+1](˜Σ)
+B ˜ω(ω),
+if ω ∈ ϕ[N/2,N/2+1](˜Σ) .
+Notice that we have B ∈ K1. By passing to a subset of ˜Σ if necessary, from Lemma 3.2 we may
+assume that σ1(A, B) < ε.
+We will see that we may choose a large N depending on η (depending on δ) such that for all
+ω ∈ ˜Σ we have
+λ1(B, ω) ≤ λ1(A, ω) + λ2(A, ω)
+2
++ δ.
+(15)
+Having ˜µ(˜Σ) > 0 implies that (15) holds for all ω in a subset ˜M ⊆ Mh with µ( ˜M) > 0, and since
+we are assuming that the flow ϕt is µ-ergodic, we have
+λ1(B) ≤ λ1(A) + λ2(A)
+2
++ δ
+as required in (12). Let us explain the effect of mixing Oseledets’ directions on the decay of the
+Lyapunov exponents.
+Fiz η > 0 to be choosed later and take ω ∈ ˜Σ with Oseledets directions
+E1
+ω ⊕ E2
+ω and N > 0 sufficiently large in order to have (13) and (14) for ω′ = ϕN/2+1(ω), that is
+ΦA (N/2, ω) · v1
+ω ≈ eλ1 N
+2 , ΦA (N/2, ω) · v2
+ω ≈ eλ2 N
+2 ,
+(16)
+8
+
+and
+ΦA
+�N/2 − 1, ω′� · v1
+ω′ ≈ eλ1( N
+2 −1), ΦA
+�N/2 − 1, ω′� · v2
+ω′ ≈ eλ2( N
+2 −1),
+(17)
+where λ1 = λ1(A) and λ2 = λ2(A) are the Lyapunov exponents of A, vi
+ω ∈ Ei
+ω and vi
+ω′ ∈ Ei
+ω′ are
+unitary vectors, and ≈ means η-close as in (13) and (14).
+Notice that, for all ω ∈ ˜Σ we have:
+ΦB(N, ω) = ΦA
+�N/2 − 1, ω′� · ΦBω(1, ϕN/2(ω)) · ΦA (N/2, ω) .
+(18)
+So, for all ω ∈ ˜Σ, we may decompose the action of the map ΦB(N, ω) in three pieces:
+• The first, between ω and ϕN/2(ω), and the third, between ω′ and ϕN(ω), that, using the
+basis induced by the Oseledets directions with respect to the splitting E1 ⊕ E2, may be
+writen as
+ΦA (N/2, ω) =
+�
+φ1,1 (N/2, ω)
+0
+0
+φ2,2 (N/2, ω)
+�
+and
+ΦA
+�N/2, ω′� =
+�
+φ1,1 (N/2 − 1, ω′)
+0
+0
+φ2,2 (N/2 − 1, ω′)
+�
+.
+Using (16) and (17) we get that ΦA (N/2, ω) is an operator that can be represented approxi-
+mately (with
+����φ1,1 − e
+λ1
+2 N���� < η and
+����φ2,2 − e
+λ2
+2 N���� < η) by the matrix
+
+e
+λ1
+2 N
+0
+0
+e
+λ2
+2 N
+
+(19)
+written, as usual, in the Oseledets basis {v1
+ω, v2
+ω}. Similarly, ΦA(N/2, ω′) can be represented
+approximately (with
+���φ1,1 − eλ1(N/2−1)��� < η and
+���φ2,2 − eλ2(N/2−1)��� < η) by the matrix
+�
+eλ1(N/2−1)
+0
+0
+eλ2(N/2−1)
+�
+(20)
+in the Oseledets basis
+�
+v1
+ω′, v2
+ω′
+�
+.
+• The second piece, between ϕN/2(ω) and ω′, with matrix in the basis induced by the Oseledets
+directions with respect to the splitting E1
+ϕN/2(ω) ⊕ E2
+ϕN/2(ω) which we denote by:
+ΦBω(1, ϕN/2(ω)) =
+�
+0
+a1,2
+a2,1
+a2,2
+�
+.
+(21)
+With this interchange of directions the largest grow for ΦA(N, ω) (given by eλ1N) can never be
+achieved for the perturbation ΦB(N, ω). Indeed, the entries of ΦB(N, ω) are bounded by:
+max
+�
+eλ1(N/2−1)e
+λ2
+2 N, eλ2(N/2−1)e
+λ1
+2 N, e
+λ2
+2 Neλ2(N/2−1)�
+× max �a1,2, a2,1, a2,2
+�
+We now estimate
+1
+N log ∥ΦB(N, ω)∥.
+By (18) we know that ΦB(N, ω) is the composition of the
+matrices (19), (21) and (20).
+For the sake of simplicity of presentation we estimate ∥ΦB(N, ω)∥
+using these three matrices which are in Oseledets basis. However, since the angle between the
+Oseledets fibers has subexponential decay as in (8), estimating ∥ΦB(N, ω)∥ by considering the
+canonical basis only increases the technicalities which we avoid in the sequel. Taking this into
+consideration and also Remark 3.1 we can ensure that:
+∥ΦB(N, ω)∥ ≤ max
+�
+e
+λ1+λ2
+2
+N−λ1, e
+λ1+λ2
+2
+N−λ2
+�
+Ce4π2.
+Therefore,
+1
+N log ∥ΦB(N, ω)∥ ≤ max
+�λ1 + λ2
+2
+− λ1
+N , λ1 + λ2
+2
+− λ2
+N
+�
++ logCe4π2
+N
+.
+Finally, taking N sufficiently large we obtain
+1
+N log ∥ΦB(N, ω)∥ ≤ λ1(A, ω) + λ2(A, ω)
+2
++ δ.
+9
+
+By (28) we have
+λ1(B, ω) = lim
+t→∞
+1
+t log ∥ΦB(t, ω)∥ = lim
+n→∞
+1
+n log ∥ΦB(n, ω)∥ = inf
+n
+1
+n log ∥ΦB(n, ω)∥
+≤ 1
+N log ∥ΦB(N, ω)∥ ≤ λ1(A, ω) + λ2(A, ω)
+2
++ δ
+and (15) is proved.
+□
+4. U���� ����-���������� ��� ��� ��� L������� ��������
+4.1. On L1-continuous dependence results. In this subsection we provide some preliminary
+results to achieve the upper-semicontinuity for the top Lyapunov exponent. We start with a basic
+measure-theoretical result that will be instrumental.
+Lemma 4.1. Let f ∈ L1(µ), f ≥ 0. For all η > 0 exists K > 0 such that for all h ∈ L1(µ) with h ≥ 0
+and ∥h − f∥1 < η, we have
+�
+{h>K}
+h dµ(ω) < 2η
+and
+µ({h > K}) < 2η
+K .
+Proof. [1, Lemma 5].
+□
+Throghout this subsection we consider A, B ∈ G1, ǫ > 0 and T ∈ N. Let us define
+f(ω) =
+� 1
+0
+∥A(ϕs(ω))∥ ds
+and
+g(ω) =
+� 1
+0
+∥B(ϕs(ω))∥ ds.
+(22)
+As we already said, by [2, Lemma 2.2.5] we have that f, g ∈ L1(µ). We set η = ε/(16(T + 2)) and
+fix B satisfying ˆσp(A, B) < η, which also implies ∥f − g∥1 < η. We use now Lemma 4.1 for f and
+h = g which gives us K > 0 (depending on η and A). Let
+E f = { f ≤ K}
+and
+Eg = {g ≤ K}.
+(23)
+Then Lemma 4.1 gives that
+�
+Ec
+h
+h dµ(ω) < 2η
+and
+µ(Ec
+h) < 2η
+K ,
+for
+h = f, g.
+(24)
+Set
+G =
+T−1
+�
+i=0
+ϕ−i(E f ∩ Eg).
+(25)
+Then Gc has small measure:
+µ(Gc) ≤
+T
+�
+i=0
+µ(ϕ−i(E f ∪ Eg)c) ≤ Tµ(E f ∪ Eg)c ≤ T 4η
+K < ε
+4K .
+(26)
+Elements belonging to the set G defined in (25) have nice estimates. Indeed we have:
+Lemma 4.2. If ω ∈ G, then for C = A, B we have
+� T
+0 ∥C(ϕs(ω))∥ ds ≤ TK and ∥ΦC(T, ω)∥ ≤ eT K.
+Proof. Once ω ∈ G we have ϕi(ω) ∈ E f for all i = 0, . . . , T, and
+� T
+0
+∥A(ϕs(ω))∥ ds =
+T−1
+�
+i=0
+� 1
+0
+∥A(ϕs+i(ω))∥ ds ≤ TK.
+By (7) we get log+ ∥ΦA(T, ω)∥ ≤
+� T
+0 ∥B(ϕs(ω))∥ ds and we are done. The other case is similar.
+□
+10
+
+Next result gives us a ‘weak form’ of continuous dependence of the solutions on its infinitesimal
+generator which will be enough for our purposes. The dependence will have an Lp flavour in the
+sense that the estimate on the separation of the two solutions will be on a set, despite huge in
+measure, not exactly the whole M and the reason we call ‘weak’.
+Lemma 4.3. For all 1 ≤ p < ∞,
+��
+G
+∥ΦA(1, ω) − ΦB(1, ω)∥p dµ(ω)
+� 1
+p
+≤ e2Kp ˆσp(A, B).
+Proof. For ω ∈ G we have
+∥ΦA(1, ω) − ΦB(1, ω)∥
+≤
+1
+�
+0
+∥A(ϕs(ω)∥
+����������������
+β(s)
+∥ΦA(s, ω) − ΦB(s, ω)∥ds
++
+1
+�
+0
+���A(ϕs(ω)) − B(ϕs(ω))
+��� ∥ΦB(s, ω)∥ds
+��������������������������������������������������������������������������������������������������
+α(t)
+.
+Taking u(t) = ∥ΦA(t, ω) − ΦB(t, ω)∥, we get u(t) ≤
+� t
+0 β(s)u(s) ds + α(t). Using Grönwall’s inequality
+we get u(t) ≤ α(t) exp
+�� t
+0 β(s) ds
+�
+, which by Lemma 4.2 implies
+∥ΦA(1, ω) − ΦB(1, ω)∥ ≤
+1
+�
+0
+���A(ϕs(ω)) − B(ϕs(ω))
+��� ∥ΦB(s, ω)∥ ds eK.
+Since by Lemma 4.2 we have ∥ΦB(s, ω)∥ ≤ esK, using Jensen’s inequality we get
+∥ΦA(1, ω) − ΦB(1, ω)∥p
+≤
+
+1
+�
+0
+���A(ϕs(ω)) − B(ϕs(ω))
+��� ∥ΦB(s, ω)∥ ds eK
+
+p
+≤
+1
+�
+0
+���A(ϕs(ω)) − B(ϕs(ω))
+���p ds e2Kp.
+Integrating in both sides and using Fubini and change of coordinates (recalling that ϕs preserves
+µ) we get:
+�
+G
+∥ΦA(1, ω) − ΦB(1, ω)∥p dµ(ω)
+≤
+e2Kp
+�
+G
+1
+�
+0
+���A(ϕs(ω)) − B(ϕs(ω))
+���p ds dµ(ω)
+≤
+e2Kp
+1
+�
+0
+�
+G
+���A(ϕs(ω)) − B(ϕs(ω))
+���p dµ(ω) ds
+=
+e2Kp
+1
+�
+0
+�
+ϕs(G)
+∥A(ω) − B(ω)∥p dµ(ω) ds
+≤
+e2Kp
+�
+M
+∥A(ω) − B(ω)∥p dµ(ω)
+=
+e2Kp ˆσp(A, B).
+□
+11
+
+4.2. On the upper semi-continuity of the top Lyapunov exponent function. In this entire
+subsection we do not assume that the flow ϕt is ergodic. We define the function
+L :
+(G1, σp)
+−→
+R
+A
+�−→
+�
+M λ1(A, ω) dµ(ω).
+Notice that under the ergodic hypothesis over the flow ϕt we have L (A) = λ1(A).
+In order to
+prove that L is upper semi-continuous when G1 is endowed with the σp metric defined in §2.3,
+we begin by given a preliminary result.
+Lemma 4.4. For all t ∈ R, ω ∈ M and A, B ∈ G1, we have
+log+ ∥ΦB(t, ω)∥ ≤ log+ ∥ΦA(t, ω)∥ + ∥ΦB(t, ω) − ΦA(t, ω)∥.
+Proof. The proof follows straightforwardly from the triangular inequality
+∥ΦB(t, ω)∥ ≤ ∥ΦA(t, ω)∥ + ∥ΦB(t, ω) − ΦA(t, ω)∥.
+and the fact that log+(x + y) ≤ log+(x) + y for all x, y ≥ 0.
+□
+In the next result the reader will find similarities with the arguments in [1, Theorem 2]. Nev-
+ertheless, we present novelties namely (i) the topology is planted in the infinitesimal generator of
+the object which provide the Lyapunov exponent (ii) this point brings several continuity depen-
+dency issues to be treated using §4.1 and (iii) as we will notice the arguments entail often the
+idiosyncrasies of the flows and so several adaptations are done accordingly.
+Proposition 4.5. For all 1 ≤ p < ∞, the function L is upper semicontinuous when we endow G1
+with the σp-topology, that is, for all A ∈ G1 and ε > 0 there is δ > 0 such that σp(A, B) < δ implies
+L (B) < L (A) + ε .
+Proof. By Proposition 2.1 we have σ1(A, B) ≤ σp(A, B), for all 1 ≤ p < ∞, thus it is enough to
+consider p = 1. Let A ∈ G1 and ε > 0 be given. We assume first that for µ almost every ω ∈ M we
+have
+λ1(A, ω) ≥ 0.
+(27)
+From the subadditive ergodic theorem, we know that the limit
+λ1(A, ω) = lim
+t→+∞
+1
+t log ∥ΦA(t, ω)∥
+holds almost everywhere and also in L1. Hence, using (27), we get
+lim
+t→+∞
+1
+t
+�
+M
+log− ∥ΦA(t, ω)∥ dµ(ω) = 0,
+where f − := min{ f, 0}. Notice that
+L (A)
+=
+lim
+t→+∞
+1
+t
+�
+M
+log ∥ΦA(t, ω)∥ dµ(ω) = lim
+n→+∞
+1
+n
+�
+M
+log ∥ΦA(n, ω)∥ dµ(ω)
+=
+inf
+n∈N
+1
+n
+�
+M
+log ∥ΦA(n, ω)∥ dµ(ω)
+(28)
+so that it is possible to find T ∈ N large enough in order to have
+1
+T
+�
+M
+log− ∥ΦA(T, ω)∥ dµ(ω) > −ε
+8
+and
+1
+T
+�
+M
+log ∥ΦA(T, ω)∥ dµ(ω) < L (A) + ε
+8
+and therefore, since f = f − + f +,
+1
+T
+�
+M
+log+ ∥ΦA(T, ω)∥ dµ(ω) ≤ L (A) + ε
+4.
+(29)
+We apply Lemma 4.1 to f as in (22) and η = ε/(16(T + 2)), giving us K as in the statement. Set
+δ′ = min
+�
+η, ηTe−K(T+2))
+�
+and δ = δ′/(1 + δ′).
+Fix B ∈ G1 satisfying σ1(A, B) < δ, which implies
+12
+
+ˆσ1(A, B) < δ′ ≤ η. We use K, T, η and B to define the sets E f , Eg and G as in (23) and (25)
+respectively. We are going to bound the expression
+1
+T
+�
+M
+log+ ∥ΦB(T, ω)∥dµ(ω) = (I) + (II),
+where
+(I) = 1
+T
+�
+Gc log+ ∥ΦB(T, ω)∥dµ(ω)
+and
+(II) = 1
+T
+�
+G
+log+ ∥ΦB(T, ω)∥dµ(ω).
+For the first part (I), and by (7) we have
+1
+T
+�
+Gc log+ ∥ΦB(T, ω)∥ dµ(ω)
+≤
+1
+T
+�
+Gc
+� T
+0
+∥B(ϕs(ω))∥ ds dµ(ω) ≤ 1
+T
+�
+Gc
+T−1
+�
+i=0
+g(ϕi(ω)) dµ(ω)
+=
+1
+T
+T−1
+�
+i=0
+�
+Gc g(ϕi(ω)) dµ(ω) = 1
+T
+T−1
+�
+i=0
+�
+ϕi(Gc)
+g(ω) dµ(ω).
+For each i = 0, . . . , T − 1 we have, by (24) and relation (26),
+�
+ϕi(Gc)
+g dµ(ω) ≤
+�
+Ecg
+g dµ(ω) +
+�
+Eg∩ϕi(Gc)
+g dµ(ω) < 2η + Kµ(Eg ∩ ϕi(Gc)) ≤ ε
+8 + Kµ(Gc) ≤ ε
+2.
+Putting all together we get
+(I) = 1
+T
+�
+Gc log+ ∥ΦB(T, ω)∥dµ(ω) ≤ ε
+2.
+(30)
+Next we estimate the second part (II). Using Lemma 4.4 and (29) we have
+(II) ≤ 1
+T
+�
+G
+log+ ∥ΦA(T, ω)∥dµ(ω) + 1
+T
+�
+G
+∥ΦB(T, ω) − ΦA(T, ω)∥dµ(ω)
+≤ L (A) + ε
+4 + 1
+T
+�
+G
+∥ΦB(T, ω) − ΦA(T, ω)∥dµ(ω).
+(31)
+To estimate the integral on the right hand side, we proceed as follows. Using the cocycle properties
+and Lemma 4.2 we have for all ω ∈ G and i = 1, . . . , T − 1:
+∥ΦB(i + 1, ω) − ΦA(i + 1, ω)∥ ≤ ∥ΦB(1, ϕi(ω))ΦB(i, ω) − ΦA(1, ϕi(ω))ΦA(i, ω)∥
+≤ ∥ΦB(1, ϕi(ω))∥∥ΦB(i, ω) − ΦA(i, ω)∥+
++ ∥ΦB(1, ϕi(ω)) − ΦA(1, ϕi(ω))∥∥ΦA(i, ω)∥
+≤ eK∥ΦB(i, ω) − ΦA(i, ω)∥ + eKi∥ΦB(1, ϕi(ω)) − ΦA(1, ϕi(ω))∥.
+Integrating over G we get by Lemma 4.3 with p = 1 that
+�
+G
+∥ΦB(i + 1, ω) − ΦA(i + 1, ω)∥dµ(ω) ≤ eK
+�
+G
+∥ΦB(i, ω) − ΦA(i, ω)∥dµ(ω) + eKi+2Kδ′
+By induction, we obtain for all i = 1, . . . , T
+�
+G
+∥ΦB(i, ω) − ΦA(i, ω)∥dµ(ω) ≤ (i + 2)eK(i+2)δ′.
+In particular if we take i = T we get
+�
+G
+∥ΦB(T, ω) − ΦA(T, ω)∥dµ(ω) ≤ (T + 2)eK(T+2)δ′ ≤ ε
+4T.
+(32)
+Finally, from (31) and (32) we get
+(II) = 1
+T
+�
+G
+log+ ∥ΦA(T, ω)∥dµ(ω) ≤ L (A) + ε
+2.
+(33)
+13
+
+To complete we consider (30) and (33) to conclude that
+L (B) ≤ 1
+T
+�
+M
+log ∥ΦB(T, ω)∥dµ(ω)
+= 1
+T
+�
+Gc log+ ∥ΦB(T, ω)∥dµ(ω) + 1
+T
+�
+G
+log+ ∥ΦB(T, ω)∥dµ(ω)
+≤ ε
+2 +
+�
+L (A) + ε
+2
+�
++ = L (A) + ε.
+Let us prove now the general case. Again, let A ∈ G1 and ε > 0 be given. For α > 0 we define the
+ϕt-invariant set
+Lα = {ω ∈ M : λ1(A, ω) < −α}.
+Consider α large enough such that
+�
+Lα
+log+ ∥ΦA(1, ω)∥ dµ(ω) < ε
+8
+and
+�
+Lα
+λ1(A, ω) dµ(ω) > −ε
+4.
+(34)
+In particular we get
+�
+Lcα
+λ1(A, ω) dµ(ω) < L (A) + ε
+4.
+(35)
+Consider the (non kinetic) infinitesimal generator defined by A + αId. We claim that the weak
+solution of (3) considering the generator A + αId is given by ΦA+αId = eαtΦA(t, ω).
+Indeed, it
+suffices to prove that
+eαtΦA(t, ω) = Id +
+� t
+0
+(A + αId)(ϕs(ω))eαsΦA(s, ω)ds
+(36)
+that is
+eαtΦA(t, ω) = Id +
+� t
+0
+eαsA(ϕs(ω))ΦA(s, ω)ds + α
+� t
+0
+eαsΦA(s, ω)ds.
+Using integrating by parts u = eαs, dv = A(ϕs(ω))ΦA(s, ω) ds, du = αeαt ds and v = ΦA(s, ω) we get
+eαtΦA(t, ω) = Id + eαtΦA(t, ω) − Id −
+� t
+0
+αeαsΦA(s, ω) ds + α
+� t
+0
+eαsΦA(s, ω) ds,
+and the claim (36) proved. Now, we define its ‘maximal Lyapunov exponent’ by
+λ1(A + αId, ω) := lim
+t→∞
+1
+t log+ ∥eαtΦA(t, ω)∥.
+Notice that λ1(A + αId, ω) = λ1(A, ω) + α and if ω ∈ Lc
+α we have λ1(A + αId, ω) ≥ 0. Define now
+˜
+L :
+(G1 , σp)
+−→
+R
+A
+�−→
+�
+M λ1(A + αId, ω)dµ(ω).
+Proceeding similarly to the previous computations for L , we have that
+˜
+L is upper semicontinuous
+if we restrict A + αId to Lc
+α. For this, notice also that Lemma 4.3 also holds if we replace ΦA and
+ΦB by the corresponding eαΦA and eαΦB. Hence there is δ > 0 such that if σp((A + αId)|Lcα, (B +
+αId)|Lcα) < δ we have
+�
+Lcα
+λ1(B + αId, ω)dµ(ω) ≤
+�
+Lcα
+λ1(A + αId, ω)dµ(ω) + ε
+4,
+that is,
+�
+Lcα
+λ1(B, ω)dµ(ω) ≤
+�
+Lcα
+λ1(A, ω)dµ(ω) + ε
+4.
+(37)
+On the other hand, since Lα is invariant, we have
+�
+La
+λ1(ΦB, ω) dµ(ω) = inf
+n
+1
+n
+�
+La
+log+ ∥ΦB(n, ω)∥ dµ(ω) ≤
+�
+La
+log+ ∥ΦB(1, ω)∥ dµ(ω)
+(38)
+14
+
+Consider T = 1 and η, K and G as before, and replace η by η′ = η/4. From Lemma 4.4, (34) and
+(32) we get
+�
+Lα∩G
+log+ ∥ΦB(1, ω)∥ dµ(ω)
+≤
+�
+Lα∩G
+log+ ∥ΦA(1, ω)∥ dµ(ω) +
+�
+Lα∩G
+∥ΦB(1, ω) − ΦA(1, ω)∥ dµ(ω)
+≤
+ε
+16 + ε
+16
+=
+ε
+8.
+(39)
+Similarly as to (30), we obtain
+�
+Lα∩Gc log+ ∥ΦB(1, ω)∥ dµ(ω)
+≤ ε
+8,
+which together with (39) leads to
+�
+Lα
+log+ ∥ΦB(1, ω)∥ dµ(ω)
+≤ ε
+4,
+(40)
+Finally, from (37), (38), (35) and (40) we have
+L (B) =
+�
+Lα
+λ1(B, ω)dµ(ω) +
+�
+Lcα
+λ1(B, ω)dµ(ω)
+≤
+�
+Lα
+λ1(A, ω)dµ(ω) + ε
+4 +
+�
+Lα
+log+ ∥ΦB(T, ω)∥dµ(ω)
+≤
+�
+L (A) + ε
+2
+�
++ ε
+4 + ε
+4 = L (A) + ε.
+□
+5. P���� �� T������ �
+The strategy applied in C0 cocycles endowed with the C0 norm (or essential bounded cocycles
+endowed with the L∞ norm) developed in [10, 11, 7, 8] to diminish Lyapunov exponents cannot
+be used in our context. Indeed, Lp norms catch information on a neighborhood of a segment of
+orbit and, contrary to the C0 norm, not from the segment itself. For this reason we must follow
+a different approach.
+We recall that since we are assuming that ϕt is ergodic, the Lyapunov exponents of a given
+A ∈ K1 are constant µ almost everywhere and referred as λ1(A) ≥ λ2(A). We define the jump map
+by:
+J :
+(K1 , σp)
+−→
+R
+A
+�−→
+λ1(A)−λ2(A)
+2
+.
+Next result is a straightforward consequence of Proposition 3.3 and is crucial to obtain the
+proof of Theorem 1.
+Lemma 5.1. Consider 1 ≤ p < ∞ and let A ∈ K1 and ε, δ > 0 be given. There exists B ∈ K1 with
+σp(A, B) < ε such that
+L (B) < δ − J (A) + L (A).
+Proof. From (i) in Proposition 2.1 and Proposition 3.3 there is B ∈ K1 with σ1(A, B) ≤ σp(A, B) < ε
+such that
+L (B) < λ1(A) + λ2(A)
+2
++ δ = δ − J (A) + L (A).
+□
+15
+
+Theorem 5.2. Consider 1 ≤ p < ∞ and the complete metric space (K1, σp). If A ∈ K1 is a continuity
+point of L , then J (A) = 0.
+Proof. We take a sequence of kinetic linear differential systems An ∈ K1 converging to A ∈ K1
+in the σp-sense. Since A is a continuity point we must have lim L (An) = L (A). By Lemma 5.1,
+given εn → 0 and δ > 0, there exists Bn ∈ K1, with σp(An, Bn) < εn, such that
+L (Bn) < δ − J (An) + L (An).
+Considering limits on n we get
+lim
+n→∞L (Bn) < δ − lim
+n→∞J (An) + L (A).
+Since A is a continuity point of L we obtain that J (An) = 0 for all n sufficiently large and thus
+J (A) = 0.
+□
+We are now in codition to finish the proof of our main result.
+Proof. (of Theorem 1) By Proposition 4.5 the function L is upper semicontinuous and by Theo-
+rem 5.2 the continuity points of L have trivial spectrum (jump equal to zero). It is well-known
+that the set of points of continuity of an upper semicontinuous function is a residual subset (see
+[18]). Thus, there exists an σp-residual subset R ⊂ K1 such that if A ⊂ R, then J (A) = 0, that is
+λ1(A) = λ2(A).
+□
+From Corollary 2.3 we have that (K1, σp) is a Baire space, hence a σp-residual is σp-dense.
+Therefore, we obtain a σp-prevalence of trivial Lyapunov spectrum among kinetic systems.
+Acknowledgements: The authors were partially supported by FCT - ‘Fundação para a Ciên-
+cia e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da
+Beira Interior, project UID/00212/2020. MB also like to thank CMUP for providing the necessary
+conditions in which this work was developed.
+R���������
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+Email address: [email protected]
+Email address: [email protected]
+Email address: [email protected]
+17
+

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+Episodes Discovery Recommendation with Multi-Source
+Augmentations
+Ziwei Fan∗
+[email protected]
+University of Illinois Chicago
+USA
+Alice Wang, Zahra Nazari
+{alicew,zahran}@spotify.com
+Spotify Research
+USA
+ABSTRACT
+Recommender systems (RS) commonly retrieve potential candidate
+items for users from a massive number of items by modeling users’
+interests based on historical interactions. However, historical in-
+teraction data is highly sparse, and most items are long-tail items,
+which limits the representation learning for item discovery. This
+problem is further augmented by the discovery of novel or cold-start
+items. For example, after a user displays interest in bitcoin financial
+investment shows in the podcast space, a recommender system
+may want to suggest e.g., a newly released blockchain episode from
+a more technical show. Episode correlations help the discovery,
+especially when interaction data of episodes is limited. Accordingly,
+we build upon the classical Two-Tower model and introduce the
+novel Multi-Source Augmentations using a Contrastive Learning
+framework (MSACL) to enhance episodes embedding learning by in-
+corporating positive episodes from numerous correlated semantics.
+Extensive experiments on a real-world podcast recommendation
+dataset from a large audio streaming platform demonstrate the ef-
+fectiveness of the proposed framework for user podcast exploration
+and cold-start episode recommendation.
+ACM Reference Format:
+Ziwei Fan and Alice Wang, Zahra Nazari. 2018. Episodes Discovery Rec-
+ommendation with Multi-Source Augmentations. In Woodstock ’18: ACM
+Symposium on Neural Gaze Detection, June 03–05, 2018, Woodstock, NY. ACM,
+New York, NY, USA, 5 pages. https://doi.org/10.1145/1122445.1122456
+1
+INTRODUCTION
+Recommender Systems (RS) have been widely applied to numerous
+web [19, 20] applications to retrieve relevant information. Collabo-
+rative filtering (CF), as the most commonly used RS [14, 18, 23, 36],
+assumes that users with similar interests prefer similar items. The
+users’ interests are typically modeled and optimized by historical
+interactions [8, 17, 26, 33]. However, as CF-based RS models interest
+based on historical interactions, these methods can only capture in-
+terests observed in training data and fails to explore topics that users
+might be interested in but may never know. Therefore, a significant
+challenge for RS is to facilitate user exploration [3]. Exploration is
+∗Work is done during the internship in Spotify Research.
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from [email protected].
+Woodstock ’18, June 03–05, 2018, Woodstock, NY
+© 2018 Association for Computing Machinery.
+ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00
+https://doi.org/10.1145/1122445.1122456
+increasingly becoming a critical problem in RS, as existing RS meth-
+ods can cause echo chambers and filter bubbles as users increasingly
+engage with RS [11]. These two coined terms [1, 27] introduce a
+phenomenon where users’ interests become self-reinforced, as only
+items matching with users’ past interests are exposed to users by RS.
+This phenomenon optimizes short-term user interests and fails to
+drive long-term user engagement. Moreover, the lack of diversity of
+recommended items also potentially reduces user satisfaction. We
+build an RS for podcast recommendation that promotes exploration
+and podcast discovery here.
+In recent years, podcast listening [24] has shown a tremendous
+increase in popularity1. This growth in user interest, as well as a
+relatively low barrier to creation compared to other media such
+as music or movies, has created an explosion of podcast creation.
+Therefore, new and diverse podcast content is increasingly and
+continuously being created. We argue that the bottleneck of the
+user podcast exploration problem for RS consists of two challenges,
+including feature sparsity and interaction sparsity [5, 7]. These two
+challenges come from the data sparsity problem in RS [12, 21],
+where limited interaction data is available for representing users
+and items. The reason for feature sparsity is that many podcast
+contents are cold-start items with few user interactions. Moreover,
+in the problem of user exploration, there is a lack of user-item
+interaction training data.
+To resolve these challenges, we propose a novel framework with
+interaction-level data augmentations from multi-sources episodes
+correlation semantics [22] in contrastive learning (MSACL), build-
+ing upon the classical backbone of various recommender systems,
+Two-Tower architecture [15]. Data augmentation [30, 31, 35] en-
+riches data with different views of similar items for learning item
+embeddings, and contrastive learning [16, 28, 37] acts as a bridge
+connecting augmented items and positively interacted items. The
+proposed MSACL framework proposes the discovery item augmen-
+tation and the discovery contrastive regularization to alleviate the
+two challenges. For the discovery item augmentation, we incorpo-
+rate similar items from different semantic item relationships [6, 29]
+as positive items to enrich scarce user-item interactions. The dis-
+covery contrastive regularization further connects the user-item
+predictions with the similarity between discovery item and positive
+item. Our contributions are summarized as follows:
+• A novel data augmentation framework MSACL is proposed to
+alleviate data sparsity in user exploration and recommendation
+for episode discovery from two perspectives: (1). discovery aug-
+mentation of different semantic similarities, and (2). discovery
+contrastive regularization for alleviating user-item explorative
+interactions sparsity with augmented discovery items.
+1https://www.edisonresearch.com/the-infinite-dial-2021-2/
+arXiv:2301.01737v1  [cs.IR]  4 Jan 2023
+
+Woodstock ’18, June 03–05, 2018, Woodstock, NY
+Trovato and Tobin, et al.
+• Extensive experiments on a large, real-world podcast recommen-
+dation dataset demonstrate the effectiveness of the proposed
+framework MSACL for episode explorative recommendation, es-
+pecially for cold episodes.
+2
+MSACL
+2.1
+Problem Definition
+In RS, we denote the user set and episode set as U and I, in which
+each user and episode are denoted as 𝑢 and 𝑖. For an episode 𝑖, it
+belongs to one podcast show 𝑝 but one podcast show has multiple
+episodes {𝑖 ∈ 𝑝}, where 𝑝 ∈ P and P denotes the set of podcast
+shows. The interactions between users and episodes are represented
+as a set {𝑟𝑢𝑖 ∈ R}, where 𝑟𝑢𝑖 is binary indicating whether the user
+𝑢 has listened to the episode 𝑖 with more than 30 seconds. For
+each user and episode, we also have feature vectors 𝑓𝑢 ∈ R𝑑𝑓𝑢
+and 𝑓𝑖 ∈ R𝑑𝑓𝑖 , where 𝑑𝑓𝑢 and 𝑑𝑓𝑖 are dimension sizes of the user
+input feature vector and the episode feature vector. The episode
+exploration recommendation is predicting the preference score of
+the user 𝑢 to an episode 𝑖 that is out of user 𝑢’s historical interests,
+i.e., 𝑖 ∉ {𝑝|𝑟𝑢𝑗 = 1, 𝑗 ∈ 𝑝, for all 𝑗 ∈ I}:
+^
+𝑟𝑢𝑖 = 𝐹u(𝑓𝑢)⊤𝐹i(𝑓𝑖),
+(1)
+where 𝐹u and 𝐹i are neural networks for learning user embeddings
+and episode embeddings. We generate the episode exploration rec-
+ommendation list by ranking the scores 𝑟𝑢𝑖 on all episodes in de-
+scending order.
+2.2
+Two-Tower Model for Recommendation
+The Eq. (1) describes the standard Two-Tower architecture for rec-
+ommendation [15], which comprises of the user tower (𝐹u) and the
+episode tower (𝐹i). To be specific, the 𝐿-th layers fully connected
+neural network output embeddings of user 𝑢 and episode 𝑖 are as
+follows:
+𝐹𝐿
+u = ReLU(𝐹𝐿−1
+u
+𝑊 𝐿
+1 + 𝑏𝐿
+1 )
+𝐹𝐿
+i = ReLU(𝐹𝐿−1
+i
+𝑊 𝐿
+2 + 𝑏𝐿
+2 ),
+(2)
+where ReLU(·) refers to the ReLU activation function, the 0-th layer
+of each tower is the input feature vector of 𝑓𝑢 or 𝑓𝑖, respectively,
+𝑊 𝐿∗ ∈ R𝑑𝐿−1×𝑑𝐿 are linear transformations and 𝑏𝐿∗ ∈ R𝑑𝐿 are bias.
+The last layer’s output embeddings will be used to make predictions
+as in Eq. (1).
+2.3
+Discovery Items Augmentation
+The scarcity of user-episode interactions is a significant issue of
+recommendation for discovery and exploration. The Two-Tower
+architecture in Eq. (1) demands feature interactions to generalize to
+unseen user-episode discovery recommendations. However, feature
+interactions based on only training interaction data are limited
+due to the data scarcity of user-episode interactions. To alleviate
+this issue, we extract more positive episodes from additional item
+similarity semantic relationships, including episodes with similar
+text content and episodes with similar knowledge information. We
+assume that a user will be more likely to explore novel episodes
+with similar content or correlated knowledge to items they have
+interacted with in the past. Note that discovery items may have
+similar semantics in knowledge and contents but are not signifi-
+cant in feature interactions. Each item has a large number of fea-
+tures in practical scenarios, and semantic information is diminished.
+Comparing directly using content and knowledge embeddings as
+features, augmenting similar episodes from different semantics pro-
+vides more information as augmented episodes have other features
+and also enrich feature interactions in the Two-Tower architecture.
+Specifically, for each episode 𝑖, we have pre-trained content em-
+beddings [25] and knowledge embeddings [32] as side information.
+The content embeddings are pre-trained with the episode script
+and title text, and the knowledge embeddings are pre-trained from
+the episode knowledge graph data. We adopt the Approximate
+Nearest Neighbors lookup with Annoy to extract the top-K sim-
+ilar episodes from each semantic relationship, which we denote
+them as 𝑆content(𝑖) = {𝑗 ∈ Annoycontent(𝐾)}, and 𝑆kg(𝑖) = {𝑗 ∈
+Annoykg(𝐾)}. We extract the top-K similar episodes by ranking
+the top-K episodes with smallest L2 distances on content embed-
+dings or knowledge embeddings, respectively. We use top-10 for
+simplicity.
+2.4
+Discovery Contrastive Regularization
+With the augmented discovery episodes from multiple sources of
+semantics, we introduce the contrastive learning loss for enriching
+feature interactions and alleviating the data sparsity in both features
+and instances. We can create ‘positive’ episodes 𝑖 from similar
+semantic relationships ({𝑖+ ∈ 𝑆content(𝑖)} or {𝑖+ ∈ 𝑆kg(𝑖)}), even
+the combination of both augmentations. To be specific, given a
+minibatch of 𝑁 user-episode exploration interactions {(𝑢𝑘,𝑖𝑘)}𝑁
+𝑘=1,
+we augment one ‘positive’ episode (augmented episode) 𝑖𝑢𝑘
++ for the
+𝑘-th interaction. In total, we have 2𝑁 episodes and we reindex all
+episodes, we obtain:
+{𝑖𝑢1
+𝑎 ,𝑖𝑢1
+𝑏 ,𝑖𝑢2
+𝑎 ,𝑖𝑢2
+𝑏 , · · · ,𝑖𝑢𝑁
+𝑎 ,𝑖𝑢𝑁
+𝑎 , },
+(3)
+and the (𝑖𝑢𝑘
+𝑎 ,𝑖𝑢𝑘
+𝑏 ) is the positive pair of the user 𝑘, 𝑎 and𝑏 subscripts
+denote the positive item 𝑎 and the semantically similar item 𝑏 for
+the user 𝑘, and other 2𝑁 − 1 pairs are considered as negative pairs.
+For each episode pair (𝑖𝑢𝑘
+𝑎 ,𝑖𝑢𝑘
+𝑏 ), their features are (𝑓𝑖
+𝑢𝑘
+𝑎 , 𝑓𝑖
+𝑢𝑘
+𝑏 ), we
+obtain the learned embeddings 𝐹𝐿
+𝑖
+𝑢𝑘
+𝑎 , 𝐹𝐿
+𝑖
+𝑢𝑘
+𝑏
+, and the learned user
+embedding 𝐹𝐿𝑢𝑘 after 𝐿-layers of episode tower layer defined in
+Eq. (1). We adopt the NT-Xent loss [4, 9, 10, 13] for optimization as
+follows:
+L𝐶𝐿 = − log
+exp
+�
+𝐹𝐿
+𝑖
+𝑢𝑘
+𝑎
+⊤𝐹𝐿
+𝑖
+𝑢𝑘
+𝑏
+�
+�2𝑁 −1
+𝑚=1 exp
+�
+𝐹𝐿
+𝑖
+𝑢𝑘
+𝑎
+⊤𝐹𝐿
+𝑖𝑚
+� .
+(4)
+Note that we maximize dot product between the positive and aug-
+mented episodes (𝐹𝐿
+𝑖
+𝑢𝑘
+𝑎 , 𝐹𝐿
+𝑖
+𝑢𝑘
+𝑏
+), which matches with the user-episode
+exploration prediction defined in Eq. (1). With the combination of
+the dot product between the positive and augmented episodes in
+Eq. (5) and the dot product between the user and episode interac-
+tion prediction in Eq. (1), we can conclude that it is equivalent to
+augment user-episode interactions with more ‘positive’ (𝑢𝑘,𝑖𝑢𝑘
+𝑏 )
+interactions.
+
+Episodes Discovery Recommendation with Multi-Source Augmentations
+Woodstock ’18, June 03–05, 2018, Woodstock, NY
+With different variants of augmentations, we investigate sev-
+eral models in this framework, including the model with only fea-
+ture dropout TT-FD, the content similar augmentation MSACL
+(Content), the knowledge similar augmentation MSACL (KG), and
+also the combination of feature dropout and knowledge similar
+augmentation MSACL (KG-FD). We adopt the sampled softmax
+cross-entropy loss function with 𝑘 sampled negative items as the
+user-episode interactions optimization and also incorporate the
+contrastive loss as follows:
+L = −
+∑︁
+(𝑢,𝑖) ∈R
+[log(𝜎( ^
+𝑟𝑢𝑖)) +
+𝑘
+∑︁
+𝑗=1
+log(1 − ^
+𝑟𝑢𝑗)] + 𝜆L𝐶𝐿,
+(5)
+where 𝜆 is a hyper-parameter for controlling the contribution from
+L𝐶𝐿.
+Note that we maximize dot product between the positive and aug-
+mented episodes (𝐹𝐿
+𝑖
+𝑢𝑘
+𝑎 , 𝐹𝐿
+𝑖
+𝑢𝑘
+𝑏
+), which matches with the user-episode
+exploration prediction defined in Eq. (1). With the combination of
+the dot product between the positive and augmented episodes in
+Eq. (5) and the dot product between the user and episode interac-
+tion prediction in Eq. (1), we can conclude that it is equivalent to
+augment user-episode interactions with more ‘positive’ (𝑢𝑘,𝑖𝑢𝑘
+𝑏 )
+interactions.
+With different variants of augmentations, we investigate sev-
+eral models in this framework, including the model with only
+feature dropout (TT-FD), the content similar augmentation (TT-
+Corr(Content)), the knowledge similar augmentation (TT-Corr(KG)),
+and also the combination of feature dropout and knowledge similar
+augmentation (TT-FD-Corr(KG)).
+2.5
+Loss and Optimization
+We adopt the sampled softmax cross-entropy loss function as the
+user-episode exploration interaction optimization and also incor-
+porate the NT-Xent contrastive loss as regularization, which are
+summarized as follows:
+L = −
+∑︁
+(𝑢,𝑖) ∈R
+[log(𝜎( ^
+𝑟𝑢𝑖)) +
+𝑘
+∑︁
+𝑗=1
+log(1 − ^
+𝑟𝑢𝑗)] + 𝜆L𝐶𝐿,
+(6)
+where we sample 𝑘 negative items in interactions optimization.
+3
+EXPERIMENTS
+In this section, we present the experimental settings and results to
+demonstrate the effectiveness of MSACL. We answer the following
+Research Questions (RQs):
+• RQ1: Is MSACL effective in episode exploration recommenda-
+tion?
+• RQ2: How does each correlation relationship contribute to the
+performance?
+• RQ3: Does MSACL achieve better performance in cold items?
+3.1
+Dataset Statistics and Preprocessing
+We conduct experiments on data collected from a large audio
+streaming platform. In this dataset, users are recommended pod-
+cast episodes. Here, our work focuses on exploration. We define
+discovery as user-episode interactions from podcast shows the user
+has never interacted with. Note that each podcast show may have
+Table 1: Dataset Statistics After Preprocessing
+Data Part
+#users
+#episodes
+#interactions
+Train
+65,756
+78,103
+77,003
+Validation
+8,230
+17,381
+9,642
+Test
+8,248
+17,758
+9,551
+multiple episodes, but we impose a stringent definition of discovery;
+user interactions with novel episodes from familiar shows are not
+considered discovery. We define positive interactions to be episode
+listening of at least 30 seconds. To this end, the dataset contains
+a sample of 82,234 users on 113,242 episodes with 96,196 user ex-
+ploration interactions, and the density is 0.001%. We separate users
+into training, validation, and test sets via the split ratio of 8:1:1 as
+users and items have features as inputs. Detailed data statistics are
+listed in Table 1. The features we used for the user tower includes
+gender, age, country, podcast topics liked in previous 90 days, user
+language, pre-trained collaborative filtering embedding vector, the
+user embedding pre-trained with podcast interactions, and aver-
+aged streaming time. For episodes, we use features including topics,
+country, collaborative filtering pre-trained embeddings, and pre-
+trained semantic embeddings of the podcast. We use two different
+pre-trained semantic embeddings, which we call KG (knowledge
+graph) and Content. The KG embeddings are learned by apply-
+ing a common KG embedding method, DistMult [32], on a graph
+that contains available metadata on the podcasts such as episode,
+topic, licensor, and publisher nodes. Edges are created between
+episode-licensor, episode-topic, episode-publisher, and topic-topic.
+The Content embeddings are obtained from using pre-trained BERT
+embeddings on the podcast titles and descriptions [25]. For discrete
+features, we encode them as one-hot or multi-hot vectors. For con-
+tinuous features, such as pre-trained embedding vectors, we directly
+use them as inputs. After encoding, we concatenate all features of
+users and items, respectively.
+3.2
+Baselines
+As the proposed framework MSACL builds upon a Two-Tower
+model, we compare with two groups of baselines, standard popularity-
+based methods and two-tower based methods. We use three popularity-
+based methods. One is Pop method, which ranks items based on the
+number of positive interactions. The other two popularity-based
+methods are Pop-Country and Pop-Age-Country, which rank items’
+number of interactions conditioned on the users’ country and the
+combination of age and country, respectively. For Two-Tower-based
+methods, the first baseline is the standard Two-Tower (TT) model
+with the full feature set. The second is the Two-Tower method with
+feature dropout (TT-FD) proposed by [34]. For parameter setting,
+we search the number of layers of Two-tower models from {1, 2, 3},
+the dropout probability from {0.1, 0.3, 0.5, 0.7}.
+3.3
+Evaluation Metrics
+We evaluate the proposed MSACL on standard ranking metrics for
+top-N recommendation, including Recall@N, NDCG@N, and MRR.
+Recall@N measures the percentage of positive items being recom-
+mended in top-N ranking lists. NDCG@N considers the positions
+
+Woodstock ’18, June 03–05, 2018, Woodstock, NY
+Trovato and Tobin, et al.
+Table 2: Performance Comparison in Recall@10, NDCG@10,
+Recall@20, NDCG@20, and MRR. The best and second-best
+results are boldfaced and underlined, respectively.
+Model
+Recall@10
+NDCG@10
+Recall@20
+NDCG@20
+MRR
+Pop
+0.02349
+0.01325
+0.03618
+0.01681
+0.01220
+Pop-Country
+0.03115
+0.01839
+0.05239
+0.02430
+0.01727
+Pop-Age-Country
+0.02898
+0.01701
+0.04707
+0.02198
+0.01560
+TT
+0.06068
+0.04039
+0.08594
+0.04726
+0.03732
+TT-FD
+0.06153
+0.04131
+0.08548
+0.04816
+0.03784
+MSACL
+0.06609
+0.04336
+0.09062
+0.05018
+0.03966
+Improv vs TT
++8.92%
++7.36%
++5.45%
++6.18%
++6.28%
+Improv vs 2nd-best
++7.41%
++4.96%
++5.45%
++4.19%
++4.81%
+of correctly ranked positive items in top-N recommendation lists.
+Mean Reciprocal Rank (MRR) is similar to NDCG@N with also the
+consideration of ranking positions but in the entire ranking list. We
+report the averaged evaluation metrics over all test users.
+3.4
+Overall Comparison (RQ1)
+We report the overall results with all baselines in Table 2. We have
+the following observations:
+• The proposed method MSACL achieves the best performance
+over the second best baseline from 4.19% to 7.41% on all metrics.
+Compared with TT, the improvements are from 5.45% to 8.92%.
+These observations demonstrate the superiority of MSACL for
+episode exploration and recommendation. The reasons for the
+improvements are twofold: (1). the feature and instance level
+data augmentation alleviates the features and interactions data
+sparsity problem for episode exploration and recommendation;
+(2). the incorporation of additional positive items from multiple
+semantics of item similarities.
+• Among TT-based methods, we can see that MSACL performs the
+best. TT-FD [34] achieves the second best performance in most
+metrics. TT-FD outperforms TT model, therefore demonstrating
+the utility of contrastive learning in episodes recommendation.
+We also observe that MSACL achieves better performance than
+TT-FD. This observation supports the superiority of instance
+augmentation in contrastive learning.
+• We can also see that the TT baseline method significantly out-
+performs the popularity baselines, which demonstrates the ef-
+fectiveness of incorporating features into the users and items
+modeling.
+• Among popularity-based baselines, we can observe that the base-
+line using users’ country information achieves the best perfor-
+mance. The second-best one is using both age and country in-
+formation. The reason why using age information degrades the
+performance is that the combination of age and country will sep-
+arate users’ interactions in extremely fine-grain buckets, where
+most items have no interaction.
+Table 3: Performance Comparisons for Different Item Simi-
+larity Semantics. The best and second-best results are bold-
+faced and underlined, respectively.
+Model
+Recall@10
+NDCG@10
+Recall@20
+NDCG@20
+MRR
+TT-FD
+0.06153
+0.04131
+0.08548
+0.04816
+0.03784
+MSACL (Content)
+0.06088
+0.04045
+0.08430
+0.04721
+0.03708
+MSACL (KG)
+0.06385
+0.04272
+0.08792
+0.04952
+0.03909
+MSACL (FD-KG)
+0.06609
+0.04336
+0.09062
+0.05018
+0.03966
+≤3
+(3, 7]
+(7, 20]
+Episodes Popularity Interval
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Count
+1752
+1021
+1336
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+NDCG@20
+TT
+MSACL(KG-FD)
+TT-FD
+MSACL(Content)
+MSACL(KG)
+(a) Cold Episodes
+(7, 20]
+(20, 50]
+≥50
+Episodes Popularity Interval
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Count
+1336
+700
+355
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.10
+NDCG@20
+TT
+MSACL(KG-FD)
+TT-FD
+MSACL(Content)
+MSACL(KG)
+(b) Popular Episodes
+Figure 1: The NDCG@20 performance on items with differ-
+ent popularity.
+3.5
+Effects of Different Correlations (RQ2)
+We incorporate positive episodes from different correlation seman-
+tics in MSACL to enhance the instance augmentation. We investi-
+gate different types of correlations, including two ways of comput-
+ing semantic similarity between episodes (one is content-based sim-
+ilarity and another is knowledge-based similarity [2]). The feature
+dropout also creates a noisy version of positive episodes, providing
+an alternative view of episodes, which can also be viewed as similar
+episodes. We show the performance results of different correlations
+in Table 3. We obtain the following observations:
+• The combination of feature dropout and knowledge graph-based
+instance augmentation perform the best among all variants. We
+can conclude that both feature level and instance level augmenta-
+tions are necessary for the episodes exploration recommendation.
+• Comparing MSACL (KG) and MSACL (Content), we can observe
+that knowledge graph-based similar correlations identify more
+informative positive episodes for matching users’ interests in ex-
+ploration. This is reasonable that knowledge embeddings encode
+more heterogeneous episodes relationships while content-similar
+episodes might bring limited unexpected episodes to users that
+are also of interest.
+3.6
+Cold Start Items Performance (RQ3)
+A major performance bottleneck for the episode exploration and
+recommendation problem is due to data sparsity, which our frame-
+work MSACL can alleviate. We separate episodes into groups based
+on popularity and visualize the average performance of each group,
+as shown in Figure 1. We visualize cold items in Figure 1a and
+popular items in Figure 1b. We can first observe that cold episodes
+comprise the majority while popular episodes are in the minority.
+Next, we have additional two observations:
+
+Episodes Discovery Recommendation with Multi-Source Augmentations
+Woodstock ’18, June 03–05, 2018, Woodstock, NY
+• For cold episodes recommendation, all variants of the proposed
+framework MSACL obtain significant improvements on extremely
+cold episodes (with no more than three interactions). It demon-
+strates the superiority of the proposed model MSACL with con-
+trastive learning and hierarchical augmentations for user ex-
+ploration recommendation. The MSACL (KG-FD) achieves the
+most consistent improvements in all cold episodes group, which
+indicates the necessity of both feature level and instance level
+augmentations.
+• Regarding popular episodes, we can see that all methods, in-
+cluding the baseline method, have the same trend, which is that
+performance increases as popularity increases and that all meth-
+ods achieve similar performance on the most popular episodes.
+4
+CONCLUSION
+In this work, we propose a novel data augmentation framework to
+solve a podcast exploration recommendation problem. To alleviate
+challenges regarding feature and instance sparsity, we propose a
+contrastive learning framework using feature and instance augmen-
+tations and achieve improvements over state-of-the-art baselines
+under the Two-Tower architecture. We also demonstrate the effec-
+tiveness of the proposed framework for cold episode recommenda-
+tion in the increasingly influential domain of podcasts.
