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+Multivariate Regression via Enhanced
+Response Envelope: Envelope Regularization
+and Double Descent
+Oh-Ran Kwon and Hui Zou
+School of Statistics, University of Minnesota
+Abstract
+The envelope model provides substantial efficiency gains over the standard multi-
+variate linear regression by identifying the material part of the response to the model
+and by excluding the immaterial part. In this paper, we propose the enhanced response
+envelope by incorporating a novel envelope regularization term in its formulation. It is
+shown that the enhanced response envelope can yield better prediction risk than the
+original envelope estimator. The enhanced response envelope naturally handles high-
+dimensional data for which the original response envelope is not serviceable without
+necessary remedies. In an asymptotic high-dimensional regime where the ratio of the
+number of predictors over the number of samples converges to a non-zero constant, we
+characterize the risk function and reveal an interesting double descent phenomenon for
+the first time for the envelope model. A simulation study confirms our main theoret-
+ical findings. Simulations and real data applications demonstrate that the enhanced
+response envelope does have significantly improved prediction performance over the
+original envelope method.
+Keywords: Double descent, Envelope model, High-dimension asymptotics, Prediction, Reg-
+ularization
+1
+arXiv:2301.04625v1  [stat.ME]  11 Jan 2023
+
+1
+Introduction
+The envelope model first introduced by Cook et al. (2010) is a modern approach to estimat-
+ing an unknown regression coefficient matrix β ∈ Rr×p in multivariate linear regression of
+the response vector y ∈ Rr on the predictors x ∈ Rp. It was shown by Cook et al. (2010) that
+the envelope estimator of β results in substantial efficiency gains relative to the standard
+maximum likelihood estimator of β. The gains arise by identifying the part of the response
+vector that is material to the regression and by excluding the immaterial part in the estima-
+tion. The original envelope model has been later extended to the envelope model based on
+excluding immaterial parts of the predictors to the regression by Cook et al. (2013). Cook
+et al. (2013) then established the connection between the latter envelope model and partial
+least squares, providing a statistical understanding of partial least squares algorithms.
+The success of the envelope models and their theories motivated some authors to propose
+new envelope models by applying or extending the core idea of envelope modeling to various
+statistical models. The two most common are the response envelope models and the predictor
+envelope models. The response envelope models (predictor envelope models) achieve estima-
+tion and prediction gains by eliminating the variability arising from the immaterial part of
+the responses (predictors) that is invariant to the changes in the predictors (responses). Pa-
+pers on response envelope models include the original envelope model (Cook et al., 2010), the
+partial envelope model (Su and Cook, 2011), the scaled response envelope model (Cook and
+Su, 2013), the reduced-rank envelope model (Cook et al., 2015), the sparse envelope model
+(Su et al., 2016), the Bayesian envelope model (Khare et al., 2017), the tensor response enve-
+lope model (Li and Zhang, 2017), the envelope model for matrix variate regression (Ding and
+Cook, 2018), and the spatial envelope model for spatially correlated data (Rekabdarkolaee
+et al., 2020). Papers on predictor envelope models include the envelope model for predictor
+reduction (Cook et al., 2013), the envelope model for generalized linear models and Cox’s
+proportional hazard model (Cook and Zhang, 2015a), the scaled predictor envelope model
+(Cook and Su, 2016), the envelope quantile regression model (Ding et al., 2020), the envelope
+model for the censored quantile regression (Zhao et al., 2022), tensor envelope partial least
+squares regression (Zhang and Li, 2017), and envelope-based sparse partial least squares
+regression (Zhu and Su, 2020). For a comprehensive review of the envelope models, readers
+2
+
+are referred to Cook (2018).
+High-dimensional data have become common in many fields. It is only natural to consider
+the performance of the envelope model under high dimensions. The likelihood-based method
+to estimate β under both the response/predictor envelope model is not serviceable for high-
+dimensional data because the likelihood-based method requires the inversion of the sample
+covariance matrix of predictors. Hence, one has to find effective ways to mitigate this issue.
+For the predictor envelope model, its connection to partial least squares provides one solution.