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+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf,len=477
+page_content='Episodes Discovery Recommendation with Multi-Source  Augmentations  Ziwei Fan∗  zfan20@uic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='edu  University of Illinois Chicago  USA  Alice Wang, Zahra Nazari  {alicew,zahran}@spotify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='com  Spotify Research  USA  ABSTRACT  Recommender systems (RS) commonly retrieve potential candidate  items for users from a massive number of items by modeling users’  interests based on historical interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' However, historical in-  teraction data is highly sparse, and most items are long-tail items,  which limits the representation learning for item discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' This  problem is further augmented by the discovery of novel or cold-start  items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' For example, after a user displays interest in bitcoin financial  investment shows in the podcast space, a recommender system  may want to suggest e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=', a newly released blockchain episode from  a more technical show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Episode correlations help the discovery,  especially when interaction data of episodes is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Accordingly,  we build upon the classical Two-Tower model and introduce the  novel Multi-Source Augmentations using a Contrastive Learning  framework (MSACL) to enhance episodes embedding learning by in-  corporating positive episodes from numerous correlated semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Extensive experiments on a real-world podcast recommendation  dataset from a large audio streaming platform demonstrate the ef-  fectiveness of the proposed framework for user podcast exploration  and cold-start episode recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  ACM Reference Format:  Ziwei Fan and Alice Wang, Zahra Nazari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Episodes Discovery Rec-  ommendation with Multi-Source Augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' In Woodstock ’18: ACM  Symposium on Neural Gaze Detection, June 03–05, 2018, Woodstock, NY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' ACM,  New York, NY, USA, 5 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='1145/1122445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='1122456  1  INTRODUCTION  Recommender Systems (RS) have been widely applied to numerous  web [19, 20] applications to retrieve relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Collabo-  rative filtering (CF), as the most commonly used RS [14, 18, 23, 36],  assumes that users with similar interests prefer similar items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The  users’ interests are typically modeled and optimized by historical  interactions [8, 17, 26, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' However, as CF-based RS models interest  based on historical interactions, these methods can only capture in-  terests observed in training data and fails to explore topics that users  might be interested in but may never know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Therefore, a significant  challenge for RS is to facilitate user exploration [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Exploration is  ∗Work is done during the internship in Spotify Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Permission to make digital or hard copies of all or part of this work for personal or  classroom use is granted without fee provided that copies are not made or distributed  for profit or commercial advantage and that copies bear this notice and the full citation  on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Copyrights for components of this work owned by others than ACM  must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' To copy otherwise, or republish,  to post on servers or to redistribute to lists, requires prior specific permission and/or a  fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Woodstock ’18, June 03–05, 2018, Woodstock, NY  © 2018 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  ACM ISBN 978-1-4503-XXXX-X/18/06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='00  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='1145/1122445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='1122456  increasingly becoming a critical problem in RS, as existing RS meth-  ods can cause echo chambers and filter bubbles as users increasingly  engage with RS [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' These two coined terms [1, 27] introduce a  phenomenon where users’ interests become self-reinforced, as only  items matching with users’ past interests are exposed to users by RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  This phenomenon optimizes short-term user interests and fails to  drive long-term user engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Moreover, the lack of diversity of  recommended items also potentially reduces user satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We  build an RS for podcast recommendation that promotes exploration  and podcast discovery here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  In recent years, podcast listening [24] has shown a tremendous  increase in popularity1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' This growth in user interest, as well as a  relatively low barrier to creation compared to other media such  as music or movies, has created an explosion of podcast creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Therefore, new and diverse podcast content is increasingly and  continuously being created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We argue that the bottleneck of the  user podcast exploration problem for RS consists of two challenges,  including feature sparsity and interaction sparsity [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' These two  challenges come from the data sparsity problem in RS [12, 21],  where limited interaction data is available for representing users  and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The reason for feature sparsity is that many podcast  contents are cold-start items with few user interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Moreover,  in the problem of user exploration, there is a lack of user-item  interaction training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  To resolve these challenges, we propose a novel framework with  interaction-level data augmentations from multi-sources episodes  correlation semantics [22] in contrastive learning (MSACL), build-  ing upon the classical backbone of various recommender systems,  Two-Tower architecture [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Data augmentation [30, 31, 35] en-  riches data with different views of similar items for learning item  embeddings, and contrastive learning [16, 28, 37] acts as a bridge  connecting augmented items and positively interacted items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The  proposed MSACL framework proposes the discovery item augmen-  tation and the discovery contrastive regularization to alleviate the  two challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' For the discovery item augmentation, we incorpo-  rate similar items from different semantic item relationships [6, 29]  as positive items to enrich scarce user-item interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The dis-  covery contrastive regularization further connects the user-item  predictions with the similarity between discovery item and positive  item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Our contributions are summarized as follows:  A novel data augmentation framework MSACL is proposed to  alleviate data sparsity in user exploration and recommendation  for episode discovery from two perspectives: (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' discovery aug-  mentation of different semantic similarities, and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' discovery  contrastive regularization for alleviating user-item explorative  interactions sparsity with augmented discovery items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='edisonresearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='com/the-infinite-dial-2021-2/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='01737v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='IR]  4 Jan 2023  Woodstock ’18, June 03–05, 2018, Woodstock, NY  Trovato and Tobin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Extensive experiments on a large, real-world podcast recommen-  dation dataset demonstrate the effectiveness of the proposed  framework MSACL for episode explorative recommendation, es-  pecially for cold episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  2  MSACL  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='1  Problem Definition  In RS, we denote the user set and episode set as U and I, in which  each user and episode are denoted as 𝑢 and 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' For an episode 𝑖, it  belongs to one podcast show 𝑝 but one podcast show has multiple  episodes {𝑖 ∈ 𝑝}, where 𝑝 ∈ P and P denotes the set of podcast  shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The interactions between users and episodes are represented  as a set {𝑟𝑢𝑖 ∈ R}, where 𝑟𝑢𝑖 is binary indicating whether the user  𝑢 has listened to the episode 𝑖 with more than 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' For  each user and episode, we also have feature vectors 𝑓𝑢 ∈ R𝑑𝑓𝑢  and 𝑓𝑖 ∈ R𝑑𝑓𝑖 , where 𝑑𝑓𝑢 and 𝑑𝑓𝑖 are dimension sizes of the user  input feature vector and the episode feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The episode  exploration recommendation is predicting the preference score of  the user 𝑢 to an episode 𝑖 that is out of user 𝑢’s historical interests,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=', 𝑖 ∉ {𝑝|𝑟𝑢𝑗 = 1, 𝑗 ∈ 𝑝, for all 𝑗 ∈ I}:  ^  𝑟𝑢𝑖 = 𝐹u(𝑓𝑢)⊤𝐹i(𝑓𝑖),  (1)  where 𝐹u and 𝐹i are neural networks for learning user embeddings  and episode embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We generate the episode exploration rec-  ommendation list by ranking the scores 𝑟𝑢𝑖 on all episodes in de-  scending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='2  Two-Tower Model for Recommendation  The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (1) describes the standard Two-Tower architecture for rec-  ommendation [15], which comprises of the user tower (𝐹u) and the  episode tower (𝐹i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' To be specific, the 𝐿-th layers fully connected  neural network output embeddings of user 𝑢 and episode 𝑖 are as  follows:  𝐹𝐿  u = ReLU(𝐹𝐿−1  u  𝑊 𝐿  1 + 𝑏𝐿  1 )  𝐹𝐿  i = ReLU(𝐹𝐿−1  i  𝑊 𝐿  2 + 𝑏𝐿  2 ),  (2)  where ReLU(·) refers to the ReLU activation function, the 0-th layer  of each tower is the input feature vector of 𝑓𝑢 or 𝑓𝑖, respectively,  𝑊 𝐿∗ ∈ R𝑑𝐿−1×𝑑𝐿 are linear transformations and 𝑏𝐿∗ ∈ R𝑑𝐿 are bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  The last layer’s output embeddings will be used to make predictions  as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='3  Discovery Items Augmentation  The scarcity of user-episode interactions is a significant issue of  recommendation for discovery and exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The Two-Tower  architecture in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (1) demands feature interactions to generalize to  unseen user-episode discovery recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' However, feature  interactions based on only training interaction data are limited  due to the data scarcity of user-episode interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' To alleviate  this issue, we extract more positive episodes from additional item  similarity semantic relationships, including episodes with similar  text content and episodes with similar knowledge information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We  assume that a user will be more likely to explore novel episodes  with similar content or correlated knowledge to items they have  interacted with in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Note that discovery items may have  similar semantics in knowledge and contents but are not signifi-  cant in feature interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Each item has a large number of fea-  tures in practical scenarios, and semantic information is diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Comparing directly using content and knowledge embeddings as  features, augmenting similar episodes from different semantics pro-  vides more information as augmented episodes have other features  and also enrich feature interactions in the Two-Tower architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Specifically, for each episode 𝑖, we have pre-trained content em-  beddings [25] and knowledge embeddings [32] as side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  The content embeddings are pre-trained with the episode script  and title text, and the knowledge embeddings are pre-trained from  the episode knowledge graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We adopt the Approximate  Nearest Neighbors lookup with Annoy to extract the top-K sim-  ilar episodes from each semantic relationship, which we denote  them as 𝑆content(𝑖) = {𝑗 ∈ Annoycontent(𝐾)}, and 𝑆kg(𝑖) = {𝑗 ∈  Annoykg(𝐾)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We extract the top-K similar episodes by ranking  the top-K episodes with smallest L2 distances on content embed-  dings or knowledge embeddings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We use top-10 for  simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='4  Discovery Contrastive Regularization  With the augmented discovery episodes from multiple sources of  semantics, we introduce the contrastive learning loss for enriching  feature interactions and alleviating the data sparsity in both features  and instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We can create ‘positive’ episodes 𝑖 from similar  semantic relationships ({𝑖+ ∈ 𝑆content(𝑖)} or {𝑖+ ∈ 𝑆kg(𝑖)}), even  the combination of both augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' To be specific, given a  minibatch of 𝑁 user-episode exploration interactions {(𝑢𝑘,𝑖𝑘)}𝑁  𝑘=1,  we augment one ‘positive’ episode (augmented episode) 𝑖𝑢𝑘  + for the  𝑘-th interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' In total, we have 2𝑁 episodes and we reindex all  episodes, we obtain:  {𝑖𝑢1  𝑎 ,𝑖𝑢1  𝑏 ,𝑖𝑢2  𝑎 ,𝑖𝑢2  𝑏 , · · · ,𝑖𝑢𝑁  𝑎 ,𝑖𝑢𝑁  𝑎 , },  (3)  and the (𝑖𝑢𝑘  𝑎 ,𝑖𝑢𝑘  𝑏 ) is the positive pair of the user 𝑘, 𝑎 and𝑏 subscripts  denote the positive item 𝑎 and the semantically similar item 𝑏 for  the user 𝑘, and other 2𝑁 − 1 pairs are considered as negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  For each episode pair (𝑖𝑢𝑘  𝑎 ,𝑖𝑢𝑘  𝑏 ), their features are (𝑓𝑖  𝑢𝑘  𝑎 , 𝑓𝑖  𝑢𝑘  𝑏 ), we  obtain the learned embeddings ����𝐿  𝑖  𝑢𝑘  𝑎 , 𝐹𝐿  𝑖  𝑢𝑘  𝑏  , and the learned user  embedding 𝐹𝐿𝑢𝑘 after 𝐿-layers of episode tower layer defined in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We adopt the NT-Xent loss [4, 9, 10, 13] for optimization as  follows:  L𝐶𝐿 = − log  exp  �  𝐹𝐿  𝑖  𝑢𝑘  𝑎  ⊤𝐹𝐿  𝑖  𝑢𝑘  𝑏  �  �2𝑁 −1  𝑚=1 exp  �  𝐹𝐿  𝑖  𝑢𝑘  𝑎  ⊤𝐹𝐿  𝑖𝑚  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  (4)  Note that we maximize dot product between the positive and aug-  mented episodes (𝐹𝐿  𝑖  𝑢𝑘  𝑎 , 𝐹𝐿  𝑖  𝑢𝑘  𝑏  ), which matches with the user-episode  exploration prediction defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' With the combination of  the dot product between the positive and augmented episodes in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (5) and the dot product between the user and episode interac-  tion prediction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (1), we can conclude that it is equivalent to  augment user-episode interactions with more ‘positive’ (𝑢𝑘,𝑖𝑢𝑘  𝑏 )  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Episodes Discovery Recommendation with Multi-Source Augmentations  Woodstock ’18, June 03–05, 2018, Woodstock, NY  With different variants of augmentations, we investigate sev-  eral models in this framework, including the model with only fea-  ture dropout TT-FD, the content similar augmentation MSACL  (Content), the knowledge similar augmentation MSACL (KG), and  also the combination of feature dropout and knowledge similar  augmentation MSACL (KG-FD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We adopt the sampled softmax  cross-entropy loss function with 𝑘 sampled negative items as the  user-episode interactions optimization and also incorporate the  contrastive loss as follows:  L = −  ∑︁  (𝑢,𝑖) ∈R  [log(𝜎( ^  𝑟𝑢𝑖)) +  𝑘  ∑︁  𝑗=1  log(1 − ^  𝑟𝑢𝑗)] + 𝜆L𝐶𝐿,  (5)  where 𝜆 is a hyper-parameter for controlling the contribution from  L𝐶𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Note that we maximize dot product between the positive and aug-  mented episodes (𝐹𝐿  𝑖  𝑢𝑘  𝑎 , 𝐹𝐿  𝑖  𝑢𝑘  𝑏  ), which matches with the user-episode  exploration prediction defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' With the combination of  the dot product between the positive and augmented episodes in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (5) and the dot product between the user and episode interac-  tion prediction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' (1), we can conclude that it is equivalent to  augment user-episode interactions with more ‘positive’ (𝑢𝑘,𝑖𝑢𝑘  𝑏 )  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  With different variants of augmentations, we investigate sev-  eral models in this framework, including the model with only  feature dropout (TT-FD), the content similar augmentation (TT-  Corr(Content)), the knowledge similar augmentation (TT-Corr(KG)),  and also the combination of feature dropout and knowledge similar  augmentation (TT-FD-Corr(KG)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='5  Loss and Optimization  We adopt the sampled softmax cross-entropy loss function as the  user-episode exploration interaction optimization and also incor-  porate the NT-Xent contrastive loss as regularization, which are  summarized as follows:  L = −  ∑︁  (𝑢,𝑖) ∈R  [log(𝜎( ^  𝑟𝑢𝑖)) +  𝑘  ∑︁  𝑗=1  log(1 − ^  𝑟𝑢𝑗)] + 𝜆L𝐶𝐿,  (6)  where we sample 𝑘 negative items in interactions optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  3  EXPERIMENTS  In this section, we present the experimental settings and results to  demonstrate the effectiveness of MSACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We answer the following  Research Questions (RQs):  RQ1: Is MSACL effective in episode exploration recommenda-  tion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  RQ2: How does each correlation relationship contribute to the  performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  RQ3: Does MSACL achieve better performance in cold items?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='1  Dataset Statistics and Preprocessing  We conduct experiments on data collected from a large audio  streaming platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' In this dataset, users are recommended pod-  cast episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Here, our work focuses on exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We define  discovery as user-episode interactions from podcast shows the user  has never interacted with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Note that each podcast show may have  Table 1: Dataset Statistics After Preprocessing  Data Part  #users  #episodes  #interactions  Train  65,756  78,103  77,003  Validation  8,230  17,381  9,642  Test  8,248  17,758  9,551  multiple episodes, but we impose a stringent definition of discovery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  user interactions with novel episodes from familiar shows are not  considered discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We define positive interactions to be episode  listening of at least 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' To this end, the dataset contains  a sample of 82,234 users on 113,242 episodes with 96,196 user ex-  ploration interactions, and the density is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='001%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We separate users  into training, validation, and test sets via the split ratio of 8:1:1 as  users and items have features as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Detailed data statistics are  listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The features we used for the user tower includes  gender, age, country, podcast topics liked in previous 90 days, user  language, pre-trained collaborative filtering embedding vector, the  user embedding pre-trained with podcast interactions, and aver-  aged streaming time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' For episodes, we use features including topics,  country, collaborative filtering pre-trained embeddings, and pre-  trained semantic embeddings of the podcast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We use two different  pre-trained semantic embeddings, which we call KG (knowledge  graph) and Content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The KG embeddings are learned by apply-  ing a common KG embedding method, DistMult [32], on a graph  that contains available metadata on the podcasts such as episode,  topic, licensor, and publisher nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' Edges are created between  episode-licensor, episode-topic, episode-publisher, and topic-topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  The Content embeddings are obtained from using pre-trained BERT  embeddings on the podcast titles and descriptions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' For discrete  features, we encode them as one-hot or multi-hot vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' For con-  tinuous features, such as pre-trained embedding vectors, we directly  use them as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' After encoding, we concatenate all features of  users and items, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='2  Baselines  As the proposed framework MSACL builds upon a Two-Tower  model, we compare with two groups of baselines, standard popularity-  based methods and two-tower based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We use three popularity-  based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' One is Pop method, which ranks items based on the  number of positive interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The other two popularity-based  methods are Pop-Country and Pop-Age-Country, which rank items’  number of interactions conditioned on the users’ country and the  combination of age and country, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' For Two-Tower-based  methods, the first baseline is the standard Two-Tower (TT) model  with the full feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The second is the Two-Tower method with  feature dropout (TT-FD) proposed by [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' For parameter setting,  we search the number of layers of Two-tower models from {1, 2, 3},  the dropout probability from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='3  Evaluation Metrics  We evaluate the proposed MSACL on standard ranking metrics for  top-N recommendation, including Recall@N, NDCG@N, and MRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Recall@N measures the percentage of positive items being recom-  mended in top-N ranking lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' NDCG@N considers the positions  Woodstock ’18, June 03–05, 2018, Woodstock, NY  Trovato and Tobin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Table 2: Performance Comparison in Recall@10, NDCG@10,  Recall@20, NDCG@20, and MRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The best and second-best  results are boldfaced and underlined, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Model  Recall@10  NDCG@10  Recall@20  NDCG@20  MRR  Pop  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='02349  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='01325  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03618  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='01681  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='01220  Pop-Country  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03115  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='01839  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='05239  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='02430  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='01727  Pop-Age-Country  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='02898  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='01701  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04707  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='02198  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='01560  TT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='06068  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04039  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='08594  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04726  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03732  TT-FD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='06153  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04131  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='08548  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04816  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03784  MSACL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='06609  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04336  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='09062  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='05018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03966  Improv vs TT  +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='92%  +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='36%  +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='45%  +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='18%  +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='28%  Improv vs 2nd-best  +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='41%  +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='96%  +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='45%  +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='19%  +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='81%  of correctly ranked positive items in top-N recommendation lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Mean Reciprocal Rank (MRR) is similar to NDCG@N with also the  consideration of ranking positions but in the entire ranking list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We  report the averaged evaluation metrics over all test users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='4  Overall Comparison (RQ1)  We report the overall results with all baselines in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We have  the following observations:  The proposed method MSACL achieves the best performance  over the second best baseline from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='19% to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='41% on all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Compared with TT, the improvements are from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='45% to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='92%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  These observations demonstrate the superiority of MSACL for  episode exploration and recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The reasons for the  improvements are twofold: (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' the feature and instance level  data augmentation alleviates the features and interactions data  sparsity problem for episode exploration and recommendation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' the incorporation of additional positive items from multiple  semantics of item similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Among TT-based methods, we can see that MSACL performs the  best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' TT-FD [34] achieves the second best performance in most  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' TT-FD outperforms TT model, therefore demonstrating  the utility of contrastive learning in episodes recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  We also observe that MSACL achieves better performance than  TT-FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' This observation supports the superiority of instance  augmentation in contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  We can also see that the TT baseline method significantly out-  performs the popularity baselines, which demonstrates the ef-  fectiveness of incorporating features into the users and items  modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Among popularity-based baselines, we can observe that the base-  line using users’ country information achieves the best perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The second-best one is using both age and country in-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The reason why using age information degrades the  performance is that the combination of age and country will sep-  arate users’ interactions in extremely fine-grain buckets, where  most items have no interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Table 3: Performance Comparisons for Different Item Simi-  larity Semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The best and second-best results are bold-  faced and underlined, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Model  Recall@10  NDCG@10  Recall@20  NDCG@20  MRR  TT-FD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='06153  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04131  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='08548  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04816  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03784  MSACL (Content)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='06088  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='08430  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04721  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03708  MSACL (KG)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='06385  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04272  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='08792  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04952  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03909  MSACL (FD-KG)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='06609  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04336  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='09062  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='05018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03966  ≤3  (3, 7]  (7, 20]  Episodes Popularity Interval  0  250  500  750  1000  1250  1500  1750  2000  Count  1752  1021  1336  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='040  NDCG@20  TT  MSACL(KG-FD)  TT-FD  MSACL(Content)  MSACL(KG)  (a) Cold Episodes  (7, 20]  (20, 50]  ≥50  Episodes Popularity Interval  0  250  500  750  1000  1250  1500  1750  2000  Count  1336  700  355  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='10  NDCG@20  TT  MSACL(KG-FD)  TT-FD  MSACL(Content)  MSACL(KG)  (b) Popular Episodes  Figure 1: The NDCG@20 performance on items with differ-  ent popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='5  Effects of Different Correlations (RQ2)  We incorporate positive episodes from different correlation seman-  tics in MSACL to enhance the instance augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We investi-  gate different types of correlations, including two ways of comput-  ing semantic similarity between episodes (one is content-based sim-  ilarity and another is knowledge-based similarity [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The feature  dropout also creates a noisy version of positive episodes, providing  an alternative view of episodes, which can also be viewed as similar  episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We show the performance results of different correlations  in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We obtain the following observations:  The combination of feature dropout and knowledge graph-based  instance augmentation perform the best among all variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We  can conclude that both feature level and instance level augmenta-  tions are necessary for the episodes exploration recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Comparing MSACL (KG) and MSACL (Content), we can observe  that knowledge graph-based similar correlations identify more  informative positive episodes for matching users’ interests in ex-  ploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' This is reasonable that knowledge embeddings encode  more heterogeneous episodes relationships while content-similar  episodes might bring limited unexpected episodes to users that  are also of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='6  Cold Start Items Performance (RQ3)  A major performance bottleneck for the episode exploration and  recommendation problem is due to data sparsity, which our frame-  work MSACL can alleviate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We separate episodes into groups based  on popularity and visualize the average performance of each group,  as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We visualize cold items in Figure 1a and  popular items in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We can first observe that cold episodes  comprise the majority while popular episodes are in the minority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Next, we have additional two observations:  Episodes Discovery Recommendation with Multi-Source Augmentations  Woodstock ’18, June 03–05, 2018, Woodstock, NY  For cold episodes recommendation, all variants of the proposed  framework MSACL obtain significant improvements on extremely  cold episodes (with no more than three interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' It demon-  strates the superiority of the proposed model MSACL with con-  trastive learning and hierarchical augmentations for user ex-  ploration recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' The MSACL (KG-FD) achieves the  most consistent improvements in all cold episodes group, which  indicates the necessity of both feature level and instance level  augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  Regarding popular episodes, we can see that all methods, in-  cluding the baseline method, have the same trend, which is that  performance increases as popularity increases and that all meth-  ods achieve similar performance on the most popular episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content='  4  CONCLUSION  In this work, we propose a novel data augmentation framework to  solve a podcast exploration recommendation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' To alleviate  challenges regarding feature and instance sparsity, we propose a  contrastive learning framework using feature and instance augmen-  tations and achieve improvements over state-of-the-art baselines  under the Two-Tower architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
+page_content=' We also demonstrate the effec-  tiveness of the proposed framework for cold episode recommenda-  tion in the increasingly influential domain of podcasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf'}
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I9FET4oBgHgl3EQfrCVC/content/tmp_files/2301.10089v1.pdf.txt

@@ -0,0 +1,1170 @@
+arXiv:2301.10089v1  [math.AP]  24 Jan 2023
+FLAT FLOW SOLUTION TO THE MEAN CURVATURE
+FLOW WITH VOLUME CONSTRAINT
+VESA JULIN
+Abstract. In this paper I will revisit the construction of a global weak
+solution to the volume preserving mean curvature flow via discrete min-
+imizing movement scheme by Mugnai-Seis-Spadaro [19]. This method
+is based on the gradient flow approach due to Almgren-Taylor-Wang
+[3] and Luckhaus-Str¨urzenhecker [14] and my aim is to replace the vol-
+ume penalization by implementing the volume constraint directly in the
+discrete scheme, which from practical point of view is perhaps more nat-
+ural. A technical novelty is the proof of the density estimate which is
+based on the second variation condition of the energy.
+1. Introduction
+A smooth family of set (Et)t≥0 is said to evolve according to volume
+preserving mean curvature flow if the normal velocity Vt is proportional to
+the mean curvature HEt as
+Vt = −(HEt − ¯HEt)
+on ∂Et,
+(1.1)
+where ¯HEt =
+ffl
+∂Et HEt dHn. Such a geometric equation has been proposed
+in the physical literature to model coarsening phenomena, where the system
+consisting on several subdomains evolve such that it decreases the interfacial
+area while keeping the total volume unchanged [7, 18]. From purely math-
+ematical point of view the equation (1.1) can be seen as the L2-gradient
+flow of the surface area under the volume constraint [18]. One has to be
+careful in this interpration as the Riemannian distance between two sets is
+in general degenerate [16]. In order to overcome this one may use the idea
+due to Almgren-Taylor-Wang [3] and Luckhaus-Str¨urzenhecker [14] and to
+view (1.1) as the gradient flow of the surface area with respect to a differ-
+ent, non-degerate, distance. Using the gradient flow structure, one may then
+construct a discrete-in-time approximation to the solution of (1.1) via the
+Euler implicit method, also known as the minimizing movements scheme.
+By letting the time step to zero, one then obtains a candidate for a weak
+solution of (1.1) called flat flow, as the convergence is measured in terms of
+”flat norm”. This method is implemented to the volme preserving setting
+in [19].
+In [19] the authors observe that from technical point of view it is easier
+to replace the volume constraint of the problem with a volume penaliza-
+tion, as this simplifies certain regularity issues at the level of the discrete
+approximation. My aim here is to show that one may construct the flat flow
+Date: January 25, 2023.
+1
+
+2
+JULIN
+solution to (1.1) by implementing the volume constraint in the minimizing
+movements scheme directly and thus avoid the volume penalization.
+Let me quickly recall the discrete minimizing movements scheme for (1.1).
+One defines a sequence of sets (Eh
+k )k, with fixed time step h > 0, iteratively
+such that Eh
+0 = E0, where E0 is the given initial set, and Eh
+k+1 is a minimizer
+of the functional
+P(E) + 1
+h ∫E
+¯dEh
+k dx
+under the constraint ∣E∣ = ∣Eh
+k ∣.
+Here P(E) denotes the perimeter (generalized surface area) of the set E
+and ¯dF is the signed distance function of the set F (see next section). One
+then defined an approximative flat flow solution to (1.1) (Eh
+t )t≥0 from the
+previous sequence by Eh
+t = Eh
+k for t ∈ [kh,(k + 1)h). Any cluster point of
+(Eh
+t )t≥0 is then defined as flat flow solution to (1.1). The advantage is that
+such a solution is defined for all times and for rough initial data. The main
+result in the paper is the existence of a flat flow solution.
+Theorem 1. Assume that E0 ⊂ Rn+1 is an open and bounded set with finite
+perimeter and let (Eh
+t )t≥0 be an approximative flat flow solution to (1.1)
+stating from E0 (see Definition 2.1). Then there exists a family of bounded
+sets of finite perimeter (Et)t≥0 and a subsequence hk → 0 such that
+lim
+hk→0 ∣Ehk
+t ∆Et∣ = 0
+for a.e. t ≥ 0
+and for every 0 < t < s it holds ∣Et∣ = ∣E0∣, P(Et) ≤ P(E0) and
+∣Et∆Es∣ ≤ C
+√
+s − t,
+where C depends on the dimension and on E0. Moreover, if the initial set
+E0 is C1,1-regular, then any such limit flow (Et)t≥0 agrees with the unique
+classical solution of (1.1) as long as the latter exists.
+The above theorem thus provides the existence of a flat flow solution and
+guarantees that this notion is consistence with the classical solution when the
+initial set is regular enough. The disadvantage of the flat flow is that it is not
+clear if it provides a solution to the original equation (1.1) in any weak sense.
+However, the conditional result in the spirit of Luckhaus-Str¨urzenhecker [14]
+holds also in this case.
+Theorem 2. Let (Eh
+t )t≥0 be an approximative flat flow solution to (1.1) and
+let (Ehk
+t )t≥0 be the converging subsequence in Theorem 1. Assume further
+that it holds
+lim
+hk→0P(Ehk
+t ) = P(Et)
+for a.e. t ≥ 0.
+Then for n ≤ 6 the flat flow (Et)t≥0 is a distributional solution to (1.1) (see
+Definition 4.5).
+One may also try to view the equation (1.1) as a mean curvature flow
+with forcing, where the forcing term depends on the flow itself. In this way
+one may try use different methods to construct a solution to the equation
+see e.g. [5, 6] . I also refer the recent work [13] for a weak-strong uniqueness
+result related to (1.1).
+
+FLAT FLOW
+3
+As I already mentioned, the flat flow is defined for all times and one may
+study its asymptotical behavior. Indeed, by using the metods from [10, 11]
+one may deduce the convergence of the flow in low dimensions.
+Remark 1.1. Assume that E0 ⊂ Rn+1, with n ≤ 2, is as in Theorem 1 and
+let (Et)t≥0 be a limit flat flow. When n = 1, the flow Et converges to a union
+of disjoint balls exponentially fast and when n = 2 the flow converges to a
+union of disjoint balls up to a possible translation of the components.
+The main technical challenge in proving Theorem 1 is to obtain the sharp
+density estimate for the discrete flow. This is also the main technical nov-
+elty of this paper. There are several techniques to deal with the volume
+constraint in variational problems, e.g. by using the argument from [2] (see
+also [15, Lemma 17.21]) or from [8] (see also [4, 9]). However, due to the
+presence of the dissipation term in the energy it is not obvious how to apply
+these arguments in order to obtain sharp density estimates in terms of the
+time step h. I will use an argument which is based on the second varia-
+tion condition of the energy to prove the density estimate in Proposition
+3.1. After this the proof of Theorem 1 follows exactly as in [14, 19] and the
+consistency follows almost directly using the argument in [12]. The proof
+also provides the dissipation inequality and therefore the results in [10, 11]
+hold and one obtains the result stated in Remark 1.1. Finally I would like
+to point out that this article is not self-consistence as several arguments are
+well-known, in particular, in Section 4.
+2. Preliminaries
+In this section I will briefly introduce the notation, the definition of the
+flat flow solution and recall some of its basic properties.
+Given a set E ⊂ Rn+1 the distance function dist(⋅,E) ∶ Rn+1 → [0,∞) is
+defined, as usual, as dist(x,E) ∶= infy∈E ∣x−y∣ and denote the signed distance
+function by ¯dE ∶ Rn+1 → R,
+¯dE(x) ∶=
+⎧⎪⎪⎨⎪⎪⎩
+−dist(x,∂E),
+for x ∈ E
+dist(x,∂E),
+for x ∈ Rn+1 ∖ E.
+Then clearly it holds dist(⋅,∂E) = ∣ ¯dE∣.
+I denote the ball with radius r
+centered at x by Br(x) and by Br if it is centered at the origin.
+For a measurable set E ⊂ Rn+1 the perimeter in an open set U ⊂ Rn+1 is
+defined by
+P(E,U) ∶= sup{∫E div X dx ∶ X ∈ C1
+0(U,Rn+1), ∥X∥L∞ ≤ 1}
+and write P(E) = P(E,Rn+1). If P(E) < ∞, then E is called a set of finite
+perimeter. For an introduction to the topic I refer to [15]. The reduced
+boundary of a set of finite perimeter E is denoted by ∂∗E and the generalized
+unit outer normal by νE. Note that it holds P(E,U) = Hn(∂∗E∩U) for open
+sets U. Recall also that if E is regular enough, say with Lipschitz boundary,
+then P(E) = Hn(∂E). For a given vector field X ∈ C1(Rn+1,Rn+1) and a set
+of finite perimeter E denote the tangential divergence on ∂E∗ as divτ X =
+div X − ⟨DXνE,νE⟩. The distributional mean curvature HE ∈ L1(∂∗E,R)
+
+4
+JULIN
+is defined via the divergence thereon such that for every test vector field
+X ∈ C1
+0(Rn+1,Rn+1) it holds
+∫∂∗E divτ X dHn = ∫∂∗E HE⟨X,νE⟩ dHn.
+I will consider a flat flow solution to (1.1) in the spirit of Almgren-Taylor-
+Wang [3] and Luckhaus-St¨urzenhecker [14].
+To this this aim for a fixed
+h ∈ (0,1) and a given (open) set F ⊂ Rn+1 I define the functional
+Fh(E,F) = P(E) + 1
+h ∫E
+¯dF dx.
+(2.1)
+The flat flow solution is defined analogously as in [19].
+Definition 2.1. Let E0 ⊂ Rn+1 be an open and bounded set of finite perime-
+ter and fix h ∈ (0,1).
+Define the sequence of sets (Eh
+k)∞
+k=0 iteratively as
+Eh
+0 = E0 and Eh
+k+1 is a minimizer of the problem
+min{Fh(E,Eh
+k ) ∶ ∣E∣ = ∣E0∣}.
+Moreover, define an approximative flat flow (Eh
+t )t≥0 for (1.1) starting from
+E0 as
+Eh
+t = Eh
+k
+for t ∈ [kh,(k + 1)h).
+One has to be carefull in the definition of the functional (2.1) if the set
+F is merely a set of finite perimeter as its value depends on the choice
+of the representative of F. One may overcome this by choosing a proper
+representative of the set F. In my case this is not necessary, as the regularity
+theorem below implies that one may in fact assume the sets Eh
+k to be open.
+The difference in the Definition 2.1 to the scheme in [19] is that here the
+minimizing problem is under volume constraint. On one hand this makes
+the minimization problem more natural, but on the other hand, it makes
+the quantitative density estimates more difficult to prove.
+For a given open and bounded set F ⊂ Rn+1 consider the minimization
+problem
+min{Fh(E,F) ∶ ∣E∣ = ∣F∣},
+(2.2)
+where Fh(��,F) is defined in (2.1). One may use an argument similar to [9]
+or [15, Lemma 17.21] to get rid of the volume constraint in (2.2) and deduce
+that a minimizer of (2.2) is a minimizer also for
+min{Fh(E,F) + ˜Λ∣∣E∣ − ∣F∣∣},
+(2.3)
+when ˜Λ is chosen large.
+Note that the constant ˜Λ may have unoptimal
+dependence on E and on h. However, the property (2.3) is enough deduce
+qualitative regularity properties since it implies that the minimizer inherits
+the regularity from the theory of the perimeter minimizers [15]. One may
+also write the Euler-Lagrange equation and by standard calculations (see e.g.
+[1]) we have the second variation condition. We state this in the following
+proposition.
+Proposition 2.2. Let F ⊂ Rn+1 be an open and bounded set, fix h ∈ (0,1)
+and let E be a minimizer of (2.2). Then E can be chosen to be open, which
+topological boundary is C2,α-regular up to a relatively closed singular set
+
+FLAT FLOW
+5
+which Hausdorff dimension is at most n − 7. In fact, the regular part is
+exactly the reduced boundary ∂∗E.
+The Euler-Lagrange equation
+dF
+h = −HE + λ,
+(2.4)
+where λ ∈ R is the Lagrange-multiplier, holds point wise on ∂∗E and in
+a distributional sense on ∂E. The quadratic form associated with the sec-
+ond variation of the energy is non-negative, i.e., for all ϕ ∈ H1(∂∗E) with
+∫∂∗E ϕdHn = 0 it holds
+∫∂∗E ∣∇τϕ∣2 − ∣BE∣2ϕ2 dHn + 1
+h ∫∂∗E⟨∇ ¯dF ,νE⟩ϕ2 dHn ≥ 0,
+(2.5)
+where BE(x) denotes the second fundamental form at x ∈ ∂∗E.
+Proof. Since the argument is standard, I will only give the outline. As I
+already mentioned, the minimizer E is also a minimizer of the problem (2.3)
+for some large constant ˜Λ, which depend on h and on E itself. This implies
+that the set E is a Λ-minimizer of the perimeter and thus the reduced bound-
+ary ∂∗E is relatively open, C1,α-regular hypersurface and the singular set
+∂E ∖∂∗E has dimension at most n−7 [15]. The C2,α-regularity then follows
+from the Euler-Lagrange equation and from standard Schauder-estimates
+for elliptic PDEs.
+One may obtain the second variation condition (2.5) by using the argu-
+ment from [1]. Indeed, given a function ϕ ∈ C1
+0(∂∗E) with ∫∂∗E ϕdHn = 0, we
+may construct a family of diffeomorphisms Φt such that Φ0 = id, ∣Φt(E)∣ = ∣E∣
+and ∂
+∂t∣t=0Φt(x)⋅ νE = ϕ. Then the inequality follows from the minimality of
+E as
+∂2
+∂t2 ∣t=0Fh(Φt(E),F) ≥ 0
+and following the standard calculation of the second variation (see e.g. [1]).
+Finally one obtains (2.5) for all ϕ ∈ H1(∂∗E) by approximation argument
+and by the fact that the singular set has zero capacity.
+□
+3. Density estimates
+This section is the theoretical core of the paper. The aim is to prove the
+following density estimate.
+Proposition 3.1. Let F ⊂ Rn+1 be an open and bounded set of finite perime-
+ter, fix h ∈ (0,1) and let E be a minimizer of (2.2). Then there is a constant
+c > 0, which depends on the dimension n, ∣F∣ and on P(F) such that for all
+r ≤
+√
+h and all x ∈ ∂E it holds
+min{∣E ∩ Br(x)∣,∣Br(x) ∖ E∣} ≥ crn+1
+and for all r ≤ C0
+√
+h, where C0 ≥ 1, it holds
+crn ≤ P(E,Br(x)) ≤ C1rn,
+where C1 depends also on C0. Moreover, the following holds
+∥HE∥L∞(∂∗E) ≤
+1
+c
+√
+h
+and
+∥ ¯dF ∥L∞(∂E) ≤ c−1√
+h.
+
+6
+JULIN
+It is interesting that in [19, Corollary 3.3] the authors obtain similar result
+for their scheme for a constant which is independent of P(F).
+I need several lemmas in order to prove Proposition 3.1 and therefore I
+postpone its proof to the end of the section. Before proceeding to technical
+details, I state a useful consequence of Proposition 3.1.
+Proposition 3.2. Let F,E ⊂ Rn+1 be as in Proposition 3.1. Then there are
+constants C ≥ 1,c > 0 and h0 > 0, depending on the dimension, ∣F∣ and P(F)
+such that E is (Λ,r)- minimizer of the perimeter for Λ =
+C
+√
+h, r = c
+√
+h and
+for h < h0. To be more precise, for sets G ⊂ Rn+1 with E∆G ⊂ Bc
+√
+h(x0) it
+holds
+P(E) ≤ P(G) + C
+√
+h
+∣E∆G∣.
+Proof. The argument is standard but I recall it for the reader’s convenience.
+Let me first show that there is x ∈ E and ˜c > 0 such that for ρ = ˜c
+√
+h it holds
+Bρ(x) ⊂ E. Fix ρ and apply Besicovich covering theorem to find disjoint
+balls {Bρ(xi)}N
+i=1 such that xi ∈ E and
+N
+∑
+i=1
+∣Bρ(xi)∣ = N∣B1∣ρn+1 ≥ c∣E∣.
+(3.1)
+I claim that for some i = 1,2,... ,N it holds Bρ/2(xi) ⊂ E. Indeed, if this is
+not the case then Proposition 3.1 implies
+P(E,Bρ(xi)) ≥ cρn
+for all i. Since the balls are disjoint, one has by the above and by (3.1)
+P(E) ≥
+N
+∑
+i=1
+P(E,Bρ(xi)) ≥ cN ρn ≥ c∣E∣
+ρ ≥
+c
+√
+h
+.
+This is a contradiction when h is small enough.
+Fix x0 ∈ ∂E and G as in the claim. Note that in general the set G does not
+have the same measure as E and one needs to modify it to ˜G with ∣ ˜G∣ = ∣E∣
+e.g. by using the argument from [9] as follows. Assume that ∣G∣ < ∣E∣ (the
+case ∣G∣ > ∣E∣ follows from similar argument). Since Bρ(x) ⊂ E, then by
+decreasing ρ and r if needed, it holds Bρ(x) ⊂ G. By continuity there is
+z ∈ Rn+1 such that ∣z − x0∣ ≥ 2ρ and ∣G ∪ Bρ(z)∣ = ∣E∣. Define ˜G = G ∪ Bρ(z).
+Then by the minimality of E and Proposition 3.1 it holds
+P(E) ≤ P( ˜G) + C
+√
+h
+∣ ˜G∆E∣.
+Arguing as in [9] one then deduces
+P( ˜G) − P(G) ≤ Hn(∂Bρ(z) ∖ G) − Hn(∂G ∩ Bρ(z))
+≤ C
+ρ ∣Bρ(z) ∖ G∣ ≤ C
+√
+h
+∣ ˜G∆E∣
+and the claim follows as ∣ ˜G∆E∣ ≤ 2∣G∆E∣ .
+□
+The first technical result which I need is the classical density estimate
+which can be found e.g. in [20].
+
+FLAT FLOW
+7
+Lemma 3.3. Assume E ⊂ Rn+1 is a set of finite perimeter with distribu-
+tional mean curvature HE which satisfies ∥HE∥L∞(B2R(x0)) ≤ Λ. Then for
+all x ∈ BR(x0) and r ≤ min{R,Λ−1} it holds
+P(E,Br(x)) ≥ cnrn,
+for a dimensional constant cn > 0.
+For a minimizer of (2.2) it holds the inverse of the isoperimetric inequality.
+Lemma 3.4. Let F and E be as in Proposition 3.1. Then for all x ∈ ∂E
+and r ≤ C0
+√
+h it holds
+P(E,Br(x)) ≤ C
+r min{∣E ∩ B2r(x)∣,∣B2r(x) ∖ E∣},
+for a constant which depends on the dimension and on C0 > 0. In, particular
+it holds P(E,Br(x)) ≤ Crn.
+Proof. Fix h ∈ (0,1), x and r > 0 as in the claim and without loss of generality
+assume that x = 0. One may also consider only the case ∣E ∩ B2r∣ ≤ ∣B2r ∖ E∣
+as the other case is similar. In particular, then it holds ∣E ∩ B2r∣ ≤ 1
+2∣B2r∣.
+Since
+∫
+2r
+0
+Hn(∂Bρ ∩ E) = ∣E ∩ B2r∣,
+there is ρ ∈ (r,2r) such that
+Hn(∂Bρ ∩ E) ≤ Cn
+∣E ∩ B2r∣
+r
+and
+∣Bρ∣ ≥ 2
+3∣B2r∣.
+(3.2)
+Consider first the set E1 = E ∖ ¯Bρ. In order to have a competing set with
+the volume of E, define ˜ρ < ρ to be a radius such that ∣B˜ρ∣ = ∣E ∩ Bρ∣ and
+define E2 = E1 ∪ B˜ρ. Then it holds by construction that ∣E2∣ = ∣E∣. By the
+minimality of E we have
+P(E) + 1
+h ∫E
+¯dF dx ≤ P(E2) + 1
+h ∫E2
+¯dF dx.
+Estimate the perimeter of E2 using (3.2) as
+P(E2) ≤ P(E,Rn+1 ∖ Bρ) + Hn(∂Bρ ∩ E) + Hn(∂B˜ρ)
+≤ P(E,Rn+1 ∖ Bρ) + Cn
+∣E ∩ B2r∣
+r
+.
+Use then E∆E2 ⊂ B2r, ∣E∣ = ∣E2∣ and the fact that the signed distance
+function is 1-Lipschitz to estimate
+∣∫E
+¯dF dx − ∫E2
+¯dF ∣ ≤ 4r∣E ∩ Bρ∣ ≤ 4r∣E ∩ B2r∣.
+Therefore one obtains by combining the three above inequalities and r ≤
+C0
+√
+h
+P(E,Br) ≤ P(E,Bρ) ≤ Cn
+∣E ∩ B2r∣
+r
++ 4r
+h ∣E ∩ B2r∣ ≤ C ∣E ∩ B2r∣
+r
+.
+□
+
+8
+JULIN
+By Lemma 3.3 and Lemma 3.4 it is clear that for Proposition 3.1 it is
+crucial to prove the curvature estimate ∥HE∥L∞ ≤
+C
+√
+h. The next lemma is a
+step towards this.
+Lemma 3.5. Let F,E and h be as in Proposition 3.1. Then it holds
+∥HE∥L2(∂∗E) ≤ C1
+√
+h
+,
+where the constant C1 depends on the dimension and on ∣F∣ and P(F).
+Proof. The proof relies on the second variation inequality in Proposition
+2.2. I would like to point out that in the case of the mean curvature flow,
+when there is no volume constraint, the proof is considerable easier as one
+could choose constant function in (2.5). In the volume preserving case I will
+choose a cut-off function for a test function.
+To this aim use first [11, Proposition 2.3] (see also [17, Lemma 2.1]) to
+find a point x0 ∈ Rn+1 and a radius r ∈ (c,1), where c = c(n,∣F∣,P(F)), such
+that
+∣E ∩ Br(x0)∣ = 1
+2∣Br∣.
+Note that the minimality of E yields P(E) ≤ P(F) ≤ C.
+Moreover, by
+the isoperimetric inequality it holds P(E) ≥ cn∣E∣
+n
+n+1 = cn∣F∣
+n
+n+1 ≥ c. These
+estimates are used repeatedly from now on without mentioning. Without
+loss of generality assume that x0 = 0. Choose ρ < r such that ∣Br∖Bρ∣ = 1
+4∣Br∣.
+Note that then r − ρ ≥ cn > 0 and
+3
+4∣Bρ∣ ≥ ∣E ∩ Bρ∣ ≥ 1
+4∣Bρ∣.
+(3.3)
+Define first a cut-off function ζ ∈ C1
+0(Rn+1) such that 0 ≤ ζ ≤ 1, ζ = 1 in Bρ,
+ζ = 0 outside Br and ∣∇ζ∣ ≤ Cn. Choose then ϕ = ζ − ¯ζ, where ¯ζ =
+ffl
+∂∗E ζ dHn,
+as a test function in (2.5), use ∣⟨∇ ¯dF ,νE⟩∣ ≤ 1 and ∣ϕ∣ ≤ 1, and obtain
+∫∂∗E ∣BE∣2(ζ − ¯ζ)2 dHn ≤ ∫∂∗E ∣∇τζ∣2 dHn + P(E)
+h
+.
+(3.4)
+Since HE = Trace(BE), it holds point wise on ∂∗E
+∣BE∣2 ≥ H2
+E
+n .
+(3.5)
+Recall that 0 ≤ ζ ≤ 1. Moreover, by the isoperimetric inequality and by (3.3)
+it holds P(E,Bρ) ≥ cn∣E ∩Bρ∣
+n
+n+1 ≥ c. Therefore ¯ζ ≥ c for c = c(n,∣F∣,P(F)).
+In particular, it holds ∣ζ(x)− ¯ζ∣ ≥ c for x ∈ ∂∗E ∖Br. Hence, we have by (3.4)
+∫∂∗E∖Br
+∣HE∣2 dHn ≤ C
+h P(F).
+We repeat the same argument by defining a cut-off function ζ ∈ C1(Rn+1)
+as ζ = 0 in Br, ζ = 1 outside BR and ∣∇ζ∣ ≤ Cn, where R > r is such that
+∣BR ∖ Br∣ = 1
+4∣Br∣. Using ϕ = ζ − ¯ζ in (2.5) and arguing as above yields
+∫∂∗E∩Br
+∣HE∣2 dHn ≤ C
+h P(F)
+and the claim follows.
+□
+
+FLAT FLOW
+9
+The last lemma I need is a bound on the Lagrange multiplier in the Euler-
+Lagrange equation (2.4).
+Lemma 3.6. Let F,E and h be as in Proposition 3.1. Then for the Lagrange
+multiplier in (2.4), i.e.,
+¯dF
+h = −HE + λ
+on ∂∗E
+it holds
+∣λ∣ ≤ C2
+√
+h
+,
+where the constant C2 depends on the dimension, on ∣F∣ and on P(F).
+Proof. Let Λ ≥ 0 be such that ∣λ∣ =
+Λ
+√
+h. Below all the constant depend on
+n,∣F∣ and P(F). I only treat the case when λ is positive as in the negative
+case the proof is the similar. Define the set
+Σ = {x ∈ ∂∗E ∶ ∣HE(x)∣ <
+ˆC
+√
+h
+}.
+I claim that we may choose ˆC > 2 such that it depends on n,∣F∣,P(F) and
+on C1 from Lemma 3.5 and it holds
+Hn(Σ) ≥ P(E)
+2
+.
+(3.6)
+Indeed, by the Euler-Lagrange equation (2.4) , by Lemma 3.5 and by ˆC > 2
+it holds
+ˆC2
+h Hn(∂∗E ∖ Σ) ≤ ∫∂∗E∖Σ H2
+E dHn ≤ ∫∂∗E H2
+E dHn ≤ C1
+h .
+By choosing ˆC large enough one then obtains Hn(∂∗E ∖ Σ) < P(E)/2 and
+(3.6) follows.
+By Besikovich covering theorem one finds disjoint balls of radius
+√
+h,
+denote them {B√
+h(xi)}N
+i=1, with xi ∈ Σ such that
+N
+∑
+i=1
+P(E,B√
+h(xi)) ≥ cnHn(Σ) ≥ cP(E).
+(3.7)
+By the Euler-Lagrange equation (2.4) and by the definition of the set Σ it
+holds for all x ∈ ∂∗E ∩ B√
+h(xi) with xi ∈ Σ that
+∣HE(x)∣ ≤ ∣
+¯dF(x)
+h
+λ∣ ≤ ∣ ¯dF (x) − ¯dF (xi)∣
+h
++ ∣
+¯dF (xi)
+h
+− λ∣
+≤
+1
+√
+h
++ ∣HE(xi)∣ ≤ 2 ˆC
+√
+h
+.
+(3.8)
+Therefore by Lemma 3.3 it holds
+P(E,B√
+h/2(xi)) ≥ P(E,B√
+h/2 ˆC(xi)) ≥ chn/2.
+On the other hand applying Lemma 3.4 first with r =
+√
+h/2 yields
+∣E ∪ B√
+h(xi)∣ ≥ c
+√
+hP(E,B√
+h/2(xi))
+
+10
+JULIN
+and then with r =
+√
+h yields h
+n+1
+2 ≥ cP(E,B√
+h(xi)). In conclusion it holds
+∣E ∪ B√
+h(xi)∣ ≥ c
+√
+hP(E,B√
+h(xi))
+(3.9)
+for all balls in the cover.
+The minimality of E implies
+1
+h ∫E∖F
+¯dF dx ≤ P(E) + 1
+h ∫E∆F ∣ ¯dF ∣ dx ≤ P(F).
+(3.10)
+Note that by (3.8), by λ =
+Λ
+√
+h and by the Euler-Lagrange equation (2.4) we
+have for all x ∈ B√
+h(xi) that
+¯dF (x) ≥ λh − ∣H(x)∣h ≥ (Λ − 2 ˆC)
+√
+h.
+Therefore either Λ ≤ 4 ˆC, in which case the claim follows trivially, or
+¯dF (x) ≥ Λ
+2
+√
+h.
+I assume the latter and show that even in this case Λ is bounded. Indeed, by
+the above discussion the balls B√
+h(xi) are in the exterior of F. Therefore
+we estimate by (3.7) and (3.9) that
+1
+h ∫E∖F
+¯dF dx ≥ 1
+h
+N
+∑
+i=1∫E∩B√
+h(xi)
+¯dF dx
+≥ 1
+h
+N
+∑
+i=1
+(Λ
+2
+√
+h∣E ∩ B√
+h(xi)∣)
+≥ cΛ
+N
+∑
+i=1
+P(E,B√
+h(xi))
+≥ cΛHn(Σ) ≥ cΛP(E).
+Since P(E) ≥ cn∣E∣
+n
+n+1 = cn∣F∣
+n
+n+1 , the above and (3.10) gives a bound for Λ
+and the claim follows.
+□
+Here is the proof of the density estimate.
+Proof of Proposition 3.1. By Lemma 3.3, Lemma 3.4, Lemma 3.6 and
+by the Euler-Lagrange equation (2.4) it is enough to prove
+∥ ¯dF ∥L∞(∂∗E) ≤ C
+√
+h.
+(3.11)
+Argue by contradiction and assume that there is x0 ∈ ∂∗E such that
+∣ ¯dF (x0)∣ = Λ
+√
+h
+for large Λ >> 1. Without loss of generality assume that x0 = 0 and consider
+only the case ¯dF (x0) > 0 as the other case is similar.
+By the Euler-Lagrange equation (2.4), by Lemma 3.3 and Lemma 3.6 it
+holds for r0 =
+√
+h
+2Λ that
+P(E,Br0) ≥ crn
+0 .
+(3.12)
+Define radii rk = k
+√
+h + r0 for k = 0,1,... ,n + 1. For every k = 0,1,... ,n + 1
+choose a cut-off function ζk ∈ C1
+0(Rn+1) such that 0 ≤ ζ ≤ 1, ζk = 1 in
+Brk, ζk+1 = 0 in Rn+1 ∖ Brk+1 and ∣∇ζk∣ ≤
+2
+√
+h. Choose ϕ = ζk − ¯ζk, where
+
+FLAT FLOW
+11
+¯ζk =
+ffl
+∂∗E ζk dHn, as a test function in the second variation condition (2.5),
+use ∣⟨∇ ¯dF ,νE⟩∣ ≤ 1 and (3.5), and obtain
+1
+n ∫∂∗E ∣HE∣2(ζk − ¯ζk)2 dHn ≤ 1
+h ∫∂∗E(ζk − ¯ζk)2 dHn + ∫∂∗E ∣∇τζk∣2 dHn.
+(3.13)
+Since ζk = 0 outside Brk+1, one may estimate
+∫∂∗E ∣∇τζk∣2 dHn ≤ 4
+hP(E,Brk+1).
+Moreover, since 0 ≤ ζk ≤ 1 it holds
+∫∂∗E(ζk − ¯ζk)2 dHn = ∫∂∗E ζ2
+k − ¯ζ2
+k dHn ≤ ∫∂∗E ζ2
+k dHn ≤ P(E,Brk+1).
+When Λ is large enough, then for all x ∈ ∂∗E ∩Brk+1 and all k ≤ n+1 it holds
+¯dF (x) ≥ Λ
+2
+√
+h. Then one may deduce from (3.6) that P(E,Brk+1) ≤ 1
+2P(E)
+when Λ is large enough. This yields
+0 ≤ ¯ζk ≤ 1
+2.
+Also by the Euler-Lagrange equation (2.4), by ¯dF (x) ≥ Λ
+2
+√
+h, and by Lemma
+3.6 it holds ∣HE∣ ≥
+Λ
+4
+√
+h on ∂∗E ∩ Brk, when Λ is large. Therefore it holds
+1
+n ∫∂∗E ∣HE∣2(ζk − ¯ζk)2 dHn ≥ 1
+n ∫∂∗E∩Brk
+∣HE∣2(1 − ¯ζk)2 dHn
+≥ cn
+Λ2
+h P(E,Brk).
+Combining the three above estimates with (3.13) yields
+cn
+Λ2
+h P(E,Brk) ≤ 5
+hP(E,Brk+1).
+For Λ large enough this implies
+ΛP(E,Brk) ≤ P(E,Brk+1).
+(3.14)
+Use (3.14) (n + 1)-times from k = 0 to k = n, use then (3.12) and recall
+that r0 =
+√
+h
+2Λ and obtain finally that
+P(E,Brn+1) ≥ Λn+1P(E,Br0) ≥ cΛn+1rn
+0 = cΛn+1 (
+√
+h
+2Λ )
+n
+= cΛhn/2.
+(3.15)
+But now since rn+1 = (n + 1)
+√
+h + r0 ≤ 2(n + 1)
+√
+h, one obtains from Lemma
+3.4 with r = 2(n + 1)
+√
+h that
+P(E,Brn+1) ≤ P(E,Br) ≤ Chn/2,
+which contradicts (3.15) when Λ is large.
+□
+
+12
+JULIN
+4. Existence of the flat flow
+Now that the density estimates are proven the proof of Theorem 1 follows
+from the arguments from [14, 19] without major changes. In this section
+I consider the approximative flat flow (Eh
+t )t≥0 and the associated sequence
+(Eh
+k)k≥0 as in the Definition 2.1 starting from an open and bounded set of
+finite perimeter E0. The proof for the following ”interpolation” result can
+be found in [14, Lemma 1.5].
+Lemma 4.1. Let (Eh
+t )t≥0 be an approximative flat flow starting from E0
+and fix h ∈ (0,1) and t > h. Then for all l ≤
+√
+h it holds
+∣Eh
+t ∆Eh
+t−h∣ ≤ C (l P(Eh
+t−h) + 1
+l ∫Eh
+t ∆Eh
+t−h
+∣ ¯dEh
+t−h∣dx)
+By the regularity result stated in Proposition 2.2 the Euler-Lagrange
+equation
+¯dEh
+t−h
+h
+= −HEh
+t + λt,h
+(4.1)
+holds point wise on ∂∗Eh
+t and in a distributional sense on ∂Eh
+t . Here λt,h is
+the Lagrange multiplier. Using the minimality of Eh
+t against the previous
+set Eh
+t−h one obtains the important inequality
+P(Eh
+t ) + 1
+h ∫Eh
+t ∆Eh
+t−h
+∣ ¯dEh
+t−h∣dx ≤ P(Eh
+t−h).
+(4.2)
+Using (4.2) and the argument [14, Lemma 2.1] (see also [19, Lemma 3.6])
+one obtains the following dissipation inequality.
+Lemma 4.2. Let (Eh
+t )t≥0 be an approximative flat flow starting from E0
+and fix h ∈ (0,1). Then for all T2 > T1 ≥ h it holds
+∫
+T2
+T1
+∥HEh
+t − λt,h∥2
+L2(∂∗Eh
+t ) dt ≤ C(P(ET1−h) − P(ET2)).
+Moreover, it holds
+∫
+T2
+T1
+(∥HEh
+t ∥2
+L2(∂∗Eh
+t ) + λ2
+t,h)dt ≤ C(1 + T2 − T1).
+The constant depends on the dimension, ∣E0∣ and P(E0).
+Proof. I will only sketch the proof. Let (Eh
+k) be the sequence of sets associ-
+ated with (Eh
+t )t≥0. For l ∈ Z with 2l ≤ 2Ch− 1
+2 set
+K(l) = {x ∈ Rn+1 ∶ 2lh < ∣ ¯dEh
+t−h∣ ≤ 2l+1h}.
+Here C is such that ∣ ¯dEh
+t−h∣ ≤ Ch on ∂Eh
+t . Proposition 3.1 yields that for
+every x ∈ ∂Eh
+t
+∣Eh
+t ∩ B2lh(x)∣ ≥ c(2lh)n+1
+and
+Hn(∂Eh
+t ∩ B2lh(x)) ≤ C(2lh)n.
+Therefore for all x ∈ ∂Eh
+t ∩ K(l) it holds
+∫B2lh(x)∩Eh
+t ∆Eh
+t−h
+∣ ¯dEh
+t−h∣dx ≥ c(2lh)n+2
+and
+∫B2lh(x)∩∂Eh
+t
+¯d2
+Eh
+t−h dHn ≤ C(2lh)n+2.
+
+FLAT FLOW
+13
+Combing these two yields
+∫B2lh(x)∩∂Eh
+t
+¯d2
+Eh
+t−h dHn ≤ C ∫B2lh(x)∩Eh
+t ∆Eh
+t−h
+∣ ¯dEh
+t−h∣dx.
+By applying Besicovitch covering theorem and summing over l ∈ Z (see [19,
+Lemma 3.6] for details) yields
+∫∂Eh
+t
+¯d2
+Eh
+t−h dHn ≤ ∫Eh
+t ∆Eh
+t−h
+∣ ¯dEh
+t−h∣dx
+which by combining with (4.1) and (4.2) implies
+h∫∂Eh
+t
+(HEh
+t − λt,h)2 dHn ≤ C(P(Eh
+t−h) − P(Eh
+t )).
+The first inequality then follows by iterating the above.
+By [11, Lemma 2.4] it holds
+∣λt,h∣ ≤ C(1 + ∥HEh
+t − λt,h∥L1(∂∗Eh
+t ))
+for a constant that depends on the dimension and on ∣E0∣ and P(E0). Note
+that then
+∥HEh
+t ∥2
+L2 + λ2
+t,h ≤ C(1 + ∥HEh
+t − λt,h∥2
+L2).
+Therefore by the first inequality one obtains
+∫
+T2
+T1
+(∥HEh
+t ∥2
+L2 + λ2
+t,h)dt ≤ C ∫
+T2
+T1
+(1 + ∥HEh
+t − λt,h∥2
+L2)dt ≤ C(1 + T2 − T1).
+□
+The third lemma we need is a quantitative bound on the diameter of the
+sets (Eh
+t ), which is essentially the same as [19, Lemma 3.8].
+Lemma 4.3. Let (Eh
+t )t≥0 be an approximative flat flow starting from E0
+for h ∈ (0,1). Then for all T > 0 there is RT , which depends on T, on the
+dimension and on the diameter of the initial set E0, such that Eh
+t ⊂ BRT for
+all t ≤ T.
+Proof. As in [19, Lemma 3.8] define rt for all t ≤ T as
+rt ∶= inf{r > 0 ∶ Eh
+t ⊂ Br}.
+Arguing as in [19, Lemma 3.8] one deduces that at the point y ∈ ∂Brt ∩ ∂Eh
+t
+it holds HEh
+t (y) ≥ 0 and therefore by (4.1)
+rt ≤ rt−h + h∣λt,h∣.
+Iterating this and using Lemma 4.2 yields
+RT − R0 ≤ ∫
+T
+0
+∣λt,h∣dt ≤ ∫
+T
+0 (1 + λ2
+t,h)dt ≤ C(1 + T).
+□
+Proof of Theorem 1. Let (Eh
+t )t≥0 be an approximative flat flow starting
+from E0 for h ∈ (0,1) and fix T ≥ 1. Then by (4.2) it holds P(Eh
+t ) ≤ P(E0)
+and by Lemma 4.3 it holds Eh
+t ⊂ BRT for all t ≤ T. I claim that for 0 < t < s
+with s − t ≥ h it holds
+∣Eh
+t ∆Eh
+s ∣ ≤ C
+√
+t − s.
+(4.3)
+
+14
+JULIN
+Once (4.3) is obtained, then the convergence of a subsequence Ehk
+t
+→ Et in
+measure follows as in [14, 19].
+Let j,k be such that s ∈ [jh,(j +1)h) and t ∈ [(j +k)h,(j +k +1)h). Then
+by applying Lemma 4.1 for l =
+h
+√
+s−t and by (4.2) one obtains
+∣Eh
+t ∆Eh
+s ∣ ≤
+j+k
+∑
+m=j
+∣Eh
+mh∆Eh
+(m+1)h∣
+≤ C
+j+k
+∑
+m=j
+(
+h
+√
+s − t
+P(Eh
+mh) +
+√
+s − t
+h
+∫Eh
+(m+1)h∆Eh
+mh
+∣ ¯dEh
+mh∣dx)
+≤ C
+j+k
+∑
+m=j
+h
+√
+s − t
+P(E0) + C
+√
+s − t
+j+k
+∑
+m=j
+(P(Eh
+mh) − P(Eh
+(m+1)h))
+≤ C
+kh
+√
+s − t
+P(E0) + C
+√
+s − tP(E0).
+Since kh ≤ 2(s − t), one obtains (4.3).
+The proof of the consistency principle for C1,1-regular initial sets follows
+using the arguments in [12]. The volume penalization is used only in [12,
+Lemma 3.2], but one may overcome this by using the lemma below.
+□
+Lemma 4.4. Let F ⊂ Rn+1 be an open and bounded set which satisfies inte-
+rior and exterior ball condition with radius r0 > 0 and let E be a minimizer
+of (2.2). There are ρ0 and h0 with the property that if G is a set of finite
+perimeter such that
+G∆E ⊂ Bρ(x) ∩ NC0h(∂F),
+for ρ ≤ ρ0 and h ≤ h0 where NC0h(∂F) = {x ∶ dist(x,∂F) < C0h}, then it
+holds
+P(E) ≤ P(G) + Cρn+1.
+Above the constant depends on the dimension, on r0,C0,∣F∣ and P(F).
+Proof. By approximation one may assume G to be smooth. Since F satisfies
+interior and exterior ball condition, then by [12, Lemma 3.1] it holds
+max
+x∈E∆F dist(x,∂F) ≤ Ch
+(4.4)
+when h ≤ h0. As in the proof of Proposition 3.2 the set G does not have
+the same measure as E and modifies it to ˜G with ∣ ˜G∣ = ∣E∣. Assume again
+that ∣G∣ < ∣E∣. Since F satisfies interior ball condition with radius r0, there
+is y ∈ G such that Br0/2(y) ⊂ G. By continuity there is z ∈ Rn+1 such that
+∣z−x∣ ≥ 2ρ0 and ∣G∪Br0/2(z)∣ = ∣E∣ when ρ0 is small. Define ˜G = G∪Br0/2(z).
+Then by the minimality of E and by (4.4) and by the assumption G∆E ⊂
+Bρ(x) ∩ NC0h(∂F) it holds
+P(E) ≤ P( ˜G) + C∣ ˜G∆E∣ ≤ P( ˜G) + Cρn+1.
+
+FLAT FLOW
+15
+Arguing as in [9] one then deduces
+P( ˜G) − P(G) ≤ Hn(∂Br0/2(z) ∖ G) − Hn(∂G ∩ Br0/2(z))
+≤ 2(n + 1)∣B1∣
+r0
+∣Br0/2(z) ∖ G∣
+≤ C∣ ˜G∆E∣ ≤ Cρn+1.
+and the claim follows.
+□
+The paper concludes with Theorem 2. To this aim I recall the definition
+of a distributional solution of (1.1) from [14].
+Definition 4.5. Family of sets of finite perimeter (Et)t≥0 is a distributional
+solution to (1.1) starting from E0 ⊂ Rn+1 if the following holds:
+(1) for almost every t > 0 the set Et has mean curvature HEt in a dis-
+tributional sense and for every T > 0
+∫
+T
+0
+∥HEt∥2
+L2(∂∗Et) dt < ∞.
+(2) There exists v ∶ Rn+1 × (0,∞) → R with v ∈ L2(0,T;L2(∂∗Et)) such
+that for every φ ∈ C1
+0(Rn+1 × [0,∞)) it holds
+− ∫
+T
+0
+∫∂∗Et
+vφdHn dt = ∫
+T
+0
+∫∂∗Et
+(HEt − ¯HEt)φdHn dt,
+∫
+T
+0
+∫Et
+∂tφdxdt + ∫E0
+φ(⋅,0)dx = −∫
+T
+0
+∫∂∗Et
+vφdHn dt.
+Proof of Theorem 2. The proof is exactly the same as [19, Theorem 2.3].
+Note that Proposition 3.2 implies that the sets Eh
+t are (Ch−1/2,c
+√
+h)-minimizers
+of the perimeter, i.e., for every F with F∆Eh
+t ⊂ Bc
+√
+h(x0) it holds
+P(Eh
+t ) ≤ P(F) + C
+√
+h
+∣Eh
+t ∆F∣.
+(4.5)
+□
+Acknowledgments
+The author is supported by the Academy of Finland grants 314227 and
+347550.
+References
+[1] E. Acerbi, N. Fusco, M. Morini. Minimality via second variation for a nonlocal
+isoperimetric problem. Comm. Math. Phys. 322 (2013), 515–557.
+[2] F. Almgren, Existence and regularity almost everywhere of solutions to elliptic vari-
+ational problems with constraints. Mem. Amer. Math. Soc. 4(165):viii+199p, (1976).
+[3] F. Almgren, J.E. Taylor, L. Wang, Curvature-driven flows: a variational ap-
+proach. SIAM J. Control Optim. 31(2), 387–438 (1993).
+[4] G. Antonelli, E. Pasqualetto, M. Pozzetta, Isoperimetric sets in spaces with
+lower bounds on the Ricci curvature. Nonlinear Anal. 220 (2022), Paper No. 112839,
+59 pp.
+[5] G. Bellettini, V. Caselles, A. Chambolle, M. Novaga, The volume preserving
+crystalline mean curvature flow of convex sets in RN. J. Math. Pures Appl. (9) 92
+(2009), 499–527.
+
+16
+JULIN
+[6] L. Bronsard, B. Stoth, Volume-preserving mean curvature flow as a limit of a
+nonlocal Ginzburg-Landau equation. SIAM J. Math. Anal. 28 (1997), 769–807
+[7] W. Carter, A. Roosen, J. Cahn, J. Taylor. Shape evolution by surface diffusion
+and surface attachment limited kinetics on completely faceted surfaces. Acta Metal-
+lurgica et Materialia 43 (1995), 4309–4323.
+[8] E. Gonzales, U. Massari, I. Tamanini, On the regularity of boundaries of sets
+minimizing perimeter with a volume constraint. Indiana Univ. Math. J. 32 (1983),
+25–37.
+[9] M. Gr¨uter, Boundary regularity for solutions of a partitioning problem. Arch. Ra-
+tional Mech. Anal. 97 (1987), 261–270.
+[10] V. Julin, M. Morini, M. Ponsiglione, E. Spadaro, The Asymptotics of the Area-
+Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions, Math.
+Ann., (2022), Early online. https://doi.org/10.1007/s00208-022-02497-3.
+[11] V. Julin, J. Niinikoski, Quantitative Alexandrov Theorem and asymptotic behavior
+of the volume preserving mean curvature flow. Preprint 2020.
+[12] V. Julin, J. Niinikoski, Consistency of the flat flow solution to the volume preserving
+mean curvature flow. Preprint 2022, arXiv:2206.05002.
+[13] T. Laux, Weak-strong uniqueness for volume-preserving mean curvature flow.
+Preprint arXiv:2205.13040.
+[14] S. Luckhaus, T. St¨urzenhecker, Implicit time discretization for the mean curva-
+ture flow equation. Calc. Var. Partial. Diff. Eq. 3 (1995), 253–271.
+[15] F. Maggi, Sets of finite perimeter and geometric variational problems. An introduc-
+tion to geometric measure theory. Cambridge Studies in Advanced Mathematics, 135.
+Cambridge University Press, Cambridge (2012).
+[16] P.W. Michor, D. Mumford, Riemannian geometries on spaces on planar curves.
+J. Eur. Mat. Soc. 8(1), 2006, 1–28.
+[17] M. Morini, M. Ponsiglione, E. Spadaro, Long time behaviour of discrete volume
+preserving mean curvature flows. J. Reine Angew. Math. 784 (2022), 27-51.
+[18] L. Mugnai, C. Seis, On the coarsening rates for attachment-limited kinetics. SIAM
+J. Math. Anal. 45 (2013), 324–344.
+[19] L. Mugnai, C. Seis, E. Spadaro, Global solutions to the volume-preserving mean-
+curvature flow. Calc. Var. Partial. Diff. Eq. 55 (2016), Art. 18, 23 pp.
+[20] L. Simon Introduction to Geometric Measure Theory. Tsinghua Lectures (2014).
+Vesa Julin, Department of Mathematics and Statistics, University of Jyv¨askyl¨a,
+P.O. Box 35, 40014 Jyv¨askyl¨a, Finland
+Email address: [email protected]
+

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@@ -0,0 +1,2286 @@
+Computational Performance of a LES Solver for
+Supersonic Jet Flow Applications
+Carlos Junqueira-Junior∗ and
+Jo˜ao Luiz F. Azevedo†
+Instituto de Aeron´autica e Espa¸co, 12228-904 S˜ao Jos´e dos Campos, SP, Brazil
+Sami Yamouni‡
+Instituto Tecnol´ogico de Aeron´autica, 12228-900 S˜ao Jos´e dos Campos, SP, Brazil
+William R. Wolf §
+Universidade Estadual de Campinas, 13083-970 Campinas, SP, Brazil
+Abstract
+An in-house large eddy simulation tool is developed in order to reproduce high fidelity
+results of compressible jet flows. The large eddy simulation formulation is written using
+the finite difference approach, with an explicit time integration and using a second order
+spatial discretization. The energy equation is carefully discretized in order to model the
+energy equation of the filtered Navier-Stokes formulation. Such numerical studies are very
+expensive and demand high performance computing. Message passage interface protocols
+are implemented into the code in order to perform parallel computations. The present work
+addresses the computational performance of the solver running on up to 400 processors in
+parallel. Different mesh configurations, whose size varies from approximately 5.9 million
+points to approximately 1.0 billion points, are evaluate in the current paper. Speedup and
+efficiency curves are evaluated in order to assess the strong scalability of the solver.
+I.
+Introduction
+Solid structure of different parts of launch vehicles and experimental apparatus on board can be da-
+maged during the take off and also during the transonic flight of such vehicles due to vibrational acoustic
+stress resulted from pressure fluctuations. Such fluctuations are originated from the complex interaction
+between the high-temperature/high-velocity exhaustion gases from the rocket engines. The acoustic design
+constraints of launch vehicles have encouraged the studies of aeroacoustic fields around compressible jet flows
+for aerospace applications. Instituto de Aeronautica e Espa¸co (IAE) has been using large eddy simulations
+(LES)1,2 coupled with the Ffowcs Williams and Hawkings approach3 in order to study the aeroacoustic of
+supersonic jet flow configurations. The LES studies are very expensive in the computational context and
+strongly demand parallel computing. The present work addresses the computational performance of the
+solver using up to 400 processors in parallel. The speedup and computational efficiency of the solver are
+measured using different mesh and partition configurations and different number of computational cores.
+∗Postdoctoral
+Research
+Fellow,
+Aerodynamics
+Division,
+Departamento
+de
+Ciˆencia
+e
+Tecnologia
+Aeroespacial,
+DCTA/IAE/ALA; E-mail: [email protected].
+†Senior Research Engineer, Aerodynamics Division, Departamento de Ciˆencia e Tecnologia Aeroespacial, DCTA/IAE/ALA;
+E-mail: [email protected]. AIAA Fellow.
+‡Postdoctoral Reasearch Fellow, Graduate Program on Computer Sciences and Electrical Engineering, Departamento de
+Ciˆencia e Tecnologia Aeroespacial, DCTA/ITA; E-mail: [email protected].
+§Assistant Professor, Faculty of Mechanical Engineering; E-mail: [email protected]. AIAA Member
+1 of 25
+arXiv:2301.00817v1  [physics.flu-dyn]  2 Jan 2023
+
+JAZzY1 is the LES solver which is used in the present work.
+It is an in-house computational tool
+developed regarding the study of unsteady turbulent supersonic jet flow configurations. The formulation is
+written using the finite difference approach. Inviscid numerical fluxes are calculated using a second order
+accurate centered scheme with the explicit addition of artificial dissipation. A five steps second order accurate
+Runge-Kutta is the chosen time marching method. A formulation based on the System I set of equations4
+is used here in order to model the filtered terms of the energy equation. Numerical simulation of perfectly
+expanded jets are performed and compared with numerical5 and experimental6 data.
+The code is written using the FORTRAN 90 standards. It uses the HDF5 7,8 and the CGNS 9–11 libraries
+for the I/O operations. The mesh is partitioned into the axial and azimuthal directions. Two layer ghost
+points are created at the surroundings of the local domain in order to carry neighbor partitions data. The
+data exchange between partitions is performed through message passing interface (MPI) protocols.12
+Simulations of a perfectly expanded jet are performed using different mesh partitioning and different
+number of processors in order to evaluate the computational performance of the code. In the present nine
+meshes are are studied running on up to 400 processors in parallel. The size of the mesh starts with 5.8
+million points and scales to 1.0 billion points. The speedup and computational efficiency curves are presented
+and compared in order to study the strong scalability of the code.
+II.
+Large Eddy Simulation Filtering
+The large eddy simulation is based on the principle of scale separation, which is addressed as a filtering
+procedure in a mathematical formalism. A modified version of the the System I filtering approach4 is used
+in present work which is given by
+∂ρ
+∂t +
+∂
+∂xj
+(ρ �uj) = 0 ,
+∂
+∂t (ρ �ui) +
+∂
+∂xj
+(ρ �ui �uj) + ∂p
+∂xi
+− ∂τij
+∂xj
++ 1
+3
+∂
+∂xj
+(δijσii) = 0 ,
+∂e
+∂t +
+∂
+∂xj
+[(e + p) �uj] −
+∂
+∂xj
+(τij �ui) + 1
+3
+∂
+∂xj
+[(δijσii) �ui] + ∂qj
+∂xj
+= 0 ,
+(1)
+in which t and xi are independent variables representing time and spatial coordinates of a Cartesian coor-
+dinate system x, respectively. The components of the velocity vector u are written as ui, and i = 1, 2, 3.
+Density, pressure and total energy per mass unit are denoted by ρ, p and e, respectively. The (·) and (˜·)
+operators are used in order to represent filtered and Favre averaged properties, respectively. The System I
+formulation neglects the double correlation term and the total energy per mass unit is written as
+e =
+p
+γ − 1 + 1
+2ρ�ui�ui .
+(2)
+The heat flux, qj, is given by
+qj = (κ + κsgs) ∂ �T
+∂xj
+.
+(3)
+where T is the static temperature and κ is the thermal conductivity, which can by expressed by
+κ = µCp
+Pr ,
+(4)
+The thermal conductivity is a function of the specific heat at constant pressure, Cp, of the Prandtl number,
+Pr, which is equal to 0.72 for air, and of the dynamic viscosity, µ. The SGS thermal conductivity, κsgs, is
+written as
+κsgs = µsgsCp
+Prsgs
+,
+(5)
+where Prsgs is the SGS Prandtl number, which is equal to 0.9 for static SGS models and µsgs is the eddy
+viscosity which is calculated by the SGS closure. The dynamic viscosity, µ, can be calculated using the
+Sutherland law,
+µ
+�
+�T
+�
+= µ∞
+� �T
+�T∞
+� 3
+2 �T0 + S1
+�T + S1
+with S1 = 110.4K .
+(6)
+2 of 25
+
+Density, static pressure and static temperature are correlated by the equation of state given by
+p = ρR �T ,
+(7)
+where R is the gas constant, written as
+R = Cp − Cv ,
+(8)
+and Cv is the specific heat at constant volume. The shear-stress tensor, τij, is written according to the
+Stokes hypothesis and includes the eddy viscosity, µsgs,
+τij = 2 (µ + µsgs)
+�
+˜Sij − 1
+3δij ˜Skk
+�
+(9)
+in which ˜Sij, components of rate-of-strain tensor, are given by
+˜Sij = 1
+2
+� ∂˜ui
+∂xj
++ ∂˜uj
+∂xi
+�
+.
+(10)
+The SGS stress tensor components are written using the eddy viscosity,13
+σij = −2µsgs
+�
+˜Sij − 1
+3
+˜Skk
+�
++ 1
+3δijσkk .
+(11)
+The eddy viscosity, µsgs, and the components of the isotropic part of the SGS stress tensor, σkk, are modeled
+by the SGS closure.
+III.
+Subgrid Scale Modeling
+The theoretical formulation of subgrid scales closures included in the present work is discussed in the
+present section. The closure models presented here are founded on the homogeneous turbulence theory, which
+is usually developed in the spectral space as an attempt to quantify the interaction between the different
+scales of turbulence.
+III.A.
+Smagorinky Model
+The Smagorinsky model14 is one of the simplest algebric models for the deviatory part of the SGS tensor
+used in large-eddy simulations. The isotropic part of the SGS tensor is neglected for Smagorinsky model in
+the current work. This SGS closure is a classical model based the large scales properties and is written as
+µsgs = (ρCs∆)2 |�S| ,
+(12)
+where
+| ˜S| =
+�
+2 ˜Sij ˜Sij
+� 1
+2 ,
+(13)
+∆ is the filter size and Cs is the Smagorinsky constant. Several attempts can be found in the literature
+regarding the evaluation of the Smagorinsky constant. The value of this constant is adjusted to improve
+the results of different flow configurations. In pratical terms, the Smagorinsky subgrid model has a flow
+dependency of the constant which takes value ranging from 0.1 to 0.2 depending on the flow. The suggestion
+of Lilly,15 Cs = 0.148, is used in the current work.
+This model is generally over-dissipative in regions of large mean strain. This is particularly true in the
+transitional region between laminar and turbulent flows. Moreover, the limiting behavior near the wall is
+not correct, and the model predictions correlate poorly with the exact subgrid scale tensor.16 However, it is
+a very simple model and, with the use of damping function and good calibration, can be successfully applied
+on large-eddy simulations.
+3 of 25
+
+III.B.
+Vreman Model
+Vreman17 proposed a turbulence model that can correctly predict inhomogeneous turbulent flows.
+For
+such flows, the eddy viscosity should become small in laminar and transitional regions. This requirement
+is unfortunately not satisfied by existing simple eddy-viscosity closures such as the classic Smagorinsky
+model.14,18,19 The Vreman SGS model is very simple and is given by
+µsgs = ρ c
+�
+Bβ
+αijαij
+,
+(14)
+with
+αij = ∂˜uj
+∂xi
+,
+(15)
+βij = ∆2
+mαmiαmj
+(16)
+and
+Bβ = β11β22 − β2
+12 + β11β33 − β2
+13 + β22β33 − β2
+23 .
+(17)
+The constant c is related to the Smagorinsky constant, Cs, and it is given by
+c = 2.5 C2
+s ,
+(18)
+and ∆m is the filter width in each direction. In the present work, the isotropic part of the SGS tensor is
+neglected for the Vreman model. The α symbol represents the matrix of first order derivatives of the filtered
+components of velocity, ˜ui. The SGS eddy-viscosity is defined as zero when αijαij equals zero. Vreman17
+affirms that the tensor β is proportional to the gradient model20,21 in its general anisotropic form.22
+The Vreman model can be classified as very simple model because it is expressed in first-order derivatives
+and it dos not involves explicit filtering, averaging, clipping procedures and is rotationally invariant for
+isotropic filter widths. The model is originally created for incompressible flows and it has presented good
+results for two incompressible flows configurations: the transitional and turbulent mixing layer at high
+Reynolds number and the turbulent channel flow.22 In both cases, the Vreman model is found to be more
+accurate than the classical Smagorinsky model and as good as the dynamic Smagorinsky model.
+III.C.
+Dynamic Smagorinsky Model
+Germano et al.23 developed a dynamic SGS model in order to overcome the issues of the classical Smagorinsky
+closure. The model uses the strain rate fields at two different scales and thus extracts spectral information
+in the large-scale field to extrapolate the small stresses.24
+The coefficients of the model are computed
+instantaneously in the dynamic model. They are function of the positioning in space and time rather than
+being specified a priori. Moin et al.24 extended the work of Germano for compressible flows. The dynamic
+Smagorinsky model for compressible flow configurations is detailed in the present section.
+The Dynamic model introduces the test filter, �
+(·), which has a larger filter width, �∆, than the one of the
+resolved grid filter, (·). The use of test filters generates a second field with larger scales than the resolved
+field. The Yoshizawa model25 is used for the isotropic portion of the SGS tensor and it is written as
+σll = 2CIρ∆2| ˜S|2 ,
+(19)
+where CI is defined by
+CI =
+�
+�
+ρ˜ul˜ul −
+�
+�
+ρ˜ul �
+ρ˜ul/�ρ
+��
+�
+2�∆2�ρ|�S|2 − 2∆2 �
+ρ|S|2
+� .
+(20)
+A volume averaging, here indicated by ⟨ ⟩, is suggest by Moin et al24 and by Garnier et al in order to avoid
+numerical issues. The eddy viscosity, µsgs, is calculated using the same approach used by static Smagorinsky
+model,
+µsgs = (ρCds∆)2 | ˜S| ,
+(21)
+4 of 25
+
+where
+| ˜S| =
+�
+2 ˜Sij ˜Sij
+� 1
+2 ,
+(22)
+and Cds is the dynamic constant of the model, which is given by
+Cds =
+��
+�
+ρ˜ui˜uj −
+�
+�
+ρ˜ui�
+ρ˜uj/�ρ
+��
+˜Sij − 1
+3 ˜Smm (Tll − �σll)
+�
+�
+2∆2
+�
+�
+ρ| ˜S| ˜Sij ˜Sij − 1
+3
+�
+ρ| ˜S| ˜Smm
+�� ˜Sll
+�
+− 2�∆2
+�
+�ρ|�˜S|�˜Sij ˜Sij − 1
+3�ρ|�˜S|�˜Smm ˜Sll
+�� .
+(23)
+The SGS Prandtl number is computed using the dynamic constant, Cds, and written as
+Prsgs = Cds
+�
+∆2
+�
+ρ| ˜S| ∂
+∼
+T
+∂xj
+��
+∂
+∼
+T
+∂xj − �∆2�ρ|�˜S| ∂
+∼
+T
+∂xj
+∂
+∼
+T
+∂xj
+�
+��
+�
+ρ˜uj
+∼
+T −
+�
+�
+ρ ˜uj
+�
+ρ
+∼
+T
+�
+/�ρ
+�
+∂
+∼
+T
+∂xj
+�
+.
+(24)
+IV.
+Transformation of Coordinates
+The formulation is written in the a general curvilinear coordinate system in order to facilitate the im-
+plementation and add more generality for the CFD tool. Hence, the filtered Navier-Stokes equations can be
+written in strong conservation form for a 3-D general curvilinear coordinate system as
+∂ ˆQ
+∂t + ∂
+∂ξ
+�
+ˆ
+Ee − ˆ
+Ev
+�
++ ∂
+∂η
+�
+ˆ
+Fe − ˆ
+Fv
+�
++ ∂
+∂ζ
+�
+ˆ
+Ge − ˆ
+Gv
+�
+= 0 .
+(25)
+In the present work, the chosen general coordinate transformation is given by
+ξ
+=
+ξ (x, y, z, t) ,
+η
+=
+η (x, y, z, t) ,
+(26)
+ζ
+=
+ζ (x, y, z, t) .
+In the jet flow configuration, ξ is the axial jet flow direction, η is the radial direction and ζ is the azimuthal
+direction. The vector of conserved properties is written as
+ˆQ = J−1 [ρ
+ρ˜u
+ρ˜v
+ρ ˜w
+e]T
+,
+(27)
+where the Jacobian of the transformation, J, is given by
+J = (xξyηzζ + xηyζzξ + xζyξzη − xξyζzη − xηyξzζ − xζyηzξ)−1 ,
+(28)
+and
+xξ = ∂x
+∂ξ ,
+xη = ∂x
+∂η ,
+xζ = ∂x
+∂ζ ,
+yξ = ∂y
+∂ξ ,
+yη = ∂y
+∂η ,
+yζ = ∂y
+∂ζ ,
+(29)
+zξ = ∂z
+∂ξ ,
+zη = ∂z
+∂η ,
+zζ = ∂z
+∂ζ .
+The inviscid flux vectors, ˆEe, ˆFe and ˆGe, are given by
+ˆEe = J−1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ρU
+ρ˜uU + pξx
+ρ˜vU + pξy
+ρ ˜wU + pξz
+(e + p) U − pξt
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+ˆFe = J−1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ρV
+ρ˜uV + pηx
+ρ˜vV + pηy
+ρ ˜wV + pηz
+(e + p) V − pηt
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+ˆGe = J−1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ρW
+ρ˜uW + pζx
+ρ˜vW + pζy
+ρ ˜wW + pζz
+(e + p) W − pζt
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(30)
+5 of 25
+
+The contravariant velocity components, U, V and W, are calculated as
+U = ξxu + ξyv + ξzw ,
+V = ηxu + ηyv + ηzw ,
+(31)
+W = ζxu + ζyv + ζzw .
+The metric terms are given by
+ξx = J (yηzζ − yζzη) ,
+ξy = J (zηxζ − zζxη) ,
+ξz = J (xηyζ − xζyη) ,
+ηx = J (yηzξ − yξzη) ,
+ηy = J (zηxξ − zξxη) ,
+ηz = J (xηyξ − xξyη) ,
+(32)
+ζx = J (yξzη − yηzξ) ,
+ζy = J (zξxη − zηxξ) ,
+ζz = J (xξyη − xηyξ) .
+(33)
+The viscous flux vectors, ˆEv, ˆFv and ˆGv, are written as
+ˆEv = J−1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+ξxτxx + ξyτxy + ξzτxz
+ξxτxy + ξyτyy + ξzτyz
+ξxτxz + ξyτyz + ξzτzz
+ξxβx + ξyβy + ξzβz
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(34)
+ˆFv = J−1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+ηxτxx + ηyτxy + ηzτxz
+ηxτxy + ηyτyy + ηzτyz
+ηxτxz + ηyτyz + ηzτzz
+ηxβx + ηyβy + ηzβz
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(35)
+ˆGv = J−1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+ζxτxx + ζyτxy + ζzτxz
+ζxτxy + ζyτyy + ζzτyz
+ζxτxz + ζyτyz + ζzτzz
+ζxβx + ζyβy + ζzβz
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(36)
+where βx, βy and βz are defined as
+βx = τxx˜u + τxy˜v + τxz ˜w − qx ,
+βy = τxy˜u + τyy˜v + τyz ˜w − qy ,
+(37)
+βz = τxz˜u + τyz˜v + τzz ˜w − qz.
+V.
+Dimensionless Formulation
+A convenient nondimensionalization is necessary in to order to achieve a consistent implementation of
+the governing equations of motion.
+Dimensionless formulation yields to a more general numerical tool.
+There is no need to change the formulation for each configuration intended to be simulated. Moreover,
+dimensionless formulation scales all the necessary properties to the same order of magnitude which is a
+computational advantage.26 Dimensionless variables are presented in the present section in order perform
+the nondimensionalization of Eq. (25)
+The dimensionless time, t, is written as function of the speed of sound of the jet at the inlet, aj, and of
+a reference lenght, l,
+t = taj
+l .
+(38)
+The dimensionless velocity components are obtained using the speed of sound of the jet at the inlet,
+u = u
+aj
+.
+(39)
+6 of 25
+
+Dimensionless pressure and energy are calculated using density and speed of the sound of the jet at the inlet
+as
+p =
+p
+ρja2
+j
+,
+(40)
+E =
+E
+ρja2
+j
+.
+(41)
+Dimensionless density, ρ, temperature, T and viscosity, µ, are calculated using freestream properties
+ρ = ρ
+ρj
+.
+(42)
+One can use the dimensionless properties described above in order to write the dimensionless form of the
+RANS equations as
+∂Q
+∂t + ∂Ee
+∂ξ + ∂Fe
+∂η + ∂Ge
+∂ζ
+= Mj
+Re
+�∂Ev
+∂ξ + ∂Fv
+∂η + ∂Gv
+∂ζ
+�
+,
+(43)
+where the underlined terms are calculated using dimensionless properties. The Mach number of the jet, Mj,
+and the Reynolds number are based on the mean inlet velocity of the jet, Uj, diamenter of the inlet, D, and
+freestream properties such as speed of sound, a∞, density, ρ∞ and viscosity, µ∞,
+Mj = Uj
+a∞
+and
+Re = ρjUjD
+µj
+.
+(44)
+VI.
+Numerical Formulation
+The governing equations previously described are discretized in a structured finite difference context for
+general curvilinear coordinate system.26 The numerical flux is calculated through a central difference scheme
+with the explicit addition of the anisotropic scalar artificial dissipation of Turkel and Vatsa.27 The time
+integration is performed by an explicit, 2nd-order, 5-stage Runge-Kutta scheme.28,29 Conserved properties
+and artificial dissipation terms are properly treated near boundaries in order to assure the physical correctness
+of the numerical formulation.
+VI.A.
+Spatial Discretization
+For the sake of simplicity the formulation discussed in the present section is no longer written using bars.
+However, the reader should notice that the equations are dimensionless and filtered. The Navier-Stokes
+equations, presented in Eq. (43), are discretized in space in a finite difference fashion and, then, rewritten as
+�∂Q
+∂t
+�
+i,j,k
+= −RHSi,j,k ,
+(45)
+where RHS is the right hand side of the equation and it is written as function of the numerical flux vectors
+at the interfaces between grid points,
+RHSi,j,k
+=
+1
+∆ξ
+�
+Ee(i+ 1
+2 ,j,k) − Ee(i− 1
+2 ,j,k) − Ev(i+ 1
+2 ,j,k) + Ev(i− 1
+2 ,j,k)
+�
+1
+∆η
+�
+Fe(i,j+ 1
+2 ,k) − Fe(i,j− 1
+2 ,k) − Fv(i,j+ 1
+2 ,k) + Fv(i,j− 1
+2 ,k)
+�
+(46)
+1
+∆ζ
+�
+Ge(i,j,k+ 1
+2 ) − Ge(i,j,k− 1
+2 ) − Gv(i,j,k+ 1
+2 ) + Gv(i,j,k− 1
+2 )
+�
+.
+For the general curvilinear coordinate case ∆ξ = ∆η = ∆ζ = 1. The anisotropic scalar artificial dissipation
+method of Turkel and Vatsa27 is implemented through the modification of the inviscid flux vectors, Ee, Fe
+and Ge. The numerical scheme is nonlinear and allows the selection between artificial dissipation terms of
+second and fourth differences, which is very important for capturing discontinuities in the flow. The numerical
+fluxes are calculated at interfaces in order to reduce the size of the calculation cell and, therefore, facilitate
+the implementation of second derivatives since the the concept of numerical fluxes vectors is used for flux
+7 of 25
+
+differencing. Only internal interfaces receive the corresponding artificial dissipation terms, and differences
+of the viscous flux vectors use two neighboring points of the interface.
+The inviscid flux vectors, with the addition of the artificial dissipation contribution, can be written as
+Ee(i± 1
+2 ,j,k) = 1
+2
+�
+Ee(i,j,k) + Ee(i±1,j,k)
+�
+− J−1d(i± 1
+2 ,j,k) ,
+Fe(i,j± 1
+2 ,k) = 1
+2
+�
+Fe(i,j,k) + Fe(i,j±1,k)
+�
+− J−1d(i,j± 1
+2 ,k) ,
+(47)
+Ge(i,j,k± 1
+2 ) = 1
+2
+�
+Ge(i,j,k) + Ge(i,j,k±1)
+�
+− J−1d(i,j,k± 1
+2 ) ,
+in which the d(i±1,j,k),d(i,j±1,k) and d(i,j,k±1) terms are the Turkel and Vatsa27 artificial dissipation terms
+in the i, j, and k directions respectively. The scaling of the artificial dissipation operator in each coordinate
+direction is weighted by its own spectral radius of the corresponding flux Jacobian matrix, which gives the
+non-isotropic characteristics of the method.26 The artificial dissipation contribution in the ξ direction is
+given by
+d(i+ 1
+2 ,j,k)
+=
+λ(i+ 1
+2 ,j,k)
+�
+ϵ(2)
+(i+ 1
+2 ,j,k)
+�
+W(i+1,j,k) − W(i,j,k)
+�
+(48)
+ϵ(4)
+(i+ 1
+2 ,j,k)
+�
+W(i+2,j,k) − 3W(i+1,j,k) + 3W(i,j,k) − W(i−1,j,k)
+�
+] ,
+in which
+ϵ(2)
+(i+ 1
+2 ,j,k)
+=
+k(2)max
+�
+νd
+(i+1,j,k), νd
+(i,j,k)
+�
+,
+(49)
+ϵ(4)
+(i+ 1
+2 ,j,k)
+=
+max
+�
+0, k(4) − ϵ(2)
+(i+ 1
+2 ,j,k)
+�
+.
+(50)
+The original article27 recomends using k(2) = 0.25 and k(4) = 0.016 for the dissipation artificial constants.
+The pressure gradient sensor, νd
+(i,j,k), for the ξ direction is written as
+νd
+(i,j,k) = |p(i+1,j,k) − 2p(i,j,k) + p(i−1,j,k)|
+p(i+1,j,k) − 2p(i,j,k) + p(i−1,j,k)
+.
+(51)
+The W vector from Eq. (48) is calculated as a function of the conserved variable vector, ˆQ, written in Eq.
+(27). The formulation intends to keep the total enthalpy constant in the final converged solution, which is
+the correct result for the Navier-Stokes equations with Re → ∞. This approach is also valid for the viscous
+formulation because the dissipation terms are added to the inviscid flux terms, in which they are really
+necessary to avoid nonlinear instabilities of the numerical formulation. The W vector is given by
+W = ˆQ + [0 0 0 0 p]T .
+(52)
+The spectral radius-based scaling factor, λ, for the i − th direction is written
+λ(i+ 1
+2 ,j,k) = 1
+2
+��
+λξ
+�
+(i,j,k) +
+�
+λξ
+�
+(i+1,j,k)
+�
+,
+(53)
+where
+λξ(i,j,k) = λξ
+�
+1 +
+�λη
+λξ
+�0.5
++
+�λζ
+λξ
+�0.5�
+.
+(54)
+The spectral radii, λξ, λη and λζ are given by
+λξ
+=
+|U| + a
+�
+ξ2x + η2y + ζ2z ,
+λξ
+=
+|V | + a
+�
+ξ2x + η2y + ζ2z ,
+(55)
+λξ
+=
+|W| + a
+�
+ξ2x + η2y + ζ2z ,
+8 of 25
+
+in which, U, V and W are the contravariants velocities in the ξ, η and ζ, previously written in Eq. (32), and
+a is the local speed of sound, which can be written as
+a =
+�γp
+ρ .
+(56)
+The calculation of artificial dissipation terms for the other coordinate directions are completely similar and,
+therefore, they are not discussed in the present work.
+VI.B.
+Time Marching Method
+The time marching method used in the present work is a 2nd-order, 5-step Runge-Kutta scheme based on
+the work of Jameson.28,29 The time integration can be written as
+Q(0)
+(i,jk,)
+=
+Q(n)
+(i,jk,) ,
+Q(l)
+(i,jk,)
+=
+Q(0)
+(i,jk,)−
+αl∆t(i,j,k)RHS(l−1)
+(i,j,k)
+l = 1, 2 · · · 5,
+Q(n+1)
+(i,jk,)
+=
+Q(5)
+(i,jk,) ,
+(57)
+in which ∆t is the time step and n and n + 1 indicate the property values at the current and at the next
+time step, respectively. The literature28,29 recommends
+α1 = 1
+4 ,
+α2 = 1
+6 ,
+α3 = 3
+8 ,
+α4 = 1
+2 ,
+α5 = 1 ,
+(58)
+in order to improve the numerical stability of the time integration. The present scheme is theoretically stable
+for CFL ≤ 2
+√
+2, under a linear analysis.26
+VII.
+Boundary Conditions
+The geometry used in the present work presents a cylindrical shape which is gererated by the rotation
+of a 2-D plan around a centerline. Figure 1 presents a lateral view and a frontal view of the computational
+domain used in the present work and the positioning of the entrance, exit, centerline, far field and periodic
+boundary conditions. A discussion on all boundary conditions is performed in the following subsections.
+(a) Lateral view of boundary conditions.
+(b) Frontal view of boundary conditions.
+Figure 1.
+Lateral and frontal views of the computational domain indicating boundary conditions.
+VII.A.
+Far Field Boundary
+Riemann invariants30 are used to implement far field boundary conditions. They are derived from the char-
+acteristic relations for the Euler equations. At the interface of the outer boundary, the following expressions
+apply
+R− = R−
+∞
+=
+qn∞ −
+2
+γ − 1a∞ ,
+(59)
+R+ = R+
+e
+=
+qne −
+2
+γ − 1ae ,
+(60)
+9 of 25
+
+FARFIELD
+FARFIELD
+EXIT
+ENTRANCE
+===
+CENTERLINE: PERIODICITY
+CENTERLINEwhere ∞ and e indexes stand for the property in the freestream and in the internal region, respectively. qn
+is the velocity component normal to the outer surface, defined as
+qn = u · ⃗n ,
+(61)
+and ⃗n is the unit outward normal vector
+⃗n =
+1
+�
+η2x + η2y + η2z
+[ηx ηy ηz]T .
+(62)
+Equation (61) assumes that the η direction is pointing from the jet to the external boundary. Solving for qn
+and a, one can obtain
+qnf = R+ + R−
+2
+,
+af = γ − 1
+4
+(R+ − R−) .
+(63)
+The index f is linked to the property at the boundary surface and will be used to update the solution at this
+boundary. For a subsonic exit boundary, 0 < qne/ae < 1, the velocity components are derived from internal
+properties as
+uf
+=
+ue + (qnf − qne)ηx ,
+vf
+=
+ve + (qnf − qne)ηy ,
+(64)
+wf
+=
+we + (qnf − qne)ηz .
+Density and pressure properties are obtained by extrapolating the entropy from the adjacent grid node,
+ρf =
+�
+ργ
+ea2
+f
+γpe
+�
+1
+γ−1
+,
+pf =
+ρfa2
+f
+γ
+.
+For a subsonic entrance, −1 < qne/ae < 0, properties are obtained similarly from the freestream variables as
+uf
+=
+u∞ + (qnf − qn∞)ηx ,
+vf
+=
+v∞ + (qnf − qn∞)ηy ,
+(65)
+wf
+=
+w∞ + (qnf − qn∞)ηz ,
+ρf =
+�
+ργ
+∞a2
+f
+γp∞
+�
+1
+γ−1
+.
+(66)
+For a supersonic exit boundary, qne/ae > 1, the properties are extrapolated from the interior of the domain
+as
+ρf
+=
+ρe ,
+uf
+=
+ue ,
+vf
+=
+ve ,
+(67)
+wf
+=
+we ,
+ef
+=
+ee ,
+and for a supersonic entrance, qne/ae < −1, the properties are extrapolated from the freestream variables as
+ρf
+=
+ρ∞ ,
+uf
+=
+u∞ ,
+vf
+=
+v∞ ,
+(68)
+wf
+=
+w∞ ,
+ef
+=
+e∞ .
+10 of 25
+
+VII.B.
+Entrance Boundary
+For a jet-like configuration, the entrance boundary is divided in two areas: the jet and the area above it.
+The jet entrance boundary condition is implemented through the use of the 1-D characteristic relations for
+the 3-D Euler equations for a flat velocity profile. The set of properties then determined is computed from
+within and from outside the computational domain. For the subsonic entrance, the v and w components of
+the velocity are extrapolated by a zero-order extrapolation from inside the computational domain and the
+angle of flow entrance is assumed fixed. The rest of the properties are obtained as a function of the jet Mach
+number, which is a known variable.
+(u)1,j,k
+=
+uj ,
+(v)1,j,k
+=
+(v)2,j,k ,
+(69)
+(w)1,j,k
+=
+(w)2,j,k .
+The dimensionless total temperature and total pressure are defined with the isentropic relations:
+Tt = 1 + 1
+2(γ − 1)M 2
+∞
+and
+Pt = 1
+γ (Tt)
+γ
+γ−1 .
+(70)
+The dimensionless static temperature and pressure are deduced from Eq. (70), resulting in
+(T)1,j,k =
+Tt
+1 + 1
+2(γ − 1)(u2 + v2 + w2)1,j,k
+and
+(p)1,j,k = 1
+γ (T)
+γ
+γ−1
+1,j,k .
+(71)
+For the supersonic case, all conserved variables receive jet property values.
+The far field boundary conditions are implemented outside of the jet area in order to correctly propagate
+information comming from the inner domain of the flow to the outter region of the simulation. However,
+in the present case, ξ, instead of η, as presented in the previous subsection, is the normal direction used to
+define the Riemann invariants.
+VII.C.
+Exit Boundary Condition
+At the exit plane, the same reasoning of the jet entrance boundary is applied. This time, for a subsonic exit,
+the pressure is obtained from the outside and all other variables are extrapolated from the interior of the
+computational domain by a zero-order extrapolation. The conserved variables are obtained as
+(ρ)IMAX,j,k
+=
+(p)IMAX,j,k
+(γ − 1)(e)IMAX−1,j,k
+,
+(72)
+(⃗u)IMAX,j,k
+=
+(⃗u)IMAX−1,j,k,
+(73)
+(ei)IMAX,j,k
+=
+(ρ)IMAX,j,k
+�
+(e)IMAX−1,j,k + 1
+2(⃗u)IMAX,j,k · (⃗u)IMAX,j,k
+�
+,
+(74)
+in which IMAX stands for the last point of the mesh in the axial direction. For the supersonic exit, all
+properties are extrapolated from the interior domain.
+VII.D.
+Centerline Boundary Condition
+The centerline boundary is a singularity of the coordinate transformation, and, hence, an adequate treatment
+of this boundary must be provided. The conserved properties are extrapolated from the ajacent longitudinal
+plane and are averaged in the azimuthal direction in order to define the updated properties at the centerline
+of the jet.
+The fourth-difference terms of the artificial dissipation scheme, used in the present work, are carefully
+treated in order to avoid the five-point difference stencils at the centerline singularity. If one considers the
+flux balance at one grid point near the centerline boundary in a certain coordinate direction, let wj denote
+a component of the W vector from Eq. (52) and dj denote the corresponding artificial dissipation term at
+the mesh point j. In the present example, (∆w)j+ 1
+2 stands for the difference between the solution at the
+interface for the points j+1 and j. The fouth-difference of the dissipative fluxes from Eq. (48) can be written
+as
+dj+ 1
+2 = (∆w)j+ 3
+2 − 2 (∆w)j+ 1
+2 + (∆w)j− 1
+2 .
+(75)
+11 of 25
+
+Considering the centerline and the point j = 1, as presented in Fig. 2, the calculation of d1+ 1
+2 demands
+the (∆w) 1
+2 term, which is unknown since it is outside the computation domain.
+In the present work a
+extrapolation is performed and given by
+(∆w) 1
+2 = − (∆w)1+ 1
+2 .
+(76)
+This extrapolation modifies the calculation of d1+ 1
+2 that can be written as
+dj+ 1
+2 = (∆w)j+ 3
+2 − 3 (∆w)j+ 1
+2 .
+(77)
+The approach is plausible since the centerline region is smooth and does not have high gradient of properties.
+Figure 2.
+Boundary points dissipation.26
+VII.E.
+Periodic Boundary Condition
+A periodic condition is implemented between the first (K = 1) and the last point in the azimutal direction
+(K = KMAX) in order to close the 3-D computational domain. There are no boundaries in this direction,
+since all the points are inside the domain. The first and the last points, in the azimuthal direction, are
+superposed in order to facilitate the boundary condition implementation which is given by
+(ρ)i,j,KMAX
+=
+(ρ)i,j,1 ,
+(u)i,j,KMAX
+=
+(u)i,j,1 ,
+(v)i,j,KMAX
+=
+(v)i,j,1 ,
+(78)
+(w)i,j,KMAX
+=
+(w)i,j,1 ,
+(e)i,j,KMAX
+=
+(e)i,j,1 .
+VIII.
+High Performance Computing
+The current section presents an overview of the LES solver and discusses the high performance computing
+implementations introduced into the code. A study on the parallel performance of JAZzY using multiple
+processors is presented and discussed in the end of the section.
+VIII.A.
+Mesh Generation
+The LES solver presents a parallel-IO feature in which each MPI partition reads its correspondent portion of
+the mesh. Therefore, a 3-D grid generator is developed in order to provide partitioned CGNS mesh files to
+12 of 25
+
+j=4
+j=3+1/2
+j=3
+j=2+1/2
+j=2
+j=1+1/2
+j=1
+12
+grid node
+X
+interfacethe LES solver. The CGNS standard9–11 is build on the HDF5 library.7,8 This library is a general scientific
+format adaptable to virtually any scientific or engineering application. It provides tools to efficiently read
+and write data structured in a binary tree fashion. This data structure can handle many types of queries
+very efficiently31,32 such as time-dependent CFD solution.
+Figure 3(a) illustrates the segmentation of the domain into the axial and azimuthal directions while Fig.
+3(b) presents the mapping of the domain. The index of each partition, indicated in Fig. 3(b), is based
+on a matrix index system in which the rows represent the position in the axial direction and the columns
+represent the position in the azimuthal direction. The partition index starts at zero to be consistent with
+the message passing interface standard. NPX and NPZ denote the number of partitions in the axial and
+azimuthal directions, respectively.
+(a) 2-D partitioning in the axial and azimuthal direction.
+(b) Mapping of the 2-D partitioning.
+Figure 3.
+2-D partitioning and mapping.
+Table 1 presents the algorithm of the mesh generator. The user can provide geometry and mesh point
+distribution parameters so the grid generator can create a 2-D mesh. A complete 2-D grid, from a different
+mesh generator, can also be provided by the user to the mesh generator. In the sequence, the 2-D grid is
+partitioned in the axial direction. After the partitioning in the axial direction, each portion of the mesh is
+13 of 25
+
+0
+NPZ-1
+1
+0
+NPZ-2
+NPX-2
+NPX-10
+NPZ
+2*NPZ
+1
+NPZ+1
+2*NPZ+1
+*(NPZ)+i
+NPZ-2
+NPZ+NPZ-2
+2*NPZ+NPZ-2
+NPZ-1
+NPZ+NPZ-1
+2*NPZ+NPZ-1
+NPX*NPZ-1extruded in the azimuthal direction respecting the positioning of the MPI partitions. Each portion of the
+mesh is written using the CGNS standard.
+Table 1.
+Mesh generator overview.
+1
+BEGIN
+2
+Read input data
+3
+Read mesh or create 2-D mesh
+4
+Perform balanced partitioning in axial direction
+5
+Perform balanced partitioning in azimuthal direction
+6
+Rotate the mesh partition in the azimuthal direction
+7
+Write a CGNS mesh file for each partition
+8
+END
+The division of the mesh in the axial and azimuthal directions is performed towards a well balanced
+distribution of points. Firstly, the total number of grid points in one direction is divided by the number of
+domains in the same direction. The remaining points are spread among the partitions in the case which the
+division is not exact. Figure 4 illustrates the balancing procedure performed in each direction during the
+partitioning of the computational grid.
+Figure 4.
+Balancing procedure performed during the patitioning of the mesh.
+VIII.B.
+JAZzY Overview
+JAZzY is the LES solver presented in the current work. Table 2 presents a brief overview of JAZzY. In the
+beginning of the calculation every MPI partition reads the same ASCII file which provides input data such
+as flow configurations and simulation settings. In the sequence, each MPI partition reads its correspondent
+CGNS mesh file. The Jacobian and the metric terms are calculated after the I-O procedure. Then, each
+processor sets the initial conditions defined in the input data and it performs an asynchronous communication
+before starting iterations in order to solve the compressible LES equations.
+The first operation, which is performed in the iteration loop, is the computation of the inviscid flux
+vectors. Then, the artificial dissipation operator is calculated. Asynchronous communications are performed
+during this computation. After the data exchange, the inviscid terms of the LES formulation is calculated
+using the convective operator and the artificial dissipation terms.
+The viscous terms are calculated in
+sequence and their contributions are added to the right-hand side of the LES equations.
+The time integration is performed using a five step Runge-Kutta time integration scheme after the
+calculation of the numerical fluxes. This time marching scheme calculates the inviscid and viscous terms
+recursively through the inner steps. Therefore, multiple communications are performed during the time
+integration. The solution, boundary conditions and fluid viscosity are updated after the time marching.
+Asynchronous communications are performed for the periodicity condition. Blocking communications are
+performed in order to calculate properties at the centerline singularity. After the updates, neighbor partitions
+14 of 25
+
+n
+n
+n
+m
+n+1n+1n+1n+1n
+nTable 2.
+The JAZzY code overview.
+1
+Read input data
+2
+Read mesh
+3
+Calculate Jacobian
+4
+Calculate metric terms
+5
+Set up initial conditions
+6
+Asynchronous communication
+7
+WHILE (it. < max nb it.)
+Compute inviscid flux vectors
+Compute artificial dissipation operator
+Calculate inviscid flux contributions to the residue
+Compute viscous flux vectors and SGS viscosity
+Calculate viscous flux contributions to the residue
+Calculate time step
+Perform multi-step explicit time integration
+Update the solution and boundary conditions
+8
+END WHILE
+9
+Output results
+10
+END
+exchange data using non blocking MPI routines. The SGS viscosity is calculated in the end of the iteration
+loop. The Vreman and the static Smagorinsky models does not request the use of communications. The
+calculation of the dynamic Smagorinsky SGS viscosity is performed using blocking data exchange in order
+to calculate properties on the second level filtering. Finally, when the requested number of iterations is
+achieved, each MPI partition appends the solution to the output CGNS file.
+VIII.C.
+Communication
+Numerical data exchange between the partitions is necessary in order to perform parallel computation. Ghost
+points are added to the boundaries of local partition mesh at the main flow direction and at the azimuthal
+direction in order to carry information of the neighbor points. The artificial dissipation scheme implemented
+in the code28 uses a five points stencil which demands information of the two neighbors of a given mesh
+point. Hence, a two layer ghost points is created at the beginning and at the end of each partition. Figure 5
+presents the layer of ghost points used in the present code. The yellow and black layers represent the axial
+and azimuthal ghost points respectively. The green region is the partition mesh.
+After the ghost points creation, each processor performs the computation. Communication between neigh-
+bor partitions are performed in order to allow data information pass through the computational domain.
+Blocking and non-blocking communications are used in the present work. In the blocking communication ap-
+proach, the partition which sends the information only restarts the computation after the neighbor partition,
+which is the receiver, has finished to read the data. The same does not occur for non-blocking communica-
+tions. The partition which has sent data does not need to wait a signal from the receiver. The developer is
+responsible to assure that data is communicated before being accessed through the use of MPI wait functions
+along the code. Most of the data exchange are performed using non-blocking MPI communication routines
+in the current research. Only the centerline boundary condition and the communications for the dynamic
+Smagorinky model24,33 are performed using blocking MPI communication routines. The first is performed in
+order to assure reproducibility of the solver. The second is performed using blocking communication because
+it represents only a small portion of the code.
+The meshes used in the current research have a singularity at the centerline. It is necessary to correctly
+treat this region for the sake of data consistency. Therefore, properties are extrapolated to the singularity in
+radial direction and, in the sequence, the master partition collects all data from the partitions that share the
+15 of 25
+
+Figure 5.
+Ghost points creation procedure.
+same singularity point and allocates into one single vector. After the allocation, the properties are averaged in
+a sequential fashion and the result is spread to the neighbors in the azimuthal direction. Figure 6 illustrates
+the singularity treatment for a configuration with 16 points in the azimuthal direction. The yellow, red,
+blue and green colors represent the four partitions in the azimuthal direction. Such procedure does not use
+collective communications in order to preserve the commutative property during the averaging. This blocking
+communication is very important in order to achieve the binary reproducibility of the computational tool.34
+The use of such communication is motivated by the work of Arteaga et. al.35
+Figure 6.
+Singularity averaging in parallel.
+Non-blocking communication is not available for the communication performed by the dynamic Smagorin-
+sky model. Only blocking communication are implemented because it represents only a small portion of the
+code. This data exchange is firstly performed in the azimuthal direction and in the sequence in the axial
+direction. The azimuthal communication is performed in four blocking steps as Figs. 7(a) and 7(b) demon-
+strate. Initially, the communication is performed in the forward direction. Even partitions send information
+of their two last local layers to the ghost points at the left of odd partitions. If the last partition is even,
+it does not share information in this step. In the sequence, odd partitions send information of their two
+last local layers to the ghost points at the left of even partitions. If the last partition is odd, it does not
+share information in this step. The third and the fourth steps are backward communications. First, odd
+16 of 25
+
+AXIALGHOST
+LAYER
+AZIMUTHAL
+GHOSTLAYER
+PARTITION3
+2
+1+2+3+4+1-2+3+4+1+2+3+4+1-2+3+4
+2
+3
+16
+4
+3
+2
+1partitions send data of their two first local layers to the ghost points at the right of even partitions. Finally,
+all even partitions, but the first, send data of their two first local layers to the ghost points at the right of
+odd partitions. Communication in the axial direction are performed using the same approach.
+(a) Forward communication between partitions.
+(b) Backward Communication between partitions.
+Figure 7.
+Forward and Backward communication schemes used in order to exchange information between neighbor
+partitions
+VIII.D.
+Computational Resources
+The current work is included into a national project know as CEPID-CeMEAI.36 This project provides
+access to a SGI cluster. The machine has 104 computational nodes and each one has two deca-core 2.8 GHz
+Intel Xeon®
+E52680v2 processors and 128 Gb DDR3 1866MHz random access memory. The entire cluster
+has 2080 computational cores available for the project members. The storage can be performed using the
+network file system (NFS) or the Lustre® file system.37 Both storage system have 175 Tb available for the
+users. The network communication is performed using Infiniband and Gigabit Ethernet. The Operational
+system is the Red Hat Enterprise Linux38 and the job scheduler is the Altair PBS Pro.39
+The Intel Composer XE Update 2 compiler, version 15.0.2.164 is used in the present work. The code
+is compiled using optimization flags in order to achieve the best computational performance.
+The best
+compilation flags were tested in the present work. The flags which provided the best results are:
+• O3: enables aggressive optimization such as global code scheduling, software pipelining, predication
+and speculation, prefetching, scalar replacement and loop transformations;
+• xHost: tells the compiler to generate instructions for the highest instruction set available on the
+compilation host processor;
+• ipo: automatic, multi-step process that allows the compiler to analyze the code and determine where
+you can benefit from specific optimizations;
+• no-prec-div: enables optimizations that give slightly less precise results than full division;
+• assume buffered io : tells the compiler to accumulate records in a buffer;
+• override-limits: deals with very large, complex functions and loops.
+VIII.E.
+Computational Performance
+Parallel computation can largely decrease the time of CFD simulations. However, the parallel upgrade of a
+serial solver must be carefully performed. The communication between partitions is not free and can affect
+the computational performance in parallel. The partitioning of the computational domain increases the
+number of communication between processors. When the number of points of a partitions becomes smaller
+17 of 25
+
+First Step Backward
+Second Step Backward
+GhostCellsFirst Step Forward
+Second Step Forward
+1111
+hostCellssimulation spend less time computing and more time performing communications. Consequently, the parallel
+performance of the solver is deteriorated.
+The speedup is one of the most common figures for the performance evaluation of parallel algorithms and
+architectures40 and it is used in the present work in order to measure the computational performance of the
+parallel solver and compare with the ideal case. Different approaches are used by the scientific community
+in order to calculate the speedup.41,42 In the present work the speedup, SpN, is given by
+SpN = Ts
+TN
+.
+(79)
+in which TN and Ts stand for the time spent to perform one thousand iterations using N processors and
+one single processor, respectively. The efficiency as function of the number of processors, ηN, is written
+considering Amdahl’s law43 as
+ηN = SpN
+N
+.
+(80)
+The performance of the solver is evaluated for a large-eddy simulation of a isothermic perfectly expanded
+turbulent jet flow without any SGS model. The impact of SGS models on the parallel performance of the
+code is evaluated in the work of Junqueira-Junior.1 Different parameters such as mesh size, partitioning
+configurations and number of processors are used to study the parallel behavior of the code in the current
+article.
+Table 3 presents the different grid configurations used in the current work. There are nine meshes whose
+total number of points doubles every time. The first column presents the name of the mesh. The second, third
+and fourth columns present the number of points in the axial, radial and azimuthal directions, respectively.
+The last column indicates the total number of points of the mesh. The grid point distribution in the azimuthal
+direction is fixed and it is equals to 361. The smallest grid, named as Mesh A, has approximately 5.9 million
+points while the biggest grid presents approximately 1.0 billion points.
+Table 3.
+Configuration of computational meshes used in the current study.
+Mesh
+No. Pt. Axial Direct.
+No. Pt. Radial Direct.
+No. Pt. Azimut. Direct.
+No. Pt.
+A
+128
+128
+361
+5.9M
+B
+256
+128
+361
+11.8M
+C
+256
+256
+361
+23.7M
+D
+512
+256
+361
+47.3M
+E
+512
+512
+361
+94.6M
+F
+1024
+512
+361
+189.3M
+G
+1024
+1024
+361
+378.5M
+H
+2048
+1024
+361
+757.1M
+I
+1700
+1700
+361
+1.0B
+The strong scalability test is used in the present paper in order to evaluate the parallel performance
+of the code. This test is a measure of the evolution of speedup and efficiency of a given problem, with
+a fixed size, as the number of processors increases. Simulations are performed using up to 400 processors
+in the present paper. Different partitioning configurations are used in order to measure its effects on the
+parallel computational efficiency.
+Table 4 presents the number of partitions in the azimuthal direction
+for each number of processors used to study the scalability of the solver. The first column indicates the
+computational resource while the second columns indicates the number of zones in the azimuthal direction
+used to evaluate the effects of the partitioning on the computation.
+Isothermic perfectly expanded jet flow simulations are performed using different grid sizes and different
+partition configurations. The Reynolds number of the jet is 1.5744 × 106 for the present simulations and
+a flat-hat profile with Mach number of 1.4 is imposed at the entrance of the computational domain. A
+stagnated flow is used as initial condition for the simulations. The time increment is 2.5 × 10−4 seconds
+for the tests performed. Numerical results of such configuration using the same code are presented in the
+work described in Ref. [2, 44]. In the present article each simulation performs 1000 iterations or 24 hours
+18 of 25
+
+Table 4.
+Number of partitions in the azimuthal direction for a given number of processors.
+No. Proc.
+No. of Part. in the Azimuthal Dir.
+1
+1
+2
+1
+2
+5
+1
+5
+10
+1
+2
+5
+10
+20
+1
+2
+4
+5
+10
+20
+40
+1
+2
+4
+5
+8
+10
+20
+40
+80
+2
+4
+5
+8
+10
+20
+40
+100
+2
+4
+5
+10
+20
+25
+200
+4
+8
+10
+20
+25
+50
+400
+8
+16
+20
+25
+50
+of computation. An average of the CPU time per iteration through the simulations is measured in order to
+calculate and compare computational cost, speed-up and efficiency of the solver.
+Tables 5 to 13 present, for a given number of processors, the partitioning configuration which provides
+the best averaged CPU time per iteration, its correspondent speedup and its correspondent computational
+efficiency for meshes A to I respectively. One can notice that meshes F,G,H and I are too big and cannot fit
+into one single node of the computational cluster used in the current paper. The scalability study started
+with 40 processors for mesh F, 80 processors for mesh G, and 200 processors for mesh H and mesh I. The
+efficiency of the solver is considered 100% at the starting point in order to have a reference for the cases in
+which is not possible to perform a simulation using one single computational core.
+Table 5.
+Computational performance of Mesh A.
+No. Proc.
+Av. CPU time
+Speedup
+Efficiency
+No. Azim. Part.
+1
+2.97E+01
+1.00E+00
+1.00E+00
+1
+2
+1.14E+01
+2.61E+00
+1.31E+00
+1
+5
+3.96E+00
+7.49E+00
+1.50E+00
+1
+10
+2.27E+00
+1.31E+01
+1.31E+00
+2
+20
+1.57E+00
+1.89E+01
+9.43E-01
+5
+40
+8.32E-01
+3.57E+01
+8.91E-01
+20
+80
+4.60E-01
+6.45E+01
+8.07E-01
+20
+100
+3.80E-01
+7.81E+01
+7.81E-01
+20
+200
+2.37E-01
+1.25E+02
+6.26E-01
+50
+400
+1.77E-01
+1.67E+02
+4.19E-01
+50
+The evolution of speedup and efficiency as function of the number of processors for all computational
+grids used in the current work are presented in Figs. 8 and 9. The code presents a good scalability, with
+an efficiency bigger than 75%, for grids which have more than 50 million points. Mesh E presented an
+efficiency of ≈ 100% and speedup of 400 when running on 400 processors. Such performance is equivalent to
+the theoretical speedup. One can notice a super linear scalability for the cases which the speedup reference
+is the sequential computation and also for mesh I. The first can be explained by the fact that there is
+not a serial version of the solver. There is only a parallel version which can run simulations using a single
+computational core. Moreover, cache memory can be the bottleneck of a simulation using a given mesh and a
+given number of processors.45 Such limitation can explain the super linear speedup of mesh I. The bottleneck
+can deteriorate the performance of the solver. When the number of processors is increased and mesh size
+conserved, the cache memory can become no longer a limitation. This effect can generate super-scalability
+which can be interpreted as computational efficiency greater than 100%.
+19 of 25
+
+Table 6.
+Computational performance of Mesh B.
+No. Proc.
+Av. CPU time
+Speedup
+Efficiency
+No. Azim. Part.
+1
+5.78E+01
+1.00E+00
+1.00E+02
+1
+2
+2.60E+01
+2.23E+00
+1.11E+02
+2
+5
+7.56E+00
+7.65E+00
+1.53E+02
+1
+10
+4.38E+00
+1.32E+01
+1.32E+02
+2
+20
+3.09E+00
+1.87E+01
+9.36E+01
+4
+40
+1.61E+00
+3.59E+01
+8.98E+01
+8
+80
+8.43E-01
+6.86E+01
+8.58E+01
+10
+100
+7.25E-01
+7.98E+01
+7.98E+01
+25
+200
+3.92E-01
+1.48E+02
+7.38E+01
+25
+400
+2.60E-01
+2.23E+02
+5.57E+01
+25
+Table 7.
+Computational performance of Mesh C.
+No. Proc.
+Av. CPU time
+Speedup
+Efficiency
+No. Azim. Part.
+1
+1.18E+02
+1.00E+00
+1.00E+02
+1
+2
+5.18E+01
+2.28E+00
+1.14E+02
+1
+5
+1.68E+01
+7.03E+00
+1.41E+02
+1
+10
+9.15E+00
+1.29E+01
+1.29E+02
+2
+20
+6.31E+00
+1.87E+01
+9.35E+01
+4
+40
+3.27E+00
+3.61E+01
+9.03E+01
+8
+80
+1.77E+00
+6.66E+01
+8.33E+01
+10
+100
+1.47E+00
+8.05E+01
+8.05E+01
+20
+200
+7.89E-01
+1.50E+02
+7.48E+01
+25
+400
+5.07E-01
+2.33E+02
+5.82E+01
+50
+Table 8.
+Computational performance of Mesh D.
+No. Proc.
+Av. CPU time
+Speedup
+Efficiency
+No. Azim. Part.
+1
+2.67E+02
+1.00E+00
+1.00E+02
+1
+2
+1.04E+02
+2.56E+00
+1.28E+02
+1
+5
+3.28E+01
+8.14E+00
+1.63E+02
+1
+10
+1.74E+01
+1.54E+01
+1.54E+02
+2
+20
+1.21E+01
+2.20E+01
+1.10E+02
+4
+40
+6.31E+00
+4.23E+01
+1.06E+02
+8
+80
+3.27E+00
+8.17E+01
+1.02E+02
+8
+100
+2.73E+00
+9.79E+01
+9.79E+01
+20
+200
+1.49E+00
+1.78E+02
+8.92E+01
+20
+400
+8.93E-01
+2.99E+02
+7.47E+01
+25
+20 of 25
+
+Table 9.
+Computational performance of Mesh E.
+No. Proc.
+Av. CPU time
+Speedup
+Efficiency
+No. Azim. Part.
+1
+7.02E+02
+1.00E+00
+1.00E+02
+1
+2
+2.18E+02
+3.22E+00
+1.61E+02
+1
+5
+7.33E+01
+9.58E+00
+1.92E+02
+1
+10
+3.68E+01
+1.91E+01
+1.91E+02
+2
+20
+2.85E+01
+2.46E+01
+1.23E+02
+2
+40
+1.27E+01
+5.53E+01
+1.38E+02
+8
+80
+6.67E+00
+1.05E+02
+1.32E+02
+2
+100
+5.54E+00
+1.27E+02
+1.27E+02
+20
+200
+3.02E+00
+2.33E+02
+1.16E+02
+20
+400
+1.70E+00
+4.14E+02
+1.03E+02
+16
+Table 10.
+Computational performance of Mesh F.
+No. Proc.
+Av. CPU time
+Speedup
+Efficiency
+No. Azim. Part.
+40
+2.47E+01
+4.00E+01
+1.00E+02
+4
+80
+1.44E+01
+6.87E+01
+8.58E+01
+8
+100
+1.04E+01
+9.54E+01
+9.54E+01
+10
+200
+5.47E+00
+1.81E+02
+9.04E+01
+8
+400
+3.12E+00
+3.17E+02
+7.92E+01
+8
+Table 11.
+Computational performance of Mesh G.
+No. Proc.
+Av. CPU time
+Speedup
+Efficiency
+No. Azim. Part.
+80
+2.61E+01
+8.00E+01
+1.00E+02
+8
+100
+2.17E+01
+9.63E+01
+9.63E+01
+10
+200
+1.17E+01
+1.78E+02
+8.92E+01
+8
+400
+6.76E+00
+3.09E+02
+7.72E+01
+20
+Table 12.
+Computational performance of Mesh H.
+No. Proc.
+Av. CPU time
+Speedup
+Efficiency
+No. Azim. Part.
+200
+2.13E+01
+2.00E+02
+1.00E+02
+8
+400
+1.13E+01
+3.75E+02
+9.37E+01
+16
+Table 13.
+Computational performance of Mesh I.
+No. Proc.
+Av. CPU time
+Speedup
+Efficiency
+No. Azim. Part.
+200
+4.09E+01
+2.00E+02
+1.00E+02
+4
+400
+1.52E+01
+5.39E+02
+1.35E+02
+16
+21 of 25
+
+Figure 8.
+Speedup curve of the LES solver for different mesh sizes.
+Figure 9.
+Parallel efficiency of the LES solver for different mesh sizes.
+22 of 25
+
+500
+A-5.9M
+B-11.8M
+C-23.7M
+400
+D-47.3M
+E-94.6M
+F-189.3M
+G-378.5M
+Speedup
+300
+H-757.1M
+I-1B
+200
+100
+100
+200
+300
+400
+NpA-5.9M
+B-11.8M
+C-23.7M
+D-47.3M
+E-94.6M
+150
+F-189.3M
+G-378.5M
+Efficiency [%]
+H-757.1M
+I-1B
+100
+50
+100
+200
+300
+400
+NpIncreasing the size of a computational problem can generate a better scalability study. The time spent
+with computation becomes more significant when compared to the time spent with communication with the
+growth of a problem. One can notice such effect for meshes A, B, C, D and E. The speedup and the efficiency
+increase with with the growth of the mesh size. However, such scalability improvement does not happen
+from mesh E to meshes F, G, H and I. This behavior is originated because the reference used to calculate
+speedup and efficiency is not the same for all grid configurations. The studies performed using meshes F, G,
+H and I does not use the serial computation as a reference, which is not the case for the scalability studies
+performed in the current paper using mesh A, B, C, D and E.
+IX.
+Concluding Remarks
+The current work is a computational performance study of a large eddy simulation solver for supersonic
+jet flow configurations. Nine strong scalability studies are performed using meshes whose size grows from
+approximately 5.9 million points to approximately 1.0 billion points. Different partitioning configurations
+are used to evaluate its effects on the computational performance of the solver. Simulations are run on one
+processor up to 400 computational cores. The speedup and the computational efficiency are calculated for
+every study performed in the present article.
+The filtered compressible large eddy simulation formulation is written using a finite-difference centered
+second-order spatial discretization with the explicit addition of artificial dissipation. The time integration is
+performed using a five-steps second order Runge-Kutta scheme. Three subgrid scale models are implemented
+into the solver. Message passing interface protocols are used in order to perform the computation in parallel.
+The code presents parallel-IO features. Each MPI partition reads its portion of the mesh. A mesh generator
+is created in order to provide balanced CGNS mesh partitions.
+The solver creates two layers of ghost
+points in the axial and in azimuthal direction for each partition in order to exchange data with neighbor
+zones. Communication between partitions are performed using non blocking data exchange towards the best
+computational performance.
+The code presented a good scalability for the calculations run in the current paper.
+The averaged
+CPU time per iteration decays with the increase of number of processors in parallel for all computation
+performed by the large eddy simulation solver evaluated in the present work. Meshes with more than 50
+million points indicated an efficiency greater than 75%. The problem with approximately 100 million points
+presented speedup of 400 and efficiency of 100% when running on 400 computational cores in parallel. Such
+performance is equivalent to theoretical behavior in parallel. It is important to remark the ability of the
+parallel solver to treat very dense meshes as the one tested in the present paper with approximately 1.0
+billion points. Large eddy simulation demand very refined grids in order to have a well representation of the
+physical problem of interest. Therefore, it is important to perform simulations of such configuration with a
+good computation efficiency. One can notice the presence of super-linear speedup in the current study. Such
+behavior can be explained by cache limitations when running simulations with low amount of computational
+resources. Moreover, there is no serial version of the code. The sequential study is a parallel version running
+on one single processor.
+Acknowledgments
+The authors gratefully acknowledge the partial support for this research provided by Conselho Nacional
+de Desenvolvimento Cient´ıfico e Tecnol´ogico, CNPq, under the Research Grants No. 309985/2013-7, No.
+400844/2014-1, No. 443839/2014-0 and No. 150551/2017-1. The authors are also indebted to the partial
+financial support received from Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo, FAPESP, under
+the Research Grants No. 2013/07375-0 and No. 2013/21535-0.
+23 of 25
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+

JtAyT4oBgHgl3EQf5_rl/content/tmp_files/2301.00816v1.pdf.txt

@@ -0,0 +1,515 @@
+ 
+ 
+1 
+ 
+Thermo-optic phase shifter based on hydrogen-doped indium oxide microheater 
+ 
+Weiyu Tong, Erqi Yang, Yu Pang, Haobo Yang, Xin Qian, Ronggui Yang, Bin Hu*, Jianji 
+Dong* and Xinliang Zhang  
+ 
+W. Tong, E. Yang, B. Hu, J. Dong, X. Zhang 
+Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and 
+Technology, Wuhan 430074, China 
+E-mail: [email protected]; [email protected] 
+Y. Pang, H. Yang, X. Qian, R. Yang  
+School of Energy and Power Engineering, Huazhong University of Science and Technology, 
+Wuhan 430074, China 
+ 
+Keywords: silicon photonics, thermo-optic, phase shifter, transparent conductive oxide, 
+hydrogen-doped indium oxide 
+ 
+Thermo-optic (TO) phase shifters are very fundamental units in large-scale active silicon 
+photonic integrated circuits (PICs). However, due to the limitation of microheater materials 
+with a trade-off between heating efficiency and absorption loss, designs reported so far typically 
+suffer from slow response time, high power consumption, low yields, and so on. Here, we 
+demonstrate an energy-efficient, fast-response, and low-loss TO phase shifter by introducing 
+hydrogen-doped indium oxide (IHO) films as microheater, and the optimized electron 
+concentration with enhanced mobility endows the IHO high conductivity as well as high near-
+infrared (NIR) transparency, which allow it to directly contact the silicon waveguide without 
+any insulating layer for efficient tuning and fast response. The TO phase shifter achieves a sub-
+microsecond response time (970 ns/980 ns) with a π phase shift power consumption of 9.6 mW. 
+And the insertion loss introduced by the IHO microheater is ~ 0.5 dB. The proposed IHO-based 
+microheaters with compatible processing technology illustrate the great potential of such 
+material in the application of large-scale silicon PICs. 
+ 
+1. Introduction 
+ 
+Silicon photonic integrated circuits (PICs) trend to lie at the heart of the communications 
+revolution owing to the low power consumption, high speed, large bandwidth and 
+
+ 
+ 
+2 
+ 
+complementary metal-oxide semiconductor (CMOS) compatibility. Recently, emerging 
+applications of PICs such as optical phased arrays (OPAs),[1-3] optical neural networks 
+(ONNs),[4-6] integrated photonic quantum,[7, 8] and so on are also attracting increased interest. 
+The phase shifter is an indispensable fundamental photonic device for its phase tuning 
+capability, and its large-scale integration is in demand in the above various applications. Due 
+to the high thermo-optic (TO) coefficient (1.84 × 10−4 K−1) of silicon,[9] phase tuning of silicon 
+waveguide can be achieved easily by heating using a microheater, and such TO tuning as an 
+efficient way in phase shifter has been widely used in large-scale silicon PICs by virtue of the 
+advantages of simple design, easy fabrication, low cost and compactness.[10-13] Unfortunately, 
+the conventional metal-based microheaters for TO tuning require a thick insulating layer (i.e., 
+SiO2 layer) to avoid large absorption loss but would result in high power consumption and slow 
+response speed, which become an obstacle in the application scenarios desiring fast-response 
+TO tuning, such as fast beam scanning of OPAs and fast switching of neurons in ONNs. 
+Recently, designs by placing the metal quite close to the waveguide have been proposed, which 
+can realize low-loss devices based on non-Hermitian system theory, thereby improving the 
+efficiency and speed of TO tuning.[14, 15] However, this approach requires a high-precision 
+fabrication process, having difficulties in large-scale PIC applications. To remove the insulating 
+layer, emerging two-dimension (2D) materials, such as graphene,[16-18] phosphorene,[19, 20] 
+MXenes,[21] and so forth, have been demonstrated to directly overlay waveguides as the 
+microheaters and contribute excellent performances in TO tuning, but the sophisticated 
+preparation and transfer processes of 2D materials also face the difficulties in practical PIC 
+applications due to the low yield. Doped-silicon microheaters can directly generate heat inside 
+the waveguide to achieve high-speed TO tuning,[22-25] but the loss induced by free carriers turns 
+into a challenging issue.  
+Transparent conductive oxides (TCOs) can be the alternative materials for microheaters, and 
+their mature and reliable mass-production processes provide the practical application in large-
+scale silicon PICs. So far, the dominant TCO known as tin-doped indium oxide (ITO) has been 
+widely used in optoelectronic devices such as photovoltaic cells, displays, touch screens, etc.[26, 
+27] Most of TCOs with relatively high free carrier concentration (n > 5 × 1020 cm-3) can be 
+transparent in the visible spectral range but always show a strong absorption due to the plasma 
+oscillations in the near-infrared (NIR) waveband,[28] which increases the insertion loss of the 
+phase shifter. Although such an issue can be alleviated by decreasing the doping amount in 
+TCOs for lower n, the enhanced NIR transmittance would accompany the decrease of electrical 
+conductivity (σ) of TCOs which may weaken the heating performance. Fortunately, the 
+
+ 
+ 
+3 
+ 
+conductivity is determined by both carrier concentration and carrier mobility (μ) and can be 
+expressed as σ = enμ (e is the elemental charge). Therefore, enhancing μ but lowering the n can 
+be an effective way to balance the conductivity and NIR transparency. Nowadays, researchers 
+have demonstrated to replace the tin doped in ITO with other metals for higher μ, such as 
+titanium,[29, 30] tungsten,[31, 32], and molybdenum,[33-35] and prove that the TCO films possess the 
+higher transparency in the NIR waveband and can maintain the comparable σ simultaneously. 
+Nevertheless, the growth temperatures of these TCOs require relatively high (> 300 ℃), which 
+exceeds the melting point of photoresists for micro-pattern manufacturing. 
+In this paper, we propose and experimentally demonstrate a TO phase shifter based on 
+hydrogen-doped indium oxide (IHO) microheater. We fabricate an IHO microheater for phase 
+shifter through room-temperature magnetron sputtering with high-temperature post-annealing 
+treatment. The deposited IHO films possess high µ and moderate n, thus can achieve 
+appropriate sheet resistance and high NIR transparency simultaneously. The compatible 
+fabrication process makes the IHO more suitable for patterning by mature lift-off process and 
+exhibits great potential for on-chip applications. The IHO microheater is designed to directly 
+contact the silicon waveguide without any insulating layer for efficient TO tuning and fast 
+response. The high NIR transparency of IHO ensures low insertion loss of the phase shifter. We 
+measure the TO tuning performance of the proposed phase shifter through the conventional 
+Mach–Zehnder interferometer (MZI) structure. Its fastest response time is 970 ns in rising time 
+and 980 ns in falling time, with a π phase shift power consumption of 9.6 mW, and the insertion 
+loss introduced by the corresponding IHO microheater is about 0.5 dB. Our work illustrates the 
+great potential of IHO to be a novel microheater material for large-scale silicon PICs. 
+ 
+2. Results 
+ 
+2.1. Chip Design and Fabrication 
+ 
+A phase shifter based on IHO microheater is designed on a silicon-on-insulator (SOI) substrate 
+with a top silicon thickness of 220 nm. Figure 1a exhibits the details of the complete fabrication 
+process of the phase shifter. First, we use electron beam lithography (EBL) and inductively 
+coupled plasma (ICP) etching to fabricate the MZI structure. The strip waveguide width is 550 
+nm and the thickness is 220 nm. Then, the IHO microheaters are directly deposited on the 
+waveguide through magnetron sputtering and the geomtery of the pattern can be well controlled 
+
+ 
+ 
+4 
+ 
+through lift-off process. The entire device is fabricated under standard silicon photonics 
+fabrication processes (no air-trench or undercut process). 
+ 
+Figure 1. a) The schematic of the device fabrication process. b) The microscope image of the 
+fabricated MZI structure. Insets: the zoom-in images of the IHO films. c) The SEM image of 
+the cross-section of the phase shifter. IHO film is false-colored. d)The thickness measurement 
+result of the IHO film. 
+ 
+Figure 1b exhibits a microscope image of the fabricated MZI structure. The insets show the 
+zoom-in images of the IHO films on the waveguides. Both arms of the MZI structure are 
+covered with IHO films to balance the loss, and two additional IHO pads are deposited on both 
+sides of the MZI tuning arm for contact with the probes. The length of the IHO film covering 
+on the waveguide is 10 μm along the light transmission direction. A pair of grating couplers 
+designed for TE polarization are used coupling to the input/output fibers. Figure 1c presents the 
+scanning electron microscope (SEM) image of the cross-section of the phase shifter. The 
+thickness of the IHO film is measured to be 66 nm with an atomic force microscope (AFM), as 
+shown in Figure 1d. 
+ 
+2.2. Chip Characterization and Measurements  
+ 
+
+Si
+Si
+Si
+SiO2
+siO2
+SiO2
+SiSubstrate
+SiSubstrate
+SiSubstrate
+Si
+si02
+siO2
+SiO2
+SiSubstrate
+SiSubstrate
+SiSubstrate
+(b)
+10μm
+(c)
+10um
+100μm
+10.0
+20.0
+um 
+ 
+5 
+ 
+ 
+Figure 2. a) Comparison of transmittance between IHO and ITO. b) Normalized transmission 
+spectra for the MZI with and without IHO. Insets: the zoom-in images around 1550 nm.c) 
+Simulated optical E field distribution for the fundamental mode at a wavelength of 1550 nm. d) 
+Simulated temperature distribution at a heating power of 9.6 mW. 
+ 
+Figure 2a compares the transmittance of the prepared IHO film and commercial ITO film. It 
+can be seen that the IHO possesses significantly higher transmittance than ITO in the NIR 
+waveband under same sheet resistance. In particular, for a 66 nm thick IHO film (corresponding 
+to the solid orange line), it shows a transmittance over 97% in the C+L band (1530 nm - 1565 
+nm and 1565 nm - 1625 nm). The measured transmission spectra of the fabricated MZI structure 
+with and without IHO microheater are depicted in Figure 2b (orange line and blue line, 
+respectively). The results are normalized to the reference strip waveguide to exclude the 
+coupling loss of the grating couplers. The IHO microheater introduces an excess loss of about 
+0.5 dB (as shown in the insect) and has little effect on the extinction ratio of the MZI structure. 
+Figure 2c, d present the simulated images of the optical E field distribution and temperature 
+distribution in a cross-section of the phase shifter based on the finite element method (FEM), 
+respectively.  
+ 
+
+100
+Without IHO -
+-With IHO
+(dB
+80
+5
+60
+L-band
+10
+C-band
+40
+15
+—IHO 80 Q/sq
+3
+- -ITO 80 Q/sq
+20
+IHO 15 Q/sq
+20
+-ITO 15 Q/sq
+1549.9
+1550
+1550.1
+25
+500
+1000
+1500
+2000
+2500
+1549
+1550
+1551
+Wavelength (nm)
+Wavelength(nm）
+Electric Field (V/m)
+×107
+TemperaturefK
+IHO
+335
+5
+330
+SiO2
+325
+4
+320
+3
+315
+310
+2
+305
+1
+300
+Si Substrate
+295 
+ 
+6 
+ 
+ 
+Figure 3. a) The transmission spectra at different heating powers. b) The waveform of the 
+driving signal (orange line) and corresponding output signal (blue line). 
+ 
+In order to measure the spectral responses, different driving power are applied to the IHO 
+microheater. The measured spectra are presented in Figure 3a. The shift of the interference dip 
+at approximately 1550.48 nm reaches 1/2 free spectral range (FSR) with a tuning power of 9.6 
+mW. The response time of the device is further characterized by driving the IHO heater with a 
+square waveform electrical signal while the input wavelength is fixed at 1550.47 nm. The 
+frequency of the driving signal is set to 50 kHz with a peak-to-peak voltage (Vpp) of 6.0 V. The 
+Vpp of 6.0 V corresponds to an applied heating power of 9.6 mW. The waveform of the output 
+signal is measured via a photodetector followed by an oscilloscope. The driving signal and 
+corresponding output signal are shown in Figure 3b. The 10–90% rising and falling times are 
+measured to be 970 and 980 ns, respectively.  
+ 
+2.3. Performance Comparison 
+We compared the performance of our proposed phase shifter with the state-of-art TO 
+modulation devices in recent years,[14-16, 18, 20, 22-25, 36-39] as shown in Figure 4. The lower π phase 
+shift power consumption (Pπ) of the phase shifter is more beneficial to reduce the overall power 
+consumption of the large-scale PICs, and the shorter response time stands for better system 
+performance. As a comparison, we focus on the cases of MZI structures, and there is no air-
+trench or undercut process during device fabrication, just as our work. Besides, the response 
+time is estimated by τ =  0.35/BW if thus parameters were not offered.[24] Compared with 
+other schemes, our work achieves both low power consumption and fast response, second only 
+to 2D material-based microheaters in terms of overall modulation performance. Furthermore, 
+
+Voltage (V)
+NormalizedPower (dB)
+Voltage (a.u.)
+970 ns
+980 ns
+25
+0mW
+-----2.1 mW -----6
+6.0mW
+-0.8mW ---.-4.0mW
+9.6mW
+30
+1549
+1550
+1551
+5
+10
+15
+20
+Wavelength(nm)
+Time (μs 
+ 
+7 
+ 
+the proposed device structure is simple to fabricate and has great potential for use in large-scale 
+PIC applications. 
+ 
+ 
+Figure 4. Performance comparison of the state-of-art TO modulation devices. 
+ 
+3. Conclusion 
+In conclusion, we propose and experimentally demonstrate a TO phase shifter based on IHO 
+microheater directly overlaying on the silicon waveguide. The phase shifter has an insertion 
+loss as low as about 0.5 dB. Based on the MZI structure, it exhibits a low π phase shift power 
+consumption of 9.6 mW, and a fast response of 970 ns in rising time and 980 ns in falling time. 
+Besides, the IHO film is easy to be patterned on chips. Therefore, our work not only 
+demonstrates an efficient, fast-response, and low-loss TO phase shifter, but also provides a 
+novel design route for optical modulation devices in large-scale PICs. 
+ 
+4. Experimental Section  
+Materials: IHO film deposited by magnetron sputtering. Firstly, a high-purity ceramic In2O3 
+target (99.99% purity, Zhongnuo Advanced Material Technology Co., Ltd) is used as a 
+sputtering target. The target is put into radio frequency (RF) magnetron sputtering system 
+(Beijing Technol Science Co., Ltd). When the base pressure is less than 10-4 Pa, the gas that 
+both argon and hydrogen argon mixture (95%Ar and 5%H2) is introduced to the system and 
+various Ar and Ar/H2. After the gas pressure stabilizes at 0.16 Pa, the sputtering power was set 
+at 150 W for 210 seconds at room temperature (300 K). Finally, the as-deposited IHO films 
+
+[39]
+[20]
+2DMaterials
+TCOs
+DopedSilicon
+Metals
+10
+[38]
+Response time
+[36]
+[37]
+.
+[18]
+[25]
+[22]
+[24][23]
+14]
+[15]+
+100
+Thiswork
+[16]
+1020
+30
+4050
+60
+(mw) 
+ 
+8 
+ 
+were annealed at 250 ℃ for 150 seconds in N2 atmosphere, respectively. All the ITO films was 
+brought by Foshan Shi Yuan Jing Mei Glass Co.,Ltd. 
+Device characterization: We use an amplified spontaneous emission (OS8143) optical 
+source to generate a C-band broadband signal, and a power supply (Rohde&Schwarz 
+HMP4040) to provide a static electrical signal, and finally record the spectral response of the 
+MZI structures by an optical spectrum analyzer (Yokokawa AQ6319). When we measure the 
+dynamic response, the power supply is replaced by an arbitrary waveform generator (RIGOL 
+DG4202), and the input light is emitted from a tunable laser source (ID-Photonics CoBriteDX4). 
+The output signal is measured via a photodetector (Discovery DSC40S) followed by the 
+oscilloscope (RIGOL DG4022). A polarization controller is used to select the TE polarization 
+state of the input light before it is injected into the chip. All the measurements are performed 
+under room temperature in ambient conditions. The specular transmittance of the samples 
+(range from 0.3 μm to 2.5 μm) was characterized by the UV-Vis-NIR spectrophotometer 
+(Shimadzu UV-3600 Plus). . 
+ 
+Acknowledgements 
+This research is supported by the Natural Science Foundation of China (U21A20511, 
+62274071), 
+the 
+National 
+Key 
+Research 
+and 
+Development 
+Program 
+of 
+China 
+(2022YFB2804200), and the Innovation Project of Optics Valley Laboratory (Grant No. 
+OVL2021BG001). The authors thank the Optoelectronic Micro&Nano Fabrication and 
+Characterizing Facility of Wuhan National Laboratory for Optoelectronics for the support in 
+device fabrication. Weiyu Tong and Erqi Yang contributed equally to this work. 
+ 
+Conflict of Interest 
+The authors declare no conflict of interest. 
+ 
+Data Availability Statement 
+The data that support the findings of this study are available from the corresponding author 
+upon reasonable request. 
+ 
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+

L9FRT4oBgHgl3EQf2Ti5/content/tmp_files/2301.13660v1.pdf.txt

@@ -0,0 +1,1059 @@
+Non-Hermitian Z2 Photonic Topological Insulators
+Rodrigo P. Câmara,1 Tatiana G. Rappoport,1, 2 and Mário G. Silveirinha1
+1Instituto Superior Técnico and Instituto de Telecomunicações,
+University of Lisbon, Avenida Rovisco Pais 1, Lisboa, 1049001 Portugal
+2Instituto de Física, Universidade Federal do Rio de Janeiro,
+C.P. 68528, 21941-972 Rio de Janeiro RJ, Brazil
+(Dated: February 1, 2023)
+Photonic platforms invariant under the action of parity, time-reversal, and duality (P · T · D)
+operators may host topological phases that are the counterpart of those supported by time-reversal
+invariant Z2 electronic systems. Here, we uncover the robustness of the Z2 photonic phases to non-
+Hermitian effects, e.g., to material dissipation. In particular, it is shown that non-Hermitian PD
+reciprocal photonic insulators may be split into two classes of topologically inequivalent materials,
+whose interfaces may support spin-polarized states. To illustrate our theory, we study in detail
+how the non-Hermitian effects may sculpt the topology of a PD symmetric parallel-plate waveguide
+(PPW). We find that there is a critical level for the plates loss that separates two different topological
+phases. We identify the phases by their spin Chern numbers and analyze the transition.
+Topological photonics1,2 determines a rather unique
+paradigm to create unidirectional channels immune to
+back-scattering.
+The topological protection of Chern
+insulators originates from an electromagnetic “quan-
+tum Hall effect” rooted in the time-reversal symmetry
+breaking3,4. In analogy with the quantum spin Hall effect
+in electronic systems5, time-reversal invariant photonic
+structures may also host topological phases6–16.
+How-
+ever, due to the different nature of fermionic and bosonic
+particles, additional symmetries must be enforced on a
+system to guarantee the Z2 topological protection, most
+notably a duality symmetry that ensures balanced elec-
+tric and magnetic responses10.
+To our best knowledge, the most general symmetry
+transformation that enables the Z2 topological protection
+in photonic systems is determined by a combination of
+the parity (P), time-reversal (T ) and duality (D) opera-
+tors, ˜T = PT D10,17–19. The ˜T operator may be regarded
+as a pseudo-time-reversal operator, as it is anti-linear and
+satisfies ˜T 2 = −1. Such a property guarantees that the
+photonic states of a PT D-symmetric system are degen-
+erate (Kramers theorem)10,17 and that there is a suitable
+basis in which the scattering matrix is anti-symmetric17.
+The latter property implies that when a PT D-symmetric
+system supports an odd number of bi-directional chan-
+nels, it is feasible to have bi-directional transport of light
+without any back-reflections17.
+Previous studies on PT D-symmetric systems dealt
+exclusively with energy conserving (Hermitian) plat-
+forms.
+The study of topological phases was recently
+extended to non-Hermitian systems through a gen-
+eralization of the notion of band gaps to complex
+spectra20,21.
+As a result, topological properties with
+no counterpart in nonreciprocal3,4,22–27 and time-reversal
+invariant6–15 closed systems have been found21,28,28–38.
+Exceptional points (parameter space points where ener-
+gies and associated modes coalesce simultaneously)39–43)
+are linked to exotic effects44–50 and unprecedented phases
+of matter21,31–38.
+Here, we study the impact of non-Hermitian effects,
+Figure 1:
+Structure of the PD-symmetric parallel-plate
+waveguide with a dielectric block (air) of width a in between
+two impedance surfaces.
+notably material dissipation, on the topological phases
+of PT D symmetric systems. We find that the Z2 topo-
+logical index is highly robust to the presence of material
+absorption, and most interestingly we identify a topo-
+logical phase transition controlled by the loss parameter
+strength.
+Amongst the different Z2 photonic topological insula-
+tors considered in previous works18,19,51–53, we are in-
+terested in a generalization of a structure introduced in
+Ref.
+18.
+It is a parallel-plate waveguide (PPW) [see
+Fig. 1] with the electromagnetic response of the top and
+bottom plates linked by electromagnetic duality. Each
+plate is modeled through an impedance (Leontovitch)
+boundary condition such that Ziˆn × H = Etan with ˆn
+the unit normal vector to the plate (directed towards
+the dielectric), Etan = E − ˆn (ˆn · E) is tangential electric
+field, and Zi is the surface impedance of the i = +, −
+(top/bottom, respectively) plate. In order that the sys-
+tem is parity-duality PD symmetric it is necessary that
+Z+Z− = Z2
+0 with Z0 the vacuum impedance.
+Similar
+to Ref.17, we define the parity transformation such that
+(x, y, z) → (x, y, −z) and the duality transformation such
+that D : (E, H) → (HZ0, −E/Z0) [see Supplementary
+Material 54]. The parity transformation exchanges the
+position of the two plates, whereas the duality mapping
+transforms Z+ → Z− and vice-versa.
+Thus, the PD
+transformation leaves the system invariant when the re-
+gion in between the plates is filled with air. The condition
+Z+Z− = Z2
+0 implies that the impedances of the plates are
+arXiv:2301.13660v1  [physics.optics]  31 Jan 2023
+
+Z+ = Zopeis
++a/2
+air
+Z_ = Zop-le-i
+22
+of the type Z+ = Z0ρeiφ and Z− = Z0ρ−1e−iφ, with ρ
+and φ dimensionless parameters.
+Thus, when the top
+plate has an inductive response (φ < 0, for a time vari-
+ation e−iωt) then the bottom plate has a capacitive re-
+sponse φ > 0, and vice-versa. In the limit ρ → 0 the top
+(bottom) plate behaves as a perfect electric (magnetic)
+conductor, i.e., a PEC (PMC). For simplicity, to avoid
+complications related to material dispersion, in this Let-
+ter we shall suppose that the surface impedances Z± are
+real-valued (φ = 0) and frequency independent. Then,
+for 0 < ρ < ∞ the waveguide is non-Hermitian as both
+the top and bottom plates absorb the fields propagating
+in the dielectric. The system is time-reversal invariant
+(and PT D-symmetric) only when ρ = 0, i.e., for PEC
+and PMC walls. For generality, we shall also consider
+the case of a negative resistivity, ρ < 0, which models the
+case of “gainy” (active) plates. Even though the propaga-
+tion of spin-polarized edge modes in related guides was
+discussed in different works18,55, this type of system still
+lacks of an appropriate topological classification, even in
+the Hermitian case.
+For PD symmetric reciprocal
+systems,
+such that
+ε(x, y, z) = µ(x, y, −z), the frequency-domain (source-
+free) Maxwell equations can be split into two independent
+sets of equations18
+±
+ic
+ε(z)
+�
+�
+0
+∂z
+∂y
+−∂z
+0
+−∂x
+∂y
+−∂x
+0
+�
+� Ψ±(−z) = ωΨ±(z),
+(1)
+that are formally equivalent to
+ˆH±Ψ±(z) ≡ ωΨ±(z)
+with
+the
+pseudospinors
+Ψ±
+written
+in
+terms
+of
+the
+electric
+and
+magnetic
+fields
+E
+and
+H
+as
+Ψ±(z)
+=
+(Ex(z) ∓ Z0Hx(−z), Ey(z) ∓ Z0Hy(−z),
+Ez(z) ± Z0Hz(−z))⊺18.
+Here, c denotes the speed of
+light in vacuum and ω is the oscillation frequency. For
+conciseness we denote a generic point of space (x, y, z) as
+(z), and the mirror-symmetric point (x, y, −z) as (−z).
+As E = 1
+2 (Ψ+ + Ψ−), the dynamics of a generic electric
+field distribution is fully controlled by the dynamics of
+the two pseudospinors [Eq. (1)]. Notably, different from
+the original Maxwell’s equations the dynamics of the
+pseudo-spinors is strongly nonlocal, as the two sides
+of Eq.
+(1) are evaluated at mirror-symmetric points.
+Interestingly, the pseudospinor decomposition holds true
+even when the dielectric response is lossy (e.g., if ε is
+frequency dependent and complex-valued) or when the
+plate walls have a non-Hermitian response.
+In the Hermitian case, when the dielectric and the
+plates are lossless, the class of eigenstates with pseu-
+dospin “+” can be transformed into eigenstates with pseu-
+dospin “-” using the time-reversal operator. Even though
+such a construction is not feasible in the non-Hermitian
+case, the key observation is that the electromagnetic reci-
+procity of the system guarantees that the total topologi-
+cal charge is zero20, similar to the fermionic case case56.
+Thereby the topological charge of the operators ˆH± in
+Eq. (1) must be exactly symmetric: C+ + C− = 0. Here,
+C± are the spin Chern numbers associated with a band
+gap of interest. Therefore, PD symmetric reciprocal sys-
+tems may host nontrivial topological phases determined
+by the invariant C+ = −C−. In the following, we apply
+the non-Hermitian topological band theory to determine
+the topological phase diagram21.
+The dielectric in Fig. 1 is taken as air (ϵ = µ = 1),
+so that non-Hermitian effects arise exclusively due to
+the plates.
+As the system is invariant under continu-
+ous translations in the xoy plane, the pseudospin states
+can be factorized as Ψ±(r, t) = Ψ±
+k (z)e−iωteik·r, where
+k is the in-plane real-valued wave vector (k · ˆz = 0) with
+magnitude k. The eigenstates are found by solving the
+Maxwell’s equations subject to the appropriate boundary
+conditions in the original guide, followed by a projection
+onto the two pseudospinor subspaces54. As the guide is
+invariant under arbitrary rotations around the z-axis it
+is convenient to write a generic eigenstate in terms of
+the vectors (ˆk, ˆz × ˆk, ˆz). The coordinates of a generic
+pseudospin state with eigenfrequency ωn = ωn(k) in this
+basis are:
+Ψ±
+k,n(z) →
+�
+�
+∓
+�
+e−iκnz + hneiκnz�
+− ωn
+cκn
+�
+eiκnz − hne−iκnz�
+∓ k
+κn
+�
+e−iκnz − hneiκnz�
+�
+�
+(2)
+where hn ≡ eiκna ωn−cρκn
+ωn+cρκn , κn = κ′
+n+iκ′′
+n is the transverse
+wavenumber, and the label n = 1, 2, ... identifies the dif-
+ferent modes. For each k real-valued, the allowed values
+for κn can be found by solving the modal condition
+e2iκna = ωn + cρκn
+ωn − cρκn
+ρωn + cκn
+ρωn − cκn
+.
+(3)
+where ωn = ω′
+n + iω′′
+n = s × c
+�
+k2 + κ2n with the square
+root branch determined by s = ±. The dispersion ω(+)
+n
+of
+the branches with positive frequencies (s = +) is linked
+to the dispersion ω(−)
+n
+of the branches with negative fre-
+quencies (s = −) as ω(+)
+n
+= −ω(−)∗
+n
+. This property is
+a consequence of the reality of the electromagnetic field
+(particle-hole symmetry). Both κn and ωn are functions
+of the in-plane momentum k and of the resistivity ρ.
+Moreover, one can always assume that Re {κn} > 0. The
+modal equation is independent of the pseudospin, and
+thereby the spectrum of the operators ˆH± is identical.
+Albeit Eq.(3) is transcendental, its solutions for k = 0
+can be analytically determined as κn = (2n − 1)π/2a −
+s × iF(ρ)/a54 with s = ± as before and F(ρ) =
+sgn(ρ) ln
+��� |ρ|+1
+|ρ|−1
+���. The profiles κn (k) for k ̸= 0 are ob-
+tained from a numerical continuation of the k = 0 solu-
+tions via a Nelder-Mead minimization scheme57.
+Figures 2(a) and 2(b) show the real and imaginary
+parts of κn(k) for the first few guided modes for a sys-
+tem with ρ = 0 (dashed lines) and ρ = 0.6 (solid lines).
+The corresponding oscillation frequencies ωn(k) are de-
+picted in Figs. 2(c) and 2(d). When the system is con-
+servative (ρ = 0), the modal equation (3) reduces to
+e2iκna = −1.
+Its solutions κn = (2n − 1)π/2a with
+
+3
+Figure 2:
+Dispersion diagrams for a PD-symmetric waveg-
+uide with the top wall characterized by ρ = 0 (dashed curves)
+or by ρ = 0.6 (solid curves). (a) Real κ′
+n and (b) imaginary κ′′
+n
+parts of the modal transverse wavenumber as functions of k
+for the modes with n = 1, 2, 3. (c) Real ω′
+n and (d) imaginary
+ω′′
+n parts of the corresponding frequency bands ω(+)
+n
+(k). The
+imaginary parts in (b) and (d) are normalized to the func-
+tion F(ρ) sketched in panel (f). (e) Projected band structure
+in the complex plane for the case ρ = 0.6. The red and blue
+curves represent the positive and negative frequency branches
+ω(+)
+n
+(k) and ω(−)
+n
+(k), respectively. The frequencies associated
+with k = 0 are represented by the solid dots.
+n = 1, 2, ... are real-valued and independent of the in-
+plane momentum k. Thus, as expected, the oscillation
+frequencies ω(±)
+n
+= ±c
+�
+k2 + κ2n are real-valued in this
+case.
+In contrast, in the dissipative (non-Hermitian) case
+(ρ = 0.6), both κn and ωn are complex-valued.
+Fur-
+thermore, as seen in Fig. 2(a), for lossy plates the real
+part of κn acquires a positive offset of π/2a as k ap-
+proaches infinity. In fact, when k → ∞ with ρ ̸= 0 the
+modal equation reduces to e2iκna = 1 (the terms involv-
+ing κn are negligible compared to the terms involving ωn
+in Eq. (3), which yields κn(∞) = 2nπ/2a. Moreover, in
+the k → ∞ limit the in-plane wavelength becomes much
+shorter than the plates distance a, and thereby the dis-
+persion of the modes approaches asymptotically that of
+the dielectric so that ωn(k) ≈ ck [see Figs. 2(c) and 2(d)].
+The imaginary part of the complex frequency ωn de-
+termines the decay rate (in time) of the mode, which is
+a clear fingerprint of the non-Hermitian dynamics. In-
+terestingly, the decay rate is boosted for ρ ∼ ±1, which
+correspond to the singularities of the function F(ρ) [see
+Fig. 2(f)]. Heuristically, this happens because for ρ ∼ 1
+there is an impedance match between the plates and the
+dielectric region. Defining δ = 1 − |ρ| as the offset of |ρ|
+from a critical point, the logarithmic nature of the singu-
+larities κn ≈ ±ia−1 ln |δ/2| and ωn ≈ ±ica−1 ln |δ/2| is
+unravelled for |δ| ≪ 1 with ρ → ±1, at k = 0. This corre-
+sponds to a time variation of the type |δ/2|±ct/a so that
+both pseudospins are either rapidly suppressed or ampli-
+fied in these resonant conditions (ρ = ±1 for lossy/gainy
+walls). It is useful to note that for ρ = ±1 the top and
+bottom plates are identical.
+In fact, replacing ρ with
+1/ρ is equivalent to interchange the the top and bottom
+plates due to the PD symmetry of the system.
+Figure 2(e) represents the projected band structure (lo-
+cus of ω(±)
+n
+(k) for all k real-valued) for ρ = 0.6. The fig-
+ure shows both the positive frequency ω(+)
+n
+(k) and neg-
+ative frequency ω(−)
+n
+(k) branches. As seen, a band gap
+separates the positive and negative frequency parts of the
+spectrum (beige rectangular region centered at ω = 0).
+Similar band structures are obtained for other values of
+ρ, provided ρ is not near the critical points ±1. The pro-
+jected band structure lies in the lower-half (upper-half)
+frequency plane for ρ > 0 (ρ < 0), i.e., for lossy (gainy)
+plates. Furthermore, due to the particle-hole symmetry
+(ω(+)
+n
+= −ω(−)∗
+n
+) there is a mirror symmetry about the
+imaginary frequency axis. Typically, different branches
+of the projected band structure are disjoint and there
+are no intersections (or self-intersections). However, as
+further discussed below, for ρ ∼ ±1 the band gap cen-
+tered about the imaginary axis can close and the different
+branches can exchange topological charge.
+The
+topological
+classification
+of
+a
+generic
+non-
+Hermitian operator
+ˆH ̸=
+ˆH† requires considering a
+biorthogonal basis of left φL
+n and right φR
+n eigenstates
+such that ˆH†φL
+n = E∗
+nφL
+n and ˆHφR
+n = EnφR
+n, with En
+the eigenvalues of ˆH. The left and right eigenstates are
+normalized as
+�
+φL
+n|φR
+m
+�
+= δnm. In our system, switch-
+ing loss into gain is equivalent to switch the sign of ρ.
+Thus, the operator ˆH† models a PPW with resistivity
+−ρ. This means that ΨL
+n(z; ρ) = ΨR
+n(z; −ρ). The eigen-
+functions of the adjoint operator can then be obtained
+from Eq. (2) using the rule ρ → −ρ (the same rule is
+applied to the dispersion equation (3); in particular, we
+have ωn(−ρ) = ω∗
+n(ρ) and κn(−ρ) = κ∗
+n(ρ)).
+We use the non-Hermitian topological band theory21
+to determine the Chern numbers C±
+n associated with the
+photonic branches of the operator ˆH±.
+As previously
+noted, for ρ not too close from the the singularities ±1
+the branches are well separated and hence their individ-
+ual topological charge is well defined. We introduce the
+Berry potential A±
+k,n = i ⟨Ψ±
+k,n(−ρ)|∇kΨ±
+k,n(ρ)⟩ con-
+structed from normalized versions Ψ±
+k,n of the pseu-
+dospinors in Eq.(2) satisfying ⟨Ψ±
+k,n(−ρ)|Ψ±
+k,n(ρ)⟩ = 121.
+The inner product of two vectors F and Q is defined
+as ⟨F (z)|Q(z)⟩ =
+� a/2
+−a/2 dz F †(z) · Q(z).
+In our sys-
+tem, the wave vector k is unbounded in the Euclidean
+plane but the latter may be compactified into the Rie-
+mann sphere, following Ref.58.
+The topology is thus
+well-defined because the eigenfunctions in Eq.(2) are ei-
+ther real-valued or pure-imaginary in the limit k = ∞,
+
+0
+(二/) /uy
+(d)1/0uy
+5
+3
+(a)
+(b)
+1
+(/ )0 /0um
+(d)10 /pum
+11
+(c)
+９７５
+S
+n
+3
+(d)
+2
+3
+1
+0
+5
+10
+15
+20
+0
+10
+20
+30
+40
+ka
+ka
+0
+(d)10 /0"m
+5
+8
+0
+个
+L
+(e)
+(f)
+k
+-20 -10
+10
+20
+0
+1
+0
+1
+p4
+rendering A±
+k,n null in that gauge.
+In other words,
+the Hamiltonian ˆH± is well-behaved at k = ∞58. For
+the same gauge, the Berry potential is smooth in the
+entire momentum space, except possibly at k = 0.
+Then, applying the Stokes’ theorem one sees that the
+Chern number of the n-th branch is given by C±
+n
+=
+(2π)−1 [limk→∞ − limk→0+] k
+� 2π
+0
+dφ A±
+k,n · ˆφ58.
+Here,
+(k, φ) ∈ [0, ∞[ × [0, 2π[ determine a system of polar co-
+ordinates in the k-space. The azimuthal component of
+the Berry potential A±
+φ,n = A±
+k,n · ˆφ is independent of
+the orientation of the wave vector k, as the waveguide
+is unchanged under arbitrary rotations about the z-axis.
+Taking into account that A±
+k,n vanishes at infinity, we
+conclude that C±
+n = − limk→0+ k A±
+φ,n(k). A detailed cal-
+culation shows that54
+C+
+n = s × (−1)n+1sgn(δ)
+with
+δ = 1 − |ρ| ,
+(4)
+where the s = ± sign is determined by the frequency
+branch (ω(s=±)
+n
+). As previously noted, due to reciprocity
+C−
+n = −C+
+n . Remarkably, the different topological phases
+are separated by the critical values ρ = ±1.
+Figures 3(a) and 3(b) illustrate how higher-order pho-
+tonic branches (n = 2, 3, ...) remain disconnected in
+the vicinity of ρ = 1.
+They also hint the divergences
+ω′′
+n → −∞ at k = 0 and ρ = 1, by approaching the
+singularity as δ varies from 10−3 to 10−6.
+In particu-
+lar, these properties explain why the topological charge
+determined by Eq. (4) changes at the critical resistivity
+ρ = 1, as the band gap closes when the branches touch
+at the divergence point.
+Importantly, the band gap that separates the posi-
+tive and negative frequency spectra [see Fig. 3(c)] closes
+for any value of the normalized resistivity in the range
+0.9434 < |ρ| < 0.9434−1 [see Figs. 3(d) and 3(e)]. In
+fact, it turns out that the n = 1 bands, ω(+)
+1
+and ω(−)
+1
+,
+may intersect for ρ near the singularity. Depending on
+the value of |ρ|, the intersection may be an isolated point
+(Fig. 3(d)) or correspond to a line degeneracy over the
+imaginary frequency axis (Fig. 3(e)). In the former case,
+the intersection occurs for a in-plane momentum on the
+order of k ≈ 1.36/a. It is relevant to note that the range
+of ρ for which the n = 1 bands intersect is determined
+uniquely by the geometry of the problem, and is totally
+independent of the plate width a54.
+Evidently, for 0.9434 < |ρ| < 0.9434−1 the topological
+charge of the n = 1 bands is undefined. This range of
+ρ links the gapped phases through an intermediate col-
+lection of exceptional points (EPs). For ρ slightly larger
+than 0.9434 there is an isolated EP which unfolds into
+a line for larger values of ρ and then collapses into an
+isolated EP again as ρ approaches 0.9434−1.
+At the
+EPs, the eigenvectors of the n = 1 spinors coalesce as
+Ψ±
+k,1 ∥ Ψ±
+k,−1.
+The modal equation (3) is invariant under the trans-
+formation ρ → ρ−1 when ρ ̸= 0. This implies that the
+band structure remains the same under the transforma-
+tion ρ → ρ−1, i.e., ωn(k; ρ) = ωn(k; ρ−1). Due to this
+symmetry, the projected band structure near the singu-
+larity ρ = 1 is to a good approximation only a function
+of |δ|. Thus, the plots in Figs. 3(c), 3(d) and 3(e) can be
+readily extended to negative values of δ. Furthermore,
+taking into account that ωn(−ρ) = ω∗
+n(ρ) it is clear that
+the topological phase transitions develop in similar way
+around the points ρ = ±1 where energy is resonantly
+dissipated or absorbed.
+The closing and reopening of the gap mediates the ex-
+change of topological charge between the negative and
+positive frequency parts of the spectrum. At the phase
+transition, each and every band flips the sign of its spin
+Chern number as predicted by Eq.(4).
+The exchange
+of topological charge between negative and positive fre-
+quencies has been discussed previously for nonreciprocal
+Chern insulators59,60.
+Figure 3(f) presents the detailed topological phase di-
+agram of C+
+n . Interestingly, for a fixed ρ the Chern num-
+bers of the different bands alternate between +1 and −1.
+This implies that, while the topology of the individual
+bands is well-defined, the topology of the band gap itself
+is not. For instance, the gap Chern number for the pseu-
+dospin “+” and for |ρ| ≪ 1 is determined by the series
+C+
+gap = −1 + 1 − 1 + ..., which is non-convergent. The
+origin of the ill-defined topology is rooted in the particle-
+hole symmetry of photonic systems61. In principle, this
+problem may be fixed with a suitable dispersive model
+for the plates that ensures that in the ω → ∞ limit their
+electromagnetic response approaches that of the vacuum,
+so modes with sufficiently large n have a trivial topo-
+logical charge61. In the Supplementary Material 54, we
+show that in the dispersive case it is reasonable to assume
+that the topological phase transition is governed by the
+exchange of the topological charge of the n = 1 bands.
+Notably,
+a
+parity
+inversion
+PI
+:
+(x, y, z)
+→
+(−x, −y, −z) exchanges the positions of the top and bot-
+tom plates.
+Since the electric and magnetic fields are
+odd/even under a parity inversion, it also exchanges the
+spin of the modes and the Chern numbers transform as
+[C±
+n ]PI = C∓
+n = −C±
+n . This result holds true for a com-
+pletely arbitrary (e.g., dispersive and inhomogeneous)
+PD-reciprocal waveguide. In particular, it justifies why
+for our geometry exchanging the position of the plates
+(ρ → 1/ρ) flips all the Chern number signs54. Moreover,
+it indicates that the modes identified in18,55 that prop-
+agate at the boundary of a guide with PEC-PMC walls
+and the corresponding PI transformed guide with PMC-
+PEC walls may be understood as topologically protected
+edge states.
+The difference between the Chern numbers of a guide
+and its PI transformed counterpart is always even, mean-
+ing there is an even number of protected unidirectional
+states at the interface. In our specific system, we expect
+two edge states per pseudospin. For a fixed pseudospin,
+the edge states can be visualized as waves attached ei-
+ther to the top or to the bottom plates62.
+While for
+conventional insulators, it is not possible to guarantee
+protection against back-scattering when C+ − [C+]PI is
+
+5
+Figure 3: (a, b) Projected band structure in the complex plane for the first few higher-order bands ωn (n = 2, 3) with
+δ = 1 − |ρ| = 10−3 and δ = 10−6, respectively. The frequencies associated with k = 0 are represented by the solid dots. (c, d,
+e) Illustration of the topological phase transition. The band gap between the positive and negative frequency branches with
+n = 1 shown in (c) in beige closes at an isolated point k ≈ 1.36/a when δ reaches δ ≈ 57 × 10−3, as depicted in (d). In panel
+(e) the degeneracy widens to a line when the resistivity gets closer to the critical point ρ = 1 (δ = 15 × 10−3). (f) Topological
+phase diagram for the PPW in Fig. 1. The spin Chern numbers C+
+n [Eq. (4)] are indicated for the range of ρ for which the
+band gap is open.
+an even number, in the case of PD-symmetric systems
+the protection against back-scattering is always guaran-
+teed by the fact that the system Hamiltonian is a direct
+sum of the (uncoupled) ˆH± Hamiltonians.
+In summary, we uncovered the topological proper-
+ties of non-Hermitian PD-symmetric and reciprocal pho-
+tonic systems.
+We found that the Maxwell’s equa-
+tions can be split into two decoupled sets of spin-up
+and spin-down states, connected by the PD-symmetry,
+even in non-Hermitian materials. By focusing on PD-
+symmetric parallel-plate guides, we discovered a robust
+zero-frequency band gap that separates the positive and
+negative frequency parts of the spectrum.
+Through the use of non-Hermitian topological band
+theory, we determined the spin Chern numbers of the
+individual bands and observed a phase transition near
+ρ = ±1, separating two distinct topological phases. At
+the transition, the positive and negative frequency spec-
+tra merge creating either isolated exceptional points or a
+continuum of EPs in the form of a line segment.
+Our findings reveal that the bulk topologies of these
+PD-symmetric photonic structures are highly resilient to
+non-Hermitian effects and explain the presence of pro-
+tected edge modes at the interfaces of different PD-
+symmetric guides.
+Acknowledgments
+This work is partially supported by the IET under
+the A F Harvey Engineering Research Prize, by the
+Simons Foundation under the award 733700 (Simons
+Collaboration in Mathematics and Physics, “Harnessing
+Universal Symmetry Concepts for Extreme Wave Phe-
+nomena”), and by Fundação para a Ciência e a Tec-
+nologia and Instituto de Telecomunicações under project
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+Anomalous flux state in higher-order topological superconductors
+Yizhi You
+Department of Physics, Northeastern University, MA, 02115 USA
+(Dated: January 13, 2023)
+We investigate the anomalous flux state of interacting higher-order topological superconduc-
+tors (HOTSC) protected by rotation symmetries. By introducing a π superconducting flux in 2D
+HOTSC, we demonstrate the existence of a robust zero mode trapped at the flux center. Remark-
+ably, the rotation symmetry and fermion parity display projective representation inside the π flux
+with N = 2 supersymmetry algebra. A similar gapless flux pattern also exists in 3D HOTSCs, the
+flux lines of which carry anomalous helical modes that cannot be realized on purely one-dimensional
+lattice models. Notably, these exotic phenomena can be manifested in a 2D frustrated quantum mag-
+net whose low energy excitation characterizes emergent Majoranas with HOTSC band structure and
+the Z2 flux exhibiting supersymmetries.
+I.
+INTRODUCTION
+Higher-order topological superconductors (HOTSC)
+are novel forms of gapped quantum matter that host
+gappable surfaces but gapless corners or hinges in-
+between[1, 2]. Since their initial discovery, HOTSC and
+their descendants have been discussed extensively and
+have become an active area of theoretical and experien-
+tial research. Recent progress has included topological
+classifications[3–8], topological field theories[8, 9], and
+experimental realization of various classes of HOTSC[10–
+19].
+Despite the rapid progress in the understanding of
+higher-order topological superconductors from a band
+structure perspective [20–31], experimentally accessible
+fingerprints for observing HOTSC still remain challeng-
+ing in strongly correlated systems.
+Notably, the ob-
+servation of gapless Majorana modes at the corners
+or hinges does not fully guarantee that the bulk is a
+higher-order topological superconductor (HOTSC), as
+some of these gapless modes can potentially be anni-
+hilated via surface gap closing, even if the bulk spec-
+trum remains gapped[30, 32]. Alternatively, some higher-
+order topological superconductors (HOTSCs) can ex-
+hibit fully gappable boundaries, including corners and
+hinges [30], while still exhibiting a non-trivial entan-
+glement structure that distinguishes them from trivial
+superconductors. Previous works have established that
+higher-order topological insulators and superconductors
+can be probed by their geometric responses, such as the
+creation of lattice defects like dislocations or disclina-
+tions.
+By creating these defects, one can observe Ma-
+jorana zero modes inside the disclination point in two
+dimensions or chiral fermion modes localized at the dislo-
+cation/disclination lines in three dimensions[9, 33–39]. In
+contrast to these approaches, which are primarily based
+on the non-interacting limit, we aim to identify the uni-
+versal fingerprints and topological responses specifically
+for strongly interacting higher-order topological super-
+conductor (HOTSC) phases.
+In this work, we seek to unravel the nature of super-
+conducting π flux and their corresponding topological re-
+sponse features in 2D and 3D HOTSC. First, we begin
+with the 2D HOTSC model on a square lattice proposed
+in Eq. [40] protected by C4 symmetry. We demonstrate
+that creating a π superconducting flux engenders a pro-
+jective symmetry between C4 and fermion parity P, so
+the resultant flux state contains a protected two-fold de-
+generacy. Remarkably, this projective symmetry within
+the flux uniquely generates the N = 2 supersymmetry
+(SUSY) algebra in quantum mechanics[41].
+Motivated by these observations, we extend our hori-
+zon into frustrated spin systems whose emergent quasi-
+particles and flux excitations exactly reproduce the topo-
+logical feature of HOTSC[42]. Namely, we construct a
+bosonic spin- 3
+2 model whose low-energy excitations con-
+tain emergent Majoranas coupling with a Z2 gauge field.
+The Majorana forms a superconducting band akin to the
+2D HOTSC, while the Z2 gauge flux excitation carries
+N = 2 SUSY structure.
+In sectionIII, we examine the role of superconducting
+π flux in 3D HOTSC with CT
+4 symmetry. One of our
+main findings is that the flux lines inside the HOTSC
+trap 1D helical Majorana modes with an intrinsic quan-
+tum anomaly. In particular, the gapless modes inside the
+flux line are anomalous in the sense that the CT
+4 sym-
+metry will inevitably be broken if we gauge the fermion
+parity inside the flux line. Under this observation, the
+helical Majorana modes inside the flux exhibit global
+anomaly, signaling the impossibility of realizing them on
+an isolated one-dimensional lattice model. Notably, such
+quantum anomaly manifested by ‘conflict of the symme-
+tries’ had been widely observed in the surface theory of
+symmetry-protected topological phase[43, 44].
+Our result provides a new route to detect HOTSC in
+numerical simulations via flux responses. The projective
+symmetry in the 2D HOTSC flux state can be detected
+from the shift of the entanglement spectrum upon flux in-
+sertion. Suppose one creates a rotational symmetric cut
+of the ground state wave function after π flux insertion;
+the entanglement spectrum will display a robust two-fold
+degeneracy in all spectrum levels. This degeneracy per-
+sists even for small system sizes, where finite-size effects
+are inevitable. More precisely, the whole entanglement
+Hamiltonian develops a projective symmetry under rota-
+tion and fermion parity after flux insertion, which results
+arXiv:2301.04703v1  [cond-mat.supr-con]  11 Jan 2023
+
+2
+in a degenerate spectrum in the entanglement Hamilto-
+nian for both the ground state and highly excited states.
+Notably, this degeneracy in the entanglement spectrum
+is not a manifestation of the corner mode, but a conse-
+quence of the projective symmetry due to the anomalous
+flux. Our result suggests that the entanglement features
+of HOTSC also reveal unique properties of topological
+flux responses, and that the projective symmetry in the
+anomalous flux state can be observed from the entangle-
+ment Hamiltonian. This paves the way for a new promis-
+ing route for exploring HOTSC in numerical simulations.
+II.
+PROJECTIVE SYMMETRY IN THE
+SUPERCONDUCTING FLUX OF 2D HOTSC
+This section investigates the topological feature of π
+flux inside a 2D higher-order topological superconductor
+(HOTSC) protected by C4 and fermion parity symme-
+try. The motivation comes from the expectation that for
+a topological quantum phase protected by symmetry G,
+one can detect its topological feature by observing the
+anomalous symmetry structure inside the symmetry de-
+fect (e.g., the gauge flux for G symmetry)[43, 45, 46].
+For instance, in a 2D p+ip superconductor with fermion
+parity symmetry, the superconducting vortex contains a
+Majorana zero mode[47]. In 3D T -invariant topological
+superconductor, the π flux line carries a 1D chiral Majo-
+rana mode with c = 1/2 central charge[48]. For a gen-
+eral G symmetry-protected topological phase, once we
+gauge the symmetry G, the symmetry flux either carries
+a fractional quantum number of G or contains anoma-
+lous gapless modes.
+This aspect provides an alterna-
+tive way to visualize the underlying quantum structure of
+the symmetry-protected topological phases. In addition,
+exploring symmetry flux offers a feasible way to detect
+topological responses via numerical simulations or exper-
+imental measurements.
+To set the stage, we begin with the 2D higher-order
+TSC proposed in Ref. [40]. The model contains four Ma-
+joranas (two complex fermions) living inside each unit
+cell on a square lattice as shown schematically in Fig. 1.
+The four Majoranas at the corners of the square mainly
+tunnel with its nearest neighbor within the plaquette
+with π flux per square. The resultant superconducting
+state is fully gapped inside the bulk and on the smooth
+edges, while the corner intersecting two boundaries con-
+tains a Majorana zero mode(MZM).
+Our model contains a plaquette-centered C4 rotation
+symmetry in addition to the fermion parity conservation
+P.
+The non-interacting Hamiltonian in the Majorana
+basis is,
+H = ηT (t + cos(kx))Γ3 + (t + cos(ky))Γ1
++ sin(kx)Γ4 + sin(ky)Γ2)η
+Γ1 = −σy ⊗ τ x, Γ2 = σy ⊗ τ y, Γ3 = σy ⊗ τ z,
+Γ4 = σx ⊗ I.
+(1)
+Figure 1: The HOTSC on the square lattice with four
+Majoranas per site. In the zero-correlation length limit,
+the four Majoranas on the four corners of the square
+hybridize within the plaquette (solid lines). The ground
+state Hamiltonian contains π flux per plaquette. By
+inserting a superconducting π flux in the center, the
+central plaquette becomes Φ = 0.
+With ηT = (η1, η2, η3, η4) being the four Majoranas on
+each site. t is the strength for intra-site Majorana cou-
+pling. For |t| < 1, the system is in the HOTSC phase.
+When t = 0, the model returns to the zero-correlation
+length limit that the four Majoranas on the four corners
+of each square only hybridize within the plaquette. The
+C4 rotation acts on η as,
+C4 :
+�
+0
+τ z
+τ x
+0
+�
+(2)
+This model was widely explored in various works of
+literature for weak and strongly interacting systems[1, 7,
+40]. The π flux inside each plaquette is essential to ac-
+quire a fully gapped bulk spectrum and the resultant C4
+symmetry has the structure with (C4)4 = −1 due to the
+π flux. In particular, the Majorana zero mode localized
+at the corner is robust against any interaction provided
+the C4 symmetry is unbroken and the SC bulk is gapped.
+It was pointed out that if one develops a symmetry de-
+fect of the C4 symmetry, namely, a π/2 disclination by
+removing a quadrant and reconnecting the disclination
+branch cut, there exists a Majorana zero mode localized
+at the disclination core[24, 36].
+Now we consider the symmetry flux of the fermion par-
+ity P by inserting an additional π flux to the center of the
+plaquette as Fig. 1. As the pairing Hamiltonian already
+contains a π flux in each plaquette, the additional flux
+insertion erases it and makes the central plaquette flux
+free Φ = 0. In the limit t = 0 in Eq.1, the flux insertion
+only changes the four Majorana coupling at the central
+plaquette, leaving a two-fold degeneracy at the center.
+Here and after, we will demonstrate that this degeneracy
+is robust against any interaction or coupling due to the
+projective symmetry.
+
+元
+元
+4
+元
+中
+元
+2
+3
+元3
+To demonstrate, we begin with the particular case with
+t = 0, but our demonstration is adaptive to more general
+circumstances, as we will elaborate on later. In this zero
+correlation-length limit, the four Majoranas at the cor-
+ners of the central plaquette coupled like a 1D ring with
+four Majoranas on four sites. The ground state contain-
+ing π flux inside the plaquette imposes an anti-periodic
+boundary condition for the 1D ring.
+The plaquette-
+centered C4 rotation symmetry performs as a ‘translation
+symmetry’ that permutes between the four Majoranas on
+the ring. Due to the π flux at the center, the C4 symme-
+try permutes the four Majorana as,
+C4(π) : γ1 → γ2, γ2 → γ3, γ3 → γ4, γ4 → −γ1
+(3)
+So (C4)4 = −1.
+If we perform this rotation to the
+fermion-parity symmetry operator P = η1η2η3η4,
+C4(π)PC−1
+4 (π) = −η2η3η4η1 = η1η2η3η4 = P
+(4)
+The fermion parity and C4 rotation commute.
+However, once we insert an additional flux in the cen-
+ter, the ‘net flux plaquette’ can be viewed as a ring of
+four Majoranas with periodic boundary conditions. The
+C4 symmetry is now defined as,
+C4(0) : γ1 → γ2, γ2 → γ3, γ3 → γ4, γ4 → γ1
+(5)
+Inside the superconducting flux, the C4 symmetry anti-
+commutes with the fermion parity operator.
+C4(0)PC−1
+4 (0) = η2η3η4η1 = −P
+(6)
+This anti-commutation relation guarantees an additional
+zero mode at the flux center.
+To this end, we demonstrate that adding a π flux at
+the rotation center engenders a projective symmetry such
+that the fermion parity and C4 symmetry anti-commute
+in the flux center.
+As a result, the flux would trap
+two degenerate modes with different fermion parity con-
+nected by a C4 rotation operation. We can label these
+degenerate modes as the even(odd) fermion parity state
+|Ψ⟩a
+0(|Ψ⟩b
+0) that is related by C4.
+C4|Ψ⟩a
+0 = |Ψ⟩b
+0
+(7)
+Our demonstration above is based on the zero-
+correlation limit in the absence of interaction. Now and
+after, we will extend this argument to a more general
+case with additional symmetry-preserving coupling and
+interaction. If the correlation length is finite, the four
+Majoranas in the central plaquette with net flux would
+unavoidably be coupled with the rest of the system. As
+long as there is no gap closing in the bulk, the flux state
+wave function |Ψ⟩ can be connected to the aforemen-
+tioned zero-correlation length limit wave function by a
+set of finite depth local unitary circuit U[49] that com-
+mutes with the C4 symmetry and fermion parity.
+U|Ψ⟩a
+0 = |Ψ⟩a, U|Ψ⟩b
+0 = |Ψ⟩b,
+(8)
+As the unitary operator commutes with all symmetries,
+the new degenerate states |Ψ⟩a, |Ψ⟩b carrying different
+fermion parity are still related by a C4 rotation. This
+concludes that the C4 symmetry and fermion parity al-
+ways anti-commute inside the flux regardless of the cor-
+relation length or additional interaction. Thus, due to
+the projective symmetry structure inside the flux, there
+is no consistent way to hybridize or lift the degeneracy
+that preserves both rotation and fermion parity.
+To summarize, we elucidate that the insertion of π flux
+in a higher-order topological superconductor engenders a
+projective symmetry between C4 and fermion parity so
+the resultant flux state contains a localized zero mode.
+In particular, inserting a gauge flux is expected to en-
+gender a projective symmetry of C2nP. Our argument
+can be generalized to other 2D higher-order topological
+superconductors with C2n symmetries[38].
+A.
+Emergent supersymmetry (SUSY) inside the
+flux
+There had been growing interest in realizing supersym-
+metry in solid-state systems, which is a highly appeal-
+ing concept from particle physics, relating bosonic and
+fermionic modes. In this section, we establish a general
+theorem that all 2D HOTSC exhibit N=2 SUSY algebra
+inside the superconducting flux. As is demonstrated in
+Sec. II, adding a flux to HOTSC creates a projective sym-
+metry between fermion parity and rotation. We will show
+that all flux states in HOTSC have an underlying N = 2
+supersymmetry and explicitly construct the generator of
+the supersymmetry[41].
+To set the stage, we first shift all the eigenvalues of
+the Hamiltonian by a constant so that they are all non-
+negative. Then we define the following fermionic, non-
+Hermitian operator based on C4 and P,
+Q =
+�
+H
+2 C4(1 + P), Q† =
+�
+H
+2 (1 + P)(C4)−1.
+(9)
+Q† commutes with the HOTSC Hamiltonian [H, Q†] = 0.
+Most importantly, due to the projective symmetry inside
+the flux, the Q, Q† obeys the algebra:
+(Q)2 = 0, (Q†)2 = 0, QQ† + Q†Q = 2H
+(10)
+Therefore, Q is the generator of an N = 2 supersymme-
+try. Such supersymmetry naturally explains the degener-
+ate modes inside the flux. By adding a flux to HOTSC, all
+energy levels are doubly degenerate, and the correspond-
+ing eigenstates can be chosen as fermion parity eigen-
+states with different parity. Q, Q† operators, assisted by
+spatial rotation, play a role of exchanging between these
+fermion parity sectors. Notably, while a wide range of
+emergent supersymmetry in condensed matter typically
+requires fine-tuned Hamiltonians or critical points[50],
+the N = 2 SUSY in HOTSC flux is guaranteed by the
+projective symmetry of C4P, and thus robust against
+perturbations.
+
+4
+B.
+Detecting flux responses from the entanglement
+Hamiltonian
+In this section, we show that the topological flux re-
+sponse in the higher-order topological superconductor
+can be assessed using a quantum information perspec-
+tive by examining the entanglement Hamiltonian. This
+method not only enables the detection of the higher-order
+topological superconductor numerically but also suggests
+that the hidden topological structure, including the topo-
+logical flux response, can be understood through an en-
+tanglement viewpoint.
+Figure 2: Tracing out the central block of the
+wavefunction arrives at the reduced density matrix ρA
+that resembles a 1D Majorana chain along the square
+cut. Each quadrant of ρA contains odd number of
+Majoranas.
+To begin with, we trace out the system’s central block
+with a size larger than the correlation length but still fi-
+nite compared to the thermal dynamical limit. The resul-
+tant reduced density matrix ρA = e−βHe can be viewed
+as a partition function of an entanglement Hamiltonian
+He that resembles a 1D ‘square frame’ along with the
+cut in Fig. 2. Such a cut contains four corners, with each
+quadrant carrying an odd number of Majoranas.
+The
+C4 rotation operator performs as a translation operator
+TL/4 on the 1D entanglement Hamiltonian that shifts the
+fermion by a quarter of the cut size.
+In the thermal
+dynamical limit, the four Majoranas at the corners of
+the ‘square frame’ generate four Majorana zero modes in
+the entanglement Hamiltonian He. However, these zero
+modes could be hybridized with finite-size cuts. Suppose
+we choose the length of the 1D entanglement Hamiltonian
+being L, the coupling strength between the four corner
+Majoranas in the entanglement spectrum scales as e−ξ/L
+so the Majorana zero-mode hybridization is inevitable
+for a finite-size system. In addition, the correspondence
+between the ‘ground state of the entanglement Hamilto-
+nian’ and the wave function correlation cannot be taken
+too literally. Since the reduced density matrix is the par-
+tition function of the entanglement Hamiltonian(EH) at
+finite temperatures, the high energy modes in the entan-
+glement spectrum(ES) also contribute to the entangled
+features of the ground state. In particular, the low-lying
+states of the ES may undergo a phase transition while
+the bulk phase remains unchanged [51].
+In terms of the ground state wave function, the central
+block in the reduced density matrix (with an odd number
+of plaquettes) contains a total π flux. The entanglement
+Hamiltonian defined on the ring with π flux inside has
+an anti-periodic boundary condition. The resultant C4
+symmetry operator of the 1D entanglement Hamiltonian
+can be defined as,
+C4(π) : γi → γN+i, γN+i → γ2N+i, γ2N+i → γ3N+i,
+γ3N+i → −γi
+(11)
+Here i labels the Majorana on each quadrant, with 4N
+being the total number of Majoranas in the effective 1D
+entanglement Hamiltonian He. It is not hard to check
+that the C4 rotation and fermion parity commute for the
+entanglement Hamiltonian He. This also agrees with the
+fact that the entanglement Hamiltonian can have a fully
+gapped spectrum for a finite-size system.
+To visualize the projective symmetry and zero modes
+inside the π superconducting flux from the entanglement
+Hamiltonian, we look into the wave function with an ad-
+ditional π flux in the center (so the central plaquette
+has net flux) and trace out the center block to get the
+entanglement Hamiltonian Hflux
+e
+. Due to the additional
+flux insertion, the 1D entanglement Hamiltonian Hflux
+e
+has net flux inside the ring with periodic boundary con-
+ditions. The resultant C4 symmetry operator of the en-
+tanglement Hamiltonian can be defined as,
+C4(0) : γi → γN+i, γN+i → γ2N+i,
+γ2N+i → γ3N+i, γ3N+i → γi
+(12)
+N is an odd number since each quadrant contains an odd
+number of Majoranas.
+After some simple algebra, we
+find that C4 rotation and fermion parity anti-commute
+C4P = −PC4 for the entanglement Hamiltonian Hflux
+e
+.
+This indicates that these two symmetries act projectively
+on Hflux
+e
+so the full entanglement spectrum should dis-
+play a robust two-fold degeneracy for all eigenstates. It is
+notable for emphasizing that this degeneracy has nothing
+to do with the corner zero modes in the original Hamil-
+tonian that can be gapped due to the finite-size effect.
+The projective symmetry-enforced degeneracy can sur-
+vive even for finite-size cuts and is robust against any
+interaction or perturbation.
+III.
+3D FLUX LINES IN HOTSC
+This section extends our discussion on anomalous flux
+states to interacting HOTSC in 3D. We begin with the
+3D HOTSC that supports chiral Majorana hinge modes
+
+BH.
+A5
+proposed in Ref. [1, 5, 40, 52],
+H = ηT [(1 − m cos kz + cos(kx))Γ3 + (1 − m cos kz
++ cos(ky))Γ1 + sin(kx)Γ4 + sin(ky)Γ2 − m sin kzΓ0]η
+Γ1 = −σy ⊗ τ x, Γ2 = σy ⊗ τ y, Γ3 = σy ⊗ τ z,
+Γ4 = σx ⊗ I, Γ0 = σz ⊗ I.
+(13)
+For −2 < m < 0, the model is in the HOTSC phase[3].
+This model exhibit a special CT
+4 symmetry that rotates
+the x-y plane along with the time-reversal operation. The
+CT
+4 acts on the Majorana field η as,
+K
+�
+0
+−τ z
+τ x
+0
+�
+(14)
+Notably, if we implement a dimension-reduction view
+by fixing the momentum kz, the momentum layer with
+kz = π resembles the aforementioned 2D HOTSC with
+C4 symmetry while the kz = 0 layer corresponds to the
+trivial one. (CT
+4 )2 = −1 indicates that the Hamiltonian
+has a π flux penetrating each tube along the z-direction
+illustrated as Fig. 3.
+Consider inserting an additional π flux along the z-
+direction, the corresponding center tube contains net
+flux. Now and after, we will demonstrate that such flux
+insertion will engender a gapless 1D mode that is anoma-
+lous and cannot be manifested in pure lower-dimensional
+systems.
+To warm up, recall our discussion in Sec. II, adding π
+flux to 2D HOTSC give rise to a projective representa-
+tion between C4 and P. Our 3D model can be treated as
+layers of 2D superconductors with fixed kz momentum so
+that we can treat different kz layers independently. We
+consider two special momentum slices kz = 0, π which
+resemble the 2D trivial superconductor and higher-order
+topological superconductor. Based on our discussion in
+Sec. II, it is clear that implementing a CT
+4 symmetry
+would change the fermion parity number P(π) = (−1)nπ
+inside the flux line that carries momentum kz = π. Like-
+wise, the fermion parity P(0) = (−1)n0 that carries mo-
+mentum kz = 0 is not affected. Based on this argument,
+we conclude that the algebra between CT
+4 and fermion
+parity inside the flux line has the form,
+CT
+4 P(π) = −P(π)CT
+4 ,
+(15)
+Unfortunately, the above argument relies on the fact that
+the fermion parity number in each momentum layer (with
+fixed kz) is a well-defined quantum number. However, as
+our HOTSC does not require a translation symmetry,
+one can add disorder along the z-direction and the corre-
+sponding kz is no longer a good quantum number. Fur-
+ther, in the presence of strong interaction, fermions with
+distinct momentum kz can hybridize and interact.
+In
+this sense, nπ again becomes ill-defined when the single-
+particle picture breaks down.
+A.
+Conflict of symmetry and quantum anomaly
+Here we provide a more detailed and systematic study
+of the flux state based on the symmetry anomaly argu-
+ment. For concreteness, we will demonstrate that the flux
+line inside the HOTSC displays a quantum anomaly that
+can be manifested as a ‘conflict of symmetry.’ If the 1D
+flux line is invariant under two independent symmetries
+G1 and G2, the theory is anomalous if gauging G1 would
+break the symmetry of G2 or vice versa[53]. Applying
+this ‘conflict of symmetry’ criteria to our case, we will
+demonstrate that after gauging the fermion parity inside
+the flux line, one observes that a fermion parity gauge
+transformation inside the flux will automatically break
+the CT
+4 symmetry.
+This conflict of symmetry alterna-
+tively suggests that the symmetry assignment of CT
+4 and
+P are incompatible with open boundaries so the corre-
+sponding 1D theory does not render a lattice realization.
+In the zero correlation length limit, the HOTSC model
+in Eq. 13 has a coupled wire construction[52]. We can
+decompose the complex fermions along each z-row into
+two up-moving and two down-moving chiral Majoranas.
+Treat the z-tube as an elementary building block; it con-
+tains four chiral Majoranas living at the four hinges of
+the tube illustrated in Fig. 3.
+Htube = ηT (kz)σ30η
+(16)
+We consider the general case where the four hinges along
+each z-tube with counter-propagating Majorana modes
+are coupled in a CT
+4 symmetric way. After inserting an
+additional π flux to the central plaquette, the CT
+4 sym-
+metry permutes the four components of the 1D Majorana
+modes in the central tube as,
+C4 : K
+�
+0
+I
+τ x 0
+�
+(17)
+with (CT
+4 )4 = 1 provided there is net flux inside the tube
+center[54]. The possible gapping terms for each z-tube
+as of Eq. 16 are:
+m1 = σ20, m2 = σ21, m3 = σ23, m4 = σ12,
+CT
+4 :m1 → m2, m2 → m1,
+m3 → m4, m4 → −m3,
+(18)
+The mass terms m3, m4 are also odd under C2 symmetry,
+so they cannot appear as a fermion bilinear mass. To
+make the theory compatible with CT
+4 , we require m1 =
+m2 and the resultant 1D flux line always remains gapless
+regardless of the strength of m. Thus, no band mass can
+fully gap out the helical modes inside the flux. This in-
+gappable condition can be generalized to the interacting
+case due to the existence of an anomalous symmetry.
+Now and after, we will demonstrate that the helical
+modes in Eq. 16 are anomalous and cannot exist in pure
+1D lattice models. This further suggests that the helical
+modes cannot be trivially gapped unless we break the CT
+4
+
+6
+Figure 3: A) 3D HOTSC top-down view from the x-y
+plane. An additional π flux penetrates the central
+plaquette so the total flux in the central plaquette is
+zero. B) Treat the z-tube as an elementary building
+block; it contains up-moving/down-moving chiral
+Majoranas living at the four hinges of the tube along
+the z-direction.
+symmetry. We would elaborate on this point by gauging
+the fermion parity symmetry inside the flux line and ex-
+amining the role of CT
+4 under such gauge transformation.
+Central to our discussion below is based on the
+bosonization picture of helical Majoranas in Eq. 16.
+Ψ†
+L = ηL
+1 + iηL
+3 = eiθ+iφ+iπ/4,
+Ψ†
+R = ηR
+2 + iηR
+4 = e−iθ+iφ+iπ/4
+ˆn = ∂zθ
+π
+(19)
+Here θ, φ are bosonic fields and the fermion charge den-
+sity ˆn is only defined modulo 2. [55]. The CT
+4 symmetry
+acts as,
+Ψ†
+L → ΨR, Ψ†
+R → −iΨ†
+L
+θ → φ, φ → −θ
+(20)
+The possible interactions that do not break CT
+4
+are
+cos(2θ)+cos(2φ) or their higher order descendants. Pre-
+cisely, the CT
+4
+symmetry exchange the role between
+particle-hole channel tunneling term cos(2θ) and particle-
+particle channel pairing term cos(2φ) by enforcing them
+with the same strength. These terms cannot symmetri-
+cally gap out the helical modes, so the resultant theory
+is either gapless or symmetry-breaking.
+If we apply a gauge transformation of P along the
+string from −∞ to z,
+G(z) = ei
+� z
+−∞ dz′πn(z′) = eiθ(z)
+(21)
+Such a gauge transformation can be viewed as the
+fermion parity operator defined on an open string with its
+half-end terminated at z. The CT
+4 symmetry transforms
+G(z) as,
+CT
+4 eiθ(z)(CT
+4 )−1 = e−iφ(z) = −G(z)e−iθ(z)−iφ(z)
+(22)
+Such gauge transformation, equivalent to the fermion
+parity defined on an open chain, is not invariant un-
+der the CT
+4 symmetry. Notably,e−iθ(z)−iφ(z) is a fermion
+operator, so the CT
+4
+transformation creates additional
+fermion at the end of the fermion parity string.
+Such
+conflict of symmetry indicates that the theory cannot be
+placed on an open 1D chain as the fermion parity oper-
+ator on the open chain is not invariant under CT
+4 . As
+a result, the helical modes inside the flux line cannot
+be realized in isolated 1D lattice models with the same
+symmetry assignment.
+It is noteworthy mentioning that the conflict of sym-
+metry was widely explored as a signature of anomalous
+surface states in symmetry-protected topological phases.
+In Ref. [53, 56], it was convinced that the conflict of the
+symmetries at the boundary of the SPT surfaces signals
+that the edge theory can never be realized as a purely
+lower dimensional lattice model. Our argument can be
+treated as a complement theorem signaling that the flux
+state inside the HOTSC also contains a gapless mode
+with anomalous symmetry action.
+IV.
+EMERGENT HOTSC FROM KITAEV SPIN
+LIQUIDS
+We conclude our discussion by extending our horizon
+into frustrated spin systems whose emergent quasiparti-
+cle excitations exactly reproduce the topological features
+of HOTSC discussed in Sec. II. Namely, we begin with a
+bosonic spin model on a honeycomb lattice. Intriguingly,
+the low energy excitations of such a bosonic system con-
+tain emergent Majoranas coupling with an emergent Z2
+gauge field. The Majoranas form a superconductor rem-
+iniscent of the 2D HOTSC in Sec. II while the emergent
+flux excitation carries N = 2 SUSY structure.
+To continue, we focus on a specific solvable honeycomb
+lattice model. However, it is worth mentioning that the
+protocol and strategy we developed here can be applied
+to a wider class of lattice models, as we will elaborate
+on later. We begin with a spin 3
+2 honeycomb model with
+strong bond anisotropy.
+H =
+�
+i∈A,j∈B
+[
+�
+ij∈green
+(Γi
+1Γj
+1 − Γi
+4Γi
+1Γj
+4Γj
+1) −
+�
+ij∈blue
+(Γi
+3Γi
+4Γj
+3Γj
+4 + Γi
+5Γi
+3Γj
+5Γj
+3) +
+�
+ij∈red
+(Γi
+2Γj
+2 − Γi
+5Γi
+2Γj
+5Γj
+2) ]
+(23)
+
+A)
+元
+n2
+B)
+元
+T
+O
+m4
+n3
+元
+元7
+Here Γa(a = 1, ..5) are the 4 × 4 Gamma matrices with
+−i �a=5
+a=1 Γa = 1. At each A/B site on the hexagon lattice,
+we color three directional bonds with red/green/blue as
+Fig. 4. Each spin only interacts with its nearest neighbor
+across the red/green/blue bond, and each colored bond
+has two preferred spin bilinear interactions.
+Figure 4: The spin 3
+2 degree of freedom and its
+Majorana representation on the A/B sublattice. The πi
+Majoranas are the itinerary fermions that only
+hybridized with their nearest neighbor within the
+hexagon (illustrated as the red dashed line.) The ηi
+Majorana plays the role of the emergent Z2 gauge
+potential. The Hamiltonian commutes with the flux
+operator defined on the blue hexagon.
+Albeit the model is non-integrable, it renders an exact
+solvable solution inherited from the spirit of the origi-
+nal Kitaev model[57]. In terms of parton construction,
+we can fermionized the spin-3/2 operator by introduc-
+ing six Majoranas π1, π2, π3, η1, η2, η3 per site as Fig. 4.
+We restricted our Hilbert space with fixed onsite parity
+iπ1π2π3η1η2η3 = 1 so the six Majoranas with even parity
+generate a four-level system per site akin to the spin-3/2
+degree of freedom.
+The spin Gamma matrices can be
+expressed as,
+Γ1 = iπ1η1, Γ2 = iπ1η2, Γ3 = iπ1η3, Γ4 = iπ2π1,
+Γ5 = iπ3π1
+(24)
+So the Clifford algebra is automatically satisfied. We can
+express the spin operators in the Hamiltonian as,
+Γ1 = iπ1η1, iΓ4Γ1 = −iπ2η1, Γ2 = iπ1η2,
+iΓ5Γ2 = −iπ3η2, iΓ3Γ4 = −iη3π2, iΓ5Γ3 = iη3π3 (25)
+The model displays a special C′
+6 symmetry that hybrid
+hexagon-centered C6 rotation with S3 spin rotation as
+C′
+6 = C6 × S3. Under the Majorana representation, the
+spin rotation becomes the S3 permutation between Ma-
+jorana flavors,
+π3 → π1, π1 → π2, π2 → π3,
+η3 → η2, η2 → η1, η1 → η3,
+(26)
+It is not hard to find a locally conserved hexagon op-
+erator illustrated in Fig .4 that commutes with all spin
+interactions in the Hamiltonian. This enables us to treat
+the ηi fermion bilinear as the gauge potential on the link,
+exp{iπAij∈green} = iηi
+1ηj
+1,
+exp{iπAij∈red} = iηi
+2ηj
+2,
+exp{iπAij∈blue} = iηi
+3ηj
+3
+(27)
+Aij denotes the Z2 gauge potential on the link between
+i-j sites (with i(j) belongs to the A(B) sublattice). The
+Z2 potential on tricolored links can be written as the
+Majorana fermion bilinears iηi
+aηj
+a(a = 1, 2, 3) that cross
+between the links.
+As a result, the total flux in each
+hexagon
+� ⃗Ad⃗l = Φ is manifested by the product of Ma-
+jorana bilinears defined in Eq. 27 across the six links
+along the hexagon loop, which returns to the hexagon
+operator in Fig. 4.
+Our argument makes it clear that, under the Majorana
+representation of spin operators, the η fermion plays a
+role as the Z2 gauge potential akin to the original Ki-
+taev model.
+Likewise, the πa(a = 1, 2, 3) fermion can
+be treated as the itinerary Majoranas that hop between
+nearest sites with minimal coupling to the gauge poten-
+tial Aij on the link. To manifest, we can decompose the
+spin interactions as
+�
+ij∈green
+Γi
+1Γj
+1 = −
+�
+ij∈green
+πi
+1ηi
+1πj
+1ηj
+1
+�
+ij∈green
+Γi
+4Γi
+1Γj
+4Γj
+1 =
+�
+ij∈green
+πi
+2ηi
+1πj
+2ηj
+1
+�
+ij∈blue
+Γi
+3Γi
+4Γj
+3Γj
+4 =
+�
+ij∈blue
+πi
+2ηi
+3πj
+2ηj
+3
+�
+ij∈blue
+Γi
+5Γi
+3Γj
+5Γj
+3 =
+�
+ij∈blue
+πi
+3ηi
+3πj
+3ηj
+3
+�
+ij∈red
+Γi
+2Γj
+2 = −
+�
+ij∈red
+πi
+1ηi
+2πj
+1ηj
+2
+�
+ij∈red
+Γi
+5Γi
+2Γj
+5Γj
+2 =
+�
+ij∈red
+πi
+3ηi
+2πj
+3ηj
+2
+(28)
+In the Majorana representation, it is clear that all bond
+interactions in Equation 23 can be treated as Majorana
+hopping between nearest sites, with minimal coupling to
+the gauge potential Aij represented by the η fermion
+bilinears.
+Since the flux operators commute with the
+Hamiltonian, we can fix the flux sector Φ when focusing
+on the ground state manifold and treat Aij as a constant.
+For net flux conditions, we can simply take Aij = 0 and
+the permutation symmetry of the π Majorana fermions
+will still hold. With π flux patterns, any specific gauge
+choice of Aij will break the permutation symmetry of the
+η Majorana fermions.
+
+n1
+T1
+T2
+M2
+m3
+元3
+T3
+元1
+T2
+2
+T1
+元3
+T3
+Y3
+n2
+T23
+T2
+Ti2
+T1
+IiT3
+n1
+IiT3
+IiI2
+I2I38
+From the Majorana construction perspective, the effec-
+tive Hamiltonian becomes a free Majorana model with
+three orbitals π1, π2, π3 persite.
+Each orbital is only
+hybridized with one of the three adjacent hexagons as
+Fig. 4. Consequently, the fermion model is decomposed
+of non-overlap clusters from all hexagons. Each hexagon
+contains six Majoranas hybridized with their nearest
+neighbor that resembles higher-order topological super-
+conductors on the honeycomb lattice[38]. In particular,
+one can easily check that the lowest energy state re-
+quires Φ = π flux per plaquette and the ground state
+is in the π-flux sector. The effective band structure for
+the itinerary Majoranas π1, π2, π3 is reminiscent of the
+higher-order topological superconductor on the honey-
+comb lattice. Remarkably, such HOTSC may not exhibit
+protected corner mode for sharp corners with 2π/3 an-
+gles.
+Nonetheless, the topological response still holds.
+By creating an additional π flux excitation in the center,
+the C′
+6 symmetry and fermion parity anti-commute so the
+flux excitations display a projective symmetry. As both
+the itinerary Majorana and the Z2 flux excitation origi-
+nate from spin models as fractionalized excitations, the
+Z2 flux should be treated as an intrinsic excitation rather
+than an external field that characterizes and probes the
+response. In particular, the flux excitation in this spin
+model contains the N=2 SUSY structure we explored in
+Eq. 9.
+Finally, the construction we adopt here can be gener-
+alized to spin models on other 2D lattices. The essence
+relies on the fact that for any HOTSC, we can introduce
+a Z2 gauge potential on the link and express them as a
+pair of ’auxiliary’ Majorana fermion bilinears across the
+link. After onsite fermion parity projection, the resultant
+onsite degree of freedom becomes a hyper-spin operator,
+and the fermion hopping term minimal coupling to the Z2
+gauge potential can be written in terms of spin-bilinear
+interactions.
+Following this protocol, one can build a
+zoology of ‘Kitaev spin liquids’ whose low energy excita-
+tion can constitute Majoranas with HOTSC band struc-
+ture and emergent Z2 gauge field. The flux excitations in
+these Kitaev-type models carry exotic SUSY structures
+with projective symmetry between spatial rotation and
+fermion parity.
+ACKNOWLEDGMENTS
+Y.Y was supported by Gordon and Betty Moore
+Foundation
+through
+Grant
+GBMF8685
+and
+Marie
+Sklodowska-Curie Actions under the new Horizon 2020.
+Y.Y acknowledges informative discussions with Taylor
+Hughes and Rui-Xing Zhang.
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@@ -0,0 +1,2461 @@
+arXiv:2301.04928v1  [math.AP]  12 Jan 2023
+Nodal bubble tower solutions to slightly subcritical
+elliptic problems with Hardy terms
+Thomas Bartsch∗, Qianqiao Guo†
+Dedicated to the 85th Birthday of Professor Dajun Guo.
+Abstract We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem
+with Hardy term
+
+
+
+
+
+−∆u − µ u
+|x|2 = |u|2∗−2−εu
+in Ω,
+u = 0
+on ∂Ω,
+in a bounded domain Ω ⊂ RN(N ≥ 7) with 0 ∈ Ω, as µ, ǫ → 0+. In [6], we obtained the existence of
+nodal solutions that blow up positively at the origin and negatively at a different point as µ = O(ǫα)
+with α > N−4
+N−2, ǫ → 0+. Here we prove the existence of nodal bubble tower solutions, i.e. superpositions
+of bubbles of different signs, all blowing up at the origin but with different blow-up order, as µ = O(ǫ),
+ǫ → 0+.
+2010 Mathematics Subject Classification 35B44, 35B33, 35J60.
+Key words Hardy term; Critical exponent; Slightly subcritical problems; Nodal solutions; Bubble towers;
+Singular perturbation methods.
+1
+Introduction
+We continue to study the possible blow-up behavior of solutions to the slightly subcritical elliptic
+problem with Hardy term
+
+
+
+
+
+−∆u − µ u
+|x|2 = |u|2∗−2−εu
+in Ω,
+u = 0
+on ∂Ω,
+(1.1)
+where Ω ⊂ RN, N ≥ 7, is a smooth bounded domain with 0 ∈ Ω; 2∗ :=
+2N
+N−2 is the critical Sobolev
+exponent. In [6], for fixed α > N−4
+N−2 and µ0 > 0, we obtain the existence of nodal solutions that blow up
+positively at the origin and negatively at a different point with µ = µ0ǫα, ǫ → 0+. In this paper we want
+to study the existence of nodal bubble tower solutions, i.e. superpositions of bubbles of different signs,
+∗Mathematisches
+Institut,
+Justus-Liebig-Universit¨at Giessen,
+Arndtstr.
+2,
+35392
+Giessen,
+Germany;
+E-mail:
+[email protected]
+†School of Mathematics and Statistics, Northwestern Polytechnical University, 710129 Xi’an, China; E-mail: gqian-
+[email protected]
+1
+
+all blowing up at the origin but with different blow-up order. For that, we need to assume α = 1, that
+is, µ = µ0ǫ.
+The blow-up phenomenon for positive and for nodal solutions to problem (1.1) for µ = 0 has been
+studied extensively, see e.g. [2, 3, 4, 5, 8, 12, 17, 21, 23, 26, 28, 29, 30, 31] and the references therein.
+However, it is well known that if µ ̸= 0, the Hardy potential
+1
+|x|2 cannot be regarded as a lower order per-
+turbation since it has the same homogeneity as the Laplace operator. This makes the analysis interesting
+and more complicated compared with the case µ = 0. The existence of positive and nodal solutions to
+the problem with Hardy type potentials and critical exponents has been studied in a number of papers,
+see e.g. [9, 10, 13, 16, 18, 19, 20, 22, 24, 32, 33, 35] and the references therein. However, few results are
+known concerning blow-up solutions. The only results we are aware of are related to the problem
+
+
+
+
+
+− ∆u −
+µ
+|x|2 u = k(x)u2∗−1,
+u ∈ D1,2(RN),
+u > 0 in RN \ {0};
+here D1,2(RN) := {u ∈ L2∗(RN) : |∇u| ∈ L2(RN)}; see [14, 15, 27].
+In order to state the main results of this paper, we first introduce some notations.
+By Hardy’s
+inequality, the norm
+∥u∥µ :=
+��
+Ω
+(|∇u|2 − µ u2
+|x|2 )dx
+� 1
+2
+is equivalent to the norm ∥u∥0 =
+��
+Ω |∇u|2dx
+�1/2 on H1
+0(Ω) provided 0 ≤ µ < µ := (N−2)2
+4
+. This will
+of course be the case for µ = µ0εα with ε > 0 small. As in [16] we write Hµ(Ω) for the Hilbert space
+consisting of H1
+0(Ω) functions with the inner product
+(u, v) :=
+�
+Ω
+�
+∇u∇v − µ uv
+|x|2
+�
+dx.
+It is known that the nonzero critical points of the energy functional
+Jε(u) := 1
+2
+�
+Ω
+�
+|∇u|2 − µ u2
+|x|2
+�
+dx −
+1
+2∗ − ε
+�
+Ω
+|u|2∗−εdx
+defined on Hµ(Ω) are precisely the nontrivial weak solutions to problem (1.1).
+We need to recall the ground states of two problems that appear as limiting problems. The ground
+states of
+
+
+
+−∆u = |u|2∗−2u
+in RN,
+u → 0
+as |x| → ∞
+(1.2)
+are the instantons
+Uδ,ξ := C0
+�
+δ
+δ2 + |x − ξ|2
+� N−2
+2
+with δ > 0, ξ ∈ RN and C0 := (N(N − 2))
+N−2
+4 ; see [1, 34]. These are the minimizers of
+S0 :=
+min
+u∈D1,2(RN )\{0}
+�
+RN |∇u|2dx
+(
+�
+RN |u|2∗dx)2/2∗
+and they satisfy
+�
+RN |∇Uδ,ξ|2dx =
+�
+RN |Uδ,ξ|2∗dx = S
+N
+2
+0 .
+2
+
+Secondly, if 0 < µ < µ then all positive solutions to
+
+
+
+
+
+−∆u − µ u
+|x|2 = |u|2∗−2u
+in RN,
+u → 0
+as |x| → ∞
+(1.3)
+are given by
+Vσ = Cµ
+�
+σ
+σ2|x|β1 + |x|β2
+� N−2
+2
+with σ > 0, β1 := (√µ − √µ − µ)/√µ, β2 := (√µ + √µ − µ)/√µ, and Cµ :=
+�
+4N(µ−µ)
+N−2
+� N−2
+4 ; see [11, 35].
+These solutions minimize
+Sµ :=
+min
+u∈D1,2(RN)\{0}
+�
+RN (|∇u|2 − µ u2
+|x|2 )dx
+(�
+RN |u|2���dx)2/2∗
+and there holds
+�
+RN
+�
+|∇Vσ|2 − µ|Vσ|2
+|x|2
+�
+dx =
+�
+RN |Vσ|2∗dx = S
+N
+2
+µ .
+The Green’s function of the Dirichlet Laplacian is given by G(x, y) =
+1
+|x−y|N−2 −H(x, y), for x, y ∈ Ω,
+where H is the regular part. These functions are symmetric: G(x, y) = G(y, x) and H(x, y) = H(y, x).
+Now we state our main result about the existence of nodal solutions that are towers of bubbles
+concentrating at the origin.
+Theorem 1.1. Let µ = µ0ε with µ0 > 0 fixed. For any given integer k ≥ 0 there exists ε0 > 0 such that
+for any ε ∈ (0, ε0), there exists a pair of solutions ±uε to problem (1.1) satisfying
+uε(x) = Cµ(−1)k
+�
+σε
+(σε)2|x|β1 + |x|β2
+� N−2
+2
++ C0
+k
+�
+i=1
+(−1)i−1
+�
+δε
+i
+(δε
+i )2 + |x − ξε
+i |2
+� N−2
+2
++ o(1) as ε → 0.
+Here the constants σε, δε
+i > 0 and ξε
+i ∈ RN are determined as follows.
+There exist λε
+i , λ
+ε > 0 and
+ζε
+i ∈ RN, i = 1, . . . , k, with η < λε
+i , λ
+ε < 1
+η and |ζε
+i | ≤ 1
+η for some η > 0 small, so that: σε = λ
+εε
+2(k+1)−1
+N−2
+,
+δε
+i = λε
+i ε
+2i−1
+N−2 , ξε
+i = δε
+i ζε
+i for i = 1, 2, . . . , k.
+The paper is organized as follows. In Section 2, we collect some notations and preliminary results.
+Section 3 is devoted to the proof of Theorem 1.1. Some useful technical lemmas are deferred to the
+appendices.
+Throughout this paper, positive constants are denoted by C, c and may vary from line to line.
+2
+Notations and preliminary results
+Here we collect some results from [6]. Let ι∗
+µ : L2N/(N+2)(Ω) → Hµ(Ω) be the adjoint operator of the
+inclusion ιµ : Hµ(Ω) → L2N/(N−2)(Ω) as in [14], that is,
+ι∗
+µ(u) = v
+⇐⇒
+(v, φ) =
+�
+Ω
+u(x)φ(x)dx,
+for all φ ∈ Hµ(Ω).
+This is continuous, so there exists c > 0 such that
+∥ι∗
+µ(u)∥µ ≤ c∥u∥2N/(N+2).
+(2.1)
+3
+
+Now the problem (1.1) is equivalent to the fixed point problem
+u = ι∗
+µ(fε(u)), u ∈ Hµ(Ω),
+(2.2)
+where fε(s) = |s|2∗−2−εs.
+Let P : H1(RN) → H1
+0(Ω) be the projection defined by ∆Pu = ∆u in Ω, Pu = 0 on ∂Ω. We need
+the following two propositions and one remark from [6].
+Proposition 2.1. Let 0 < µ < µ be fixed, and let Λi, i = 1, 2, . . . , be the eigenvalues of
+
+
+
+
+
+−∆u − µ u
+|x|2 = Λ|Vσ|2∗−2u
+in RN,
+|u| → 0
+as |x| → +∞,
+in increasing order. Then Λ1 = 1 with eigenfunction Vσ and Λ2 = 2∗ − 1 with eigenfunction ∂Vσ
+∂σ .
+Setting dinf := inf{|x| : x ∈ ∂Ω} and dsup := sup{|x| : x ∈ ∂Ω} we have
+Proposition 2.2. Let 0 < µ < µ be fixed. Then for σ > 0 the function ϕσ := Vσ − PVσ satisfies
+0 ≤ ϕσ ≤ Vσ
+and ϕσ(x) = Cµ(d(x))
+√µ−√µ−µH(0, x)σ
+N−2
+2
++ ℏσ(x);
+with
+dinf ≤ d ≤ dsup and ℏσ = O(σ
+N+2
+2 ),
+∂ℏσ
+∂σ = O(σ
+N
+2 ) as σ → 0.
+Remark 2.3.
+a) If µ → 0+, then
+ϕσ(x) = C0H(0, x)σ
+N−2
+2
++ O(µσ
+N−2
+2 ) + ℏµ,σ(x),
+(2.3)
+where ℏµ,σ satisfies ℏµ,σ(x) = O(σ
+N+2
+2 ), ∂ℏµ,σ(x)
+∂σ
+= O(σ
+N
+2 ) as σ → 0.
+b) Let us recall the similar results for Uδ,ξ obtained in [30], that is
+0 ≤ ϕδ,ξ := Uδ,ξ − PUδ,ξ ≤ Uδ,ξ, ϕδ,ξ = C0H(ξ, ·)δ
+N−2
+2
++ O(δ
+N+2
+2 ),
+(2.4)
+as δ → 0, uniformly in compact subsets of Ω.
+3
+Solutions with tower of bubbles concentrating at the origin
+3.1
+The finite-dimensional reduction
+Let the integer k ≥ 0 be fixed. For ε > 0 small, λ = (λ1, . . . , λk, λ) ∈ Rk+1
++
+and ζ = (ζ1, . . . , ζk) ∈
+(RN)k we set σ := λε
+2(k+1)−1
+N−2
+, δi := λiε
+2i−1
+N−2 ,
+ξ = (ξ1, . . . , ξk) := (δ1ζ1, . . . , δkζk) ∈ Ωk,
+and define
+Wε,λ,ζ :=
+k
+�
+i=1
+Ker
+�
+−∆ − (2∗ − 1)U 2∗−2
+δi,ξi
+�
++ Ker
+�
+−∆ −
+µ
+|x|2 − (2∗ − 1)V 2∗−2
+σ
+�
+.
+4
+
+By Proposition 2.1 and [7] we know
+Wε,λ,ζ = span
+�
+Ψ, Ψ0
+i , Ψj
+i : i = 1, 2, . . ., k, j = 1, 2, . . ., N
+�
+,
+where for i = 1, 2, . . . , k and j = 1, 2, . . . , N:
+Ψj
+i := ∂Uδi,ξi
+∂ξi,j
+,
+Ψ0
+i := ∂Uδi,ξi
+∂δi
+,
+Ψ := ∂Vσ
+∂σ
+(3.1)
+with ξi,j the j−th component of ξi.
+We also need the spaces
+Kε,λ,ζ := PWε,λ,ζ,
+and
+K⊥
+ε,λ,ζ := {φ ∈ Hµ(Ω) : (φ, PΨ) = 0, for all Ψ ∈ Wε,λ,ζ},
+as well as the (·, ·)µ-orthogonal projections
+Πε,λ,ζ : Hµ(Ω) → Kε,λ,ζ,
+and
+Π⊥
+ε,λ,ζ := Id − Πε,λ,ζ : Hµ(Ω) → K⊥
+ε,λ,ζ.
+For ε > 0 small we want to find solutions of (1.1) close to
+Vε,λ,ζ :=
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ,
+where
+(λ, ζ) ∈ Oη :=
+�
+(λ, ζ) ∈ Rk+1
++
+× (RN)k : λi ∈ (η, η−1), λ ∈ (η, η−1), |ζi| ≤ 1
+η , i = 1, . . . , k
+�
+for some η ∈ (0, 1).
+This is equivalent to finding η > 0, (λ, ζ) ∈ Oη and φε,λ,ζ ∈ K⊥
+ε,λ,ζ such that
+Vε,λ,ζ + φε,λ,ζ solves (2.2), hence
+Π⊥
+ε,λ,ζ
+�
+Vε,λ,ζ + φε,λ,ζ − ι∗
+µ(fε(Vε,��,ζ + φε,λ,ζ))
+�
+= 0
+(3.2)
+and
+Πε,λ,ζ
+�
+Vε,λ,ζ + φε,λ,ζ − ι∗
+µ(fε(Vε,λ,ζ + φε,λ,ζ))
+�
+= 0.
+Now we solve (3.2) first for φε,λ,ζ. Let us introduce the operator Lε,λ,ζ : K⊥
+ε,λ,ζ → K⊥
+ε,λ,ζ defined by
+Lε,λ,ζ(φ) = φ − Π⊥
+ε,λ,ζι∗
+µ(f ′
+0(Vε,λ,ζ)φ).
+Take ρ > 0 small enough and let
+Ak+1 := B(0,
+�
+δk+1δk),
+Ai := B(0,
+�
+δiδi−1) \ B(0,
+�
+δiδi+1) for i = 1, . . . , k;
+here δ0 = ρ2
+δ1 , δk+1 = σ; cf. [26].
+Proposition 3.1. For any η ∈ (0, 1), there exist ε0 > 0 and c > 0 such that for every (λ, ζ) ∈ Oη and
+for every ε ∈ (0, ε0),
+∥Lε,λ,ζ(φ)∥µ ≥ c∥φ∥µ
+for all φ ∈ K⊥
+ε,λ,ζ.
+In particular, Lε,λ,ζ is invertible with continuous inverse.
+5
+
+Proof. Following the same line as in [25] we argue by contradiction. Suppose that there exist η > 0,
+sequences εn > 0, (λn, ζn) ∈ Oη, φn ∈ Hµ(Ω) satisfying
+εn → 0, λn
+i → λi, λ
+n → λ, ζn
+i → ζi,
+as n → ∞, and such that
+φn ∈ K⊥
+εn,λn,ζn,
+∥φn∥µ = 1,
+and
+Lεn,λn,ζn(φn) = hn with ∥hn∥µ → 0;
+(3.3)
+here λn = (λn
+1 , . . . , λn
+k, λ
+n), ζn = (ζn
+1 , . . . , ζn
+k ), and for ε > 0 small: σn = λ
+nε
+2(k+1)−1
+N−2
+, ξn = (ξn
+1 , . . . , ξn
+k ) =
+(δn
+1 ζn
+1 , δn
+2 ζn
+2 , . . . , δn
+k ζn
+k ) ∈ Ωk, δn
+i = λn
+i ε
+2i−1
+N−2 for i = 1, 2, . . ., k. Consider the sets
+An
+k+1 := B
+�
+0, �δn
+k+1δn
+k
+�
+,
+An
+i := B
+�
+0, �δn
+i δn
+i−1
+�
+\ B
+�
+0, �δn
+i δn
+i+1
+�
+, i = 1, 2, . . ., k,
+where δn
+0 := ρ2
+δn
+1 and δn
+k+1 := σn. Thus we have:
+φn − ι∗
+µ (f ′
+0(Vεn,λn,ζn)φn) = hn − Πεn,λn,ξn �
+ι∗
+µ(f ′
+0(Vεn,λn,ξn)φn)
+�
+.
+(3.4)
+As in the proof of [6, Proposition 4.1], we obtain
+wn := −Πεn,λn,ξn(ι∗
+µ(f ′
+0(Vεn,λn,ξn)φn)) =
+k
+�
+i=1
+N
+�
+j=0
+cn
+i,jP(Ψj
+i)n + cn
+0P(Ψ)n
+for some coefficients cn
+i,j, cn
+0, where (Ψj
+i)n, j = 1, . . . , N, (Ψ0
+i )n, and (Ψ)n are defined analogously to (3.1).
+We argue in three steps.
+Step 1. We claim that
+lim
+n→∞ ∥wn∥µ = 0.
+(3.5)
+Multiplying (3.4) by ∆P(Ψh
+l )n + µ P (Ψh
+l )n
+|x|2
+, using Lemma B.1, Lemma A.1, Lemma B.2, and arguing as
+in the proof of [6, Proposition 4.1], we deduce cn
+l,h → 0, for l = 1, . . . , k, h = 0, 1, . . . , N, and cn
+0 → 0, as
+n → ∞. Thus the claim lim
+n→∞ ∥wn∥µ = 0 follows.
+Step 2. As in [26] we use cut-off functions χn
+i , i = 1, . . . , k + 1, with the properties
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+χn
+i (x) = 1
+if
+�
+δn
+i δn
+i+1 ≤ |x| ≤
+�
+δn
+i δn
+i−1;
+χn
+i (x) = 0
+if |x| ≤
+�δn
+i δn
+i+1
+2
+or |x| ≥ 2
+�
+δn
+i δn
+i−1;
+|∇χn
+i (x)| ≤
+1
+�δn
+i δn
+i−1
+and|∇2χn
+i (x)| ≤
+4
+δn
+i δn
+i−1
+,
+for i = 1, . . . , k, and
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+χn
+k+1(x) = 1,
+if |x| ≤
+�
+δn
+k+1δn
+k ;
+χn
+k+1(x) = 0,
+if |x| ≥ 2
+�
+δn
+k+1δn
+k ;
+|∇χn
+k+1(x)| ≤
+1
+�δn
+k+1δn
+k
+,
+and
+|∇2χn
+k+1(x)| ≤
+4
+δn
+k+1δn
+k
+.
+6
+
+The functions φn
+i defined by
+φn
+i (y) := (δn
+i )
+N−2
+2 φn(δn
+i y)χn
+i (δn
+i y),
+for y ∈ Ωn
+i := Ω
+δn
+i
+, i = 1, . . . , k + 1.
+are bounded in D1,2(RN). Therefore we may assume, up to a subsequence,
+φn
+i ⇀ φ∞
+i
+weakly in D1,2(RN), i = 1, 2, . . ., k + 1.
+Now we prove
+φ∞
+i
+= 0
+for i = 1, . . . , k + 1.
+(3.6)
+Again as in the proof of Proposition 4.1 in [6], using (3.4), (3.3) and (3.5), we obtain for any ψ ∈ C∞
+0 (RN)
+and i = 1, . . . , k:
+�
+Ωn
+i
+∇φn
+i (y)∇ψ(y) = (δn
+i )
+2−N
+2
+�
+Ω
+∇ι∗
+µ (f ′
+0(Vεn,λn,ζn(x))φn(x)) ∇
+�
+χn
+i (x)ψ
+� x
+δn
+i
+��
++ o(1)
+= (δn
+i )
+2−N
+2
+�
+Ω
+f ′
+0 (Vεn,λn,ζn(x)) φn(x)χn
+i (x)ψ
+� x
+δn
+i
+�
++ o(1)
+= (δn
+i )2
+�
+Ωn
+i
+f ′
+0 (Vεn,λn,ζn(δn
+i y)) φn
+i (y)ψ(y) + o(1)
+=
+�
+RN f ′
+0 (U1,ζi(y)) φ∞
+i (y)ψ(y) + o(1).
+Hence φ∞
+i
+is a weak solution to
+−∆φ∞
+i
+= f ′
+0(U1,ζi)φ∞
+i , in D1,2(RN).
+Setting Ψj
+1,ζi :=
+∂U1,ζi
+∂ζi,j , for j = 1, . . . , N, and Ψ0
+1,ζi :=
+∂Uδ,ζi
+∂δ
+|δ=1, we deduce as in [26, Lemma 3.1]:
+�
+RN ∇φ∞
+i (x)∇Ψj
+1,ζi(x) = 0,
+j = 0, 1, 2, . . ., N, i = 1, 2, . . . , k.
+Consequently (3.6) holds for i = 1, . . . , k. The proof of φ∞
+k+1 = 0 is similar.
+Step 3. A contradiction arises as in the proof of [6, Proposition 4.1] and [25].
+□
+Proposition 3.2. For every η ∈ (0, 1), there exist ε0 > 0, c0 > 0 such that for every (λ, ζ) ∈ Oη and
+every ε ∈ (0, ε0), there exists a unique solution φε,λ,ζ ∈ K⊥
+ε,λ,ζ of equation (3.2) satisfying
+∥φε,λ,ζ∥µ ≤ c0(ε
+N+2
+2(N−2) + ε
+2k+3
+4 ).
+Moreover, the map Φε : Oη → K⊥
+ε,λ,ζ defined by Φε(λ, ζ) := φε,λ,ζ is of class C1.
+Proof. As in [3], solving (3.2) is equivalent to finding a fixed point of the operator Tε,λ,ζ : K⊥
+ε,λ,ζ → K⊥
+ε,λ,ζ
+defined by
+Tε,λ,ζ(φ) = L−1
+ε,λ,ζΠ⊥
+ε,λ,ζ(ι∗
+µ(fε(Vε,λ,ζ + φ) − f ′
+0(Vε,λ,ζ)φ) − Vε,λ,ζ).
+Now we prove that Tε,λ,ζ is a contraction mapping. As in the proof of [6, Proposition 4.2], using Propo-
+7
+
+sition 3.1, (2.1) and Lemma B.3 we have
+∥Tε,λ,ζ(φ)∥µ ≤ C∥fε(Vε,λ,ζ + φ) − fε(Vε,λ,ζ) − f ′
+ε(Vε,λ,ζ)φ∥2N/(N+2)
++ C∥(f ′
+ε(Vε,λ,ζ) − f ′
+0(Vε,λ,ζ))φ∥2N/(N+2)
++ C∥fε(Vε,λ,ζ) − f0(Vε,λ,ζ)∥2N/(N+2)
++ C
+�����f0(Vε,λ,ζ) −
+� k
+�
+i=1
+(−1)i−1f0(Uδi,ξi) + (−1)kf0(Vσ)
+������
+2N/(N+2)
++
+k
+�
+i=1
+O(µδi) + O
+��
+µσ
+N−2
+2
+� 1
+2 �
+.
+Using Lemma B.4 and observing that
+∥fε(Vε,λ,ζ + φ) − fε(Vε,λ,ζ) − f ′
+ε(Vε,λ,ζ)φ∥2N/(N+2) ≤ C∥φ∥2∗−1
+µ
+,
+we have
+∥Tε,λ,ζ(φ)∥µ ≤ C∥φ∥2∗−1
+µ
++ Cε∥φ∥µ + Cε + O
+�
+ε
+N+2
+2(N−2)
+�
++
+k
+�
+i=1
+O(µδi) + O
+��
+µσ
+N−2
+2
+� 1
+2 �
+= C∥φ∥2∗−1
+µ
++ Cε∥φ∥µ + O
+�
+ε
+N+2
+2(N−2)
+�
++ O
+�
+ε
+2k+3
+4
+�
+.
+The remaining part of the argument is standard and will therefore be left to the reader.
+□
+For λ = (λ1, . . . , λk, λ) and ζ = (ζ1, . . . , ζk) we now consider the reduced functional
+Iε(λ, ζ) = Jε(Vε,λ,ζ + φε,λ,ζ).
+Proposition 3.3. If (λ, ζ) ∈ Oη is a critical point of Iε then Vε,λ,ζ + φε,λ,ζ is a solution of problem (1.1)
+for ǫ > 0 small.
+Proof. We omit it since it is similar to the proof of [6, Proposition 4.3].
+□
+3.2
+Proof of Theorem 1.1
+We assume µ = µ0ǫ with µ0 > 0 fixed, and use the following notations from the above subsection.
+For
+ε > 0 small, λ = (λ1, . . . , λk, λ) ∈ Rk+1
++
+, ζ = (ζ1, . . . , ζk) ∈ (Rn)k,
+we set
+σ = λε
+2(k+1)−1
+N−2
+, δi = λiε
+2i−1
+N−2 , ξ = (ξ1, . . . , ξk) = (δ1ζ1, . . . , δkζk), i = 1, . . . , k.
+For convenience, we denote λk+1 := λ in this subsection.
+Lemma 3.4. For ε → 0+, there holds
+Iε(λ, ζ)
+=
+a1 + a2ε − a3ε ln ε + ψ(λ, ζ)ε + o(ε)
+(3.7)
+C1-uniformly with respect to (λ, ζ) in compact sets of Oη. The constants are given by
+a1 = k + 1
+N
+S
+N
+2
+0 , a2 = (k + 1)
+2∗
+�
+RN U 2∗
+1,0 ln U1,0 − k + 1
+(2∗)2 S
+N
+2
+0 − 1
+2S
+N−2
+2
+0
+Sµ0, a3 = (k + 1)2
+2 · 2∗
+�
+RN U 2∗
+1,0.
+8
+
+The function ψ is given by
+ψ(λ, ζ) = b1λN−2
+1
++
+k
+�
+i=1
+b2(λi+1
+λi
+)
+N−2
+2 h1(ζi) −
+k
+�
+i=1
+b3h2(ζi) − b4 ln(λ1 . . . λk+1)
+N−2
+2
+with
+b1 = 1
+2C0
+�
+RN U 2∗−1
+1,0
+, b2 = C2∗
+0 , b3 = 1
+2C2
+0µ0, b4 = 1
+2∗
+�
+RN U 2∗
+1,0,
+and
+h1(ζi) =
+�
+RN
+1
+|y + ζi|N−2(1 + |y|2)
+N+2
+2
+,
+h2(ζi) =
+�
+RN
+1
+|y + ζi|2(1 + |y|2)N−2 .
+Proof. Observe that
+Jε(Vε,λ,ζ) = 1
+2
+�
+Ω
+�
+|∇Vε,λ,ζ|2 − µ|Vε,λ,ζ|2
+|x|2
+�
+− 1
+2∗
+�
+Ω
+|Vε,λ,ζ|2∗
++
+� 1
+2∗
+�
+Ω
+|Vε,λ,ζ|2∗ −
+1
+2∗ − ε
+�
+Ω
+|Vε,λ,ζ|2∗−ε
+�
+= (I) + (II) + (III).
+For k ≥ 1, Lemma B.5 and Lemma A.4 yield:
+(I) = 1
+2(k + 1)S
+N
+2
+0 − N
+4 S
+N−2
+2
+0
+Sµ0ε − 1
+2C2∗
+0 H(0, 0)λN−2
+1
+�
+RN
+1
+(1 + |z|2)
+N+2
+2
+· ε
+− 1
+2
+k
+�
+i=1
+µ0C2
+0
+�
+RN
+1
+|y|2(1 + |y − ζi|2)N−2 · ε
+− C2∗
+0
+�λk+1
+λk
+� N−2
+2
+�
+RN
+1
+(1 + |y|2)
+N+2
+2
+·
+1
+(1 + |ζk|2)
+N−2
+2
+· ε
+−
+k−1
+�
+i=1
+C2∗
+0
+�λi+1
+λi
+� N−2
+2
+�
+RN
+1
+(1 + |y|2)
+N+2
+2
+·
+1
+(1 + |ζi|2)
+N−2
+2
+· ε + o(ε).
+(3.8)
+By Lemma B.6 and Lemma A.4, we obtain:
+(II) = − 1
+2∗ (k + 1)S
+N
+2
+0 + N − 2
+4
+S
+N−2
+2
+0
+Sµ0ε + C2∗
+0 H(0, 0)λN−2
+1
+�
+RN
+1
+(1 + |z|2)
+N+2
+2
+· ε
++ C2∗
+0
+k
+�
+i=1
+�λi+1
+λi
+� N−2
+2
+�
+RN
+1
+|y|N−2(1 + |y − ζi|2)
+N+2
+2
+· ε
++ C2∗
+0
+k
+�
+i=1
+(λi+1
+λi
+)
+N−2
+2
+�
+RN
+1
+(1 + |y|2)
+N+2
+2
+1
+(1 + |ζi|2)
+N−2
+2
+· ε + o(ε).
+(3.9)
+Finally, Lemma B.7 and Lemma A.4 imply:
+(III) = −
+ε
+(2∗)2 (k + 1)S
+N
+2
+0 − (N − 2)ε
+2 · 2∗
+�
+RN U 2∗
+1,0 · ln(δ1 . . . δkσ)
++ (k + 1)ε
+2∗
+�
+RN U 2∗
+1,0 ln U1,0 + o(ε)
+= −
+ε
+(2∗)2 (k + 1)S
+N
+2
+0 − (N − 2)ε
+2 · 2∗
+�
+RN U 2∗
+1,0 · ln(λ1 . . . λkλ)
+− (k + 1)2
+2 · 2∗
+�
+RN U 2∗
+1,0 · ε ln ε + (k + 1)
+2∗
+�
+RN U 2∗
+1,0 ln U1,0 · ε + o(ε).
+(3.10)
+9
+
+On the other hand, we deduce from Proposition 3.2, (2.3), (2.4), and Lemma B.4 that
+Jε(Vε,λ,ζ + φε,λ,ζ) − Jε(Vε,λ,ζ) = o(ε).
+(3.11)
+Now for k ≥ 1, (3.8), (3.9), (3.10) and (3.11) imply (3.7).
+The case k = 0 can be easily dealt with by using Lemma A.2, Lemma A.3 and (3.11). That (3.7)
+holds C1-uniformly with respect to (λ, ζ) in compact sets of Oη can be seen as in [26, Lemma 7.1]. We
+omit the details here.
+□
+As a corollary of Lemma 3.4 we obtain the following.
+Corollary 3.5. If (λ, ζ) is a stable (e.g. non-degenerate) critical point of ψ(λ, ζ), then Iǫ has for ǫ > 0
+small a critical point (λǫ, ζǫ) that converges towards (λ, ζ) as ǫ → 0.
+Proof. The proof is standard.
+□
+Proof of Theorem 1.1. By the change of variables
+λ
+N−2
+2
+1
+= s1,
+�λ2
+λ1
+� N−2
+2
+= s2, . . . ,
+�λk+1
+λk
+� N−2
+2
+= sk+1,
+ψ(λ, ζ) can be rewritten as
+�ψ(s, ζ) = b1s2
+1 +
+k
+�
+i=1
+b2si+1h1(ζi) −
+k
+�
+i=1
+b3h2(ζi) − b4 ln(sk+1
+1
+sk
+2 . . . sk+1),
+where s = (s1, s2, . . . , sk+1).
+For fixed ζ the equation ∇s �ψ(s, ζ) = 0 has the unique solution �s(ζ) = (�s1(ζ), . . . , �sk+1(ζ)) with
+�s1 =
+�
+(k + 1)b4
+2b1
+,
+�s2 =
+kb4
+b2h1(ζ1), . . . , �sk+1 =
+b4
+b2h1(ζk).
+It is easy to see that �s(ζ) is non-degenerate. Plugging it into �ψ(s, ζ) gives
+�ψ(�s(ζ), ζ) = (k + 1)2b4
+2
+−
+k
+�
+i=1
+b3h2(ζi) − b4(k + 1
+2
+ln (k + 1)b4
+2b1
++
+k
+�
+i=1
+i ln ib4
+b2
+) +
+k
+�
+i=1
+b4(k + 1 − i) ln h1(ζi)
+= C1 +
+k
+�
+i=1
+gi(ζi),
+where
+C1 = (k + 1)2b4
+2
+− b4
+�
+k + 1
+2
+ln (k + 1)b4
+2b1
++
+k
+�
+i=1
+i ln ib4
+b2
+�
+,
+and
+gi(ζi) = b4(k + 1 − i) ln
+�
+RN
+1
+|y + ζi|N−2(1 + |y|2)
+N+2
+2
+− b3
+�
+RN
+1
+|y + ζi|2(1 + |y|2)N−2 .
+A direct computation shows that ζi = 0 is a critical point of gi(ζi) such that
+∂2gi(ζi)
+∂ζi,j∂ζi,l
+���
+ζi=0 = 0
+if j ̸= l;
+10
+
+and
+∂2gi(ζi)
+∂(ζi,j)2
+���
+ζi=0 = 2N − 8
+N
+�
+RN
+b3
+|y|4(1 + |y|2)N−2 > 0.
+Consequently ζi = 0 is a nondegenerate local minimum of gi. Hence ζ = 0 is a C1-stable critical point of
+�ψ(�s(ζ), ζ). Thus we conclude by Corollary 3.5 and Proposition 3.3.
+□
+A
+Some lemmas from [6]
+In this part we collect some lemmas from [6]. We define for η ∈ (0, 1):
+Tη :=
+�
+(λ, ξ) ∈ Rk+1
++
+× Ωk : λi ∈ (η, η−1), λ ∈ (η, η−1), dist(ξi, ∂Ω) > η,
+|ξi| > η, |ξi1 − ξi2| > η, i, i1, i2 = 1, 2, . . ., k, i1 ̸= i2
+�
+.
+Lemma A.1. (i) For i = 1, 2, . . . , k, and j = 0, 1, . . . , N, there holds
+∥PΨj
+i − Ψj
+i∥2N/(N−2) =
+
+
+
+
+
+O
+�
+δ
+N−2
+2
+i
+�
+if j = 1, 2, . . ., N,
+O
+�
+δ
+N−4
+2
+i
+�
+if j = 0
+as δi → 0 uniformly for ξi in a compact subset of Ω.
+(ii) There holds
+∥PΨ − Ψ∥2N/(N−2) = O
+�
+σ
+N−4
+2
+�
+as σ → 0, uniformly for 0 < µ < µ.
+Lemma A.2. For i = 1, 2, · · · , k, the following estimates hold uniformly for (λ, ξ) ∈ Tη:
+(i) For µ, σ → 0:
+�
+Ω
+|∇PVσ|2 − µ|PVσ|2
+|x|2
+= S
+N
+2
+µ − C0C2∗−1
+µ
+H(0, 0)σN−2
+�
+RN
+1
+(|z|β1 + |z|β2)
+N+2
+2
++ O(µσN−2) + O(σN).
+(ii) For δi, σ → 0:
+�
+Ω
+|∇PUδi,ξi|2 = S
+N
+2
+0 − C2∗
+0 H(ξi, ξi)δN−2
+i
+�
+RN
+1
+(1 + |z|2)
+N+2
+2
++ o(δN−2
+i
+).
+Lemma A.3. For µ, σ → 0 there holds
+�
+Ω
+|PVσ|2∗ = S
+N
+2
+µ − 2∗C0C2∗−1
+µ
+H(0, 0)σN−2
+�
+RN
+1
+(|z|β1 + |z|β2)
+N+2
+2
++ O(µσN−2) + O(σN ).
+Lemma A.4. For µ → 0+ there holds
+�
+RN V p
+1 =
+�
+RN U p
+1,0 + o(1) and
+�
+RN V p
+1 ln V1 =
+�
+RN U p
+1,0 ln U1,0 + o(1)
+for p > 1 as well as
+Cµ = C0 −
+C0
+N − 2µ + O(µ2) and Sµ = S0 − Sµ + O(µ2),
+for some positive constant S independent of µ.
+11
+
+B
+Proof of the lemmas from Section 3
+Lemma B.1. For i, l = 1, 2, . . ., k, and j, h = 0, 1, . . . , N, with i ̸= l or j ̸= h, there are constants
+�c0 > 0, �ci,j > 0 such that the following estimates hold uniformly for 0 < µ < µ:
+(PΨ, PΨ)µ = �c0
+1
+σ2 + o(σ−2) as σ → 0,
+(PΨ, PΨj
+i)µ = o(σ−2)o(δ−2
+i
+) as σ → 0, δi → 0, uniformly for ξi in a compact subset of Ω,
+(PΨj
+i, PΨj
+i)µ = �ci,j
+1
+δ2
+i
++ o(δ−2
+i
+) as δi → 0, uniformly for ξi in a compact subset of Ω,
+(PΨj
+i, PΨh
+l )µ = o(δ−2
+i
+) as δi → 0, uniformly for ξi, ξl in a compact subset of Ω.
+Proof. We omit the proof since it is similar to [6, Lemma A.1].
+□
+Lemma B.2. (i) For i, l = 1, 2, · · · , k there holds
+�����
+�
+f ′
+0
+� k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ
+�
+− f ′
+0(Uδl,ξl)
+�
+Ψh
+l
+�����
+2N/(N+2)
+= o
+�
+δ
+− 2N
+N+2
+l
+�
+as σ, δi, δl → 0 uniformly for 0 < µ < µ and ξi in a compact subset of Ω.
+(ii) There holds
+�����
+�
+f ′
+0
+� k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ
+�
+− f ′
+0(Vσ)
+�
+Ψ
+�����
+2N/(N+2)
+= o
+�
+σ− 2N
+N+2
+�
+as σ, δi → 0 uniformly for 0 < µ < µ and ξi in a compact subset of Ω.
+Proof. We only prove (i) for h ̸= 0.
+�
+Ω
+|(f ′
+0(
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ) − f ′
+0(Uδl,ξl))Ψh
+l |2N/(N+2)
+=
+k+1
+�
+i=1
+�
+Ai
+|(f ′
+0(
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ) − f ′
+0(Uδl,ξl))Ψh
+l |2N/(N+2)
++
+�
+Ω\B(0,ρ)
+|(f ′
+0(
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ) − f ′
+0(Uδl,ξl))Ψh
+l |2N/(N+2).
+As in [26, Lemma A.3], by (2.3) and (2.4) we have
+�
+Al
+|(f ′
+0(
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ) − f ′
+0(Uδl,ξl))Ψh
+l |2N/(N+2)
+≤ C
+�
+Al
+|U 2∗−3
+δl,ξl ϕδl,ξlΨh
+l |2N/(N+2) + C
+�
+i̸=l
+�
+Al
+|U 2∗−3
+δl,ξl Uδi,ξiΨh
+l |2N/(N+2)
++ C
+�
+Al
+|U 2∗−3
+δl,ξl VσΨh
+l |2N/(N+2)
+≤ o
+�
+δ
+− 2N
+N+2
+l
+�
+,
+(B.1)
+where we use
+�
+Al
+|U 2∗−3
+δl,ξl ϕδl,ξlΨh
+l |2N/(N+2) ≤ C
+�
+Al
+| δ
+N+2
+2
+l
+(xh − ξh
+l )
+(δ2
+l + |x − ξl|2)3 |2N/(N+2) = O
+�
+δ
+2N(N−3)
+N+2
+l
+�
+,
+12
+
+for i ̸= l,
+�
+Al
+|U 2∗−3
+δl,ξl Uδi,ξiΨh
+l |2N/(N+2) = C
+�
+Al
+|
+δ2
+l (xh − ξh
+l )
+(δ2
+l + |x − ξl|2)3
+δ
+N−2
+2
+i
+(δ2
+i + |x − ξi|2)
+N−2
+2
+|2N/(N+2)
+≤ C(
+�
+Al
+|
+δ2
+l (xh − ξh
+l )
+(δ2
+l + |x − ξl|2)3 |
+N
+2 )
+4
+N+2 (
+�
+Al
+|
+δ
+N−2
+2
+i
+(δ2
+i + |x − ξi|2)
+N−2
+2
+|2N/(N−2))
+N−2
+N+2 = o
+�
+δ
+− 2N
+N+2
+l
+�
+,
+and similarly,
+�
+Al
+|U 2∗−3
+δl,ξl VσΨh
+l |2N/(N+2) = o
+�
+δ
+− 2N
+N+2
+l
+�
+.
+The same arguments as for (B.1) yield for i ̸= l:
+�
+Ai
+�����
+�
+f ′
+0(
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ) − f ′
+0(Uδl,ξl)
+�
+Ψh
+l
+�����
+2N/(N+2)
+= o
+�
+δ
+− 2N
+N+2
+l
+�
+.
+Finally,
+�
+Ω\B(0,ρ)
+�����
+�
+f ′
+0(
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ) − f ′
+0(Uδl,ξl)
+�
+Ψh
+l
+�����
+2N
+(N+2)
+=
+
+
+
+
+
+
+
+O
+�
+δ
+N(N−2)
+N+2
+l
+� �
+O
+�
+σ
+4N
+N+2
+�
++
+k�
+i=1
+O
+�
+δ
+4N
+N+2
+i
+��
+if h = 1, 2, . . . , N,
+O
+�
+δ
+N(N−4)
+N+2
+l
+� �
+O
+�
+σ
+4N
+N+2
+�
++
+k�
+i=1
+O
+�
+δ
+4N
+N+2
+i
+��
+if h = 0.
+Then (i) follows.
+□
+Lemma B.3. There holds
+�����ι∗
+µ
+� k
+�
+i=1
+(−1)i−1f0(Uδi,ξi) + (−1)kf0(Vσ)
+�
+− Vε,λ,ζ
+�����
+µ
+≤
+k
+�
+i=1
+O(µδi) + O
+��
+µσ
+N−2
+2
+� 1
+2 �
+as µ, σ, δi → 0 uniformly for ξi in a compact subset of Ω.
+Proof. It is similar to [6, Lemma A.4].
+□
+Lemma B.4. For ǫ → 0, the following estimates hold uniformly for 0 < µ < µ and (λ, ζ) ∈ Oη:
+∥(f ′
+ε(Vε,λ,ζ) − f ′
+0(Vε,λ,ζ))φ∥2N/(N+2) = O(ε)∥φ∥µ,
+∥fε(Vε,λ,ζ) − f0(Vε,λ,ζ)∥2N/(N+2) = O(ε),
+∥f0(Vε,λ,ζ) − (
+k
+�
+i=1
+(−1)i−1f0(Uδi,ξi) + (−1)kf0(Vσ))∥2N/(N+2) = O
+�
+ε
+N+2
+2(N−2)
+�
+.
+Proof. The first two are from [3]. The last one can be proved as (4.5) in [26].
+□
+Lemma B.5. Let k ≥ 1. Assume without loss of generality that 1 ≤ i < j ≤ k. Then the following
+estimates hold uniformly for (λ, ζ) ∈ Oη:
+(i) For ǫ → 0:
+�
+Ω
+|∇PVσ|2 − µ|PVσ|2
+|x|2
+= S
+N
+2
+µ + o(ε).
+13
+
+(ii) For ǫ → 0:
+�
+Ω
+∇PVσ∇PUδi,ξi − µPVσPUδi,ξi
+|x|2
+=
+
+
+
+
+
+C2∗
+0 ( λ
+λk )
+N−2
+2
+�
+RN
+1
+(1+|y|2)
+N+2
+2
+1
+(1+|ζk|2)
+N−2
+2
+· ε + o(ε)
+if i = k,
+o(ε)
+if i ̸= k.
+(iii) For ǫ → 0:
+µ
+�
+Ω
+|PUδi,ξi|2
+|x|2
+= µC2
+0
+�
+RN
+1
+|y|2(1 + |y − ζi|2)N−2 + o(ε).
+(iv) For ǫ → 0:
+µ
+�
+Ω
+PUδi,ξiPUδj,ξj
+|x|2
+= o(ε),
+i ̸= j.
+(v) For ǫ → 0:
+�
+Ω
+|∇PUδi,ξi|2 =
+
+
+
+
+
+S
+N
+2
+0 − C2∗
+0 H(0, 0)λN−2
+1
+�
+RN
+1
+(1+|z|2)
+N+2
+2
+· ε + o(ε)
+if i = 1,
+S
+N
+2
+0 + o(ε)
+if i ̸= 1.
+(vi) For ǫ → 0:
+�
+Ω
+∇PUδi,ξi∇PUδj,ξj =
+
+
+
+
+
+C2∗
+0 ( λi+1
+λi )
+N−2
+2
+�
+RN
+1
+(1+|y|2)
+N+2
+2
+1
+(1+|ζi|2)
+N−2
+2
+· ε + o(ε)
+if j = i + 1,
+o(ε)
+otherwise.
+Proof. (i) and (v) follow from Lemma A.2.
+Now we prove (ii). Using (2.4), integration by parts yields
+�
+Ω
+∇PVσ∇PUδi,ξi − µPVσPUδi,ξi
+|x|2
+=
+�
+Ω
+V 2∗−1
+σ
+(Uδi,ξi − ϕδi,ξi) + µ
+�
+Ω
+ϕσ(Uδi,ξi − ϕδi,ξi)
+|x|2
+.
+(B.2)
+It is easy to show, using (2.3) and (2.4), that
+�
+Ω
+V 2∗−1
+σ
+ϕδi,ξi ≤ Cδ
+N−2
+2
+i
+�
+Ω
+σ
+N+2
+2
+(σ2|x|β1 + |x|β2)
+N+2
+2
+≤ Cσ
+µ
+√µ−µ δ
+N−2
+2
+i
+�
+RN
+1
+(|y|β1 + |y|β2)
+N+2
+2
+= O(σ
+N−2
+2 δ
+N−2
+2
+i
+),
+(B.3)
+and
+µ
+�
+Ω
+ϕσ(Uδi,ξi − ϕδi,ξi)
+|x|2
+≤ µ
+�
+Ω
+ϕσUδi,ξi
+|x|2
+≤ O(µσ
+N−2
+2 ).
+(B.4)
+On the other hand,
+�
+Ω
+V 2∗−1
+σ
+Uδi,ξi =
+k+1
+�
+j=1
+�
+Aj
+V 2∗−1
+σ
+Uδi,ξi + O(σ
+N+2
+2 δ
+N−2
+2
+i
+).
+(B.5)
+If i = k, j = k + 1, then
+�
+Ak+1
+V 2∗−1
+σ
+Uδk,ξk = C2∗−1
+µ
+C0σ
+N+2
+2 δ
+N−2
+2
+k
+�
+Ak+1
+1
+(σ2|x|β1 + |x|β2)
+N+2
+2
+1
+(δ2
+k + |x − ξk|2)
+N−2
+2
+= C2∗−1
+µ
+C0
+σ
+µ
+√µ−µ
+δ
+N−2
+2
+k
+�
+Ak+1
+σ
+õ
+√µ−µ
+1
+(|y|β1 + |y|β2)
+N+2
+2
+1
+(1 + | σ
+õ
+√µ−µ
+δk
+y − ζk|2)
+N−2
+2
+= C2∗
+0 ( λ
+λk
+)
+N−2
+2
+�
+RN
+1
+(1 + |y|2)
+N+2
+2
+1
+(1 + |ζk|2)
+N−2
+2
+· ε + o(ε).
+(B.6)
+14
+
+If i ̸= k or j ̸= k + 1, then similar arguments as for (B.6) yield
+�
+Aj
+V 2∗−1
+σ
+Uδi,ξi = o(ε).
+(B.7)
+Then we conclude by (B.2)-(B.7).
+Next (iii) follows from:
+µ
+�
+Ω
+|PUδi,ξi|2
+|x|2
+= µ
+�
+Ω
+|Uδi,ξi|2
+|x|2
++ O(µδN−2
+i
+)
+= µC2
+0
+�
+Ω
+δi
+1
+|y|2(1 + |y − ζi|2)N−2 + O(µδN−2
+i
+)
+= µC2
+0
+�
+RN
+1
+|y|2(1 + |y − ζi|2)N−2 + o(ε).
+Similar arguments imply (iv).
+For the proof of (vi), without loss of generality let 1 ≤ i < j ≤ k. Then
+�
+Ω
+∇PUδi,ξi∇PUδj,ξj =
+�
+Ω
+U 2∗−1
+δj,ξj Uδi,ξi + o(ε)
+=
+
+
+
+
+
+C2∗
+0 ( λi+1
+λi )
+N−2
+2
+�
+RN
+1
+(1+|y|2)
+N+2
+2
+1
+(1+|ζi|2)
+N−2
+2
+· ε + o(ε)
+if j = i + 1,
+o(ε)
+otherwise.
+(B.8)
+□
+Lemma B.6. Let k ≥ 1. For ǫ → 0 there holds, uniformly for (λ, ζ) ∈ Oη, setting λk+1 = λ:
+�
+Ω
+�����
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ
+�����
+2∗
+= kS
+N
+2
+0 + S
+N
+2
+µ − 2∗C2∗
+0 H(0, 0)λN−2
+1
+�
+RN
+1
+(1 + |z|2)
+N+2
+2
+· ε
+− 2∗C2∗
+0
+k
+�
+i=1
+(λi+1
+λi
+)
+N−2
+2
+�
+RN
+1
+|y|N−2(1 + |y − ζi|2)
+N+2
+2
+· ε
+− 2∗C2∗
+0
+k
+�
+i=1
+(λi+1
+λi
+)
+N−2
+2
+�
+RN
+1
+(1 + |y|2)
+N+2
+2
+1
+(1 + |ζi|2)
+N−2
+2
+· ε + o(ε),
+Proof. It is easy to see that
+�
+Ω
+�����
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ
+�����
+2∗
+=
+k+1
+�
+j=1
+�
+Aj
+|
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ|2∗ + O(δN
+1 ).
+Observe that
+�
+Ak
+VσU 2∗−1
+δk,ξk = CµC2∗−1
+0
+σ
+N−2
+2 δ
+N+2
+2
+k
+�
+Ak
+1
+(σ2|x|β1 + |x|β2)
+N−2
+2
+1
+(δ2
+k + |x − ξk|2)
+N+2
+2
+= CµC2∗−1
+0
+σ
+N−2
+2
+δ
+√µ−µ
+k
+�
+Ak
+δk
+1
+(( σ
+δk )2|y|β1 + |y|β2)
+N−2
+2
+1
+(1 + |y − ζk|2)
+N+2
+2
+= C2∗
+0 ( λ
+λk
+)
+N−2
+2
+�
+RN
+1
+|y|N−2(1 + |y − ζk|2)
+N+2
+2
+· ε + o(ε),
+15
+
+and
+�
+Aj
+VσU 2∗−1
+δi,ξi = o(ε), if i ̸= k, or j ̸= k.
+From [26], for 1 ≤ i < j ≤ k we also have
+�
+Al
+Uδj,ξjU 2∗−1
+δi,ξi + o(ε) =
+
+
+
+
+
+C2∗
+0 ( λi+1
+λi )
+N−2
+2
+�
+RN
+1
+|y|N−2(1+|y−ζi|2)
+N+2
+2
+· ε + o(ε)
+if j = i + 1, i = l,
+o(ε)
+otherwise.
+Using (B.6), (B.8) and the above three equalities, the proof of (B.9) is contained in Lemma 6.2 in [26].
+□
+Lemma B.7. For µ, σ, δi → 0 there holds, uniformly in compact subsets of Ω,
+�
+Ω
+|
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ|2∗ ln |
+k
+�
+i=1
+(−1)i−1PUδi,ξi + (−1)kPVσ|
+= −N − 2
+2
+ln σ ·
+�
+RN V 2∗
+1
+− N − 2
+2
+ln(δ1δ2 . . . δk) ·
+�
+RN U 2∗
+1,0
++
+�
+RN V 2∗
+1
+ln V1 + k
+�
+RN U 2∗
+1,0 ln U1,0 + o(1).
+Proof. The proof is similar to the one of [6, Lemma A.9].
+□
+Acknowledgements: The authors would like to thank Professor Daomin Cao for many helpful discus-
+sions during the preparation of this paper. This work was carried out while Qianqiao Guo was visiting
+Justus-Liebig-Universit¨at Gießen, to which he would like to express his gratitude for their warm hospi-
+tality.
+Funding: Qianqiao Guo was supported by the National Natural Science Foundation of China (Grant
+No. 11971385) and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No.
+2019JM275).
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+18
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+arXiv:2301.03604v1  [astro-ph.GA]  9 Jan 2023
+MNRAS 000, 000–000 (2022)
+Preprint 11 January 2023
+Compiled using MNRAS LATEX style file v3.0
+PISN-explorer: hunting the descendants of very massive first stars⋆
+D. S. Aguado1,2,5†, S. Salvadori1,2, Á. Skúladóttir1,2, E. Caffau3, P. Bonifacio3,
+I. Vanni1,2, V. Gelli1,2, I. Koutsouridou1,2, and A. M. Amarsi4
+1Dipartimento di Fisica e Astronomia, Universit á degli Studi di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
+2INAF/Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy
+3GEPI, Observatoire de Paris, Université PSL, CNRS, 5 Place Jules Janssen, 92190 Meudon, France.
+4Theoretical Astrophysics, Department of Physics and Astronomy, Uppsala University, SE-751 20 Uppsala, Sweden
+5Instituto de Astrofísica de Canarias, Vía Láctea, 38205 La Laguna, Tenerife, Spain
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The very massive first stars (m > 100 M⊙) were fundamental to the early phases of reion-
+ization, metal enrichment, and super-massive black hole formation. Among them, those with
+140 ≤ m/M⊙ ≤ 260 are predicted to evolve as Pair Instability Supernovae (PISN) leaving a
+unique chemical signature in their chemical yields. Still, despite long searches, the stellar de-
+scendants of PISN remain elusive. Here we propose a new methodology,the PISN-explorer,
+to identify candidates for stars with a dominant PISN enrichment. The PISN-explorer is
+based on a combination of physically driven models, and the FERRE code; and applied to data
+from large spectroscopic surveys (APOGEE, GALAH, GES, MINCE, and the JINA database).
+We looked into more than 1.4 million objects and built a catalogue with 166 candidates of PISN
+descendants. One of which, 2M13593064+3241036, was observed with UVES at VLT and
+full chemical signature was derived, including the killing elements, Cu and Zn. We find that
+our proposed methodology is efficient in selecting PISN candidates from both the Milky Way
+and dwarf satellite galaxies such as Sextans or Draco. Further high-resolution observations
+are highly required to confirm our best selected candidates, therefore allowing us to probe the
+existence and properties of the very massive First Stars.
+Key words: stars: abundances – stars: Population III – stars: Population II– Galaxy: halo–
+cosmology: early universe
+1
+INTRODUCTION
+The first (Pop III) stars formed out of primordial composition gas,
+i.e. in environments where the fragmentation process was less ef-
+ficient then in local star-forming regions, thus enabling the for-
+mation of more massive stars than those that we observe today:
+from a few tens, up to thousands of solar masses (e.g. Susa et al.
+2014; Hirano et al. 2015). Their characteristic mass, furthermore,
+was likely ≥ 10 M⊙ (e.g. Bromm 2013). These theoretical findings
+have strong physical grounds and have been reported across decades
+by means of different analytical calculations (see e.g. Silk 1977;
+McKee & Tan 2008), numerical simulations (see e.g. Abel et al.
+2002; Hosokawa et al. 2011), and studies of stellar archaeology
+(e.g. Ishigaki et al. 2018; Rossi et al. 2021). According to cosmo-
+logical models, these primitive stars were likely hosted by the bulge
+and the stellar halo of the Milky Way (see e.g. Tumlinson 2010;
+⋆ Based on observations made with the ESO Very Large Telescope at the
+La Silla Paranal Observatory under program ID 108.23N5.001
+† E-mail: david.aguado@unifi.it
+Salvadori et al. 2010; Starkenburg et al. 2017a) and by Local group
+dwarf galaxies (e.g. Salvadori et al. 2015). From an observational
+point of view, however, we are still lacking key probes of first stars in
+the high-mass regime (m∗ > 140 M⊙) very recently some indirect
+hints have been provided (Pagnini et al. sumbitted).
+Very massive first stars, 140 ≤ m∗/M⊙ ≤ 260, are predicted
+to end their lives as energetic Pair Instability Supernovae (PISN),
+which inject 50% of their mass in the form of metals into the in-
+terstellar medium (ISM). Thus, a single PISN will strongly enrich
+the primordial gas with a probability distribution function peak-
+ing at [Fe/H] ≈ −2.0 (Karlsson et al. 2008; de Bennassuti et al.
+2017; Salvadori et al. 2019), leaving a unique chemical signa-
+ture with a strong odd-even effect (e.g. Heger & Woosley 2002;
+Takahashi et al. 2018). Unfortunately, pure PISN descendants are
+predicted to be extremely rare: even at the peak metallicities,
+[Fe/H] ≈ −2.0, they are thought to represent < 0.1% of Milky
+Way stars (de Bennassuti et al. 2017). The traditional approach of
+searching only for stars showing 100% PISN enrichment has thus
+not proven successful. Rather, a large fraction of PISN descendants
+are expected to also have a significant contribution from normal
+© 2022 The Authors
+
+2
+D. Aguado et al.
+Pop II stars, which can form early on in the Pop III-enriched ISM
+and then evolve as core-collapse supernovae (CCSN).
+The abundances of the so-called killing elements Cu and Zn
+(Salvadori et al. 2019) may be the smoking gun for a true PISN
+descendant. These elements are barely produced by PISNe, but
+yielded by all other supernovae (see also Vanni et al. in prep). The
+extreme sub-solar abundances of Zn and Cu with respect to Fe from
+PISNe, persist even in environments polluted by other sources up
+to a 50% level (Salvadori et al. 2019). But given the rarity of such
+stars, until now, only two descendants of PISN have been reported
+(Aoki et al. 2014; Salvadori et al. 2019). We are thus in the situation
+where even a few more identified PISN descendants would greatly
+advance the field.
+In the recent years there has been a breakthrough in large
+spectroscopic surveys targeting stars in and around the Milky
+Way (e.g. Gaia, APOGEE, GALAH, GES). Millions of stars
+have been observed with intermediate- to high-resolution spectra
+(R ≳ 10, 000), and these numbers will further increase with two
+large upcoming spectroscopic surveys, starting operations in the
+next two years: WEAVE in the Northern hemisphere (Dalton et al.
+2016), and 4MOST in the South (de Jong et al. 2019). For the first
+time, we are thus able to have the statistics to identify and charac-
+terise rare populations, such as PISN descendants.
+Searching through such large databases, however, requires a
+focused and dedicated effort, and with this goal in mind we have
+developed an innovative methodology, the PISN-explorer. With
+a combination of theoretical models (Salvadori et al. 2019), and
+the FERRE code (Allende Prieto et al. 2006), the PISN-explorer
+exploits all the chemical abundances measured by large surveys
+(e.g. APOGEE, GALAH, GES, or MINCE) to identify stars that
+have likely been dominantly enriched by PISN (≳ 50%). Here we
+present this new method, allowing us for the first time to use large
+databases of chemical abundances to systematically search for the
+elusive PISN descendants.
+2
+THE THEORETICAL MODELS
+The models used by the PISN-explorer are adopted from the
+general parametric study presented in Salvadori et al. (2019). This
+study provides predictions for the chemical properties of an ISM
+predominantly imprinted by very massive first stars exploding as
+PISNe, i.e. where the PISNe products account for ≥ 50% of metals
+in the ISM. The model is general because it condenses the un-
+known physical processes related to early cosmic star-formation,
+metal diffusion, and mixing, into three free parameters: 1) the star
+formation efficiency, f∗, which provides the fraction of ISM gas
+condensed into stars; 2) the dilution factor, fdil, which parametrises
+the amounts of metals effectively retained into the ISM; 3) and the
+mass fraction of ISM metals contributed by PISNe, fPISN. Predic-
+tions for a PISN-imprinted ISM are provided by exploring the full
+parameter space. Hence the predictions are general and essentially
+model independent.
+The ansatz of this approach is to assume that a single very
+massive first star exploding as PISN can form in the primordial
+star-forming (mini-)haloes. This assumption is strongly supported
+by the results of hydrodynamical cosmological simulations follow-
+ing the formation of the first stars (e.g. Hirano et al. 2014). The
+chemical enrichment of the ISM is evaluated after the injection of
+metals by a single PISN (fPISN = 100%) with different progeni-
+tor masses, mPISN = [140 − 260]M⊙. It turns out that the final
+metallicity (or [Fe/H]) of the ISM is settled by the PISN mass and
+by the ratio between the star-formation efficiency and the dilution
+factor, f∗/fdil. Furthermore, it is always ZISM > 10−3Z⊙, which
+implies that “normal” Pop II stars can form out of this medium and
+thus contribute to the subsequent ISM enrichment (see Fig. 2 of
+Salvadori et al. 2019).
+The abundance ratios of each element X (from C to Zn)
+with respect to iron, are computed by varying the relative con-
+tribution of PISN and Pop II stars to the chemical enrichment
+(fPISN = [50 − 100]%) and by assuming that Pop II stars form
+according to a standard Larson Initial Mass Function (IMF):
+φ(m⋆) = m−2.35
+⋆
+exp(−mch/m⋆) with mch = 0.35M⊙ (Larson
+1998). The calculations are made by adopting the yields by
+Heger & Woosley (2002) for very massive first stars exploding
+as PISNe, YPISN
+X
+(mPISN), and of Woosley & Weaver (1995) for
+Pop II stars with initial masses m∗ = [8−40]M⊙ and metallicities,
+Z∗ = [10−4; 1]Z⊙, which explode as CCSN, YXII(m∗, Z∗). Ac-
+cording to Woosley & Weaver (1995) we assume that stars between
+40 and 140 M⊙ collapse directly into a black hole thus not contribut-
+ing to the chemical enrichment. Note that since we are integrating
+over the entire Pop II IMF the derived yields are not so different from
+those obtained with models assuming that m∗ > 20M⊙ stars pro-
+duce a negligible amount of heavy elements (e.g. Limongi & Chieffi
+2018). Salvadori et al. (2019) demonstrated that the ISM abundance
+ratios, [X/Fe], or the chemical abundance pattern of the stars formed
+out of it, depend upon fPISN, the yields of PISN and of Pop II stars;
+but they are not directly affected by f∗/fdil although this ratio
+controls the metallicity of the Pop II stars contributing to the ISM
+enrichment.
+3
+EXPLOITED DATASETS
+The goal of this work is to efficiently select candidates for PISN
+descendants from existing and publicly available data. Suitable
+databases should provide reliable elemental abundances for FGK
+stars together with stellar parameters: effective temperature, sur-
+face gravity, and metallicity (Teff, log g, and [Fe/H]). From the
+available spectroscopic surveys, we restrict our selection to the
+large surveys that have the best chemical abundances information
+(APOGEE, GALAH, Gaia-ESO, and the MINCE survey). In ad-
+dition, we exploit high-quality literature data available through the
+JINA database. Here, we will mainly focus on elements from C to
+Zn but sometimes abundances of heavier elements (Ba, Ce) could be
+also used to impact the selection, since these are expected to be low
+in PISN descendants. Finally, in the following we assumed the solar
+abundances each survey used at that time. Future -improved- mea-
+surements of solar abundances are easily applicable to our method-
+ology.
+3.1
+APOGEE
+The Apache Point Observatory Galactic Evolution Experiment 2
+(APOGEE-2; Majewski et al. 2017) is a near-infrared high-
+resolution survey in the H-band (1.514-1.696 µm at R ∼ 22, 500)
+that provides stellar parameters (Teff, log g , [Fe/H]), radial veloc-
+ities, and elemental abundances. Solar abundances are those from
+Asplund et al. (2005). The high quality of APOGEE measurements
+at metallicities down to [Fe/H] ∼ −2 clearly recommends the use
+of this spectroscopic survey. For our purpose, we use the final all-
+Star catalogue of APOGEE Data Release (DR) 17 (Abdurro’uf et al.
+2022) which includes up to 20 elemental abundances (C, N, O, Na,
+Mg, Al, Si, P, S, K, Ca, Sc, Ti, Cr, Mn, Fe, Co, Ni, Cu, and Ce) for
+MNRAS 000, 000–000 (2022)
+
+3
+Figure 1. Example of the PISN-explorer analysis performed on an APOGEE star.
+(a) The measured chemical signature of 2M13593064+3241036 from
+APOGEE (blue dots) together with the best fit to a PISN model computed
+with the PISN-explorer (red line). Derived PISN parameters and main
+stellar parameters are also shown. Finally, for comparison we show interpo-
+lated models with ±20M⊙ of PISN mass (dotted lines) and ±15% of PISN
+contamination (dashed lines).
+(b) fPISN−mPISN space of solutions when applying and MCMC algorithm
+computed with FERRE over the chemical signature of 2M13593064+3241036
+including 10 chains of 50,000 experiments each. Color bar represents the
+likelihood of each solution.
+879,437 spectra (733,901 objects)1. Following the recommendation
+of the APOGEE team, we discard both Ti i and Ti ii from further
+analysis. Unfortunately, the Cu absorption lines in the infrared are
+extremely week and therefore only detectable for metal-rich stars,
+[Fe/H] > −1.
+3.2
+GALAH
+The
+GALactic
+Archaeology
+with
+HERMES
+(GALAH,
+De Silva et al. 2015) survey is a large spectroscopic survey
+in the optical at a resolving power R ∼ 28, 000, covering four
+wavelength windows within 4713 to 7887 Å. The third data
+release (DR3; Buder et al. 2021), including the main_allspec_v2
+catalogue2, provides for 588,571 stars: stellar parameters, radial
+velocities, and chemical abundances for up to 30 elements (Li, O,
+C, Na, Mg, Al, Si, K, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Cu, Zn, Rb,
+Sr, Y, Zr, Mo, Ru, Ba, La, Ce, Nd, Sm, Eu). GALAH provides
+chemical abundances at [Fe/H] ≳ −2, but not all elements are
+available at the lowest metallicities. Solar abundances are from
+Grevesse et al. (2007).
+3.3
+Gaia-ESO
+The Gaia-ESO Survey (GES, Gilmore et al. 2012, 2022) has ob-
+served 115,000 stars from the main components of the Milky Way,
+including star clusters, using FLAMES at the VLT. The employed
+instruments (GIRAFFE and UVES) provide high-resolution spec-
+tra, 16, 200 < R < 47, 000, covering various wavelength ranges,
+depending on the configuration. We retrieved3 the latest available
+DR5 (Randich et al. 2022) with 114,324 stars. The solar abundances
+1 The
+APOGEE
+catalogue
+can
+be
+retrived
+here:
+https://www.sdss.org/dr17/irspec/spectro_data/
+2 GALAH catalogue:https://cloud.datacentral.org.au/teamdata/GALAH/
+3 Gaia-ESO catalogue:https://www.eso.org/qi/catalogQuery/index/393
+used for this survey are those from Asplund et al. (2009). A signif-
+icant fraction of GES pointings is dominated by metal-rich stars
+[Fe/H] > −1, which are less likely to be PISN descendants. These
+fields can, however, also contain some metal-poor stars, so we in-
+cluded the entire dataset. Including the entire GES sample also
+ensures that our search is as unbiased as possible. The GES DR5
+dataset includes stellar parameters, radial velocities, and chemical
+abundances for up to 31 elements (He, Li, C, N, O, Na, Mg, Al, Si,
+S, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Cu, Zn, Sr, Y, Zr, Mo, Ba, La, Ce,
+Pr, Nd, Sm, Eu)
+3.4
+MINCE
+Measuring at Intermediate metallicity Neutron-Capture Elements
+(MINCE, Cescutti et al. 2022) is a project aiming to study abun-
+dances for neutron-capture elements using different facilities such
+as HARPS, UVES, or FIES. Conveniently, the targets selection
+is designed to observe intermediate and very metal-poor stars
+(−0.7 ≳ [Fe/H] ≳ −2.5). For this work we employed the first
+year sample including 43 stellar targets and high quality elemental
+abundances O, Na, Mg, Al, Si, S, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Cu,
+Zn. The employed solar abundances are those from Lodders et al.
+(2009) and Caffau et al. (2011) for O and S. The following data
+releases in subsequent years will be a valuable source of PISN
+descendant candidates.
+3.5
+JINA
+The Joint Institute for Nuclear Astrophysics (JINA) database
+(Abohalima & Frebel 2018) is a collection of literature papers with
+high-resolution analysis of 1658 metal-poor stars [Fe/H] < −2,
+including a variable range of elemental abundances (from Li to U).
+We downloaded the whole database4, set the solar abundances to
+4 JINA database: https://jinabase.pythonanywhere.com/
+MNRAS 000, 000–000 (2022)
+
+LOW
+HIGH
+likelihood
+210
+205
+[Ma]
+200
+195
+190
+50
+56
+62
+69
+75
+fpisn [%]4
+D. Aguado et al.
+Figure 2. Metallicity distribution functions for the 1.4 million stars anal-
+ysed (blue), and the 166 PISN selected candidates (red). Note that the red
+distribution has been arbitrarily scaled for easier comparison.
+Asplund et al. (2009), and treat each entry in the same way regard-
+less of its origin (e.g. halo, bulge, or dwarf galaxy members).
+4
+METHODOLOGY
+We have developed the PISN-explorer, a systematic methodology
+to efficiently select PISN-polluted candidates from observational
+data (Sec. 3) using the FERRE code5 from Allende Prieto et al.
+(2006) and theoretical predictions from Salvadori et al. (2019)
+(Sec. 2). FERRE is a FORTRAN code optimised for spectral analysis
+(e.g. Allende Prieto et al. 2014; Aguado et al. 2017) but capable of
+performing fitting of any class of data to a rectangular grid of theo-
+retical models. The PISN-explorer methodology implies several
+steps:
+(i) Packaging of models: A grid of theoretical predictions for
+the chemical abundances of PISN descendants (50-100% PISN
+pollution) is created with a FERRE-friendly header. Key information
+included in the header: a) the number and label for each free
+parameter included in the models (fPISN, log10 (f∗/fdil), and
+mPISN); b) the minimum value for each parameter (0.5, -4, and
+140 M⊙ respectively); c) the steps for each parameter (0.1, 1,
+9M⊙, respectively); d) the number of elements (“pixels”) within
+the models (25, counting from C to Zn), and their atomic numbers
+(“λ-array”). Finally, the grid is complete with the predictions
+([Xi=6,30/Fe]) in rows just below the header. Here, as explained
+in Sec. 2, we use Pop II yields from (Woosley & Weaver 1995) to
+be fully consistent with Salvadori et al. (2019). However, if the
+potential user of PISN-explorer wishes to use a different set
+of models (e.g. Nomoto et al. 2013; Limongi & Chieffi 2018, or
+any other that can be published in the future) the grid has to be
+re-computed as explained along these lines.
+(ii) Preparing the data: Following the recommendation of
+each survey, we discard stars with suboptimal flags, STAR_BAD
+ASPCAPFLAG for APOGEE and flag_sp̸= 0 for GALAH, while
+all the sample was taken from Gaia-ESO, MINCE, and JINA.
+Additionally, we only considered elemental abundances with
+5 FERREis availablefrom http://github.com/callendeprieto/ferre
+x_fe_flag = 0 in both APOGEE and GALAH, meaning that
+only reliable values are considered. For each star in our sample we
+prepared three different files with 25-columns length each: 1) the
+measured chemical ratios ([Xi=6,30/Fe]obs) from C to Zn; 2) the
+uncertainties (σ[Xi=6,30/Fe]obs); and 3) the weight we gave to each
+measurement (ω[Xi=6,30/Fe]obs). The ω values are equal to 0 if
+there is no measurement for element X, 1 for Cu and Zn, and 0.5
+for the rest. Thus, we give double weight to the killing elements
+since they are the smoking gun of PISN pollution. This tuning
+implies that the χ2
+0 will heavily depend on the Cu and Zn if they
+are available but no effect otherwise.
+(iii) Launching the code: FERRE is able to look for the best
+fit by interpolating between the nodes of the grid. We used
+quadratic interpolation and Nash’s truncated Newton algorithm
+(Nash & Sofer 1990). After the minimum χ20 is found, the code
+produces the set of best parameters with errors, along with an
+interpolated model. In Fig. 1a we show an example of a fit with the
+observed values ([Xi=6,30/Fe]obs, blue dots) from the literature for
+a single APOGEE object, and the best fit ([Xi=6,30/Fe]fit, red line)
+derived with FERRE, giving a set of parameters: fPISN = 66.1%;
+f∗/fdil = 10−3.1; and mPISN = 193.8M⊙. We also calculated
+a reduced χ2 as χ2 = χ2
+◦/(N − M), where χ2
+◦ is the one
+defined in Equation 14 of Salvadori et al. (2019), N the number of
+fitting points, and M the number of free parameters of the model.
+Therefore, we will have reduced χ2 values > 0. In the following
+we will refer to this reduced χ2.
+(iv) Interpreting the results and caveats: The results of the
+PISN-explorer are interpreted, i.e. the best set of parameters
+(fPISN, log10 (f∗/fdil), and mPISN), and an interpolated model
+([Xi=6,30/Fe]fit). When a star’s chemical signature is very far from
+the theoretical predictions (the vast majority of the cases), the code
+will not be able to find a good fit and will usually end up on a
+solution with a high χ2 at the limit of the grid (mPISN = 140 M⊙,
+for example). This is because the code is trying to find something
+beyond the ranges of the grid, and should therefore be interpreted
+as a non-reliable solution. Unless the quality of the fit is very
+high, solutions with fPISN and mPISN close to the limit of the
+grid are discouraged. However, the log10 (f∗/fdil) parameter is
+less sensitive to model-to-model variations because the predicted
+abundance patterns do not directly depend upon this quantity
+(see Sec. 2). Therefore good fits can appear at the limits of the
+log10 (f∗/fdil) range. Objects with a very low number of derived
+elemental abundances (≤ 5) lead to no recommended solutions.
+Additionally, there are some elements that are more affected by
+stellar processes in evolved stars. C, for instance, is converted into
+N during the CN cycle, and brought to the surface during the red
+giant branch phase. Thus, it is highly recommended to correct
+for these stellar evolutionary effects (see, e.g. Placco et al. 2014).
+Finally, any available corrections due to inhomogeneities in the
+three-dimensional (3D) stellar atmosphere and the non-local ther-
+modynamic equilibrium (NLTE) state of the matter for elemental
+abundances will significantly improve the results of the analysis.
+(v) Looking for asymmetric signatures: The well known
+odd-even bias is a remarkable feature of measured stellar chemical
+compositions (see, e.g. Payne 1925; Russell 1941). Elements with
+an even atomic number (C, O, Mg, Si, Ca, Ti) are more easily
+detected in the atmospheres of metal-poor stars and therefore most
+of the surveys are able to provide them for a large number of
+objects with good precision. Consequently, the available chemical
+MNRAS 000, 000–000 (2022)
+
+5
+signatures are asymmetric, i.e. biased towards even-elements. This
+is unfortunate, since a stark contrast between the abundances of
+odd and even elements is a key signature of PISN yields (see
+e.g. 1a). Therefore, the higher is the number of available odd
+elemental abundances the better performance can be obtained with
+this methodology.
+(vi) MCMC validation: To ensure that degeneracy in the pa-
+rameters determination is not affecting our methodology we used
+a Markov-Chain Monte-Carlo (MCMC) analysis based on self-
+adaptive randomised subspace sampling (Vrug et al. 2009). This
+algorithm is also implemented in the FERRE code and can help to
+understand the solution space, and offer a statistical validation for
+the evaluation of uncertainties. We launched 10 chains of 50,000
+experiments each and let the code burn up to 500 experiments in
+each chain. In Fig. 1b we show the likelihood in the fPISN −mPISN
+space of solutions. Although the employed algorithm looking for
+the best solution is different, unsurprisingly the best solution is
+compatible within the uncertainties with the one shown in Fig. 1a
+(fPISN = 65%; f∗/fdil = 10−3.1; and mPISN = 195.3 M⊙). The
+likelihood distribution (Fig. 1b) shows solution areas with similar
+probability that are smaller than the step of the grid. Furthermore,
+the smooth behaviour of the likelihood distribution shows that no
+evident degeneracies are playing a critical role in our calculations.
+Therefore, this MCMC test demonstrates that the solution FERRE is
+finding is stable and the nodes corresponding to the space parame-
+ters are small enough to account for the required sensitivity.
+5
+TARGET SELECTION
+Our sample mainly consists of nearby stars from the disk, the bulge,
+and the halo of the Milky Way (Sec. 3). We analyzed 1,438,497 stars
+following the methodology explained in Sec. 4 with no further cuts
+applied at this stage. In Fig. 2 we show in blue the metallicity distri-
+bution function of the whole sample, including GALAH, APOGEE,
+GES, andJINAstarswithavailablemeasurementsofchemical abun-
+dances. The target selection based on the PISN-explorer analysis
+is a two step procedure based first on hard cuts applied blindly
+over the entire sample, systematic selection, and secondly on more
+specific and survey-dependent selection, individual selection. We
+finally end up with our golden catalogue of candidates for further
+high-resolution follow-up. The three stages are detailed below:
+• Systematic Selection. The MDF for the entire considered sam-
+ple is shown in red in Fig. 2. Following the known caveats (discussed
+in Sec. 4), we discarded objects with less than 5 available chemical
+abundances. In addition, since we focus on FGK type metal-poor
+stars we only include stars with 4000 K<Teff < 7000 K and [Fe/H]
+< −0.7. Hotter stars have much weaker metallic absorption lines,
+and it is in general not possible to derive a complete chemical sig-
+nature. On the other hand, the early M type stars are dominated
+by molecular bands, and thus cooler stars would also prevent from
+determining accurate chemical abundances. The remaining sample
+of 385202 stars is run through the PISN-explorer. Subsequently,
+we discard fits with χ2 > 0.35 and remove out after inspection
+solutions with mPISN < 145M⊙ and/or fPISN < 55% since they
+are close to the limit of the grid.
+• Individual Selection. Hard cuts themselves do not guaranty
+an optimal selection. First of all, the killing elements, Cu and Zn
+are important guidelines and, if available, we discard objects with
+[Cu, Zn/Fe] > 0. In addition, we maximised, when possible, odd-
+even effect by selecting objects with [Mg/Al] > 0.7. We also
+preferred objects with [Mg/Ca] < 0.0 that is also a desirable
+sign according to our models. After applying all of this criteria we
+end-up in a catalogue of 166 candidates from all included surveys
+(Sec. 3). In Fig. 2 we also show in red our selection. It is quite clear
+that different datasets contribute in a different manner. While the
+most metal-poor bump ([Fe/H] ≲ −2.5) is mainly from the JINA
+database, the most metal-rich peak ([Fe/H] ≈ −1) is dominated
+by GES stars. However, the bulk of our selection based on GALAH
+and APOGEE is peaked at [Fe/H] = −1.7, close to the most likely
+metallicity region where PISN descendants are predicted to live
+(e.g. Karlsson et al. 2008; Salvadori et al. 2019). We note that the
+red MDF showed in Fig. 2 is, at first order, independent of the
+selection criteria we applied.
+• Golden catalogue We included extra criteria before consider
+any star for future follow-up. At this stage more careful direct in-
+spection is recommended, analysing the suitability of each can-
+didate individually. Therefore, we included two objects that did
+not completely achieve the general criteria χ2 < 0.35 but have
+[Zn/Fe] < −0.8. In addition to it, we removed objects with clear
+s-process enrichment (Ba or Ce) to discard possible contribution
+from AGB stars. In addition to it, to select objects with large num-
+ber of abundances (N≥ 10) is obviously a desirable strategy so we
+prioritise those to others with better fit but less complete chemi-
+cal information. Finally, we also include suitable observability cuts
+(Vmag ≤ 16) ending up in a 45 objects golden catalogue that
+contains the most promising candidates to be enriched by PISN
+pollution.
+Following the selection criteria described in this section we
+ended up with our golden catalogue of 45 objects6. One of the most
+promising candidates, 2M13593064+3241036 from APOGEE, is
+shown in Fig. 1a. For this object the quality of the fit is re-
+markable: measurements for two odd elements were available (Al
+and K), and the odd-even effect is clear, [Mg/Al] = +0.92,
+[Mg/Ca] < −0.18, and [Ca/K] < +0.75 . APOGEE does pro-
+vide NLTE corrections for Mg, K, and Ca (Osorio et al. 2020).
+Unfortunately, APOGEE does not measure Zn and provides very
+few Cu abundance measurements. Thus, the smoking gun of PISN
+pollution is missing for this interesting star. Therefore, we applied
+for an ESO-DDT proposal of one hour to try to accurately measure
+both killing elements, Zn and Cu.
+6
+OBSERVATIONS AND ANALYSIS
+6.1
+UVES observations
+Our target, 2M13593064+3241036, was observed with UVES at
+the 8.2 m VLT Kueyen Telescope in a single observing block (OB)
+of one hour in service mode, during the night of the 28th of March
+2022, under program ID 108.23N5.001. A 1.′′2 slit was used with
+1 × 1 binning in grey sky conditions and an airmass of ∼ 2.0.
+The seeing was 1.′′2 after corrected by airmass. Our settings used
+dichroic #Dic1 (390 + 564) and provided a spectral coverage be-
+tween 330 and 660 nm. We corrected each spectrum for the barycen-
+tric velocity. The signal-to-noise ratio (SNR) per pixel in the spectra
+was ∼19 at 390 nm, 48 at 510 nm, and 55 at 660 nm. The resolving
+power for this set up is R∼ 45, 000 for the blue part of the spectrum
+(330 − 452 nm) and R∼ 41, 500 for the red (480 − 680 nm). The
+6 The catalogue will be published in a forthcoming paper including a set of
+new high-resolution observations.
+MNRAS 000, 000–000 (2022)
+
+6
+D. Aguado et al.
+data were reduced using the REFLEX environment (Freudling et al.
+2013) within the ESO Common Pipeline Library.
+6.2
+Stellar Parameters
+APOGEE analysis pipeline (ASPCAP; García Pérez et al. 2016)
+used spectral profile fitting to derive for 2M13593064+3241036:
+Teff =5106 K, log g =2.60, and [Fe/H] = −1.79 (Majewski et al.
+2017). We perform a similar analysis for our UVES optical spec-
+trum by fitting the data with the FERRE. FERRE is able to match
+observations with a library of stellar models by interpolating be-
+tween the nodes of the grid for different parameters. The synthetic
+models were computed originally by Aguado et al. (2021), covering
+the following range of parameters:
+• 4500 K < Teff < 7000 K, ∆Teff = 250 K
+• 1.0 < log g < 5.0, ∆ log g = 0.5
+• −4.0 < [Fe/H] < +1.0, ∆[Fe/H] = 0.5
+• −1.0 < [C/Fe] < +3.0, ∆[C/Fe] = 1.0
+The models were computed in one dimension local thermody-
+namical equilibrium (1D-LTE) with the ASSET code and the AT-
+LAS12 stellar atmospheres (Sbordone et al. 2007; Kurucz 2005).
+The atomic and molecular data were taken from Kurucz webpage7
+and enriched with the literature as explained in (Allende Prieto et al.
+2018). For the analysis we first smoothed and resampled the models
+to the UVES resolution and performed a running-mean normal-
+ization with a 150-pixel window. Then we also normalised the
+data accordingly and launched the code with the Nash’s truncate
+Newton search algorithm (Nash & Sofer 1990), and cubic order
+interpolation. Thus we derived for 2M13593064+3241036: Teff
+= 5036 ± 105 K, log g = 2.59 ± 0.20, [Fe/H] = −2.29 ± 0.10,
+and [C/Fe]= +0.05 ± 0.15. Atmospheric parameters are in ex-
+cellent agreement with those derived from the H-band, however,
+the measured [Fe/H] is 0.5 dex lower than the ASPCAP value.
+This discrepancy is certainly unusual and cannot be easily ex-
+plained since the APOGEE spectrum is of high quality (SNR∼130).
+We re-analysed the APOGEE spectrum with consistent models
+from Allende Prieto et al. (2018) and found compatible metallicity
+[Fe/H] ∼ −1.9. Therefore, we attribute the discrepancy to a prob-
+lem that could be related with a problem to the sky or background
+subtraction in the APOGEE spectrum. Additionally, we found per-
+sisting_high flag (residual signal in the detector) within AS-
+PCAP documentation that, in principle, giving the brightness of
+this object should not be a problem. Yet, we argue that it could
+possibly play a role and contribute to explain such a difference. Un-
+fortunately, the impact of this metallicity discrepancy significantly
+influences the abundances, [X/Fe], derived by APOGEE for this
+star, as is shown in the following Sec. 6.3. Thus, for sake of caution
+we will not use APOGEE abundances for this object.
+As sanity check, we compared our stellar parameters
+from spectroscopy with the ones we derived with the Gaia
+colours (Gaia Collaboration et al. 2021), applying the calibration
+by Mucciarelli et al. (2021) and assuming [Fe/H] = −2.3, Teff
+= 5144 ± 100 K and log g = 2.5 ± 0.2. Both set of parameters are
+in good agreement also with the APOGEE ones. Furthermore, we
+note that in different visits APOGEE finds different radial veloci-
+ties: MJD 56389: -15.6
+,km s−1; MJD 56405: -9.3 km s−1; and MJD 56412 -8.8 km s−1
+with S/N > 30 for all visits. In addition, Gaia RVS (Cropper et al.
+7 http://kurucz.harvard.edu/
+Figure 3. UVES spectrum (black lines) of the star 2M13593064+3241036,
+around the absorption lines: Cu i at 5105 Å (top); and Zn i at 4810 Å (bot-
+tom). Two models at different elemental abundances are also shown (blue
+and red lines). Other metallic absorption lines in the spectra are also labelled.
+2018) is giving vrad = −13.5 ± 3.2 km s−1 with 38 transits.
+Finally, from the UVES spectrum we derive vrad = −15.8 ±
+1.2 km s−1. Therefore we cannot exclude that this star is a binary
+system.
+6.3
+Elemental abundances
+Detailed chemical abundances analysis has also been performed
+taking advantage of the FERRE code. First, we identified all the re-
+solved lines present in the UVES spectrum and measure their central
+wavelength by using splot routine within IRAF8 (Tody 1993). Sec-
+ondly, we built for each chemical species a FERRE-readable mask
+that includes all the observed lines. These spectral windows include
+the vast majority of the information of each elemental abundance.
+Thirdly, we fixed the stellar parameters to the values derived in the
+Sec. 6.2 and launched FERRE again leaving only as a free parame-
+ter the chemical abundances. The code calculates and averages the
+abundance of each element and gives the associated uncertainty.
+The result of this analysis is presented in Table 1, and summarised
+in the following subsections.
+6.3.1
+CNO-elements
+The carbon abundance is derived, as explained in Sec. 6.2, by using
+specific models with C as a free parameter. The CH features are
+clear, resolved, and spread all over the UVES spectrum, especially
+around the G-band at 4385 Å. The obtained abundance ratio is al-
+most solar [C/Fe] = +0.05 ± 0.11, and no significant correction
+is needed to account for the star’s evolutionary stage (0.01 dex ac-
+cording to Placco et al. 2014). However, for nitrogen we are only
+able to provide an upper limit from the CN molecular band at
+3885 Å. Finally, since we do not have access to the oxygen triplet
+at ∼ 7775 Å we directly take the APOGEE value from the H-band.
+8 IRAF is distributed by the National Optical Astronomy Observatory,
+which is operated by the Association of Universities for Research in As-
+tronomy (AURA) under cooperative agreement with the National Science
+Foundation
+MNRAS 000, 000–000 (2022)
+
+7
+Table 1. Stellar parameters, abundances, ratios, errors and number of de-
+tected lines for individual species derived in 2M13593064+3241036 from
+UVES and APOGEE data.
+Gaia DR2 id 1457695618046140800
+RA(deg) = 209.877547, DEC(deg) = 32.684315
+vrad = −13.45 km s−1, Teff = 5035 K, log g =2.59
+Species log ǫ (X)1
+⊙ log ǫ (X) [X/Fe]2 σ[X/Fe]
+N
+[X/Fe]NLTE
+C (CH)
+8.39
+0.05
+0.11
+–
+N (CN)
+7.78
+<0.10
+–
+–
+O i
+8.66
+1.103
+0.15
+2
+Na i
+6.17
+4.22
+0.36
+0.12
+2
+−0.13
+Mg i
+7.53
+5.77
+0.28
+0.09
+9
+0.28
+Al i
+6.37
+3.86
+−0.20
+0.11
+2
+0.06
+Si i
+7.51
+5.48
+0.40
+0.10
+1
+0.34
+K i
+5.08
+5.08
+0.243
+0.10
+1
+Ca i
+6.31
+4.53
+0.39
+0.12
+12
+0.49
+Sc ii
+3.05
+0.87
+−0.07
+0.08
+3
+Ti i
+4.90
+3.10
+−0.19
+0.23
+4
+Tiii
+4.90
+3.01
+−0.05
+0.08
+16
+−0.12
+V i
+4.00
+1.85
+−0.02
+0.08
+8
+Cr i
+5.64
+3.23
+−0.27
+0.11
+6
+Mn i
+5.39
+2.68
+−0.25
+0.08
+3
+Fe i
+7.45
+5.31
+−2.314
+0.06
+231
+−2.27
+Fe ii
+7.45
+5.16
+−2.385
+0.09
+7
+Co i
+4.92
+2.83
+0.08
+0.12
+6
+Ni i
+6.23
+3.84
+−0.37
+0.13
+5
+Cu i
+4.21
+–
+<−0.40
+–
+–
+Zn i
+4.60
+3.84
+0.17
+0.13
+2
+Sr ii
+2.92
+0.12
+0.12
+0.09
+2
+Y ii
+2.21
+0.63
+−0.63
+0.09
+4
+Zr ii
+2.59
+0.24
+0.24
+0.09
+3
+Ba ii
+2.17
+0.13
+0.13
+0.10
+3
+Eu ii
+0.52
+0.55
+0.03
+0.10
+2
+1Solar abundances adopted from Asplund et al. (2005)
+2LTE and NLTE ratios are referred to [Fe/H]LTE and [Fe/H]NLTE.
+3Abundance derived from the APOGEE spectrum and not used here.
+4[Fe/H] from Fe i is given instead of [X/Fe]
+4[Fe/H] from Fe ii is given instead of [X/Fe]
+Taking into account the metallicity from iron lines was off in the AS-
+PCAP calculation we corrected the oxygen ratio accordingly, from
+[O/Fe]apogee = 0.57 to [O/Fe]assumed = 1.10. Unfortunately, the
+forbidden oxygen lines at 6300 and 6363 Å are dominated by strong
+sky lines in the UVES spectra so we could not check this high oxy-
+gen value. Therefore, we consider [O/Fe]assumed only tentatively
+and not used in the following analysis.
+6.3.2
+Odd elements: Na, Al, Sc, and K
+Absorption lines from odd elements are well resolved in the UVES
+spectrum, and FERRE provides a good fit for Na (2 lines), Al (2
+lines), and Sc (3 lines). We detect several features from interstellar
+Na around ∼ 5985 Å. Luckily, they are well separated from the
+stellar component and we are able to provide an accurate [Na/Fe]
+value. The strongest K lines are outside of the range of the available
+UVES spectrum, so we use the corrected value from the H-band in
+APOGEE, [K/Fe]apogee = −0.24 to [K/Fe]assumed = 0.24. We
+consider this value cautiously for a number of reasons: a) K lines
+in the H-band are weak in this metallicity regime; b) K is derived
+by ASPCAP with the metallicity as a free parameter which could
+potentially introduce some deviation since we know that [Fe/H] was
+overestimated. Therefore, as explained in Sec. 6.1 we do not use K
+in our analysis (i.e. we give weight equal zero).
+6.3.3
+Alpha-elements: Mg, Si, S, Ca, and Ti
+The α−elements show strong absorptions, even in very metal-poor
+stars. In particular, we detect 9, 1, 14, and 20 lines for Mg, Si,
+Ca, and Ti, respectively. We do not use the saturated Ca ii lines at
+3933 Å and 3968 Å since the size of the spectral window would be
+gigantic and many other lines actually do live in their wings. We
+included two ionised states of titanium, Ti i and Ti ii, with 4 and 16
+detected lines respectively. The measured abundances of the two Ti
+species agreed with in errors, and the assumed value in our fits is
+the average of the two (See Table 1). Finally, we discard the infrared
+sulphur measurement from APOGEE, plotted in Fig. 1a, since no
+appropriate line is available in the APOGEE spectrum and no strong
+absorption lines populate the optical spectrum.
+6.3.4
+Iron peak-elements: V, Cr, Mn, Fe, Co, and Ni
+The relatively high SNR and resolution of the UVES spectrum al-
+lowed us to derive iron peak-elements with high accuracy (number
+of lines): V (8), Cr (6), Mn (3), Co (6), and Ni (5). For Fe i we iden-
+tified up to 231 lines with a mean metallicity of [Fe i/H] = −2.31,
+and we adopted this as the metallicity of 2M13593064+3241036.
+On the other hand, we obtained [Fe ii/H] = −2.38 from 7 isolated
+lines.
+6.3.5
+Killing elements: Cu and Zn
+As previously explained, Cu and Zn are key elements for our PISN
+analysis. Thus the required SNR in the UVES spectrum was based
+on these two elements, so that a minimum detection threshold of
+[Cu/Fe] = −0.40 and [Zn/Fe] = −0.70 could be potentially
+detected from the Cu i (5105 Å) and Zn i (4810 Å) lines. In Fig. 3
+the targeted lines for Cu and Zn, and their derived values are shown.
+6.3.6
+Neutron-capture elements: Sr, Y, Zr, Ba, and Eu
+We also derived ionised species for neutron-capture elements from
+the UVES spectrum. We detect several of the typical s-process
+elements (number of lines): Sr (2), Y (4), Zr (3), and Ba (3). Addi-
+tionally, corresponding to a r-process production, the strongest Eu
+lines (4129 Å and 4205 Å) were detected and measured.
+6.4
+NLTE corrections
+We note that the way we derived elemental abundances with spec-
+tral windows does not allow direct NLTE corrections since we are
+fitting several lines at the same time. However, we estimated an
+overall NLTE correction by averaging the individual corrections
+over the relevant lines. We employed NLTE corrections provided in
+the literature:
+• Mashonkina et al. (2016)9 to calculate both Fe i and Ti ii. The
+average Fe i NLTE correction is 0.04 dex and the dispersion line
+to line is also low with the majority of the lines between 0.02-
+0.05 dex. Ti ii corrections go in the other way but are also small
+with an average of −0.03 dex.
+9 http://spectrum.inasan.ru/nLTE/
+MNRAS 000, 000–000 (2022)
+
+8
+D. Aguado et al.
+• (b) Lind et al. (2012)10 for Na. The lines/individual corrections
+are 5889 Å/−0.42 dex and 5895 Å/−0.45 dex, giving a final value
+of 0.44 dex. These are, as expected, the largest in this giant star.
+• Mashonkina et al. (2007); Bergemann et al. (2013, 2017)11
+for Ca i, Si i, and Mg i, respectively. Ca i are comparatively large
+(0.13 dex), while Si i, and Mg i are small and at the level of −0.02
+and 0.04, respectively. This is not surprising for a very metal-poor
+K type star (e.g. Alexeeva et al. 2018).
+• We also applied a +0.30 dex correction for Al based on NLTE
+calculations by Nordlander & Lind (2017).
+7
+PISN DESCENDANTS CANDIDATES
+7.1
+Our observed candidate: 2M13593064+3241036
+In Sec. 4 we reanalysed with FERRE the chemical signature
+of our PISN candidate. As explained in Sec. 6.2 the fact that
+2M13593064+3241036 is more metal-poor than previously pub-
+lished with APOGEE has a direct impact on the analysis and makes
+the killing ratios [Cu/Fe] and [Zn/Fe] higher than expected, with
+catastrophic consequences. Additionally, we consider the K mea-
+surement as tentative with lower weight in the analysis (see Sec.
+6.3). Furthermore, the [Mg/Al] ratio has dramatically increased by
+0.69 dex which clearly attenuates the odd-even effect. Still some key
+features remain, such as high [Si/Fe] value, and [Ca/Mg] > 0, but
+the importance of those is relatively smaller when identifying PISN
+descendants.
+In Fig. 4a we show the elemental abundances from Table 6.1
+together with the best PISN-explorer fit and parameters. The best
+fit gives significantly different results than the original analysis:
+mPISN = 204.0 M⊙ (prev. 193.8 M⊙), and fPISN = 50% (prev.
+66%), and f∗/fdil = 10−2.1 (prev. 10−2). The quality of the fit
+is now worse (See Fig. 1a), in particular the killing element Zn
+is not anymore well reproduced by the models. Additionally, the
+odd-even effect is attenuated and the best fit (red line) is unable
+to reproduce it which is far from we could expect from a PISN
+descendant. That is consistent with the fact that now the best fit is
+giving a fPISN value that is in the limit of the grid, which could be
+an indication of even lower PISN contribution. Three key results
+therefore forced us to be cautious on reaching strong conclusions
+on whether 2M13593064+3241036 is a PISN descendant: (a) the
+quality of the fit has significantly decreased with the analysis of
+the higher quality UVES spectrum; (b) the odd-even effect is now
+clearly reduced; and (c) the high Zn value does not allow us to
+clearly identify the smoking gun of PISN production. The candi-
+date 2M13593064+3241036 could therefore still be a regular halo
+star. The following is an additional test we propose to verify this
+hypothesis.
+The authors in Cayrel et al. (2004b) derived precise elemental
+abundances for 35 very metal-poor halo stars, considered chem-
+ically unmixed (Spite et al. 2005). These abundances adequately
+averaged and NLTE corrected by Andrievsky et al. (2010) can be
+understood as the mean chemical signature for a regular (C-normal)
+giant halo star polluted almost exclusively by CCSN. This “repre-
+sentative” abundance pattern is not polluted by PISN at the 99.9%
+level, and is therefore useful as a comparison sample to test our
+methodology. Thus, following the methodology described in Sec.
+4, we analyzed this average abundance pattern for a regular giant
+10 http://www.inspect-stars.com/
+11 https://nlte.mpia.de/gui-siuAC_secE.php
+halo star having as result fPISN = 50%; f∗/fdil = 10−2.0; and
+mPISN = 231M⊙. In Fig. 4b we show the comparison between
+2M13593064+3241036 and the regular giant halo star pattern. As
+in the case of our candidate we obtained the minimum value of PISN
+contribution while the mass of the progenitor is slightly higher. In
+fact, both objects and their best fits seem similar. Main differences
+are concentrated in lighter elements such us the CNO family where
+the regular halo star pattern shows smother trend. In addition to it,
+the odd-even effect is even less clear in 2M13593064+3241036 in
+the Na-Mg-Al-Si range. Furthermore, the super solar Zn abundance
+in the regular halo star pattern (as in our APOGEE candidate) is a
+strong indication of no PISN contribution. The super-solar [Zn/Fe]
+makes it difficult to establish an unambiguous difference between
+this star and the average halo star. Therefore we could conclude that
+2M13593064+3241036 is likely like other halo stars with no major
+contribution from PISN pollution.
+7.2
+Previously reported candidates
+In the literature, two stars have been reported as being probable
+descendants of PISNe: BD+80◦245 (Salvadori et al. 2019), and
+SDSS J0018−0939 (Aoki et al. 2014). To verify our methods, and
+compare our approach to their published results, we also analysed
+these stars with the PISN-explorer, and the result is shown in
+Figs 4d and 4c.
+The star BD+80◦245 (Carney et al. 1997; Fulbright et al.
+2010; Ivans et al. 2003; Roederer et al. 2014), is a low-α halo
+star (Fig. 4c), which was initially proposed as a PISN descendant
+(Salvadori et al. 2019) based on the low abundance of the killing
+elements (Cu and Zn). Their originally derived PISN parameters
+were fPISN = 50%; f∗/fdil = 10−4.0; and mPISN = 223M⊙.
+Our analysis, as explained, is based on the same set of models
+(Heger & Woosley 2002; Salvadori et al. 2019) but the way we fit
+the data with FERRE is different (see Sec. 4). The best solution we
+get is fPISN = 50%; f∗/fdil = 10−2.0; and mPISN = 213.8M⊙
+with a χ2 = 0.17, which is in good agreement with Salvadori et al.
+(2019). Interestingly, the f∗/fdil factor is significantly different in
+the two analysis while the quality of the fit is similar. The reason
+for that is that f∗/fdil is the least sensitive parameter (see Sec. 2)
+within the models and the FERRE code tends to go to the limit of the
+grid.
+SDSS J0018−0939 is another interesting star that was pro-
+posed to be a PISN descendant by Aoki et al. (2014). This star has
+remarkably low α-abundances ([Mg/Fe] = −0.52, [Ca/Fe] =
+−0.26), see Fig. 4d. The authors excluded that the peculiar chem-
+ical pattern could not be produced by CCSN or SN Ia. They con-
+cluded that the most likely origin is PISN and they attribute a
+fPISN = 100%value. Unfortunately, noinformative upperlimitsfor
+Cu and Zn were measured in this star. We used the PISN-explorer
+to derive PISN parameters from the measured abundances by
+Aoki et al. (2014) and found fPISN = 50%; f∗/fdil = 10−2.0;
+and mPISN = 256M⊙ with χ2 = 0.44. Although the χ2 is ele-
+vated, we are able to reproduce the chemical signature of lighter
+elements (See Fig. 4d) but the models failed for iron-peak elements.
+The low α ratios lead to high mass of the progenitor (256M⊙)
+in agreement to what was claimed by Aoki et al. (2014). However,
+the fPISN = 50% value suggests that this star could be polluted
+by PISN but a higher contribution should be attributed to normal
+Pop II stars exploding as CCSN. Further observations in order to
+detect and measure killing elements is highly required to confirm
+the percentage of PISN contamination we see in this star.
+The Pristine survey (Starkenburg et al. 2017b) derives metal-
+MNRAS 000, 000–000 (2022)
+
+9
+(a) 2M13593064+3241036
+(b) Halo abundance pattern
+(c) BD+80◦245
+(d) SDSS J0018−0939
+(e) TYC 4267-2023-1
+(f) BD −07 3523
+(g) Sextans S58
+(h) Draco 361
+Figure 4. Results of the PISN-explorer analysis. Points are measured chemical abundances while lines are best fits. Listed on the plot are parameters of the
+fit, along with Teff, log g, and [Fe/H]. Top row: Our new analysis is presented (blue filled circles) and compared to the APOGEE results (a, blue open circles);
+and the average abundance pattern and the best fit of the Cayrel et al. (2004a) sample (b, green open diamonds). Second row: Previously published candidates
+for dominant PISN enrichment (green squares), SDSS J0018−0939 (c; Aoki et al. 2014) and BD+80◦245 (d; Salvadori et al. 2019). Third row: Candidates
+selected from the MINCE survey (e and f; Cescutti et al. 2022). Bottom row: Promising candidates for PISN descendants (green star symbols) in the Sextans
+(g; Shetrone et al. 2001) and Draco (h; Cohen & Huang 2009) dwarf spheroidal galaxies.
+MNRAS 000, 000–000 (2022)
+
+10
+D. Aguado et al.
+licities for halo stars based on narrow band filter photometry. In this
+context, there is also a remarkable observational effort to derive Cu
+and Zn in metal-poor stars (Caffau et al. 2022). The authors found,
+among others, three interesting candidates to be polluted by PISN
+production using also the PISN-explorer. In particular, in their
+Fig. 12, they show the chemical pattern of TYC1118-595-1, a very
+metal-poor star with [Fe/H] = −2.12, and derived fPISN = 50%;
+f∗/fdil = 10−2.3; and mPISN = 193M⊙. TYC 2207–992–1 and
+TYC1194–507–1 are also reported together with a best fit that sug-
+gested that they could be enriched by PISN up to fPISN = 83%
+and 90%, respectively. However, the probability of such is signifi-
+cantly smaller than for TYC1118-595-1 due to the lower quality of
+the fit. According to what is explained in Sec. 4, we consider that
+TYC1118-595-1 is likely reflecting the theoretical predictions in
+their chemical pattern in a percentage that could be to the order of
+fPISN = 50% or slightly smaller. TYC 2207–992–1 is also a very
+promising candidate with a progenitor mass mPISN = 170M⊙
+while TYC1194–507–1 has more uncertain origin due to the low
+quality of the fit.
+We also mined the bibliography including high-resolution
+analysis of halo stars looking for interesting candidates. We found
+that the vast majority of interesting stars published before 2018 are
+already included in the JINA database. However, Xing et al. (2019)
+found an interesting star, J1124+4535, with [Fe/H] = −1.27 and
+[Zn/Fe] = −0.37. We also analyzed its chemical signature with
+the PISN-explorer and found fPISN = 50%; f∗/fdil = 10−2.4;
+and mPISN = 231M⊙. The quality of the fit is relatively good
+(χ2 = 0.29) but the absence of Cu measurement or upper limit pre-
+vent us to conclude this object is polluted by PISN. We propose to
+re-observe this interesting object with peculiar chemistry to confirm
+its origin.
+7.3
+PISN candidates in classical dwarf galaxies
+It is commonly accepted that the classical dwarf spheroidal galaxies
+are massive enough to retain the chemical products of PISN explo-
+sions (see e.g. Bromm & Loeb 2003; Salvadori et al. 2008). How-
+ever, their total number of stars is much lower compared to more
+massive systems. Consequently, the expected fraction of PISN de-
+scendants in classical dwarf galaxies is expected to be significantly
+higher than in the Milky Way. Therefore, classical dwarf galax-
+ies are interesting places to search for PISN descendants. Luckily,
+within the JINA database there are ∼ 50 stars from Fornax, Sex-
+tans, Draco, and other dwarf satellites. In Figs. 4g and 4h two
+interesting examples of the FERRE analysis are shown, Sextans S58
+(Shetrone et al. 2001) and Draco 361 (Cohen & Huang 2009). The
+quality of both fits are remarkably high with χ2 = 0.10, 0.20, re-
+spectively. Additionally, the killing element Zn, is largely subsolar
+in the atmosphere of the two stars [Zn/Fe] ∼ −0.45. Actually, the
+case of Sextans S58 with fPISN = 55% suggests that the majority
+of the material this star formed from was polluted mostly by PISN
+with a mass of mPISN = 253 M⊙.
+Finally, we also identified another star from JINA database,
+Draco 3053 (Cohen & Huang 2009). This star is also Zn-poor with
+[Zn/Fe] = −0.3 and the best set of parameters derived with
+the PISN-explorer are fPISN = 50%; f∗/fdil = 10−2.0; and
+mPISN = 218M⊙; and χ2 = 0.19. In this case, the high quality of
+the fit and the low value of Zn suggest that this star could be indeed
+polluted by PISN in a percentage close to fPISN = 50%. Further
+observations of all of the three candidates in Draco and Sextans
+galaxies are required.
+8
+DISCUSSION
+In the previous sections we have validated and applied our proposed
+PISN-explorer methodology to find PISN descendants. Although
+the 2M13593064+3241036 candidate seems not to be as interesting
+as expected (mostly due to the initial overestimation of metallicity in
+APOGEE), its study was useful to understand the capabilities of our
+methodology. As shown, its chemical signature, and in particular
+the killing elements Cu and Zn, is not significantly different from
+that of an average pattern for regular giant halo stars. Indeed, when
+the contribution of Pop II stars exploding as normal CCSN becomes
+> 50%, the peculiar chemical signatures left by different Pop III star
+progenitors are essentially lost (Vanni et al. in prep.)
+A different situation is presented for BD+80◦245, shown
+in Fig. 4c. For this star we confirm the result provided by
+Salvadori et al. (2019) and we conclude that it is at least par-
+tially (but genuinely) polluted by PISN. The clearly low value of
+the killing elements that this stars shows is indeed the smoking
+gun of PISN contamination. On the other hand, the analysis of
+SDSS J0018−0939 shows that while very different from a regular
+halo star (see Sec. 7.1), the percentage of material produced by PISN
+may not be as high as originally suggested by Aoki et al. (2014). In
+any case, some further high-resolution observations of this interest-
+ing star are needed to better constrain the amount of Cu and Zn and
+shed light over the exact amount of material that effectively come
+from PISN production. At this point we propose using facilities that
+will be available in the next generation of 30 m telescopes. More-
+over, the agreement of our analysis with previous works suggests
+that the PISN-explorer is efficient in characterising candidates.
+A very challenging issue when identifying the descendants of
+the very massive first stars is how to discriminate between them and a
+regular halo star (mostly polluted by Pop II exploding as CCSN). By
+comparing Fig. 4b with 4c, 4d, 4g, and 4h it is clear that the chemical
+signatures that correspond to good fits in this work (χ2 ≲ 0.20) are
+very different than the mean abundance pattern for a regular giant
+halo star (Cayrel et al. 2004b). The much higher CNO abundances
+together with high α’s (i.e. Mg, Ti) led to a significantly higher
+χ2. Obviously, this is deeply related with the fact that the regular
+halo star pattern is not Zn-poor, which is the most robust indicator.
+Additionally, as shown in Fig. 4b, the odd-even effect is clearly lost
+for atomic numbers higher than Z=19, i.e. for most of the iron-
+peak elements. According to our novel PISN-explorer, the most
+efficient way to separate PISN candidates from other halo field stars
+is a combination of all the criteria presented in Sec. 5, low values of
+χ2, and the killing elements deficiency [Cu, Zn/Fe] < 0. In Figs.
+4e and 4f two interesting examples of promising candidates from
+MINCE survey with subsolar Cu and Zn abundances are shown.
+Moreover, the PISN-explorer allowed us to successfully
+identify very interesting PISN descendants candidates in classi-
+cal dwarf galaxies (see Fig. 2). Note that none of them are found
+in the less massive ultra-faint dwarf galaxies. These systems, which
+are smaller and thus have lower binding energy than classical dSph
+galaxies, are probably not able to retain the chemical products of
+energetic PISN (Rossi et al. in prep). Unfortunately, since those
+from classical dwarfs are typically distant and therefore faint, the
+next generation of 30 m class telescopes will be required to derive
+systematically their killing elements.
+A final remark concerns the MDF of our 166 selected can-
+didates shown in Fig. 2. It is indeed extremely interesting to see
+that its main peak is located at [Fe/H] = −1.7, i.e. exactly at the
+level predicted by the cosmological models for the MW formation
+of de Bennassuti et al. (2017) (see their Fig. 9) and close to the
+MNRAS 000, 000–000 (2022)
+
+11
+Figure 5. Our catalogue of 166 candidate PISN descendants, shown in the fPISN − mPISN plane, colour-coded by metallicity. Histograms of the candidates
+are also shown on top and right side panels.
+values provided by the parametric study of Salvadori et al. (2019),
+[Fe/H] ≈ −2, and by the inhomogeneous chemical enrichment
+model by Karlsson et al. (2008), [Fe/H] ≈ −2.5. This result is
+quite remarkable since the PISN-explorer does not use the iron
+abundance to select PISN candidates. As explained in Sec. 5, indeed,
+our selection is solely based on the predicted chemical abundance
+ratios, [X/Fe]. We should also note that it is very likely that our
+sample, and thus our selected candidates, are biased towards higher
+[Fe/H]. In fact, both APOGEE and GALAH rapidly decrease in ac-
+curacy in the very metal-poor regime, which is possibly the origin
+of the second peak at higher [Fe/H]≈ −1. Finally, we recall that
+the iron abundances of PISN descendants is expected to vary in a
+broad range, −5 < [Fe/H] < 0.5 (see Fig. 7 of Salvadori et al.
+2019). Therefore, it is quite normal that in our MDF we find a
+second and smaller peak at [Fe/H] = −3.0. This peak is indeed
+made by PISN candidates selected from the JINA database, which
+is naturally biased towards Fe-poor stars.
+In Fig. 5 we display the fPISN − mPISN distribution for our
+166 selected candidates colour coded by metallicity. While the his-
+tograms of fPISN are peaked at lower contribution and monoton-
+ically decreases, which is expected, the distribution of mPISN is
+more condensed around 200 M⊙. We also see a more metal-poor
+population lying at fPISN = 50%. As explained in Sec. 5 the vast
+majority of those values should be avoided when selecting candi-
+dates. However, as we pointed out, after visual inspection we could
+consider them as a candidates whether they show a remarkably good
+fit and Cu and Zn abundances are subsolar. This population was se-
+lected following this approach and some of them (due its visibility
+and/or special features) are included in our golden sample. How-
+ever, some others are quite faint and we propose to observe at higher
+SNR when possible with the future high-resolution spectrographs.
+9
+CONCLUSIONS
+We have presented a new methodology to identify candidates that
+have been significantly polluted by PISN in the early Universe. We
+mined different datasets in order to find chemical patterns that match
+with what is expected to find if significant PISN production took
+place. We summarise here the main conclusions of this work:
+• Current and upcoming large spectroscopic surveys (APOGEE,
+GALAH, GES, 4MOST, WEAVE), and existing databases such as
+JINA, are an invaluable tool to unveil the origin and characteristics
+of the very massive first stars, which exploded as PISN.
+• The PISN-explorer, using the FERRE code in combination
+withtheoretical predictions(Salvadori et al. 2019), isaveryefficient
+methodology when selecting PISN descendant candidates in large
+databases.
+• The MDF of the selected candidates is a confirmation of
+the predicted peak in PISN production at [Fe/H]
+∼
+−1.7
+(de Bennassuti et al. 2017).
+• killing elements (Cu and Zn), low values of α-elements (Mg
+and Ti), and clear odd-even effect (e.g. high [Mg/Na], and/or high
+[Mg/Al]), are indicators of PISN production. Therefore, they could
+be used to efficiently select candidates.
+• It is possible that 2M13593064+3241036 was mostly polluted
+by PISN production but the presence of Zn on its atmosphere does
+not allow us to confirm this hypothesis. BD+80◦245 contains sig-
+nificant material that was formed during a PISN event. The fPISN =
+value previously reported for SDSS J0018−0939 (100%) could be
+significantly lower according to its chemical signature.
+• Sextans S58 is the most promising candidate and maybe mostly
+formed out of PISN material. To confirm this hypothesis further
+high-resolution follow-up is needed to complete the chemical sig-
+MNRAS 000, 000–000 (2022)
+
+12
+D. Aguado et al.
+nature already reported in the literature. Draco 361 and Draco 3053
+are also excellent candidates selected from classical dwarf galaxies.
+• From our selected 166 candidates we propose 45 of them as
+a golden catalogue with the most promising and visible stars for
+future follow-up.
+Future high-resolution facilities mounted in the Extremely
+Large Telescope such as ANDES (Marconi et al. 2022) and other
+facilities will provide and unbeatable opportunity to observe stars
+in dwarf satellites, where it is more likely to find PISN descendants.
+In addition to it, the next generation of large spectroscopic surveys
+such as WEAVE (Jin et al., in press) and 4MOST (de Jong et al.
+2019) with high resolution observations will be an unbeatable place
+to test the PISN-explorer and finally finding the descendants of
+the very massive first stars.
+10
+DATA AVAILABILITY AND ONLINE MATERIAL
+All the spectroscopic data reduced and analyzed for the present
+article are fully available under request to the corresponding au-
+thor12. A table with the detected lines in the UVES spectrum of
+2M13593064+3241036 is included as online material.
+ACKNOWLEDGEMENTS
+We thanks the anonymous referee for positive and constructive com-
+ments. The authors of this work really thank Carlos Allende Pri-
+eto (Instituto de Astroísica de Canarias) for insightful discussion
+about the capabilities of the FERRE code. We warmly thank all
+the members of the NEFERTITI group of the University of Flo-
+rence for insightful discussions. We also thank the streams group at
+the University of Cambridge for fruitful interactions. These results
+are based on VLT/UVES observations collected at the European
+Organisation for Astronomical Research (ESO) in the Southern
+Hemisphere under program ESO ID 108.23N5.001. This project
+has received funding from the European Research Council (ERC)
+under the European Union’s Horizon 2020 research and innovation
+program (grant agreement No. 804240). DA, SS, AS, IV, VG and IK
+acknowledge support from the European Research Council (ERC)
+Starting Grant NEFERTITI H2020/808240. SS acknowledges sup-
+port from the PRIN-MIUR2017, prot. n. 2017T4ARJ5. EC and PB
+gratefully acknowledge support from the French National Research
+Agency (ANR) funded project “Pristine” (ANR-18-CE31-0017).
+AMA acknowledges support from the Swedish Research Council
+(VR 2020-03940).
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+

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+Leveraging the SCION Internet Architecture to
+Accelerate File Transfers over BitTorrent
+Marten Gartner
+Otto-von-Guericke University
+Magdeburg, Germany
+[email protected]
+Thorben Kr¨uger
+Otto-von-Guericke University
+Magdeburg, Germany
+[email protected]
+David Hausheer
+Otto-von-Guericke University
+Magdeburg, Germany
+[email protected]
+Abstract—As the needs of Internet users and applications
+significantly changed over the last decade, inter-domain routing
+became more important to fulfill these needs. The ways how
+data flows over the Internet are still completely in the hand of
+network operators, who optimize traffic according to their own,
+local view of the network. We observe two potential limitations
+from this: Optimizing according to the local view may a) result in
+unused capacities in the global network and b) not meet the actual
+needs of users and applications. To identify and overcome these
+limitations, we present our BitTorrent over SCION approach,
+which enables multipath communication and intelligent path
+selection for endhosts in global torrent networks. We compare
+our implementation against BitTorrent over BGP and BGP-M in
+a small-scale Internet topology, observing an increase in goodput
+of 48% through multipathing compared to BitTorrent over BGP
+and 33% compared to the BGP-M candidate. Furthermore, we
+show that our proposed disjoint path selection algorithm is able
+to improve traffic flow in the network with a low number of
+outgoing connections to unchoked peers.
+Index Terms—Peer-to-Peer, Path-aware networking, Path Se-
+lection, SCION, BitTorrent
+I. INTRODUCTION
+The Internet was designed decades ago, and some of the
+original design decisions have had unfortunate side-effects and
+security implications that persist to this day. So far, from
+the perspective of a mere host, there has been no general
+mechanism for splitting traffic across multiple different paths
+to a specific destination, which would be beneficial bandwidth-
+intensive applications.
+As the de facto standard protocol for inter-domain routing,
+BGP [32] is used to disseminate routes between Autonomous
+Systems (ASes), forming the Internet as we know it today.
+Within each AS on a given path through the Internet, packet
+forwarding is affected by (sometimes highly complex) local
+traffic engineering preferences and policies. In general how-
+ever, there is just a single packet forwarding path between
+two hosts in a BGP-based Internet. To also add support for
+multiple forwarding paths, BGP-M [24] was proposed, which
+adds support for load sharing across multiple inter-domain
+links, subject to configurable preferences. However, BGP-M is
+does little to help optimizing traffic flow in the global Internet:
+Firstly, it’s impact is necessarily of limited, local scope.
+Secondly, BGP-M is not adaptive and can not dynamically
+react to changing network conditions. Given these limitations
+and given the existence of alternative inter-domain paths in the
+Internet [2], we hypothesize that the current Internet still has
+unused capacities that cannot be exploited entirely by using
+BGP or BGP-M as the inter-domain routing protocol.
+Path-aware networking architectures promise to overcome
+these limitations by offering deeper insight into and better
+control over packet forwarding in the network. Different
+approaches to enable path-control in networks have been
+proposed. Some try to work within the limits of the current In-
+ternet architecture [18], [41], other attempts involve a complete
+redesign of the Internet architecture from scratch. While there
+are a number of path-aware approaches [34], [42] with their
+own merits, our work focuses on the (arguably) most mature
+and most widely-deployed open-source SCION architecture
+[7], [44].
+To leverage unused capacities in the network, especially in
+the backbone, Peer-to-Peer (P2P) applications promise unique
+opportunities through their globally distributed nature [8], [36],
+[39]. As the most well-researched and understood protocol,
+we selected BitTorrent as the foundation of our work, adding
+support for active endhost-based path-selection via SCION.
+We compare our augmented BitTorrent implementation with a
+non path-aware BitTorrent implementation in comparable BGP
+and BGP-M-based inter-domain network topologies.
+We anticipate that any achieved improvements will easily
+translate to other P2P networks, e.g. IPFS [38]. While BitTor-
+rent itself is famous for improving bandwidth utilization on the
+last mile, we hypothesize that in combination with SCION, it
+could also unlock unused capacities in the network at large.
+To this end, we contribute the following:
+• We discuss the existence of unused capacities in the
+backbone with BGP or BGP-M as utilized inter-domain
+routing protocol
+• We show why path-aware networking is able to unlock
+these capacities and discuss the impact of host-based path
+selection
+• We present our BitTorrent over SCION design intro-
+ducing the notion of path-level peers and propose an
+algorithm for disjoint path selection
+• Finally, we analyze the performance improvements of
+BitTorrent over SCION aggregating unused capacities in
+the network compared to BitTorrent over BGP and
+BGP-M
+arXiv:2301.13499v1  [cs.NI]  31 Jan 2023
+
+The remainder of this work is structured as follows: In
+Section II we provide background for SCION and BitTorrent,
+followed by a discussion about limitations of BGP-based de-
+ployments and impacts of host-based path selection in Section
+III. We present the design and implementation of multipath
+support for BitTorrent over SCION in Section IV, followed
+by a presentation of our disjoint path selection algorithm.
+Afterwards, we show our virtualized Internet-scale testbed in
+Section V and the experimental results of comparing Multipath
+BitTorrent over SCION against an unmodified implementation
+in BGP and BGP-M-based scenarios in Section VI. Finally,
+we discuss related work in Section VII, conclude and provide
+outlook for future work in Section VIII.
+II. BACKGROUND
+In the following, we provide a brief overview on BGP, BGP-
+M and the SCION architecture as well as on BitTorrent.
+A. BGP and BGP-M
+BGP plays a unique role in the current Internet. While
+it can also enable in intra-domain routing (iBGP), we will
+exclusively refer to BGP’s more important role as the Internet’s
+predominant inter-domain routing protocol in the rest of this
+work. The principle task of a BGP border router is to inform
+and update its neighbours about specific IP address ranges
+(i.e., IP prefixes), to which the router’s AS is able to forward
+traffic to. The router announces the prefixes that its AS has
+learned from its other neighbors. Routing loops are prevented
+by means of the AS PATH, a list of hops to which each
+router prepends its own AS Number (ASN) before sending
+it on as part of a route announcement. BGP routers maintain
+the respective AS PATHs as well as the announced prefixes
+in a routing table. The forwarding destination is determined
+by referencing this table based on the destination address of
+an incomping packet. By default, BGP selects a single path
+for each match in the routing table according to configurable
+policies. These policies may reflect e.g., local forwarding
+preferences, filters constraining the number of exported routes,
+etc.
+To account for the fact that there often are multiple matching
+paths to a particular destination, multipath BGP (BGP-M) [15]
+was introduced. For certain cases, BGP-M adds support for
+traffic load splitting1 over multiple outgoing AS links. In the
+most simple case, if several possible forwarding AS PATHs
+are of the same length, Equal-Cost Multipath Routing (ECMP)
+can be applied, with a configurable maximum number of
+parallel paths2. In Addition, BGP-M also allows for more
+advanced policies on how to combine multiple paths, e.g.,
+some Cisco routers offer Unequal-Cost Load Sharing [21],
+which allows load splitting over paths with different length by
+using a dedicated configurable weight.
+1Load splitting is based on flow hashes of the 5 tuple (src address, dst
+address, src port, dst port, layer 4 protocol).
+2In some border router implementations, the relevant settings are: a) bgp
+bestpath as-path multipath-relax to enable load splitting for
+equal length paths, and b) maximum-paths to set the maximum number of
+paths over which load splitting will be performed.
+B. SCION
+The SCION architecture [44] has been designed to over-
+come the limitations of BGP and BGP-M in the future
+Internet, addressing modern threat models at the fundamental
+protocol level and endeavouring to avoid many current issues
+with hijacking attacks and single roots of trust. SCION also
+promises communication guarantees and path control capa-
+bilities, allowing applications to use two or more paths in
+parallel to a given destination, generally enabling multipath
+communication.
+For traditional multipath approaches like MPTCP [29] and
+MPQUIC [9] to work as intended, a host must provide multiple
+network interfaces. With these protocols, multipath communi-
+cation implies sending data over multiple interfaces in parallel,
+beyond which no further influence on the data forwarding
+paths is possible. SCION on the other hand is based around
+”packet carried forwarding state” (PCFS), where every packet
+contains the complete inter-AS path to the intended destination
+(in the form of hop fields) in its packet header. Packets can
+thereby be easily directed via different paths, by changing the
+SCION header alone. Our work heavily relies upon this precise
+path awareness property of the SCION architecture.
+Unlike the flatter organizational hierarchy of today’s BGP-
+based Internet, in SCION, collections of ASes are combined
+into Isolation Domains, (ISDs), which are envisioned to corre-
+spond to, e.g., geographical regions or legislative domains (like
+a single country), but can also be made up of company or re-
+search networks. One or multiple ASes in an ISD form the ISD
+Core, which collectively manages a cryptographic trust root
+(TRC) on behalf of the other ISD members, enabling service
+authentication and other cryptographic functions within the
+ISD, simultaneously avoiding many of the notorious problems
+that plague globally centralized cryptographic trust systems
+with single points of failure that are beyond the control of
+the ISD. The ISD Core also manages the exchange of path
+information with other ISDs, while independent peering is also
+possible among non-Core ASes of different ISDs.
+C. BitTorrent
+The BitTorrent protocol specifies file transfer as a distributed
+mechanism between peers without the need for any central
+coordination. Some initial way to exchange network addresses
+among peers is nevertheless required, e.g., by means of a
+tracker or, alternatively, a distributed hash table (DHT). In
+BitTorrent, files are typically not transferred in the usual
+form of a single, continuous byte stream that contains the
+complete file. Instead, large files are split into equal-sized
+pieces (typically with a size between 32KB and 256KB). It
+is a key feature of BitTorrent that exchange of these pieces
+can be easily parallelized. BitTorrent peers that already have
+a complete local copy of a file are referred to as seeders, as
+opposed to leechers, which still have to obtain some or all of
+the constituent pieces of the file from other peers. For each
+new file uploaded to the BitTorrent network, there must be at
+least one seeding peer that initiates the distribution of the file.
+
+III. HOST-BASED PATH SELECTION
+A. Limitations of the BGP-Based Inter-Domain
+Generally, two critical aspects of BGP and BGP-M for inter-
+domain routing may lead to unused capacities in the global
+Internet: Firstly, AS operators inherently only have limited
+insight into network conditions beyond their local network and
+can also only manage traffic locally. Secondly, BGP and BGP-
+M do not provide features to dynamically adapt routing to
+different needs. In this section, we further characterize these
+limitations before discussing possible improvements that path-
+aware networking could bring to the table.
+Despite the large benefits of traffic engineering of AS-
+operators on intra-domain level, the potential of optimizing
+inter-domain routing is limited in the current Internet. Each
+operator can only optimize the traffic flow in their own AS
+until it reaches its local destination or the neighbour AS. This
+may especially impact performance for flows that traverse
+multiple hops before they reach their destination, since each
+hop performs its own, local optimization. In case one of the
+first hops performs a non-optimal routing decision for the
+flow (e.g. routing to particular neighbour interfaces that are
+already under heavy load), the overall performance of the flow
+is affected.
+As discussed in Section II, BGP-M provides several options
+to perform load sharing on multiple links. However, these
+options need to be configured statically in the network. Conse-
+quently, the network can not always fulfill the varying needs of
+different participants, e.g., endhosts who prefer to optimize for
+different criteria. BGP-M reflects the anticipated needs of AS
+operators, not the actual needs that endhosts have, which may
+differ significantly from those anticipated by the AS operators.
+Path-aware networking promises to overcome many of these
+limitations and their impacts on performance, and, (in the case
+of SCION,) gives endhosts the opportunity to freely choose
+suitable inter-domain paths for their traffic.
+B. Implications of Host-Based Path Selection
+Traffic engineering on the Internet is performed by network
+operators attempting to locally optimize data flows for various
+factors. With host-based path selection, operators hand over
+this control to endhosts. While this promises to help endhosts
+to optimize their traffic, it comes with potential implications
+for network operators and may be at odds with their own
+interests, especially with respect to the inter-domain. Operators
+tend to prefer the use of peering links over that of transit links
+to avoid costs, while endhosts do not have such an incentive
+to avoid transit links. Additionally, host-based path selection
+ideally requires up-to-date insight into network conditions on
+all hosts. It also needs to ensure that hosts do not change paths
+too often, which could result in undesirable oscillation. In
+simulation, Scherrer et al. show that the impact of oscillation
+through host-based path selection is low [37]. Moreover, in
+SCION, endhosts can only dictate the ingress and egress router
+interfaces of ASes, allowing network operators to still optimize
+their internal traffic within these constraints. Overall, while
+Piece 1
+Piece 2
+Piece 3
+Piece 4
+Piece 5
+Piece 6
+1
+0
+1
+0
+0
+0
+Connection 1 
+(peer1, path1)
+Connection 2 
+(peer1, path2)
+Torrent
+Feedback which
+pieces to download
+next
+Piece assigned
+Piece downloaded
+Result
+Bitmap
+Fig. 1.
+Multipath implementation by downloading a torrent file over multiple
+paths (connections).
+the benefits, drawbacks and trade-offs of path-awareness are
+certainly not yet fully understood, there is significant potential
+for it in the the future Internet.
+IV. DESIGN AND IMPLEMENTATION OF BITTORRENT OVER
+SCION
+In this section, we present our approach of path-level
+peers to enable multipath support for BitTorrent over SCION,
+followed by our disjoint path selection algorithm.
+A. Multipath through Path-Level Peers
+As one of its features, SCION provides path control for
+inter-AS traffic while guaranteeing that the traffic flows along
+the chosen paths. This opens up the potential to aggregate
+capacities in the network, enabling applications to leverage
+multipath communication and parallel data processing to in-
+crease performance. In this work, we choose BitTorrent as
+a suitable, already parallelized application as a foundation
+for our experiments on bandwidth aggregation via multiple
+SCION paths.
+By default, a BitTorrent peer is identified solely by its
+network address and port. This address could be either an
+IPv4, IPv6 or, in our case, a SCION address. We will refer
+to peers that are only described by their address as address-
+level peers for the rest of this work. To distinguish between
+multiple SCION paths to a particular peer, we introduce a new
+representation that we will refer to as a path-level peer. They
+are represented by the tuple (addr, path), consisting of the
+peer’s SCION address (including the port), together with one
+possible path to this address.
+We introduce this notion of path-level peers to a path-aware
+BitTorrent implementation which we will henceforth refer to
+as BitTorrent over SCION, and which treats different paths to
+the same peer as several distinct, path-level peers.
+Generally, peers that are returned from a tracker or that are
+added via static bootstrapping3 to BitTorrent are address-level
+3I.e., providing a list of peer addresses as arguments to the client binary.
+
+peers, since they may not be SCION peers to begin with and
+may not have path information associated with them. Thus,
+to generate path-level peers from a given SCION address, an
+additional path lookup is required. For a resulting number of
+possible paths that are available to a peer, the same number
+of corresponding path-level peers can be generated.
+BitTorrent over SCION is configured with an upper bound
+for the number of different path-level peers, as the default
+BitTorrent client is, too. Path-level peers are obtained from
+address-level peers through a path selection algorithm.
+After this obtaining of path-level peers, the usual BitTorrent
+P2P algorithms operate over each path independently. For each
+path-level peer, a QUIC4 [23] connection is established to
+ensure reliable transfer of data. Figure 1 shows the piece
+download of a particular file. Each successfully established
+connection fetches piece information from a queue and re-
+quests the particular piece by sending request messages and
+wait for peers sending back the requested pieces. After each
+received piece, its integrity is verified. For this, a hash is
+computed and verified against the one referenced in the torrent
+file for that piece. Retrieved pieces are stored in main memory
+until the file is downloaded completely.
+Since requests and retrievals of pieces over different con-
+nections are handled in their own dedicated threads, pieces
+can easily be downloaded concurrently, which speeds up the
+process and improves the overall download bandwidth. It is
+the responsibility of the main thread to iteratively check the
+result bitmap for missing pieces. Once all pieces are retrieved,
+the main thread closes all connections to all still connected
+peers that serve pieces of the current torrent and assembles
+the complete file to disk.
+B. Upload-based Disjoint Path Selection
+Although BitTorrent over SCION may simply use all avail-
+able paths to each connected peer, there are reasons to limit
+the number of path-level peers. When this limit is set suit-
+ably high, all available paths to each peer are considered.
+However, multiple paths may share the same bottleneck,
+making it pointless to aggregate them in hope of increasing
+overall performance. Since each BitTorrent peer has an upper
+limit of outgoing connections to neighbours, a performance
+increase could also be expected by simply increasing this
+upper limit to exchange pieces with more peers. Consequently,
+using the shortest path or simply aggregating all paths does
+not promise to have significant impacts on BitTorrent over
+SCION’s performance compared to path-unaware BitTorrent
+implementations. To address this, we use built-in SCION
+capabilities to implement an improved, disjoint path selection
+strategy, aiming to avoid such shared bottlenecks.
+In BitTorrent over SCION, our improved path selection
+strategy relies on two core assumptions: 1) Different paths
+that share the same hop may also share a bottleneck at this
+hop and 2) Peers that offer pieces to others are in a good
+4We choose QUIC, because TCP is currently not yet implemented for
+SCION.
+AS1
+AS2
+AS3
+1
+2
+3
+4
+AS1-1
+AS2-2
+AS2-3
+AS3-4
+List-based
+Representation
+Visual
+Representation
+Fig. 2. List-based representation of hop interfaces using unique ids to perform
+conflict detection.
+position for path selection decisions with knowledge about all
+downloading peers, allowing them to strategically distribute
+their outgoing traffic via disjoint paths.
+We implement a disjoint path selection strategy by searching
+for overlaps in all paths and discard all but one of the paths that
+share the same hops, following assumption 1). With respect
+to assumption 2), we decide to delegate path selection exclu-
+sively to the uploading peer. Both in combination promise to
+outperform na¨ıve path selection approaches.
+As presented in Section II, a SCION path consists of
+multiple hops. Each hop contains ingress and egress interfaces
+of the respective AS. In SCION, the interfaces are represented
+as numeric IDs, that are unique within the given AS. The
+combination of the AS identifier with the interface number
+results in a globally unique interface ID. In Figure 2, we show
+an intuitive mapping of a visual representation of hops and
+their interfaces to a list of such interface IDs. The interface IDs
+are used to determine disjointness between paths by counting
+the number of identical interface IDs.
+Data: peers, maxOutgoingConns
+Result: pathLevelPeers ← [ ];
+allPaths ← [ ];
+for p ∈ peers do
+paths ← lookupPaths(peer);
+allPaths ← append(allPaths, paths);
+end
+for path1 ∈ allPaths do
+for path2 ∈ allPaths do
+if path1! = path2 then
+confs ← numConflicts(path1, path2);
+path.conflicts+ = confs;
+end
+end
+end
+allPaths ← sortByConflictsAndHops(allPaths)
+i ← 0;
+while i ≤ maxOutgoingConns do
+pathLevelPeer ← fromPath(allPaths[i])
+pathLevelPeers ←
+append(pathLevelPeers, pathLevelPeer);
+i ← i + 1
+end
+Algorithm 1: Disjoint path selection
+
+Based on the interface IDs in the SCION paths, the up-
+loading peer is able to perform disjoint path selection to all
+connected peers. To achieve this, an interested peer connects
+over the first available path to the uploading peer and waits for
+it to connect back. The uploading peer now applies its disjoint
+path selection and connects back to the interested peer over
+the selected path set. Algorithm 1 depicts the procedure of
+finding the least disjoint paths to all connected address-level
+peers and returning a proper list of path-level peers. At first,
+the paths to each address-level peers are determined in a loop
+and aggregated in the allPaths variable. Afterwards, each path
+in allPaths is checked against all other paths calculating the
+number of conflicts (i.e. conflicting interface IDs), which is
+saved in the path. Next, the allPath list is sorted in ascending
+order by the number of conflicts and the number of hops.
+Finally, until the number of maxOutgoingConns is reached,
+the algorithm iterates over allPaths and transforms each path
+into a path-level peer, which is stored in the return variable
+pathLevelPeers.
+V. BENCHMARK SETUP: A REPRESENTATIVE
+SMALL-SCALE INTERNET TESTBED
+To compare our BitTorrent over SCION implementation
+against a BGP-based setup, we choose to run a virtualized
+network of multiple ASes that reflects the topology of the
+current Internet in a small-scale setup. Figure 3 shows the
+designed topology that we used to evaluate BitTorrent over
+SCION.
+The topology consists of two core layers which represent
+Tier-1 and Tier-2 ASes in the Internet, connected via peering
+and transit links. With the growing popularity of IXPs, the
+number of Tier-3 ASes decreased significantly over time. Con-
+sequently, our testbed does not contain any Tier-3 ASes. The
+topology design follows actual AS and peering data, obtained
+from CAIDA data [4]–[6] and peeringdb [28], with random-
+ized AS numbers. We limit the network link capacities to 15
+Mbit/s for Tier-1 links and 10 Mbit/s for all remaining links
+to make all candidates network bound. Otherwise, the CPU
+could limit candidates resulting in potentially biased results.
+We derive two torrent networks from our proposed topology:
+The first network 5ASes consists of the 5 ASes marked dark
+green (AS102, AS1002, AS1004, AS103, AS1006) and the
+second network consists of the 10 ASes marked dark and light
+green (AS102, AS1002, AS1004, AS103, AS1006, AS1001,
+AS101, AS04, AS105, AS1009). ASes are connected either
+with transit or peering links. Our topology follows the valley-
+free routing [30]. Consequently, peering links can only be used
+by the peering neighbours and their customers. In BGP and
+BGP-M, this is implemented with prefix lists on each AS that
+has peering links. Since filtering support of peering links in
+SCION is currently in progress, we apply a static path filter
+to each AS to filter out valley-free violations. Since ISD’s are
+a concept exclusively for SCION, we locate all SCION ASes
+in the same ISD, to achieve better comparability.
+In our evaluations, we run three different BitTorrent candi-
+dates: The first one is BitTorrent over BGP, which is a usual
+BitTorrent client implemented in Go based on inter-domain
+routing performed by BGP. The second candidate is BitTorrent
+over BGP-M, which uses the same BitTorrent client but with
+configured load splitting in BGP in each AS. Despite other
+interesting approaches for load splitting, we apply ECMP-
+based load splitting for BGP-M in our testbed, since it is
+the most deployed approach in the current Internet. Our third
+candidate is BitTorrent over SCION, which implements the
+disjoint path selection based on the presented idea of path-
+level peers.
+The complete topology is running on a bare-metal server
+in a virtualized environment based on Docker. Each AS runs
+one or more containers (routers, hosts). Multiple ASes are
+connected via Docker Bridge Networks [13]. The server is
+running an Intel(R) Xeon(R) Gold 6150 CPU @ 2.70GHz
+with 36 threads and 500GB of main memory.
+VI. EVALUATION
+After presenting our virtualized Internet topology, we eval-
+uate our BitTorrent over SCION approach in this section.
+A. Terminology
+The following parameters are used to create and evaluate
+different BitTorrent experiments:
+• MaxPeers: Number of available peers in the torrent
+• OutgoingConns: Number of outgoing connections (to
+peers) that each peer instantiates
+• NumASes: Number of involved ASes in the experiment
+To design our experiments, we stick to findings from Hamra
+et al. [19] measuring BitTorrent’s performance in real-world
+torrents. In most cases, MaxPeers is significantly higher than
+OutgoingConns, meaning each peer exchanges pieces with
+a subset of all availables peers. Hamra et al. show that
+setting OutgoingConns to the half of MaxPeers is a good
+tradeoff. Furthermore, the MaxPeers parameter changes over
+time in real-world torrents. This number increases often at
+the beginning of the torrent when many peers are interested
+in downloading the file and decreases after the majority of
+peers finished downloading and leave the torrent. We adapt
+this behaviour for our experiments.
+In the following, we present the results of our two exper-
+iments: At first, we evaluate how heavy multipathing imple-
+mented in BitTorrent over SCION can aggregate bandwidth in
+the network that is not available for BitTorrent over BGP/BGP-
+M. Afterwards, we compare BitTorrent over SCION against
+BitTorrent over BGP/BGP-M with a varying number of Out-
+goingConns to evaluate the effect of our disjoint path selection
+approach.
+B. Bandwidth Aggregation
+In our first experiment, we run 20 BitTorrent peers in the
+torrent networks 5ASes and 10ASes exchanging a 100Mbyte
+file. By choosing 2 different torrent network sizes with a
+fixed-size topology, we can evaluate the impact of density
+
+AS103
+AS101
+AS102
+Tier 1 
+Networks
+AS1001
+AS1002
+AS1003
+Tier 2
+Networks
+AS1004
+AS104
+AS105
+AS1005
+AS1007
+AS1008
+AS1009
+AS1010
+Light green + Green: 10 AS experiment 
+Green: 5 AS Experiment
+Peering Link
+Transit Link
+AS1006
+Fig. 3.
+Virtualized testbed of a representative Internet topology
+of peers and the number of additional ASes that are not
+participating in the torrent, and consequently may provide
+additional capacities. We set the OutgoingConns parameter to
+infinity in this experiment (in detail it is set to the maximum
+number of path-level peers that one peer can connect to) to
+evaluate the maximum possible goodput that all candidates can
+achieve.
+Figure 4a) shows the aggregated goodput in percent of all
+peers for the three candidates, while BGP serves as baseline
+with 100%. We decide to compare the goodput instead of the
+overall bandwidth, since SCION packets have a larger header
+than plain IP packets.
+For the torrent network 5ASes, we observe an aggregated
+goodput of 111% for BGP-M compared to BGP, while SCION
+achieves 148% compared to BGP. From these results, we
+observe that enabling multipath BGP via ECMP increases
+the overall goodput by 11%. We conclude that in the 5ASes
+torrent network, the number of equal length BGP paths are
+comparatively low, leading to a small increase of goodput.
+However with BitTorrent over SCION’s approach of path-
+level peers, a 48% increase of goodput is achieved compared
+to BGP and a 33% increase compared to BGP-M. In the
+10ASes torrent network, we observe an increase of goodput
+for BitTorrent over SCION of still 38% compared to BitTor-
+rent over BGP, despite that the 10AS torrent network has a
+smaller number of not participating ASes that may provide
+additional network capacities. Consequently, BitTorrent over
+SCION is able to aggregate also heterogeneous paths and we
+confirm our hypothesis of BitTorrent over SCION’s capability
+of aggregating bandwidth in the network that is unused in
+BGP/BGP-M.
+To verify that increased goodput has a direct impact on
+the actual download time of peers, we measure the average
+download time of all peers in the 5ASes and 10ASes torrent
+networks, shown in Figure 4b). Reflected by the lowest overall
+goodput, we observe the highest average download time for
+peers in BitTorrent over BGP, which is our baseline at 100%.
+BitTorrent over BGP-M results in 91% average download time
+for the 5ASes torrent network. In BitTorrent over SCION, the
+average download time is around 69%. Also for the 10ASes
+torrent network, we observe that the goodput reflects the
+average download times, resulting in 73% for BitTorrent over
+SCION and 88% for BitTorrent over BGP-M. As expected,
+the increased goodput through aggregating unused capacities
+in the network directly translates into lower download times
+for peers, increasing the overall performance of the system.
+In addition to the goodput and download times, we also
+measure the overall CPU usage of all candidates running the
+5ASes and 10ASes torrent network, shown in Figure 4c). Since
+we observed an equal distribution of load over all threads,
+we present the CPU usage as average CPU usage per thread.
+While BitTorrent over BGP and BGP-M only use 5% and
+8%, respectively, BitTorrent over SCION uses around 31%
+of the available CPU usage for the 5ASes torrent network.
+We observe an expectable increase of resource usage of all
+candidates running 10ASes, with SCION using around 47%
+of each thread. Despite the higher throughput, BitTorrent over
+SCION achieves, the increase of CPU usage is not inreasing in
+the same amount. We reason this increase by the open-source
+SCION stack [26] (there also exists a closed-source SCION
+stack optimized for performance [1]). Especially the SCION
+Border Router implementation has potential to be optimized
+for performance, while the framework to route BGP and BGP-
+M (frrouter [25]) is heavily tuned. Consequently, we assume
+that using the closed-source SCION stack, we can decrease
+the CPU usage to a level comparable to BGP and BGP-M.
+From this experiment, we confirm our hypothesis that
+BitTorrent over SCION can aggregate otherwise unused ca-
+pacities in the network through multipath usage. We observe
+a significantly higher CPU usage for BitTorrent over SCION,
+which can potentially be strongly reduced by using the high-
+performance SCION stack, which is closed source.
+
+5
+10
+80
+100
+120
+140
+160
+Number of ASes
+Relative Goodput over all peers [%]
+a)
+BGP (baseline)
+BGP-M
+SCION
+5
+10
+40
+60
+80
+100
+Number of ASes
+Relative Download time [%]
+b)
+BGP (baseline)
+BGP-M
+SCION
+5
+10
+0
+20
+40
+60
+80
+100
+Number of ASes
+Average CPU usage per thread [%]
+c)
+BGP
+BGP-M
+SCION
+Fig. 4. Comparison of BitTorrent over BGP/BGP-M and SCION in the 5AS and 10AS torrent network with 4 peers per AS. a) shows the aggregated goodput
+of all peers in % with BGP as 100% baseline, b) shows the download time in % with BGP as 100% and c) the aggregated CPU usage of all threads in %
+C. Peer Selection
+Always setting the OutgoingConns parameter sufficiently
+high may create unfair advantages for BitTorrent over SCION.
+Therefore, we compare all 3 candidates with a varying upper
+limit of OutgoingConns in this experiment. We again run 20
+peers exchanging pieces of a 100Mbyte file in our 5ASes
+and 10ASes torrent network. As discussed before, setting
+OutgoingConns to the half of MaxPeers is a good tradeoff,
+we decide to vary OutgoingConns between 3 and 10. We
+expect that BitTorrent over BGP and BGP-M stagnates with a
+low number of OutgoingConns, meaning simply adding more
+connected peers to BitTorrent over BGP and BGP-M does
+not directly lead to improved performance, while BitTorrent
+over SCION handles an increasing number of OutgoingConns
+better.
+Figure 5 shows the average download time of all peers with
+a varying number of OutgoingConns for the 5ASes torrent
+network. We observe that BitTorrent over BGP and BGP-M
+start to stagnate after 5 OutgoingConns, while BitTorrent over
+SCION results in decreased download times until 8 Outgoing-
+Conns. With OutgoingConns greater than 7, the results are
+matching the ones presented in the bandwidth aggregation
+experiment.
+We conclude that between 3 and 7 OutgoingConns, Bit-
+Torrent over SCION is still able to find disjoint paths between
+peers, while with more than 7 OutgoingConns, the additionally
+used path-level peers share bottlenecks. However, we assume
+that in real-world, Internet-scale torrent networks, the number
+of disjoint paths is significantly higher, leading to better results
+for higher numbers of OutgoingConns.
+In Figure 6, we show the average download time of all
+peers with a varying number of OutgoingConns for the 10ASes
+torrent network. We observe a high decrease of download
+times with less than 5 OutgoingConns for all candidates, while
+BitTorrent over SCION is still able to decrease the download
+time for up to 7 OutgoingConns. All three candidates have
+reached their minmum download times for greater than 7
+OutgoingConns, in contrast to 5 OutgoingConns for the 5ASes
+3
+4
+5
+6
+7
+8
+9
+10
+80
+100
+120
+140
+160
+180
+200
+Number of OutgoingConns
+Average Download Time [s]
+BGP
+BGP-M
+SCION
+Fig. 5.
+Average download time in seconds for BitTorrent over BGP, BGP-M
+and SCION in the 5ASes torrent network.
+torrent network. This is reasoned by the higher percentage of
+participating ASes in the torrent compared to the total number
+of ASes, and for BitTorrent over SCION especially by the
+lower number of additional network capacities.
+From this experiment, we observe that BitTorrent over
+SCION also outperforms BitTorrent over BGP and BGP-M
+with a limited number of connected peers, disproving the
+intuitive argument that the path-level peer approach only works
+for a sufficiently high number of OutgoingConns.
+VII. RELATED WORK
+The mature P2P mechanisms behind BitTorrent have made
+the protocol an attractive target for networking research in the
+past. Ren et al. present TopBt [33], an adaption of BitTorrent
+that uses proximities in addition to transmission rates to detect
+peers to collaborate with. Catro et al. propose BestPeer [3],
+
+3
+4
+5
+6
+7
+8
+9
+10
+40
+60
+80
+100
+120
+Number of OutgoingConns
+Average Download Time [s]
+BGP
+BGP-M
+SCION
+Fig. 6.
+Average download time in seconds for BitTorrent over BGP, BGP-M
+and SCION in the 10ASes torrent network.
+a peer selection algorithm that supports multipath in a multi-
+radio, multi-channel wireless mesh network.
+Recent works cover analyses of BitTorrent’s locality [8],
+[36], [39], concluding that the majority of BitTorrent’s traffic
+is still running globally. Furthermore, Decker et al. analyze
+behavioral patterns and topologies in existing torrent networks
+[10], while Cuevas et al. [8] analyze how BitTorrent’s locality
+impacts transit costs in existing networks.
+A lot of research investigates IP multicast as an efficient
+way to distribute content to multiple peers without duplicating
+the traffic [11]. Since IP multicast requires expensive dedicated
+support in network equipment, it has so far only seen localized
+deployment [12], [31]. As an alternative to IP multicast,
+overlay approaches are considered: Bullet by Kosti´c et al
+[22] is an overlay approach to efficiently distribute files from
+a single source to a large number of receivers. Also IPFS
+[38] shares similarities with BitTorrent through its P2P based
+nature. Finally, Fujinoki et al. provide an approach to unlock
+private peering links for inter-domain routing in the Internet
+[16], which provides interesting potential for our multipath
+approach.
+Next to SCION, several other approaches to enable path
+control on the host exist. PathLet Routing by Godfrey et al.
+[17] is an approach based on segmentation of inter-domain
+routes into path fragments. Establishing multipath data transfer
+can also be realized completely on the application level: Yu
+et al. proposed mpath [41], an algorithm and implementation
+to leverage proxies to create multiple paths to a particular end
+host.
+To detect shared bottlenecks, different approaches have been
+proposed for MPTCP, some via active measurements [14],
+[40], others via passive shared bottleneck detection [20]. These
+approaches are constrained by an inherent lack of information
+about the actual path that the data uses through the Internet.
+With Espresso [43], Google presented a BGP-based approach
+for traffic distribution and bottleneck avoidance at the edge of
+their network, rather than on the endhosts.
+Finally, demonstrating SCION’s high-performance capabil-
+ities, Neukom et al. propose Hercules [27], a protocol for very
+high performance bulk data transfer over SCION and de Ruiter
+et al. present a SCION Border Router implementation in P4
+[35].
+VIII. CONCLUSIONS AND FUTURE WORK
+In this work, we developed the notion of path-level peers,
+which allowed us to enhance BitTorrent with SCION support
+to add multipath features, with minimal modifications to the
+underlying file-sharing algorithms. Furthermore, we propose
+an algorithm for disjoint selection of path-level peers to im-
+prove the usage of network capacities. We evaluate BitTorrent
+over SCION in a virtualized inter-domain testbed comparing
+it to BitTorrent over BGP and BGP-M. We observe a 48% im-
+provement of goodput for BitTorrent over SCION compared to
+BGP and 38% compared to BGP-M, which reflects in smaller
+average download times for participating peers. Furthermore,
+we show that our proposed disjoint path selection algorithm is
+able to improve traffic flow in the network with a low number
+of outgoing connections to unchoked peers. Consequently, we
+confirm our hypothesis that BitTorrent over SCION is capable
+of aggregating capacities in the network that are unused when
+BGP or BGP-M is utilized for inter-domain routing.
+As future improvement for BitTorrent over SCION, we plan
+to extend the tracker implementation to pre-select particular
+path-level peers. Peers may actively communicate their se-
+lected path sets to the tracker, which can improve the location
+of shared bottlenecks based on the knowledge about path usage
+of all known peers.
+Furthermore, we plan to evaluate the impact of peers
+actively communicating the selected path set to other peers.
+We assume that peers can improve the location and avoidance
+of shared bottlenecks with this approach.
+Finally, we plan to extend our disjoint path selection to
+allow multiple peers to reuse the same hops without creating
+shared bottlenecks, by observing variation in bandwidth to
+all connected peers when adding new paths containing shared
+hops.
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+

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+arXiv:2301.00849v1  [cs.SI]  2 Jan 2023
+Small-World Formation via Local Information
+Soroush Alamdari
+[email protected]
+Abstract
+It is observed that in a society almost anyone is acquainted with almost anyone else through
+only a few intermediary links. This is known as the small-world phenomenon. In this paper
+we investigate this observation from a theoretical stand-point by imagining each individual as a
+greedy agent satisfying a drive for knowledge by acquiring links that cost to maintain. We show
+that in such a setting small-world properties emerge naturally.∗
+∗The results in this paper were initially announced in 35th ACM SIGACT-SIGOPS Symposium on Principles of
+Distributed Computing (PODC 2016).
+1
+
+1
+The small-world phenomenon
+The drives that move us towards each other are far more complicated than to be mathematically
+understandable, and yet, there could be clues that may help us paint an abstract picture. For
+this, we focus on one particular observation, that is, the small-world phenomenon: The idea that
+any two people know each other through only a few links.
+The small-world phenomenon was confirmed in an experiment carried by Milgram [3], where
+randomly selected individuals in Nebraska and Kansas received an information packet with in-
+struction asking them to help the packet reach a certain individual in Boston Massachusetts. In
+case they did not know the target individual personally, the recipients were instructed to pass the
+packet on to someone they knew personally who was most likely to personally know the target.
+So each initial recipient started a chain of correspondence, some of which ended before reaching
+the target due to lack of participation. Among the chains that reached the target, the average
+length of a chain was just bellow six. An observation we have been trying to understand since.
+Explaining this observation is challenging, particularly because. Consider the network whose
+nodes are residents of united states at the time of the experiment, and where there is a link from a
+node v to a node u, if v has the address of u and claims to personally know them. Minus oddities,
+the experiment essentially shows that in this network any two nodes are connected via a short
+chain of links. This is particularly fascinating as each individual only knows a very small group
+of others compared to the total population. Satisfying these two constraints simultaneously is a
+challenge even in an abstract mathematical setting:
+1. Degree: The number of others that any node has links to, that is the degree of that node,
+is small relative to the total population.
+2. Diameter: For any recipient and any target, the length of the shortest chain of links that
+starts from the recipient and ends in the target is very small. The length of the longest
+such chain in a network is referred to as the diameter of that network.
+To address this, Watts and Strogatz [5] showed that in a society with a population of n that
+are sitting around a ring, if each individual has links to the nε others that are sitting on each side
+of it plus nε random other nodes, then each pair of nodes are connected via a chain of expected
+length smaller than 1/ε. While this simple model satisfies both Constraints 1 and 2, it does
+not explain how such a chain is actually found by the forwarded packages in the experiment.
+There each recipient is unaware of the structure of this network and cannot decide who among
+the people it has a link to is on the shortest chain to the target. The fact that such a short
+chains are actually observed by the experiment implies a much more restricting constraint than
+Constraint 2. This observation allows us to strengthen Constraint 2:
+3. Limited information: Each node is unaware of the links maintained by other nodes. We
+refer to this condition as the limited information constraint.
+4. Greedy diameter: For any recipient and any target, a very short chain connecting the
+recipient to the target can be found if each recipient chooses who to forward the packet to
+based on similarity with the target. We refer to the length of the longest such chain in a
+network as the greedy diameter of that network.
+Closer analysis of the experiment has shown that participants favor geographical information
+when deciding the next recipient [4]. Kleinberg [2] argues that such information is sufficient
+for satisfying Constraint 4 by showing that in a population that is distributed homogeneously
+over the plane, if each node has a link to a node of geographical distance l with probability
+1/lα, then iteratively choosing the next recipient based on geographical proximity to the target
+almost always produces a short chain. Particularly, when α = 2, both the expected length of the
+2
+
+α < d + 1
+α = d + 1
+d + 1 < α
+Maximum degree
+o(n1−δ
+α
+d+1 log log n)
+O(log n)
+O(log
+α
+d+1 log n) for d = 1
+Observable diameter
+O(log d+1
+α log n)
+O(log n)
+o(n1−
+δ
+α−d log log n)
+Table 1: The bounds on maximum degree and observable diameter at stability, for different values of
+α relative to the dimension d. Here δ can be any constant smaller than 1.
+chain that is found and the expected number of links that each node creates can be bounded by
+O(log n).
+Although Kleinberg’s model satisfies all of the constraints with a natural explanation, it does
+so by reducing the behavior of the individuals to a random process. It remains curious whether
+the small-world phenomenon exists by coincidence, or if it is designed by our collective minds.
+While it is possible that our social experience is dominated by random encounters, it seems
+more plausible that a selfish drive connects us in this particular way. And so we add one more
+constraint, although it is not immediately verifiable:
+5. Strategic choice: Each node create links by making choices in order to satisfy some
+objective.
+To address this, researchers have been trying to find the right game-theoretic model that
+would be able to explain the astonishing properties of our social networks. However, by consid-
+ering a game-theoretic model where individuals make choices strategically based on the choices of
+others, the limited information constraint is immediately violated. However, if a game-theoretic
+model that reproduces the small-world properties is established based on the correct objective
+function, then the small-world properties should persist even when the full information assump-
+tion is relaxed.
+Among various game-theoretic models of network formation, our result highlights the work
+of Even-Dar and Kearns [1], where the population is distributed uniformly on the plane and
+each agent v chooses its links in order to minimize the total cost of maintaining the links plus
+the average distance to other nodes through the network. In their model, a link to a node of
+geographical distance l induces a cost of lα. Particularly, they show that in this model when
+α < 2, the diameter can be bounded by a constant.
+When attempting to derandomize the work of Kleinberg [2] at the critical point where loga-
+rithmic bounds appear on the expected degrees and greedy diameter, we come upon a strategic
+model of network creation that achieves similar bounds and satisfies all of our constraints. When
+analyzing the objective function that gives this we see a natural interpretation that hints towards
+the role of curiosity in emergence of small-world phenomenon.
+In Section 2 we present the model in details and discuss the components of the cost function
+that the agents are trying to minimize. In Section 3 we analyze the model on a d-dimensional
+grid†.We focus on a notion of equilibrium where agents cannot further improve their cost by
+adding or deleting a link, showing that α = d + 1 is a threshold where the upper bounds on the
+performance of the greedy routing and maximum degree match to O(log n). We also analyze
+the maximum degree and the performance of greedy routing for other values of α, particularly
+showing that when α < d + 1 the number of iterations that are required for greedy routing to
+find its destination can be bounded by O(log d+1
+α log(n)). We conclude in Section 4 by discussing
+the implications of our results.
+†Throughout the paper we assume that d is a constant.
+3
+
+2
+The Model
+Consider a d-dimensional circular grid of n nodes representing the underlying structure upon
+which the network is formed. For nodes u and v we use d(u, v) to refer to the distance‡ of u and
+v. Also, for a node u and a set of nodes V we use d(u, V ) to refer to the distance of u to its
+closest member in V , that is, d(u, V ) = minv∈V d(u, v).
+A network N assigns to each node v a set N(v) of nodes that v has a links to. To model
+the creation of such a network we introduce a cost that each agent incurs at a given setting.
+Let cN(v) be such a function that calculates the cost induced to a node v in N. There are two
+components to this cost function; the cost of maintaining the links that v has to other nodes and
+the cost induced by the information that v is missing regarding the nodes that it does not have
+a link to. For a node u, let lN(v, u) be the former and sN(v, u) be the latter. We have
+cN(v) =
+�
+u∈N(v)
+lN(v, u) +
+�
+u/∈N(v)
+sN(v, u).
+For the cost that v incurs for maintaining a link to an agent at distance d(v, u), it is natural
+to assume a dependence between this cost and distance. Let us define lN(v, u) = dα(v, u) for
+some number α ∈ R. The exact choice of α is discussed in section 3. We will refer to this cost
+as the link-cost.
+It remains to define sN(v, u), that is, the cost that v pays for what it does not know about
+u. We capture this by assuming that each node u induces a cost to v that is proportional to the
+distance between u and the node closest to u that v has a link to, that is, d(u, N(v)). We refer
+to this as the curiosity-cost.
+We are ready to express our cost function cN(v) such that it only depends on N(v) and the
+position of the nodes on the underlying grid:
+cN(v) = β
+�
+u∈N(v)
+dα(v, u) +
+�
+u/∈N(v)
+d(u, N(v)).
+(1)
+3
+Analysis
+In this section we analyze the network constructed by agents who seek to reduce their costs
+by adding and deleting links to other nodes. In particular, we discuss the trade-off between
+the greedy routing diameter of the network and the maximum degree among nodes.
+When
+α <
+1
+logβ−1 n every link has a cost smaller than 1 and therefore the diameter is 1 and all of the
+degrees are equal to n − 1, and when α > log(β−1n2) no agent can justify a link to a node of
+distance > 1, and therefore the degrees and diameter are 2d and dn1/d respectively. Here we
+provide analysis for intermediary values of α.
+To establish our bounds, we do not require the assumption that each agent chooses their links
+to minimize their cost. In each of our cases, the assumption that no agent can reduce its cost by
+adding a link (add-stability) is sufficient to prove the upper bound on the greedy diameter of the
+network. When in addition to add-stability we also assume that no agent can improve its cost
+by removing a link (toggle-stability) we are able to also prove upper bounds on the maximum
+degree. Since each agent reduces its cost with each these operations of addition and deletion
+of a link, therefore, performing these operations in any order will eventually result in stability.
+It is also worth noting that these assumptions are much more general than, for example, the
+assumption of a Nash equilibrium.
+‡In this paper, unless specified otherwise, by distance we are referring to the distance through the grid, that is,
+grid-distance
+4
+
+v
+l
+l − l′
+v
+l
+l − l′′
+Figure 1: Items 3 (left) and 4 (right) of Observation 3.1. In each case l′ is the diameter of the smaller
+squares.
+We start with a set of observations on the properties of the grid that will be used throughout
+this section in order to bound the greedy diameter and maximum degree among the nodes. For
+this, let us define S(v, l) to be the set of all nodes of distance exactly l from v, and B(v, l) be
+the set of all nodes u with d(u, v) ≤ l. The following observation lists some properties regarding
+S(v, l) and B(v, l).
+Observation 3.1. For any node v on the d-dimensional grid and l ∈ N the following holds:
+1. |S(v, l)| ∈ Θ(ld−1).
+2. |B(v, l)| ∈ Θ(ld).
+3. The nodes in B(v, l)\B(v, l − l′) can be covered with O(( l
+l′ )
+d−1) balls of the form B(u, l′)
+with u ∈ B(v, l)\B(v, l − l′).
+4. The nodes in B(v, l)\B(v, l − l′′) can be covered by O(( l
+l′ )
+d) balls of the form B(u, l′) with
+u ∈ B(v, l)\B(v, l − l′′).
+3.1
+Balance: α = d + 1
+Here we focus on the case when α = d+1 and show that both maximum degree and the iterations
+required for greedy routing to find its destination are bounded by O(log n).
+Let us start with a useful lemma that provides a lower bound on the drop in the curiosity-cost
+when adding a link.
+Lemma 3.2. If a node v has no link in B(u, l), then adding a link to u will decrease the total
+curiosity-cost of v by Ω(ld+1).
+5
+
+Proof. Consider the set of nodes S with distance l′ from u where l′ ≤ 1
+3l. Note that any node
+in S induces a curiosity-cost of at least 2
+3l. If v adds a link to u, then the curiosity-cost of each
+node in S will be reduced to at most 1
+3l. Therefore, by Item 2 of Lemma 3.1 the total decrease
+in curiosity-cost is at least c l
+3
+d × l
+3 = c l
+3
+d+1 which concludes the lemma.
+Consider an add-stable node v, that is, a node v that does not benefit from adding any more
+links to any other nodes. Next we provide an upper bound on the distance that a node u with
+d(u, v) = l may have from the neighbors of v, that is, d(u, N(v)). This is done by showing that
+if the distance is too large then v would benefit from adding a link to u.
+Lemma 3.3. If a node v in N is add-stable, then for any u with d(v, u) ≤ l there exists a node
+u′ of distance at most O(l
+α
+d+1) from u that v has a link to, that is, N(v) ∩ B(u, cl
+α
+d+1) ̸= ∅ for
+some constant c.
+Proof. For the sake of contradiction, suppose we have a node u such that N(v)∩B(u, cl
+α
+d+1) = ∅.
+By Lemma 3.2 adding a link from v to u decreases the curiosity-cost by at least c′lα for some
+constant c′. When β < c′, this implies that the reduction in the curiosity-cost is larger than the
+cost of a link to u which is at most βlα, contradicting add-stability.
+Now we can prove an upper bound on the number of iterations required for the greedy routing
+algorithm to reach its destination.
+Theorem 3.4. If a network N is add-stable and α = d + 1, then any node u is reachable from
+any node v via greedy routing in O(log(n)) steps.
+Proof. When α = d + 1, by Lemma 3.3 we know that in each iteration of greedy routing the
+distance to destination is reduced by a constant factor, resulting in logarithmic bound on the
+number of iterations until the destination is reached.
+Next we turn our focus towards bounding the maximum degree when α = d + 1 by showing
+that the number of links that v has to all the nodes of distance between l and cl for some constant
+c, is bounded by a constant.
+Lemma 3.5. When α ≤ d + 1, if a node v in N is toggle-stable then v has at most a constant
+number of links to any ball of radius l′ ≤ ǫl
+α
+d+1 ≤ ǫl centered at a node u with d(u, v) = l, that
+is, |N(v) ∩ B(u, l′)| ≤ c.
+Proof. Any node within B(u, l′) has distance at most l + l′ from v, and since v in N is toggle-
+stable, by Lemma 3.3 we can argue that any node within the ball can serve only nodes of
+distance at most c′(l + l′)
+α
+d+1 for some constant c′. Therefore any node that is served by a node
+in B(u, l′) must also lie within the ball B(u, l′ + c′(l + l′)
+α
+d+1 ). Since l′ ≤ ǫl
+α
+d+1 ≤ ǫl, we have
+l′ + c′(l + l′)
+α
+d+1 ≤ c′′l
+α
+d+1 for some constant c′′. Let C(u′) be the set of the nodes whose closest
+node in N(v) is u′. By Item 2 of Lemma 3.1 we can argue that
+�
+u′∈N(v)∩B(u,l′)
+|C(u′)| ≤ c∗(l
+α
+d+1 )d.
+for some constant c∗.
+By removing a link to a node u′ ∈ B(u, l′), the curiosity-cost induced to v is increased by
+at most c′′l
+α
+d+1|C(u′)|, and therefore there is a node u′ ∈ N(v) ∩ B(u, l′) the removal of which
+decreases the total cost induced to v by at least
+β(l − l′)α−c′′l
+α
+d+1|C(u′)| ≥ β(l − l′)α−c′′l
+α
+d+1 ×
+c∗l
+α
+d+1)d
+|N(v) ∩ B(u, l′)| ≥ β(1−ǫ)αlα−
+c⋆lα
+|N(v) ∩ B(u, l′)|.
+6
+
+Therefore if |N(v) ∩ B(u, l′)| >
+c⋆
+β(1−ǫ)α = c, v would be able to decrease its total cost by
+removing u′, contradicting stability.
+Theorem 3.6. If a node v in N is toggle-stable and α = d + 1, then the degree of v is bounded
+by O(log n) links.
+Proof. For any integer 0 ≤ i ≤ log1+ǫ(n) let P(i) be the set of grid points that are of distance
+l from v such that (1 + ǫ)i < l ≤ (1 + ǫ)i+1. By Lemma 3.5 for any p ∈ P(i) we know that the
+number of links that v has to the members of the ball B(p, ǫ(1 + ǫ)i) is bounded by constant.
+Also, by Item 3 of Lemma 3.1 the nodes in P(i) can be covered using c(1+ǫ)i+1
+ǫ(1+ǫ)i
+d−1
+such balls for
+some constant c. Therefore, for each 0 ≤ i ≤ log1+ǫ(n) we can bound the size of N(v) ∩ P(i) by
+a constant, concluding the theorem.
+3.2
+Sub-logarithmic diameter: α < d + 1
+In this section we consider cases when
+1
+logβ−1 n < α < d + 1 and prove upper bounds on the
+number of iterations that greedy routing requires and maximum degree for this range of α.
+Theorem 3.7. If a network N is add-stable and α < d + 1, then any node u is reachable from
+any node v via greedy routing in O(log d+1
+α log(n)) steps.
+Proof. By Lemma 3.3 if in an iteration of greedy routing the distance to destination is l, then
+in next iteration the distance is at most cl
+α
+d+1 for some constant c. Therefore, after
+log
+α
+d+1 (
+1
+log(n)) = log d+1
+α log(n)
+iterations the distance to destination will be constant.
+Theorem 3.8. If a node v in N is toggle-stable and α < d + 1, then the degree of v is bounded
+by o(log log(n)n1−δ
+α
+d+1) links for any δ < 1.
+Proof. For some positive constant 0 < a < 1 and some integer 0 ≤ i ≤ loga−1 log(n) let P(i) be
+the set of grid points that are of distance l from v such that nai+1/d < l ≤ nai/d. By Lemma
+3.5 for any p ∈ P(i) we know that the number of links that v has to the members of the ball
+B(p, ǫ(nai+1/d)
+α
+d+1 ) is bounded by constant.
+By Item 4 of Lemma 3.1 P(i) can be covered with O(nai−ai+1
+α
+d+1) balls B(p, ǫ(nai+1/d)
+α
+d+1)
+where p ∈ P(i).
+Therefore the total number of links of v can be bounded by
+|N(v)| ≤ c +
+i≤loga−1 log(n)
+�
+i=0
+nai−ai+1
+α
+d+1 ≤ c + loga−1 log(n)n1−a
+α
+d+1
+for some constant c.
+3.3
+Sub-logarithmic degrees: α > d + 1
+Now let us turn our focus to cases when d + 1 < α < log(β−1n2) and prove bounds analogous to
+those of previous sections. Here we need to modify some of our lemmas, as in this case the cost
+of links increases relative to length so fast that it no longer benefits v to establish a link to the
+middle of a region it wishes to cover with that link.
+7
+
+v
+u
+u′
+u′′
+l − ǫl
+1
+α−d
+l − 2ǫl
+1
+α−d
+ǫl
+1
+α−d
+ǫl
+1
+α−d
+4
+Figure 2: For Lemma 3.9. Adding a link from v to u′ (resp. u′′) decreases the curiosity-cost of each
+of the nodes in the area indicated by red (resp. blue) hashing by at least ǫl
+1
+α−d
+4
+.
+Lemma 3.9. If a node v in N is add-stable, then for any u with d(v, u) ≤ l there exists a node
+u′ ∈ N(v) with d(u, u′) ≤ l − ǫl
+1
+α−d .
+Sketch of proof: For the sake of contradiction assume that v has no link to any node in B(u, l −
+ǫl
+1
+α−d ) and let u′ be arg min
+u′∈B(u,l−2ǫl
+1
+α−d ) d(u′, v). Then, the node v benefits from adding a link
+to u′, as u′ will be able to reduce the curiosity of Ω((l
+1
+α−d )d−1l) nodes by at least Ω(l
+1
+α−d ) (See
+Figure 2), and therefore reduce its total curiosity-cost by Ω(l1+
+d
+α−d ). We can argue that
+ǫ′l1+
+d
+α−d = ǫ′l
+α
+α−d > β(2ǫl
+1
+α−d )α
+for proper value of ǫ and β, and therefore it is beneficial for v to add a link to u′.
+Theorem 3.10. If a network N is add-stable and α > d + 1, then any node u is reachable from
+any node v via greedy routing in o(log log(n)n1−
+δ
+α−d ) steps where δ is any constant smaller than
+1.
+Proof. By induction we can argue that starting from a node v, the worst case of our analysis of
+greedy routing is realized when the destination is the furthest node from v. By Lemma 3.9 in
+each step if the distance to the furthest node is l, in the next step this distance will be at most
+l−ǫl
+1
+α−d . Let us consider the number of iterations it takes to get from distance nai of destination
+to get to distance lai+1 for some a < 1. This is bounded by at most
+lai
+ǫl
+ai+1
+α−d
+= ǫ−1lai− ai+1
+α−d .
+8
+
+Therefore, the total number of iterations is bounded by
+i≤log log n
+�
+i=0
+ǫ−1lai− ai+1
+α−d ≤
+i≤log log n
+�
+i=0
+ǫ−1l1−
+a
+α−d ≤ log log(n)ǫ−1l1=
+a
+α−d
+Here we show that for d = 1, when α > d + 1 the degrees are restricted by a sub-logarithmic
+upper bound. Equivalent proof for higher dimensions remains a challenge.
+Theorem 3.11. If α > d + 1 = 2 and a network N is toggle-stable, then any node v has at most
+O(log
+α
+d+1 log(n)) links.
+Proof. Think of each node of distance l from v to initially have a budget of l of curiosity-cost
+that can be spent to justify maintenance of a link. We will show that if v were to have more
+than c log
+α
+d+1 log(n) links, then there would be a set of nodes that would have insufficient budget
+in order to justify a set of links that serve a subset of them, and therefore, N would not be
+toggle-stable.
+Consider a node v and let P(i) be the set of nodes with distance l′ from v such that n
+ai+1
+d
+<
+l′ ≤ n
+ai
+d where a = d+1
+α . We show that the size of P(i) ∩ N(v) is bounded by a constant. For
+this, note that any link that v has in P(i) costs at least βn
+αai+1
+d
+.
+Noting that d = 1, let S be the set P(i) ∩ N(v) minus the elements that are furthest from v
+on each side, and hence |S| ≥ |P(i)∩N(v)|−2. No node in S can serve a node of distance larger
+than n
+ai
+d from v, and by Item 2 of Lemma 3.1 there are at most n
+ai
+d
+d
+nodes that can contribute
+to justify S, and each such a node has a budget of at most n
+ai
+d . Therefore, we have
+|S| ≤ n
+ai
+d
+d+1
+βn
+αai+1
+d
+= n
+(d+1)i+1
+dαi
+βn
+(d+1)i+1
+dαi
+= β−1
+.
+We conjecture that the bound from Theorem 3.11 can be extended to d > 1.
+4
+Discussion
+Our main contribution is to provide a simple explanation for the small-world phenomenon that
+matches previously established theoretical observations.
+While the questions that we are concerned with here are inherently inspired by nature, this
+work was initially carried out as a purely mathematical mission to reestablish the logarithmic
+bounds on maximum degree and greedy diameter in a setting where individuals have only local
+information. However, the model that came out of this research seems to be providing insight
+into drives that connect us.
+Imagine a population distributed over a plane, where each individual maintains the status
+(say current address) of some of the others in order to gain some knowledge through them. There
+is a real effort here, which is mainly the effort and attention that is required to maintain the
+status of another, and there is an implicit drive that can be described as a desire to know all.
+Our results suggest that such a population ends up forming a small-world network.
+It is imaginable that α (the exponent of the distance in cost of maintaining a link) has been
+reduced by the progresses made through history. Particularly with writing and a postal service,
+9
+
+the cost of maintaining the status of another seems to increase relative to distance with a small
+exponent. It would be interesting to gain an understanding of the real value of α through analysis
+of empirical data.
+As for applicability our model may be useful to engineers whenever robust networks with
+nodes of bounded degree are required, particularly when nodes are to make decisions indepen-
+dently regarding their connections.
+It must be noted that the our cost function can be modified in a number of ways while still
+yielding networks with small-world properties. For instance, if our model is used to connect a
+swarm of drones, if each drone v chooses the set N(v) of those whose status it watches for by
+minimizing:
+cN(v) = β
+�
+u∈N(v)
+dα(v, u) +
+�
+u/∈N(v)
+wv(u)d(u, N(v))
+(2)
+where wv(u) represents the relative importance of u for v, then some degree of small-world
+properties are established as a side effect depending on parameters α and β, and the correlation
+between relative importance and proximity.
+At this point, many directions remain unexplored.
+Particularly, it is curious as to what
+happens to the threshold for other values of β. When we set β = n, our cost function closely
+resembles the objective function upon which the model of Even-Dar and Kearns [1] is based.
+Although their model is inherently different, a similar threshold with features of its own appears
+there. All of this hints towards a bigger picture, perhaps a more general framework that is yet
+to be discovered.
+10
+
+References
+[1] E. Even-Dar and M. J. Kearns. A small world threshold for economic network formation. In
+Advances in Neural Information Processing Systems 19, pages 385–392. MIT Press, 2007.
+[2] J. Kleinberg. The small-world phenomenon: An algorithmic perspective. In Proceedings of
+the Thirty-second Annual ACM Symposium on Theory of Computing, pages 163–170, New
+York, NY, USA, 2000. ACM.
+[3] S. Milgram. The Small World Problem. Psychology Today, 2:60–67, 1967.
+[4] J. Travers and S. Milgram. An experimental study of the small world problem. Sociometry,
+32:425–443, 1969.
+[5] D. J. Watts and S. H. Strogatz.
+Collective dynamics of’small-world’networks.
+Nature,
+393(6684):409–10, 1998.
+11
+

R9AyT4oBgHgl3EQf7_qI/content/tmp_files/load_file.txt

@@ -0,0 +1,262 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf,len=261
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='00849v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='SI]  2 Jan 2023  Small-World Formation via Local Information  Soroush Alamdari  sorush@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='com  Abstract  It is observed that in a society almost anyone is acquainted with almost anyone else through  only a few intermediary links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' This is known as the small-world phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' In this paper  we investigate this observation from a theoretical stand-point by imagining each individual as a  greedy agent satisfying a drive for knowledge by acquiring links that cost to maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We show  that in such a setting small-world properties emerge naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='∗  ∗The results in this paper were initially announced in 35th ACM SIGACT-SIGOPS Symposium on Principles of  Distributed Computing (PODC 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  1  1  The small-world phenomenon  The drives that move us towards each other are far more complicated than to be mathematically  understandable, and yet, there could be clues that may help us paint an abstract picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For  this, we focus on one particular observation, that is, the small-world phenomenon: The idea that  any two people know each other through only a few links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  The small-world phenomenon was confirmed in an experiment carried by Milgram [3], where  randomly selected individuals in Nebraska and Kansas received an information packet with in-  struction asking them to help the packet reach a certain individual in Boston Massachusetts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' In  case they did not know the target individual personally, the recipients were instructed to pass the  packet on to someone they knew personally who was most likely to personally know the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  So each initial recipient started a chain of correspondence, some of which ended before reaching  the target due to lack of participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Among the chains that reached the target, the average  length of a chain was just bellow six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' An observation we have been trying to understand since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Explaining this observation is challenging, particularly because.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Consider the network whose  nodes are residents of united states at the time of the experiment, and where there is a link from a  node v to a node u, if v has the address of u and claims to personally know them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Minus oddities,  the experiment essentially shows that in this network any two nodes are connected via a short  chain of links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' This is particularly fascinating as each individual only knows a very small group  of others compared to the total population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Satisfying these two constraints simultaneously is a  challenge even in an abstract mathematical setting:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Degree: The number of others that any node has links to, that is the degree of that node,  is small relative to the total population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Diameter: For any recipient and any target, the length of the shortest chain of links that  starts from the recipient and ends in the target is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' The length of the longest  such chain in a network is referred to as the diameter of that network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  To address this, Watts and Strogatz [5] showed that in a society with a population of n that  are sitting around a ring, if each individual has links to the nε others that are sitting on each side  of it plus nε random other nodes, then each pair of nodes are connected via a chain of expected  length smaller than 1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' While this simple model satisfies both Constraints 1 and 2, it does  not explain how such a chain is actually found by the forwarded packages in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  There each recipient is unaware of the structure of this network and cannot decide who among  the people it has a link to is on the shortest chain to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' The fact that such a short  chains are actually observed by the experiment implies a much more restricting constraint than  Constraint 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' This observation allows us to strengthen Constraint 2:  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Limited information: Each node is unaware of the links maintained by other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We  refer to this condition as the limited information constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Greedy diameter: For any recipient and any target, a very short chain connecting the  recipient to the target can be found if each recipient chooses who to forward the packet to  based on similarity with the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We refer to the length of the longest such chain in a  network as the greedy diameter of that network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Closer analysis of the experiment has shown that participants favor geographical information  when deciding the next recipient [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Kleinberg [2] argues that such information is sufficient  for satisfying Constraint 4 by showing that in a population that is distributed homogeneously  over the plane, if each node has a link to a node of geographical distance l with probability  1/lα, then iteratively choosing the next recipient based on geographical proximity to the target  almost always produces a short chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Particularly, when α = 2, both the expected length of the  2  α < d + 1  α = d + 1  d + 1 < α  Maximum degree  o(n1−δ  α  d+1 log log n)  O(log n)  O(log  α  d+1 log n) for d = 1  Observable diameter  O(log d+1  α log n)  O(log n)  o(n1−  δ  α−d log log n)  Table 1: The bounds on maximum degree and observable diameter at stability, for different values of  α relative to the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Here δ can be any constant smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  chain that is found and the expected number of links that each node creates can be bounded by  O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Although Kleinberg’s model satisfies all of the constraints with a natural explanation, it does  so by reducing the behavior of the individuals to a random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' It remains curious whether  the small-world phenomenon exists by coincidence, or if it is designed by our collective minds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  While it is possible that our social experience is dominated by random encounters, it seems  more plausible that a selfish drive connects us in this particular way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' And so we add one more  constraint, although it is not immediately verifiable:  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Strategic choice: Each node create links by making choices in order to satisfy some  objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  To address this, researchers have been trying to find the right game-theoretic model that  would be able to explain the astonishing properties of our social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' However, by consid-  ering a game-theoretic model where individuals make choices strategically based on the choices of  others, the limited information constraint is immediately violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' However, if a game-theoretic  model that reproduces the small-world properties is established based on the correct objective  function, then the small-world properties should persist even when the full information assump-  tion is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Among various game-theoretic models of network formation, our result highlights the work  of Even-Dar and Kearns [1], where the population is distributed uniformly on the plane and  each agent v chooses its links in order to minimize the total cost of maintaining the links plus  the average distance to other nodes through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' In their model, a link to a node of  geographical distance l induces a cost of lα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Particularly, they show that in this model when  α < 2, the diameter can be bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  When attempting to derandomize the work of Kleinberg [2] at the critical point where loga-  rithmic bounds appear on the expected degrees and greedy diameter, we come upon a strategic  model of network creation that achieves similar bounds and satisfies all of our constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' When  analyzing the objective function that gives this we see a natural interpretation that hints towards  the role of curiosity in emergence of small-world phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  In Section 2 we present the model in details and discuss the components of the cost function  that the agents are trying to minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' In Section 3 we analyze the model on a d-dimensional  grid†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='We focus on a notion of equilibrium where agents cannot further improve their cost by  adding or deleting a link, showing that α = d + 1 is a threshold where the upper bounds on the  performance of the greedy routing and maximum degree match to O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We also analyze  the maximum degree and the performance of greedy routing for other values of α, particularly  showing that when α < d + 1 the number of iterations that are required for greedy routing to  find its destination can be bounded by O(log d+1  α log(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We conclude in Section 4 by discussing  the implications of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  †Throughout the paper we assume that d is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  3  2  The Model  Consider a d-dimensional circular grid of n nodes representing the underlying structure upon  which the network is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For nodes u and v we use d(u, v) to refer to the distance‡ of u and  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Also, for a node u and a set of nodes V we use d(u, V ) to refer to the distance of u to its  closest member in V , that is, d(u, V ) = minv∈V d(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  A network N assigns to each node v a set N(v) of nodes that v has a links to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' To model  the creation of such a network we introduce a cost that each agent incurs at a given setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Let cN(v) be such a function that calculates the cost induced to a node v in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' There are two  components to this cost function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' the cost of maintaining the links that v has to other nodes and  the cost induced by the information that v is missing regarding the nodes that it does not have  a link to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For a node u, let lN(v, u) be the former and sN(v, u) be the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We have  cN(v) =  �  u∈N(v)  lN(v, u) +  �  u/∈N(v)  sN(v, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  For the cost that v incurs for maintaining a link to an agent at distance d(v, u), it is natural  to assume a dependence between this cost and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Let us define lN(v, u) = dα(v, u) for  some number α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' The exact choice of α is discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We will refer to this cost  as the link-cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  It remains to define sN(v, u), that is, the cost that v pays for what it does not know about  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We capture this by assuming that each node u induces a cost to v that is proportional to the  distance between u and the node closest to u that v has a link to, that is, d(u, N(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We refer  to this as the curiosity-cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  We are ready to express our cost function cN(v) such that it only depends on N(v) and the  position of the nodes on the underlying grid:  cN(v) = β  �  u∈N(v)  dα(v, u) +  �  u/∈N(v)  d(u, N(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  (1)  3  Analysis  In this section we analyze the network constructed by agents who seek to reduce their costs  by adding and deleting links to other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' In particular, we discuss the trade-off between  the greedy routing diameter of the network and the maximum degree among nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  When  α <  1  logβ−1 n every link has a cost smaller than 1 and therefore the diameter is 1 and all of the  degrees are equal to n − 1, and when α > log(β−1n2) no agent can justify a link to a node of  distance > 1, and therefore the degrees and diameter are 2d and dn1/d respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Here we  provide analysis for intermediary values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  To establish our bounds, we do not require the assumption that each agent chooses their links  to minimize their cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' In each of our cases, the assumption that no agent can reduce its cost by  adding a link (add-stability) is sufficient to prove the upper bound on the greedy diameter of the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' When in addition to add-stability we also assume that no agent can improve its cost  by removing a link (toggle-stability) we are able to also prove upper bounds on the maximum  degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Since each agent reduces its cost with each these operations of addition and deletion  of a link, therefore, performing these operations in any order will eventually result in stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  It is also worth noting that these assumptions are much more general than, for example, the  assumption of a Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  ‡In this paper, unless specified otherwise, by distance we are referring to the distance through the grid, that is,  grid-distance  4  v  l  l − l′  v  l  l − l′′  Figure 1: Items 3 (left) and 4 (right) of Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' In each case l′ is the diameter of the smaller  squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  We start with a set of observations on the properties of the grid that will be used throughout  this section in order to bound the greedy diameter and maximum degree among the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For  this, let us define S(v, l) to be the set of all nodes of distance exactly l from v, and B(v, l) be  the set of all nodes u with d(u, v) ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' The following observation lists some properties regarding  S(v, l) and B(v, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For any node v on the d-dimensional grid and l ∈ N the following holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' |S(v, l)| ∈ Θ(ld−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' |B(v, l)| ∈ Θ(ld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' The nodes in B(v, l)\\B(v, l − l′) can be covered with O(( l  l′ )  d−1) balls of the form B(u, l′)  with u ∈ B(v, l)\\B(v, l − l′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' The nodes in B(v, l)\\B(v, l − l′′) can be covered by O(( l  l′ )  d) balls of the form B(u, l′) with  u ∈ B(v, l)\\B(v, l − l′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='1  Balance: α = d + 1  Here we focus on the case when α = d+1 and show that both maximum degree and the iterations  required for greedy routing to find its destination are bounded by O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Let us start with a useful lemma that provides a lower bound on the drop in the curiosity-cost  when adding a link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If a node v has no link in B(u, l), then adding a link to u will decrease the total  curiosity-cost of v by Ω(ld+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Consider the set of nodes S with distance l′ from u where l′ ≤ 1  3l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Note that any node  in S induces a curiosity-cost of at least 2  3l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If v adds a link to u, then the curiosity-cost of each  node in S will be reduced to at most 1  3l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Therefore, by Item 2 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='1 the total decrease  in curiosity-cost is at least c l  3  d × l  3 = c l  3  d+1 which concludes the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Consider an add-stable node v, that is, a node v that does not benefit from adding any more  links to any other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Next we provide an upper bound on the distance that a node u with  d(u, v) = l may have from the neighbors of v, that is, d(u, N(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' This is done by showing that  if the distance is too large then v would benefit from adding a link to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If a node v in N is add-stable, then for any u with d(v, u) ≤ l there exists a node  u′ of distance at most O(l  α  d+1) from u that v has a link to, that is, N(v) ∩ B(u, cl  α  d+1) ̸= ∅ for  some constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For the sake of contradiction, suppose we have a node u such that N(v)∩B(u, cl  α  d+1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='2 adding a link from v to u decreases the curiosity-cost by at least c′lα for some  constant c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' When β < c′, this implies that the reduction in the curiosity-cost is larger than the  cost of a link to u which is at most βlα, contradicting add-stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Now we can prove an upper bound on the number of iterations required for the greedy routing  algorithm to reach its destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If a network N is add-stable and α = d + 1, then any node u is reachable from  any node v via greedy routing in O(log(n)) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' When α = d + 1, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='3 we know that in each iteration of greedy routing the  distance to destination is reduced by a constant factor, resulting in logarithmic bound on the  number of iterations until the destination is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Next we turn our focus towards bounding the maximum degree when α = d + 1 by showing  that the number of links that v has to all the nodes of distance between l and cl for some constant  c, is bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' When α ≤ d + 1, if a node v in N is toggle-stable then v has at most a constant  number of links to any ball of radius l′ ≤ ǫl  α  d+1 ≤ ǫl centered at a node u with d(u, v) = l, that  is, |N(v) ∩ B(u, l′)| ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Any node within B(u, l′) has distance at most l + l′ from v, and since v in N is toggle-  stable, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='3 we can argue that any node within the ball can serve only nodes of  distance at most c′(l + l′)  α  d+1 for some constant c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Therefore any node that is served by a node  in B(u, l′) must also lie within the ball B(u, l′ + c′(l + l′)  α  d+1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Since l′ ≤ ǫl  α  d+1 ≤ ǫl, we have  l′ + c′(l + l′)  α  d+1 ≤ c′′l  α  d+1 for some constant c′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Let C(u′) be the set of the nodes whose closest  node in N(v) is u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' By Item 2 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='1 we can argue that  �  u′∈N(v)∩B(u,l′)  |C(u′)| ≤ c∗(l  α  d+1 )d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  for some constant c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  By removing a link to a node u′ ∈ B(u, l′), the curiosity-cost induced to v is increased by  at most c′′l  α  d+1|C(u′)|, and therefore there is a node u′ ∈ N(v) ∩ B(u, l′) the removal of which  decreases the total cost induced to v by at least  β(l − l′)α−c′′l  α  d+1|C(u′)| ≥ β(l − l′)α−c′′l  α  d+1 ×  c∗l  α  d+1)d  |N(v) ∩ B(u, l′)| ≥ β(1−ǫ)αlα−  c⋆lα  |N(v) ∩ B(u, l′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  6  Therefore if |N(v) ∩ B(u, l′)| >  c⋆  β(1−ǫ)α = c, v would be able to decrease its total cost by  removing u′, contradicting stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If a node v in N is toggle-stable and α = d + 1, then the degree of v is bounded  by O(log n) links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For any integer 0 ≤ i ≤ log1+ǫ(n) let P(i) be the set of grid points that are of distance  l from v such that (1 + ǫ)i < l ≤ (1 + ǫ)i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='5 for any p ∈ P(i) we know that the  number of links that v has to the members of the ball B(p, ǫ(1 + ǫ)i) is bounded by constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Also, by Item 3 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='1 the nodes in P(i) can be covered using c(1+ǫ)i+1  ǫ(1+ǫ)i  d−1  such balls for  some constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Therefore, for each 0 ≤ i ≤ log1+ǫ(n) we can bound the size of N(v) ∩ P(i) by  a constant, concluding the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='2  Sub-logarithmic diameter: α < d + 1  In this section we consider cases when  1  logβ−1 n < α < d + 1 and prove upper bounds on the  number of iterations that greedy routing requires and maximum degree for this range of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If a network N is add-stable and α < d + 1, then any node u is reachable from  any node v via greedy routing in O(log d+1  α log(n)) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='3 if in an iteration of greedy routing the distance to destination is l, then  in next iteration the distance is at most cl  α  d+1 for some constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Therefore, after  log  α  d+1 (  1  log(n)) = log d+1  α log(n)  iterations the distance to destination will be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If a node v in N is toggle-stable and α < d + 1, then the degree of v is bounded  by o(log log(n)n1−δ  α  d+1) links for any δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For some positive constant 0 < a < 1 and some integer 0 ≤ i ≤ loga−1 log(n) let P(i) be  the set of grid points that are of distance l from v such that nai+1/d < l ≤ nai/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' By Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='5 for any p ∈ P(i) we know that the number of links that v has to the members of the ball  B(p, ǫ(nai+1/d)  α  d+1 ) is bounded by constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  By Item 4 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='1 P(i) can be covered with O(nai−ai+1  α  d+1) balls B(p, ǫ(nai+1/d)  α  d+1)  where p ∈ P(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Therefore the total number of links of v can be bounded by  |N(v)| ≤ c +  i≤loga−1 log(n)  �  i=0  nai−ai+1  α  d+1 ≤ c + loga−1 log(n)n1−a  α  d+1  for some constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='3  Sub-logarithmic degrees: α > d + 1  Now let us turn our focus to cases when d + 1 < α < log(β−1n2) and prove bounds analogous to  those of previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Here we need to modify some of our lemmas, as in this case the cost  of links increases relative to length so fast that it no longer benefits v to establish a link to the  middle of a region it wishes to cover with that link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  7  v  u  u′  u′′  l − ǫl  1  α−d  l − 2ǫl  1  α−d  ǫl  1  α−d  ǫl  1  α−d  4  Figure 2: For Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Adding a link from v to u′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' u′′) decreases the curiosity-cost of each  of the nodes in the area indicated by red (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' blue) hashing by at least ǫl  1  α−d  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If a node v in N is add-stable, then for any u with d(v, u) ≤ l there exists a node  u′ ∈ N(v) with d(u, u′) ≤ l − ǫl  1  α−d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Sketch of proof: For the sake of contradiction assume that v has no link to any node in B(u, l −  ǫl  1  α−d ) and let u′ be arg min  u′∈B(u,l−2ǫl  1  α−d ) d(u′, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Then, the node v benefits from adding a link  to u′, as u′ will be able to reduce the curiosity of Ω((l  1  α−d )d−1l) nodes by at least Ω(l  1  α−d ) (See  Figure 2), and therefore reduce its total curiosity-cost by Ω(l1+  d  α−d ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We can argue that  ǫ′l1+  d  α−d = ǫ′l  α  α−d > β(2ǫl  1  α−d )α  for proper value of ǫ and β, and therefore it is beneficial for v to add a link to u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If a network N is add-stable and α > d + 1, then any node u is reachable from  any node v via greedy routing in o(log log(n)n1−  δ  α−d ) steps where δ is any constant smaller than  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' By induction we can argue that starting from a node v, the worst case of our analysis of  greedy routing is realized when the destination is the furthest node from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='9 in  each step if the distance to the furthest node is l, in the next step this distance will be at most  l−ǫl  1  α−d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Let us consider the number of iterations it takes to get from distance nai of destination  to get to distance lai+1 for some a < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' This is bounded by at most  lai  ǫl  ai+1  α−d  = ǫ−1lai− ai+1  α−d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  8  Therefore, the total number of iterations is bounded by  i≤log log n  �  i=0  ǫ−1lai− ai+1  α−d ≤  i≤log log n  �  i=0  ǫ−1l1−  a  α−d ≤ log log(n)ǫ−1l1=  a  α−d  Here we show that for d = 1, when α > d + 1 the degrees are restricted by a sub-logarithmic  upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Equivalent proof for higher dimensions remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' If α > d + 1 = 2 and a network N is toggle-stable, then any node v has at most  O(log  α  d+1 log(n)) links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Think of each node of distance l from v to initially have a budget of l of curiosity-cost  that can be spent to justify maintenance of a link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We will show that if v were to have more  than c log  α  d+1 log(n) links, then there would be a set of nodes that would have insufficient budget  in order to justify a set of links that serve a subset of them, and therefore, N would not be  toggle-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Consider a node v and let P(i) be the set of nodes with distance l′ from v such that n  ai+1  d  <  l′ ≤ n  ai  d where a = d+1  α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' We show that the size of P(i) ∩ N(v) is bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For  this, note that any link that v has in P(i) costs at least βn  αai+1  d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Noting that d = 1, let S be the set P(i) ∩ N(v) minus the elements that are furthest from v  on each side, and hence |S| ≥ |P(i)∩N(v)|−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' No node in S can serve a node of distance larger  than n  ai  d from v, and by Item 2 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='1 there are at most n  ai  d  d  nodes that can contribute  to justify S, and each such a node has a budget of at most n  ai  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Therefore, we have  |S| ≤ n  ai  d  d+1  βn  αai+1  d  = n  (d+1)i+1  dαi  βn  (d+1)i+1  dαi  = β−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  We conjecture that the bound from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='11 can be extended to d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  4  Discussion  Our main contribution is to provide a simple explanation for the small-world phenomenon that  matches previously established theoretical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  While the questions that we are concerned with here are inherently inspired by nature, this  work was initially carried out as a purely mathematical mission to reestablish the logarithmic  bounds on maximum degree and greedy diameter in a setting where individuals have only local  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' However, the model that came out of this research seems to be providing insight  into drives that connect us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Imagine a population distributed over a plane, where each individual maintains the status  (say current address) of some of the others in order to gain some knowledge through them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' There  is a real effort here, which is mainly the effort and attention that is required to maintain the  status of another, and there is an implicit drive that can be described as a desire to know all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Our results suggest that such a population ends up forming a small-world network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  It is imaginable that α (the exponent of the distance in cost of maintaining a link) has been  reduced by the progresses made through history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Particularly with writing and a postal service,  9  the cost of maintaining the status of another seems to increase relative to distance with a small  exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' It would be interesting to gain an understanding of the real value of α through analysis  of empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  As for applicability our model may be useful to engineers whenever robust networks with  nodes of bounded degree are required, particularly when nodes are to make decisions indepen-  dently regarding their connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  It must be noted that the our cost function can be modified in a number of ways while still  yielding networks with small-world properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' For instance, if our model is used to connect a  swarm of drones, if each drone v chooses the set N(v) of those whose status it watches for by  minimizing:  cN(v) = β  �  u∈N(v)  dα(v, u) +  �  u/∈N(v)  wv(u)d(u, N(v))  (2)  where wv(u) represents the relative importance of u for v, then some degree of small-world  properties are established as a side effect depending on parameters α and β, and the correlation  between relative importance and proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  At this point, many directions remain unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Particularly, it is curious as to what  happens to the threshold for other values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' When we set β = n, our cost function closely  resembles the objective function upon which the model of Even-Dar and Kearns [1] is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  Although their model is inherently different, a similar threshold with features of its own appears  there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' All of this hints towards a bigger picture, perhaps a more general framework that is yet  to be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content='  10  References  [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Even-Dar and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' Kearns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' A small world threshold for economic network formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
+page_content=' In  Advances in Neural Information Processing Systems 19, pages 385–392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
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+page_content=' Kleinberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf'}
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+Cross-domain recommendation via user interest alignment
+Chuang Zhao
+College of Management and
+Economics, Tianjin University
+Tianjin, China
+[email protected]
+Hongke Zhao
+College of Management and
+Economics, Tianjin University
+Tianjin, China
+[email protected]
+Ming HE
+AI Lab at Lenovo Research
+Beijin, China
+[email protected]
+Jian Zhang
+School of Cyberspace Security,
+Hangzhou Dianzi University
+Hangzhou, China
+[email protected]
+Jianping Fan
+AI Lab at Lenovo Research
+Beijin, China
+[email protected]
+ABSTRACT
+Cross-domain recommendation aims to leverage knowledge from
+multiple domains to alleviate the data sparsity and cold-start prob-
+lems in traditional recommender systems. One popular paradigm
+is to employ overlapping user representations to establish domain
+connections, thereby improving recommendation performance in
+all scenarios. Nevertheless, the general practice of this approach is
+to train user embeddings in each domain separately and then aggre-
+gate them in a plain manner, often ignoring potential cross-domain
+similarities between users and items. Furthermore, considering
+that their training objective is recommendation task-oriented with-
+out specific regularizations, the optimized embeddings disregard
+the interest alignment among user’s views, and even violate the
+user’s original interest distribution. To address these challenges,
+we propose a novel cross-domain recommendation framework,
+namely COAST, to improve recommendation performance on dual
+domains by perceiving the cross-domain similarity between entities
+and aligning user interests. Specifically, we first construct a uni-
+fied cross-domain heterogeneous graph and redefine the message
+passing mechanism of graph convolutional networks to capture
+high-order similarity of users and items across domains. Targeted
+at user interest alignment, we develop deep insights from two more
+fine-grained perspectives of user-user and user-item interest in-
+variance across domains by virtue of affluent unsupervised and
+semantic signals. We conduct intensive experiments on multiple
+tasks, constructed from two large recommendation data sets. Exten-
+sive results show COAST consistently and significantly outperforms
+state-of-the-art cross-domain recommendation algorithms as well
+as classic single-domain recommendation methods.
+KEYWORDS
+Cross-domain similarity, Interest alignment, Recommender system
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from [email protected].
+Conference’17, July 2017, Washington, DC, USA
+© 2018 Association for Computing Machinery.
+ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00
+https://doi.org/XXXXXXX.XXXXXXX
+Knowledge Transfer
+Rich Domain
+Sparse Domain
+Watch
+Read
+User Interest
+Source
+Target
+Improve
+（a）
+（b）
+Figure 1: (a) illustrates that the behavior of an overlapping
+user in different domains is driven by the same distribution
+of interests. (b) illustrates that we improve the recommen-
+dation performance of the two domains through knowledge
+transfer of overlapping users.
+1
+INTRODUCTION
+In an effort to alleviate information overload [6, 27], various well-
+known platforms such as Netflix [11] and Amazon [23] deploy
+recommender systems to capture users’ personalized preferences.
+Despite their excellent performance, data sparsity and cold-start
+problems, as two serious challenges, pose obstacles to model user
+interests accurately and efficiently [36].
+To address these headaches, researchers put their insights into
+cross-domain recommender systems (CDR), i.e., transfer knowledge
+from informative recommendation scenarios (source domain) to
+scenarios with sparse interactions (target domain) via transfer learn-
+ing techniques [34]. This directed transfer essentially enhances the
+knowledge of the target domain and achieves promising results
+on multiple recommendation data sets [46]. Further, several re-
+searchers engage in bidirectional cross-domain recommendation,
+arguing that reasonable model structures can facilitate the mutual
+transfer of source and target domain knowledge [22]. For instance,
+user Jack searches and browses a large number of computer cost-
+effective related posts in the online community (source domain),
+and we can simultaneously recommend various types of comput-
+ers to him in the online mall (target domain), and vice versa. This
+dual recommendation paradigm can not only alleviate the negative
+arXiv:2301.11467v1  [cs.IR]  26 Jan 2023
+
+##Stephen Knight
+Form&IdeologyANNASMITH
+Jhe Brile
+groomDLIVER STONE ANG PETER KUZNICK
+UNTOLI
+HISTORY
+WITESTHE HISTORY OF
+LOVEPanipatWUUTHARREISON
+MCCOIMUGHEY
+BI
+DETECTITHEGIRLHE
+DRAGONTATTOOConference’17, July 2017, Washington, DC, USA
+Trovato and Tobin, et al.
+transfer phenomenon, but also promote the upper bound on the
+target domain by improving the recommendation ability of the
+model in the source domain [43].
+To our best knowledge, the mainstream taxonomy of dual cross-
+domain recommendation can be separated into collective matrix fac-
+torization, mapping-based methods, graph neural network-based ap-
+proaches, and representation combination of overlapping entities [34].
+This paper strives to kick the last paradigm upstairs, the general
+practice of which is to train user and item representations separately
+in the two domains, and then perform specific aggregations (concat,
+dot, pooling) on them for knowledge transfer [41]. Even with the re-
+markable results [37], they still encounter three serious challenges.
+Firstly, vast majority of these studies conduct experiments on ex-
+plicit data sets with fully overlapping users, which significantly
+pole apart from rich implicit content and partial user overlap in
+real-world scenarios [26]. Secondly, the general practice of inde-
+pendently training entity representations in each domain struc-
+turally isolates the interactions among users-items, thereby failing
+to perceive higher-order similarities between entities. Thirdly, con-
+sidering the recommendation task-oriented optimization objective,
+these work cannot guarantee the alignment of overlapping users’
+interests across domains [1]. In other words, we argue that the plain
+aggregations of entity representations across domains without any
+regularizations are incapable of distinguishing users’ personal pref-
+erences at the instance level, nor can it ensure that users’ interests
+in items are consistent, or even cause conflicting interests among
+users’ views.
+With the aim of addressing these challenges, we propose a Cross-
+domain recOmmendation viA uSer inTerest alignment, i.e. COAST,
+which endeavors to improve cross-domain recommendation with
+partial user overlap, as shown in Figure 1(b). Unlike previous stud-
+ies, we extract enough features from the affluent content data
+(comments, tags, user/item profiles) to form an implicit data set
+to capture more feedback. Meanwhile, we modernize the previ-
+ous approach of separately training representations into a unified
+cross-domain heterogeneous graph to assimilate the cross-domain
+similarity of users and items. Targeted at overlapping users’ inter-
+est alignment across multiple domains, we gain in-depth insights
+from both user-user and user-item perspectives. Specifically, for
+user-user interest alignment, we believe that users’ behaviors in
+different domains are driven by the same interest distribution, thus
+encouraging all views of the user to possess similar interest distribu-
+tions over K interest representations, as shown in Figure 1(a). This
+not only allows the model to distinguish users at the instance level,
+but also mitigates conflicting interests in views of the same user.
+For user-item interests alignment, we contend that interacted items
+are a observation of user interests, and all user views should exhibit
+consistent preferences for them. Particularly, benefiting from the
+rich semantics of gradients [10], we employ gradient alignment
+to encourage higher-order projections across views to follow the
+same optimization path.
+In this paper, we make the following contributions:
+• To the best of our knowledge, we make significant efforts in
+cross-domain recommendation by considering cross-domain
+similarity and user interest alignment. Our framework performs
+dual knowledge transfer on basis of partial user overlap to im-
+prove recommendation performance.
+• Instead of training entity representations separately, we con-
+struct a unified cross-domain heterogeneous graph, and cor-
+respondingly develop a novel message passing mechanism to
+capture the cross-domain similarity between entities.
+• We resort to contrastive learning and gradient alignment to con-
+strain user-user and user-item interest alignment, respectively,
+thereby enhancing the interest consistency across views.
+• We compare COAST to state-of-the-art algorithms for real-world
+recommendations, achieving significant improvements on all
+tasks. We promise the code and data sets will be released for
+further comparison after acceptance 1.
+The rest of this paper is organized as follows. Section 2 briefly
+introduces related work, and then introduces the details of our
+proposed model. The experimental results and analysis are given
+in Section 4. Finally, we summarize the paper in the fifth section.
+2
+RELATED WORK
+Our proposed framework stems from two research areas: cross-
+domain recommendation [34] and contrastive learning [17]. We
+respectively summarize their main research paradigms, pros and
+cons, and close links with our research.
+2.1
+Cross-domain Recommendation
+Cross-domain recommendation strives to explore data from multi-
+ple domains to simultaneously improve the recommendation per-
+formance of the model in all scenarios [18].
+A rudimentary idea is to incorporate several constraints of cross-
+domain knowledge to decompose the user-item interaction matri-
+ces in both domains simultaneously [14, 28, 39]. This genre can be
+extended on a large number of matrix factorization-based single-
+domain recommendations [31], whereas its performance is inferior
+to deep learning approaches. Another paradigm is to customize a
+mapping function whose optimization objective is that the trans-
+formed cold-start user representation generalizes well in the target
+domain [29, 45]. The effiency of this paradigm depends on the rea-
+sonableness and representational power of the mapping function
+and whether enough overlapping entities are available for training,
+which limits the generalizability of the model. The third paradigm
+resorts to the popular knowledge graph technology [35], which
+builds shared graphs to represent the relationships among users,
+items, and attributes, and learns entity representations through
+graph embeddings [4, 20]. Despite the excellent extraction capabil-
+ity of graph structure, the high demands of computational resources
+make the scalability of these methods potentially limited. Recently,
+algorithms utilizing overlapping user representations and combi-
+nations is trendy, and their standard practice is to learn entity
+representations from various domains, and then combine overlap-
+ping entity representations to enrich the knowledge of each do-
+main [9, 44]. Apparently, the lack of cross-domain similarity and the
+rough combination way limit their recommendation performance.
+Our approach falls within the last paradigm, but strives to con-
+quer the proposed drawbacks. The closest algorithm to ours in this
+paradigm is GADTCDR [42], but they are fundamentally different.
+1https://github/anonymous/COAST
+
+Cross-domain recommendation via user interest alignment
+Conference’17, July 2017, Washington, DC, USA
+First, at the data level, apart from explicit interactions, we attach
+exploration of content information. Second, at the algorithm level,
+we construct a unified cross-domain heterogeneous graph and user
+interest alignment for training, which enhances the generalization
+of the model. Finally, at the optimization level, we optimize in an
+end-to-end manner, avoiding the potential target inconsistency
+brought by two-stage training.
+2.2
+Contrastive Learning
+Contrastive learning emphasizes learning common features be-
+tween different views of an instance, with the intention of instance-
+level discrimination [24]. In contrast to supervised learning, it learns
+in a self-supervised manner.
+Early contrastive learning architectures favored large batch sizes
+to aggregate enough negative examples, but the scalability of such
+methods was limited by GPU memory [3]. Aiming to improve on
+this, Wu et al. [33] applied a memory bank to store a large number
+of sample representations as negative examples, thus avoiding the
+common out-of-memory. Despite the approximate performance, a
+potential pitfall of this approach is that representation updates in
+the memory bank can be computationally expensive as it becomes
+outdated quickly within a few iterations. Consequently, He et al.
+[12] further improved the form of the static repository, using a mo-
+mentum encoder to generate a dictionary as a queue for encoding
+keys, the current mini-batch is enqueued, and the oldest mini-batch
+is dequeued. This approach eliminates the need to use two separate
+models for feature extraction, and dynamic queues avoid exces-
+sive memory consumption. All of the above architectures place
+insight into using specific metrics to measure sample similarity,
+i.e., encouraging different views of the same entity to be closer
+in the projected space and vice versa [32]. Recently, Caron et al.
+[2] abandoned the traditional comparison of positive and negative
+examples, and launched a new exploration of contrastive learning
+from the perspective of clustering.
+Inspired by contrastive learning, we intend to discriminate user
+representations at the instance level. Particularly, following the
+idea of clustering, we encourage different views of the same user to
+aggregate into the same interest center, thereby generating better
+user interest representations.
+3
+PROPOSED METHOD
+In this section, we introduce the proposed COAST framework.
+Specifically, we first elaborate the definition of the general CDR
+problem, then outline our framework, and finally detail the sub-
+modules and optimization methods.
+3.1
+Problem Formulation
+This work considers a general CDR scenario with two domains S
+(source) and T (target), where the former contains rich and informa-
+tive interactions and the latter is relatively sparse. Suppose source
+domain DS = (US, VS, ES, XS), target domain DT = (UT, VT,
+ET, XT), where U, V, E, X are user set, item set and edge set, at-
+tribute set in each domain, respectively. In particular, the user sets
+US and UT contain an overlapping user subset U𝑜. Then, the
+user set can be redefined as US = {U𝑠, U𝑜}, UT = {U𝑡, U𝑜},
+where U𝑠 and U𝑡 are non-overlapping/distinct user sets in the
+two domains. For simplicity of exposition, we further introduce
+two binary matrices to store user-item interactions, namely AS =
+{0, 1}|US |×|VS |, AT = {0, 1}|UT |×|VT |, where element 𝐴𝑖𝑗 in each
+domain denotes whether the user 𝑢𝑖 ∈ U and item 𝑣𝑗 ∈ V have an
+interaction in the edge set E. The definition of dual cross-domain
+recommendation is as follows,
+Given the observed interaction and content of S and T, dual CDR
+aims to leverage knowledge transfer from overlapping users to im-
+prove recommendation performance in both domains. Formally, given
+AS, AT, XS, XT , we expect to recommend 𝑣𝑖 ∈ VS, 𝑣𝑗 ∈ VT
+respectively in domains S and T.
+Important mathematical notes can be found in Appendix A.
+3.2
+Overview of COAST Framework
+Distinct-I
+Overlap-U
+Overlap-U
+Distinct-U
+Distinct-U
+Distinct-I
+𝐹𝒮
+𝐹𝒯
+𝑧𝒮
+𝑧𝒯
+𝑄𝒮
+𝑄𝒯
+U
+U
+I
+I
+MLP
+MLP
+MLP
+MLP
+MLP
+MLP
+MLP
+MLP
+ො𝑦𝑠
+ො𝑦𝑟
+gradient
+gradient
+Target
+Source
+overlap
+swap
+K
+Figure 2: Overall framework of COAST.
+In this section, we outline the proposed cross-domain recommen-
+dation framework COAST, whose architecture is shown in Figure 2.
+First, we construct a unified cross-domain heterogeneous graph,
+and improve the message passing mechanism of graph convolu-
+tional network to capture the cross-domain similarity of users and
+items. Then, for each overlap user, we utilize contrastive learning
+and gradient alignment from both user-user and user-item perspec-
+tives to ensure the alignment of user interests. Finally, following
+previous studies, we adopt a negative sampling mechanism to cal-
+culate the supervision loss of the two domains, which is jointly
+optimized with the above two losses for alignment.
+3.3
+Cross-domain Graph Convolution
+We argue that previous separately trained representations can only
+capture single-domain information; therefore we construct a unified
+cross-domain heterogeneous graph and a novel message passing
+mechanism to capture cross-domain similarity.
+3.3.1
+Construction. We determine nodes and edges in the hetero-
+geneous graph G on basis of AS and AT. Note that for items from
+
+Conference’17, July 2017, Washington, DC, USA
+Trovato and Tobin, et al.
+both domains, we treat them as nodes of the same type, the differ-
+ence being the type of edges users interact with them. For the initial
+embeddings of nodes, we generate them in the following data pre-
+processing manner. Specifically, for common numerical attributes
+and category attributes, we perform normalization and one-hot
+encoding respectively. For text attributes (tags, comments, profiles,
+etc.), we first aggregate the text associated with the entity into
+a large document, which is then converted into semantic vectors
+using doc2vec technique [5]. Note that we perform joint encoding
+on users of both domains. Finally, we get the initial embedding for
+each user and item, i.e., 𝑒𝑢 ∈ HU, 𝑒𝑣
+S ∈ HS, 𝑒𝑣
+T ∈ HT. Formally,
+𝑒𝑢 =
+
+
+𝑒𝑢
+S,
+𝑖𝑓 𝑢 ∈ U𝑠
+𝑒𝑢
+T,
+𝑖𝑓 𝑢 ∈ U𝑡
+𝑒𝑢
+S ⊗ 𝑒𝑢
+T,
+𝑖𝑓 𝑢 ∈ U𝑜
+,
+(1)
+where ⊗ is max pooling operation. Overlapping users have behav-
+iors in both domains, so we aggregate their representations in both
+domains. Without loss of generality, we adopt max pooling here.
+We experimented with operations such as sum and averaging, and
+found no significant improvement.
+3.3.2
+Propagation. To capture the high-order cross-domain sim-
+ilarity of users and items, we improve upon the message passing
+mechanism of graph convolution networks [30]. Formally,
+𝑚𝑢←𝑣 =
+1
+√︃
+|N𝑢||NS𝑣 ||NT𝑣 |
+(𝑊1𝑒𝑢 +𝑊2(𝑒𝑣
+S ⊙𝑒𝑢)+𝑊3(𝑒𝑣
+T ⊙𝑒𝑢)), (2)
+where N represents set of 1-hop neighbors,𝑊 is a trainable parame-
+ter, and ⊙ denotes the element-wise product. We add cross-domain
+user-item interactions to the message passing mechanism of graph
+convolution operation, expecting to capture historical interaction
+information. This approach not only enriches the embedding rep-
+resentation, but also enhances the capture of cross-domain collabo-
+rative signals. Formally, the user embedding propagation is,
+𝑒𝑢 (𝑙+1) = LeakyReLU(𝑚(𝑙)
+𝑢←𝑢 +
+∑︁
+𝑣∈N𝑢
+𝑚(𝑙)
+𝑢←𝑣),
+(3)
+where 𝑙 represents the 𝑙-th GNN layer. We also support stacking of
+GNN layers to perceive higher-order similarities. Formally,
+𝐸(𝑙) = 𝜎((𝐿 + 𝐼)𝐸(𝑙−1)𝑊 (𝑙)
+1
++ 𝐿𝐸(𝑙−1) ⊙ 𝐸(𝑙−1)𝑊 (𝑙)
+2
++ 𝐿𝐸(𝑙−1) ⊙ 𝐸(𝑙−1)𝑊 (𝑙)
+3
+),
+(4)
+where 𝜎 is activation function 𝑅𝑒𝑙𝑢, 𝐸 is the representations for
+users and items, 𝐼 denotes an identity matrix. 𝐿 represents the
+Laplacian matrix for the graph. Formally,
+𝐿 = D− 1
+2 𝐴D− 1
+2 and 𝐴 =
+�
+0
+𝑅
+𝑅⊤
+0
+�
+,
+(5)
+where D is the diagonal degree matrix, 𝐴 is the adjacency matrix,
+𝑅 is the user-item interaction matrix and 0 is all zero matrix. We
+concat the user and item representations of each layer, i.e., 𝑒𝑢 =
+𝑒𝑢 (0) ⊕ · · · ⊕ 𝑒𝑢 (𝑙),𝑒𝑣
+S = 𝑒𝑣(0)
+S
+⊕ · · · ⊕ 𝑒𝑣(𝑙)
+S ,𝑒𝑣
+T = 𝑒𝑣(0)
+T
+⊕ · · · ⊕ 𝑒𝑣(𝑙)
+T .
+Our approach has several advantages. On the one hand, we
+form a unified graph structure for user-item interactions in differ-
+ent domains, which is intuitive and easy to capture cross-domain
+similarity. On the other hand, we generalize the message passing
+mechanism to cross-domain scenarios, enhancing the practicality
+of traditional graph convolution operators.
+3.4
+User Interest Alignment
+Previous studies applied plain representation aggregation to trans-
+fer knowledge of both domains; however we argue that this ap-
+proach ignores the alignment of user interests. Consequently, we
+align user interests from user-user and user-item perspectives to
+constrain user representation.
+3.4.1
+User-User Alignment.
+To discriminate users at the instance
+Neighbors-U
+View1
+U-feature
+View2
+U-feature
+Neighbors-U
+View1
+View2
+item
+user
+feature
+rich
+sparse
+overlap
+Figure 3: User-User interest alignment.
+level, we separately aggregate users’ second-order neighbors’ repre-
+sentation in different domains to obtain corresponding contrastive
+views, as shown in Figure 3. The motivation behind it is that the
+user’s context can enhance the user’s interest representation in this
+domain, which is widely used in single-domain graph recommen-
+dation [25]. Formally, for 𝑢 ∈ U𝑜
+𝑒𝑢
+S =
+∑︁
+𝑖 ∈NS
+𝑢,2
+𝛼𝑖𝑒𝑢𝑖,
+𝑒𝑢
+T =
+∑︁
+𝑖 ∈NT
+𝑢,2
+𝛼𝑖𝑒𝑢𝑖,
+(6)
+where N𝑢,2 is the 2-hop neighbors of 𝑢 and 𝛼𝑖 =
+exp(𝑠 (𝑢𝑖,𝑢))
+�
+𝑗∈N𝑢,2 exp(𝑠 (𝑢𝑗,𝑢)) .
+𝑠(·) represents the scoring function, and without loss of generality,
+we use the dot product.
+Then we feed 𝑒𝑢
+S and 𝑒𝑢
+T into the feature extractors 𝐹S and
+𝐹T respectively, and get their representations 𝑧S, 𝑧T. We assume
+that overlapping users have a total of K interests, i.e., {𝑐1, · · · ,𝑐𝐾 }.
+According to our assumption, the distribution of interests among
+different views of the same user should be consistent. Formally,
+ℓ(𝑧T,𝑞S) = −
+∑︁
+𝑘
+𝑞(𝑘)
+S log𝑝 (𝑘)
+T
+𝑝 (𝑘)
+T
+=
+exp( 1
+𝜏 𝑧⊤
+T𝑐𝑘)
+�
+𝑘′ exp( 1
+𝜏 𝑧⊤
+T𝑐𝑘′)
+,
+(7)
+where 𝑞 is the higher-order projection through the Q extractor and
+𝜏 is a temperature parameter. In other words, we encourage the
+contrastive views of 𝑢 to posses the same clustering results over
+interest distribution. The user-user alignment loss is as follows,
+LU,U = ℓ(𝑧T,𝑞S) + ℓ(𝑧S,𝑞T),
+(8)
+Moreover, we follow the same solution in swav [2], which restricts
+the transportation of tensors in the mini-batch to ensure that the
+model is memory efficient.
+
+Cross-domain recommendation via user interest alignment
+Conference’17, July 2017, Washington, DC, USA
+3.4.2
+User-Item Alignment. To ensure consistent user interest in
+items, we encourage different views of 𝑢 to be closer to the inter-
+acted item representation, as shown in Figure 4. A straightforward
+motivation of this insight is that both user views and interacted
+items can represent the user’s real interests; therefore they should
+be close in the projected space, even if the views and items are in
+different domains. Consequently, benefit from the rich semantics of
+gradients [10], we introduce gradient alignment to induce different
+views to follow the same optimization path for interacted items.
+Formally, we define 𝑔S and 𝑔T to represent the expected gradients
+on the user’s source and target views.
+𝑔S =
+E
+(𝑢,𝑣)∼(U𝑜,VS)[∇𝜃𝑓 𝑢
+𝑠 ℓ𝑐𝑒 (𝐹𝑢
+𝑠 (𝑒𝑢) · (𝐹 𝑣
+𝑠 (𝑒𝑣))⊤,𝑦𝑢,𝑣)],
+(9)
+where 𝐹𝑢𝑠 , 𝐹 𝑣𝑠 are tower structures for extracting the representations
+of users and items in the source domain, both composed of Multi-
+Layer Perceptrons (MLPs).
+𝑔T =
+E
+(𝑢,𝑣)∼(U𝑜,VT)[∇𝜃𝑓 𝑢
+𝑡 ℓ𝑐𝑒 (𝐹𝑢
+𝑡 (𝑒𝑢) · (𝐹 𝑣
+𝑡 (𝑒𝑣))⊤,𝑦𝑢,𝑣)],
+(10)
+We aim to minimize discrepancy between 𝑔S and 𝑔T. Without loss
+of generality, we use cosine similarity as the discrepancy measure.
+L𝑈,𝐼 = 1 −
+𝑔⊤
+S · 𝑔T
+∥𝑔S∥2∥𝑔T ∥2
+,
+(11)
+where || · ||2 represents the 2-Norm.
+Overlap-U
+Overlap-U
+I
+I
+ො𝑦𝑟
+ො𝑦𝑟
+gradient
+𝑦𝑟
+A
+B
+C
+Figure 4: User-Item interest alignment.
+Overall, we constrain user representations from a more fine-
+grained perspective, i.e., user interest alignment. On the one hand,
+this approach acts as a regularizer to prevent overfitting of user
+representations. On the other hand, contrastive learning utilizes
+unsupervised information and gradient alignment utilizes semantic
+information, both of which further enrich the transfer of cross-
+domain knowledge.
+3.5
+Model Optimization
+In this section, we first elaborate the supervised prediction of
+COAST, and then illustrate the joint optimization process.
+3.5.1
+Supervised Estimation. Similar to the previous work [21], we
+adopt a dual-tower structure to capture high-order representations
+of users and items, where the tower structure is composed of MLPs.
+The structure of MLPs uses [𝐷, 2𝐷, 4𝐷, 8𝐷, 4𝐷, 2𝐷, 𝐷], which has
+been shown to be effective in feature extraction [42].
+ˆ𝑦𝑠 =
+𝐹𝑢𝑠 (𝑒𝑢
+S) · (𝐹 𝑣𝑠 (𝑒𝑣
+S))⊤
+||𝐹𝑢𝑠 ||||𝐹 𝑣𝑠 ||
++ 𝜆1(||𝑒𝑢|| + ||𝑒𝑣
+S||)
+ˆ𝑦𝑡 =
+𝐹𝑢
+𝑡 (𝑒𝑢
+T) · (𝐹 𝑣
+𝑡 (𝑒𝑣
+T))⊤
+||𝐹𝑢
+𝑡 ||||𝐹 𝑣
+𝑡 ||
++ 𝜆1(||𝑒𝑢|| + ||𝑒𝑣
+T||)
+,
+(12)
+where ||𝑒𝑢|| is the embedding regularizer. To avoid our model over-
+fitting 𝑌 + (ground truth), we randomly select a certain number of
+unobserved user-item interactions as negative instances, denoted
+𝑌 −, 𝑦 = {𝑌 +,𝑌 −}. This negative sampling-based training strategy
+has been widely used in existing algorithms [40]. Formally, we
+optimize using binary cross-entropy,
+ℓ(𝑦, ˆ𝑦) = 𝑦 log ˆ𝑦 + (1 − 𝑦) log(1 − ˆ𝑦),
+(13)
+The supervised loss is optimized in both domains simultaneously,
+L𝑠 = ℓ(𝑦𝑠, ˆ𝑦𝑠) + ℓ(𝑦𝑡, ˆ𝑦𝑡)
+(14)
+3.5.2
+Total Loss. Loss functions for each part are added together
+for joint optimization. The overall loss function is
+L = L𝑠 + 𝜆2(L𝑈,𝑈 + L𝑈,𝐼 ),
+(15)
+where 𝜆2 is the weight of the two interest alignment constraints.
+Overall, we propose an end-to-end solution for dual cross-domain
+recommendation, which can improve the recommendation perfor-
+mance of both domains while ensuring the alignment of overlapping
+user interests. The overall optimization process of the algorithm is
+shown in Algorithm 1 in Appendix B.
+4
+EXPERIMENTS
+To demonstrate the state-of-the-art and robustness of our model, we
+conduct extensive experiments to answer the following questions:
+• RQ1: How does COAST perform on common metrics compared
+to state-of-the-art algorithms?
+• RQ2: How do overlapping user ratios and sub-modules affect
+model performance?
+• RQ3: What impact do several key parameters have on model
+performance?
+4.1
+Experimental Settings
+In this section, we present the statistics of the data sets, necessary
+parameter settings for the model, and state-of-the-art algorithms
+for comparison.
+4.1.1
+Data Sets. We conduct extensive experiments using large-
+scale anonymized data sets obtained from Douban and a well-
+known industrial platform. They both allow users to rate and review
+a range of items from different domains, each of which represents
+the user’s interests. On that account, the combination of explicit
+user feedback and implicit domain knowledge is unique and valu-
+able for cross-domain recommendation.
+• Douban data set. We choose a subset containing the three
+largest domains, including books, movies, and music. They are
+linked together by a shared user ID that identifies the same user.
+Correspondingly, we construct three cross-domain recommen-
+dation tasks: movie-book, movie-music, and book-music.
+• Industrial data set. This platform has two scenarios, mall and
+community, which are connected by a shared user id. Conse-
+quently, we constructed a task mall-community, expecting to
+improve the recommendation performance in both domains.
+Statistics on the two data sets can be found in Table 1. For both
+data sets, the user’s content features are aggregated by user com-
+ments, user tags, and user profiles, and the item’s content features
+are composed of its profile and the comments below it. Note that
+
+Conference’17, July 2017, Washington, DC, USA
+Trovato and Tobin, et al.
+Table 1: Statistics of data sets.
+data sets
+Douban
+Industrial Platform
+Domains
+Movie
+Music
+Book
+Mall
+Community
+Users
+2,712
+1,672
+2,110
+35,233
+29,355
+Items
+34,893
+5,567
+6,777
+1,749
+2,452
+Interactions 1,278,401 69,709 96,041 319,795
+175,802
+Density
+1.35%
+0.75%
+0.67%
+0.52%
+0.24%
+Tasks
+Richer
+Sparser
+Overlap
+Douban
+Task1
+Movie
+Book
+2,106
+Task2
+Movie
+Music
+1,666
+Task3
+Book
+Music
+1,566
+Industrial Platform Task4
+Mall
+Community
+3,146
+each user may interact with items from different domains, but each
+item belongs to only one domain. To improve data quality, we filter
+all data sets to keep users and items with at least 5 interactions [42].
+We normalize the scoring range from 0 to 1.
+4.1.2
+Parameter Settings. Our framework is implemented using
+Pytorch. Except for the necessary concat operation, the embedding
+size is 64. We adopt Kaiming method [13] for parameter initial-
+ization. For gradient descent, we take Adam [19] with the initial
+learning rate 5e-4 for model optimization. In our proposed model,
+we set batch size to 4096 and the training maximum epoch to 100.
+We initialize the user’s interest K to 256, set the regularization
+weight 𝜆1 and alignment weight 𝜆2 to 1e-2 and 1e-3, respectively.
+Similar to previous work [7], we adopt a leave-one-out approach
+to evaluate model performance. Specifically, for each user in the
+test set, we randomly sample 99 items that the user has not inter-
+acted with as negative examples, and calculate the ground truth
+hit rate and ranking position. The results of model and baselines
+are evaluated by Hit Ratio (Hit) and Normalized Discounted Cu-
+mulative Gain (NDCG) values, where HR measures whether the
+test item is ranked on the Top-N list while NDCG measures the
+specific ranking quality that assigns high scores to hits at top posi-
+tion ranks [16]. Note that this paper is evaluated with @10 unless
+otherwise specified.
+4.1.3
+Baselines. To verify the effectiveness of cross-domain recom-
+mendation and the superiority of our model, we choose the classic
+single-domain recommendation algorithms and cross-domain rec-
+ommendation approaches for comparison.
+• NMF [16]: NMF aims to find a reasonable user-item interaction
+function for recommendation by combining the linearity of MF
+and the nonlinearity of MLP.
+• LightGCN [15]: LightGCN only obtains node embeddings by
+neighborhood aggregation because it believes that feature trans-
+formation and nonlinear activation have little effect on collabo-
+rative filtering, and even damage recommendation performance.
+• MVDNN [8]: MVDNN maps users and items from multiple do-
+mains into a common latent space, and optimizes by maximizing
+the similarity between users and their preferred items.
+• DTCDR [41]: DTCDR extends NMF to cross-domain recom-
+mendation, leveraging the textual and rating representations of
+overlapping users from both domains for knowledge transfer.
+• DDTCDR [21]: DDTCDR seeks to learn a latent orthogonal
+mapping function between domains to obtain cold-start user
+representations in other domains.
+• DML [22]: DML further extends DDTCTR based on dual metric
+learning, which exploits multiple orthogonal mapping functions
+to explore the transfer of cold-start user representations.
+• GADTCDR [42]: GADTCDR adds user-user and item-item
+edges to heterogeneous graphs based on content similarity to
+improve representation capabilities.
+• CDRIB [1]: CDRIB uses the information bottleneck principle
+to debias recommendations in two domains.
+Please note that NMF and LightGCN are single-domain recom-
+mendation algorithms, and experiments are performed on the two
+domains separately. The others are cross-domain recommenda-
+tion algorithms, where DDTCDR and DML are mapping-based
+methods, while MVDNN, DTCDR, GADTCDR, and CDRIB are
+representation-combination-based approaches. To be fair, we tune
+the hyper-parameters of each model to achieve the best results.
+4.2
+Comparison with Baselines (RQ1)
+The results of all algorithms on the four tasks are shown in Ta-
+ble 2, with the last row representing the improvement of our model
+over the best baseline for that task. To summarize, benefiting from
+perception of cross-domain similarity and user interest alignment,
+COAST achieved 0.32%-10.22% improvement compared to the best
+performance on different tasks.
+These experiments reflect some interesting findings: (1) Cross-
+domain algorithms outperform single-domain algorithms in most
+tasks, demonstrating the importance of knowledge transfer in cross-
+domain recommendation. Underperforming cross-domain baselines,
+especially those based on mapping genres, over-rely on overlapping
+user ratios such as DDTCDR, DML. (2) Algorithms incorporating
+implicit features outperform models using only explicit interac-
+tions, indicating the importance of capturing content similarity.
+(3) The representation-combination-based models outperform the
+mapping-based approaches, proving that a custom simple map-
+ping function cannot reflect the complex transformation of user
+representations across domains. (3) The improvement of the tar-
+get domain is greater than that of the source domain. On the one
+hand, the source domain can provide more information, and on the
+other hand, the improvement of the recommendation capability
+of the source domain leads to a further promotion in the upper
+bound of the recommendation performance of the target domain.
+(4) Furthermore, we observe that the proposed model improves the
+movie-book task larger than the movie-music task. The possible rea-
+sons are differences in data set size and the number of overlapping
+users, which determine the richness of knowledge and the caliber
+of transfer. We plan to leave this as a topic for future research.
+4.3
+Robust Testing (RQ2, RQ3)
+We perform overlap ratio tests, ablation experiments, and hyper-
+parameter tests to verify the robustness of our model.
+4.3.1
+Length N. We also examine the performance of COAST as
+well as the most competitive algorithms in single-domain, cross-
+domain baselines, i.e., LightGCN, GADTCDR, on different recom-
+mendation list lengths, as shown in Figure 5.
+
+Cross-domain recommendation via user interest alignment
+Conference’17, July 2017, Washington, DC, USA
+Table 2: Performance comparison for cross-domain recommendation.
+Algorithm
+Task1
+Task2
+Task3
+Task4
+Movie
+Book
+Movie
+Music
+Book
+Music
+Mall
+Community
+Hit
+NDCG
+Hit
+NDCG
+Hit
+NDCG
+Hit
+NDCG
+Hit
+NDCG
+Hit
+NDCG
+Hit
+NDCG
+Hit
+NDCG
+NMF
+0.5445
+0.3154
+0.3916
+0.2224
+0.5445
+0.3154
+0.3959
+0.2206
+0.3916
+0.2224
+0.3959
+0.2206
+0.5850
+0.3265
+0.3793
+0.2048
+LightGCN
+0.6174
+0.3492
+0.3805
+0.2226
+0.6174
+0.3492
+0.3528
+0.2023
+0.3805
+0.2226
+0.3528
+0.2023
+0.5848
+0.2933
+0.5119
+0.2490
+MVDNN
+0.6382
+0.3689
+0.4654
+0.2575
+0.6414
+0.3641
+0.3965
+0.2238
+0.5104
+0.2947
+0.3923
+0.2390
+0.5963
+0.3002
+0.5211
+0.2507
+DTCDR
+0.6420
+0.3794
+0.4302
+0.2394
+0.6197
+0.4278†
+0.3593
+0.2211
+0.5108†
+0.3263†
+0.2848
+0.2017
+0.5580
+0.3109
+0.3632
+0.2643
+DDTCDR
+0.5937
+0.3558
+0.4436
+0.2511
+0.5921
+0.3722
+0.3467
+0.2189
+0.4540
+0.2666
+0.3086
+0.2042
+0.5135
+0.2884
+0.3729
+0.1886
+DML
+0.6060
+0.3638
+0.4662
+0.2662
+0.6093
+0.4059
+0.3821
+0.2287
+0.4521
+0.2616
+0.4253†
+0.2548†
+0.5491
+0.3181
+0.4283
+0.2124
+GADTCDR
+0.6817†
+0.4205†
+0.4882†
+0.3026†
+0.6818†
+0.4276
+0.4383†
+0.2498†
+0.4492
+0.2761
+0.3571
+0.1933
+0.6654†
+0.4055†
+0.5173†
+0.2907†
+CDRIB
+0.6114
+0.3301
+0.4630
+0.2772
+0.6411
+0.3578
+0.4103
+0.2272
+0.5021
+0.2654
+0.2866
+0.2038
+0.5744
+0.3007
+0.4802
+0.2814
+COAST
+0.6905
+0.4271
+0.5052
+0.3174
+0.6938
+0.4292
+0.4497
+0.2515
+0.5138
+0.3293
+0.4688
+0.2712
+0.6769
+0.4073
+0.5503
+0.3195
+Improvement
+1.2909%
+1.5696%
+3.4821%
+4.8909%
+1.7600%
+0.3273%
+2.6001%
+0.6805%
+0.5873%
+0.9194%
+10.2280%
+6.4364%
+1.7283%
+0.4439%
+6.3793%
+9.9071%
+† means the strongest baseline’s performance.
+1
+3
+5
+7
+10
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Top-N Hit (movie)
+COAST
+GADTCDR
+LightGCN
+(a) Hit of Douban-movie.
+1
+3
+5
+7
+10
+0.20
+0.25
+0.30
+0.35
+0.40
+Top-N NDCG (movie)
+COAST
+GADTCDR
+LightGCN
+(b) NDCG of Douban-movie.
+1
+3
+5
+7
+10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+Top-N Hit (book)
+COAST
+GADTCDR
+LightGCN
+(c) Hit of Douban-book.
+1
+3
+5
+7
+10
+0.125
+0.150
+0.175
+0.200
+0.225
+0.250
+0.275
+0.300
+0.325
+Top-N NDCG (book)
+COAST
+GADTCDR
+LightGCN
+(d) NDCG of Douban-book.
+Figure 5: Top-N performance.
+Obviously, the performance of all algorithms increases as the
+recommendation list grows, because the longer the list, the higher
+the fault tolerance. Meanwhile, compared with the LightGCN and
+GADTCDR algorithms, our algorithm achieves the best perfor-
+mance in all scenarios, especially in the difficult 𝑁 = 3 scenario
+with the greatest improvement, which shows our superiority.
+4.3.2
+Overlap Ratio M. To investigate the robustness of our model,
+we experiment with scaling the number of overlapping users.
+Table 3 reports the recommendation performance of COAST,
+GADTCDR trained on corresponding cross-domain scenarios with
+overlapping users of 25%, 50%, 75%, and 100%, respectively. From
+Table 3, we have the following observations. (1) With the increase
+of the overlapping user training ratio,the recommendation perfor-
+mance of all algorithms steadily improves, which demonstrates that
+overlapping ratio is effective to enhance the correlation across do-
+mains. (2) Our model shows robust performance to make recommen-
+dations for both domains than the strongest baseline GADTCDR,
+even with only 25% user overlap. This is attributed to the unified
+Table 3: Overlap ratio test.
+Task
+Ratio
+COAST
+GADTCDR
+source
+target
+source
+target
+Hit
+NDCG
+Hit
+NDCG
+Hit
+NDCG
+Hit
+NDCG
+Task1
+25%
+0.6606
+0.4013
+0.4531
+0.2735
+0.6067
+0.3447
+0.3933
+0.2412
+50%
+0.6824
+0.4242
+0.4801
+0.2976
+0.6193
+0.3711
+0.4402
+0.2709
+75%
+0.6831
+0.4132
+0.5019
+0.3078
+0.6263
+0.3766
+0.4474
+0.2889
+100%
+0.6905
+0.4271
+0.5052
+0.3174
+0.6817
+0.4205
+0.4882
+0.3026
+Task2
+25%
+0.6875
+0.4188
+0.4055
+0.2238
+0.5997
+0.3527
+0.2805
+0.1512
+50%
+0.6881
+0.4161
+0.4372
+0.2445
+0.6186
+0.3601
+0.3301
+0.1297
+75%
+0.6872
+0.4143
+0.4382
+0.2469
+0.6101
+0.3712
+0.3445
+0.1808
+100%
+0.6938
+0.4292
+0.4497
+0.2515
+0.6818
+0.4276
+0.4383
+0.2498
+Taks3
+25%
+0.4763
+0.2958
+0.3929
+0.2148
+0.4080
+0.2484
+0.2805
+0.1512
+50%
+0.4845
+0.3081
+0.3954
+0.2181
+0.4338
+0.2698
+0.3367
+0.1812
+75%
+0.5014
+0.3115
+0.4192
+0.2293
+0.4350
+0.2619
+0.3375
+0.1891
+100%
+0.5138
+0.3293
+0.4688
+0.2712
+0.4492
+0.2761
+0.3571
+0.1933
+Task4
+25%
+0.6453
+0.3883
+0.5240
+0.3454
+0.6380
+0.3713
+0.5037
+0.3000
+50%
+0.6550
+0.3912
+0.5236
+0.2998
+0.6470
+0.3863
+0.5069
+0.2916
+75%
+0.6590
+0.3945
+0.5439
+0.3242
+0.6493
+0.3874
+0.5010
+0.2922
+100%
+0.6769
+0.4073
+0.5503
+0.3195
+0.6654
+0.4055
+0.5173
+0.2907
+graph message passing mechanism and user interest alignment,
+which enable the model to perceive the cross-domain similarity
+between entities and ensure consistent interests across views. (3)
+Further, we observe that the overlap ratio has little improvement on
+75%→100% than 25% → 50%, as the absolute number of overlapping
+users is large enough to ensure basic knowledge transfer.
+4.3.3
+Ablation Studies. We further compare COAST with several
+ablation variants to demonstrate the effectiveness and advance-
+ment of different sub-modules. For fairness, other settings are kept
+unchanged except for the specified ablation module.
+• COAST-NF: This variant uses only explicit interactions.
+• COAST-NS: Instead of constructing cross-domain heteroge-
+neous graphs, each domain trains representations separately.
+• COAST-NM: No user-item interaction in section 3.3.2.
+• COAST-NU: This variant is not subject to user-user consis-
+tency.
+• COAST-NI: This variant is not subject to user-item consistency.
+As reported in Figure 6, COAST-NF has the worst performance
+but is still stronger than the vast majority of baselines (except for
+GADTCDR), illustrating that our model structure is able to mine
+structural similarities from explicit data. Regarding COAST-NS and
+COAST-NM, as ablations of the cross-domain graph module, both
+decrease compared with COAST. The former cannot capture cross-
+domain similarity due to the isolation of user-item cross-domain
+
+Conference’17, July 2017, Washington, DC, USA
+Trovato and Tobin, et al.
+COAST-NF
+COAST-NS COAST-NM COAST-NU
+COAST-NI
+COAST
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Hit (movie)
+(a) Hit@10 of Douban-movie.
+COAST-NF
+COAST-NS COAST-NM COAST-NU
+COAST-NI
+COAST
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+NDCG (movie)
+(b) NDCG@10 of Douban-movie.
+COAST-NF
+COAST-NS COAST-NM COAST-NU
+COAST-NI
+COAST
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Hit (book)
+(c) Hit@10 of Douban-book.
+COAST-NF
+COAST-NS COAST-NM COAST-NU
+COAST-NI
+COAST
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+NDCG (book)
+(d) NDCG@10 of Douban-book.
+Figure 6: Ablation studies.
+16
+32
+64
+128
+0.683
+0.684
+0.685
+0.686
+0.687
+0.688
+0.689
+0.690
+Hit
+NDCG
+0.414
+0.416
+0.418
+0.420
+0.422
+0.424
+0.426
+Hyper-Emb (movie)
+(a) Hit@10 of Douban-movie.
+16
+32
+64
+128
+0.498
+0.499
+0.500
+0.501
+0.502
+0.503
+0.504
+0.505
+Hit
+NDCG
+0.312
+0.313
+0.314
+0.315
+0.316
+0.317
+Hyper-Emb (book)
+(b) Hit@10 of Douban-book.
+Figure 7: The impact of 𝐷.
+interactions at the graph structure level, while the latter is insuffi-
+cient to characterize the collaborative filtering relationship due to
+ignoring the collaborative signal of user-item. Meanwhile, with the
+same structure, COAST improve over COAST-NU, COAST-NI. This
+demonstrates that using user interest alignment as a constraint can
+not only effectively prevent overfitting, but also, as a fine-grained
+knowledge utilization, significantly enhance the generalization of
+user representations across domains. From a deeper perspective,
+contrastive learning and gradient alignment leverage the poten-
+tial unsupervised signals and semantic features in the data, which
+greatly facilitates the extraction of domain-invariant features. In
+general, each submodule of COAST plays an indispensable role and
+contributes significantly to the model performance.
+4.3.4
+Hyper-testing. In this subsection, we present the tuning of
+several key hyper-parameters in our framework.
+Embedding size D. Embedding size is one of the most important
+hyper- parameters in deep learning and is closely related to model
+capacity [38]. To improve the performance of the proposed COAST,
+we perform a hyper-parameter search on the embedding size.
+64
+128
+256
+512
+0.680
+0.682
+0.684
+0.686
+0.688
+0.690
+Hit
+NDCG
+0.414
+0.416
+0.418
+0.420
+0.422
+0.424
+0.426
+Hyper-Interest (movie)
+(a) Hit@10 of Douban-movie.
+64
+128
+256
+512
+0.492
+0.494
+0.496
+0.498
+0.500
+0.502
+0.504
+Hit
+NDCG
+0.304
+0.306
+0.308
+0.310
+0.312
+0.314
+0.316
+0.318
+Hyper-Interest (book)
+(b) Hit@10 of Douban-book.
+Figure 8: The impact of 𝐾.
+1e-3
+5e-3
+1e-2
+5e-2
+0.683
+0.684
+0.685
+0.686
+0.687
+0.688
+0.689
+0.690
+Hit
+NDCG
+0.4100
+0.4125
+0.4150
+0.4175
+0.4200
+0.4225
+0.4250
+0.4275
+Hyper- 2 (movie)
+(a) Hit@10 of Douban-movie.
+1e-3
+5e-3
+1e-2
+5e-2
+0.496
+0.498
+0.500
+0.502
+0.504
+Hit
+NDCG
+0.310
+0.312
+0.314
+0.316
+Hyper- 2 (book)
+(b) Hit@10 of Douban-book.
+Figure 9: The impact of 𝜆2.
+As shown in Figure 7, our algorithm performs best when 𝐷 = 64
+on any metrics. The larger the embedding size, the more expressive
+the model is, but too high embedding size will slow down the
+convergence speed and lead to overfitting. In consequence, we
+choose 𝐷 = 64 as the embedding size in COAST.
+Number of Interests K. In section 3.4.1, we constrain users’ con-
+trasting views to belong to the same cluster center. In view of this,
+we perform a test on the number of interest cluster centers 𝐾.
+As shown in Figure 8, our model is sensitive to 𝐾. We argue that
+this phenomenon arises because 𝐾 represents an abstract interest
+center rather than a concrete interest. Meanwhile, we propose
+that higher 𝐾 can be chosen to characterize the distribution of
+user interests when the number of items and users increases. This
+is intuitive, as the number of users increases, the interests will
+obviously become more diverse. Consequently, we choose 𝐾 = 256.
+Consistency weight 𝜆2. The consistency weight 𝜆2 is a trade-off
+between task interest and user interest alignment. To improve the
+recommendation effect, we have tuned it.
+The larger 𝜆2 is, the stronger the constraint on user interest
+consistency is, but hinders domain-specific user representation,
+thereby impairing recommendation performance on that domain.
+Conversely, the smaller 𝜆2 is, our model will degenerate into a
+general representation combination model, which cannot solve
+user interest alignment. Experimentally, we set 𝜆2 = 0.01.
+5
+CONCLUSION
+In this work, we propose the COAST framework, which aims to
+improve model performance in dual cross-domain recommenda-
+tion scenarios. This work represents an attempt to leverage rich
+content information and user interest alignment for bidirectional
+knowledge transfer. Specifically, we model the interaction of users
+and items in two domains as a unified cross-domain heterogeneous
+
+1
+11Cross-domain recommendation via user interest alignment
+Conference’17, July 2017, Washington, DC, USA
+graph, and improve the message passing mechanism of graph con-
+volution to capture the cross-domain similarity of users and items.
+Further, we utilize contrastive learning and gradient alignment to
+constrain overlapping user interest alignment from both user-user
+and user-item perspectives. Overall, our solution has several ad-
+vantages. First, at the data level, our task is constructed on data
+sets with partial user overlap and exploits both explicit and implicit
+information, which has a wider range of application scenarios. Sec-
+ond, at the algorithm level, we learn better representations from
+high-order cross-domain similarity and user interest alignment com-
+pared to previous plain combinations. Finally, at the experimental
+level, we conduct extensive experiments, all of which demonstrate
+the state-of-the-art and superiority of our model.
+There are still several limitations of our study for future work.
+First, how to extend our work to more complex scenarios, such as
+the case of overlapping items or multi-domain recommendation.
+Second, how to integrate data from other modalities or integrate
+more complex interactions, such as images, attribute nodes, in fea-
+ture extraction module. Finally, we should validate the robustness
+of COAST on more large cross-domain recommendation data sets.
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+Cross-domain recommendation via user interest alignment
+Conference’17, July 2017, Washington, DC, USA
+A
+IMPORTANT NOTATIONS
+Table 4: Mathematical Notation
+Symbol
+Notation
+S, T
+source/target domain
+U
+user set
+V
+item set
+A
+user-item interaction matrix
+X, H
+features before/after preprocessing
+G = (U, V, E, H)
+heterogeneous graph of user-item interactions
+𝑁
+neighbors set
+𝑚
+message passing function
+𝐾
+the total number of interests in the user set
+𝑔
+gradient calculation
+𝑦
+whether the user clicked on the item
+B
+ALGORITHM
+Algorithm 1 The Algorithm of COAST
+Input: Interaction matrix AS,AT,XS,XT;
+Output: Parameters Θ;
+1: Random initialize model parameters Θ,
+2: Data preprocessing 𝑒𝑢 ∈ HU, 𝑒𝑣
+S ∈ HS, 𝑒𝑣
+T ∈ HT
+3: Graph construction G = (U, V, E, H)
+4: while not converged do
+5:
+Sample a batch of training data
+6:
+Graph propagation, getting 𝑒𝑢, 𝑒𝑣
+7:
+for 𝑢 ∈ U𝑜 do
+8:
+User-User interest alignment L𝑈 ,𝑈
+9:
+User-Item interest alignment L𝑈 ,𝐼
+10:
+end for
+11:
+Supervise loss L𝑠
+12:
+Joint optimization L
+13:
+Update the parameters
+14: end while
+15: return Parameters Θ
+