+Partial least squares (De Jong, 1993) can be used for estimating β for the predictor envelope
+model (Cook et al., 2013). The partial least squares algorithm is an iterative moment-based
+algorithm involving the sample covariance of predictors and the sample covariance between
+the response vector and predictors, which does not require inversion of the sample covariance
+matrix of predictors. In addition, the algorithm provides the root-n consistent estimator of β
+in the predictor envelope model with the number of predictors fixed (Chun and Kele¸s, 2010;
+Cook et al., 2013) and can yield accurate prediction in the asymptotic high-dimensional
+regime when the response is univariate (Cook and Forzani, 2019). Motivated by this, Zhu and
+Su (2020) introduced envelope-based sparse partial least squares and showed the consistency
+of the estimator for the sparse predictor envelope model. Zhang and Li (2017) proposed a
+tensor envelope partial least squares algorithm, which provides the consistent estimator for
+the tensor predictor envelope model. Another way to apply predictor envelope models for
+high-dimensional data is by selecting the principal components of predictors and then using
+likelihood-based estimation on the principal components. This simple remedy is adapted by
+Rimal et al. (2019) to compare the prediction performance of the likelihood-based predictor
+envelope method, principal component regression, and partial least squares regression for
+high-dimensional data. Their extensive numerical study showed that this simple remedy
+produced better prediction performance than principal component regression and partial
+least squares regression. The impact of high dimensions is more severe for the response
+envelope. There is far less work on making the response envelope model serviceable for high-
+dimensional data. The Bayesian approach for the response envelope model (Khare et al.,
+2017) can handle high-dimensional data. The sparse envelope model (Su et al., 2016) which
+performs variable selection on the responses can handle data with the sample size smaller
+3
+
+than the number of responses, but still requires the number of predictors smaller than the
+number of sample size.
+In this paper, we propose the enhanced response envelope for high-dimensional data by
+incorporating a novel envelope regularization term in its formulation. The envelope regu-
+larization term respects the fundamental idea of the original envelope model by considering
+the presence of the material and immaterial parts of the response in the model. The en-
+hancements are twofold. First, our enhanced response envelope estimator can handle both
+low- and high-dimensional data, while the original envelope estimator can only handle low-
+dimensional data where the sample size n is smaller than the number of predictors p. From
+the connection between the original envelope estimator and the enhanced response envelope
+estimator in low-dimension, we extend the definition of the original envelope estimator to
+high-dimensional data by considering the limiting case of the enhanced response envelope es-
+timator with a vanishing regularization parameter; see the discussion in Section 2.3. Second,
+we prove that the enhanced response envelope can reduce the prediction risk relative to the
+original envelope for all values of n and p. Moreover, we study the asymptotics of the predic-
+tion risk for the original envelope estimator and the enhanced response envelope estimator
+when both n, p → ∞ and their ratio converges to a nonzero constant p/n → γ ∈ (0, ∞).
+This kind of asymptotic regime has been considered in high-dimensional machine learning
+theory (El Karoui, 2018; Dobriban and Wager, 2018; Liang and Rakhlin, 2020; Hastie et al.,
+2022) for analyzing the behavior of prediction risk of certain predictive models. We derive an
+interesting asymptotic prediction risk curve for the envelope estimator. The risk increases as
+γ increases, and then decreases after γ > 1. This phenomenon is known as the double descent
+phenomenon in the machine learning literature. Although the double descent phenomenon
+has been observed for neural networks and ridgeless regression (Belkin et al., 2019; Hastie
+et al., 2022), this is the first time that such a phenomenon is shown for the envelope models.
+The rest of the paper is organized as follows. We review the original envelope model
+and the corresponding envelope estimator in Section 2.1. In Section 2.2, we introduce a
+new regularization term called the envelope regularization based on which we propose the
+enhanced response envelope in section 2.3. The enhanced response envelope estimator nat-
+urally provides a definition for the envelope estimator when p > n. Section 2.4 describes
+4
+
+how to implement this new method in practice. In Section 3.1, we prove that the enhanced
+response envelope can yield better prediction risk than the original envelope for any (n, p)
+pair. Considering n, p → ∞ and p/n → γ ∈ (0, ∞), we derive the limiting prediction risk
+result of the original envelope and the enhanced response envelope in Section 3.2. This result
+along with our simulation study in Section 4 verify the double descent phenomenon. Real
+data analyses are presented in Section 5. Proofs of theorems are provided in Appendix A.
+2
+Enhanced response envelope
+2.1
+Review of envelope model
+Envelope model
+Let us begin with the classical multivariate linear regression model of a
+response vector y ∈ Rr given a predictor vector x ∈ Rp:
+y = βx + ε, ε ∼ N(0, Σ),
+(1)
+where ε is the error vector with a positive definite Σ and independent to x. β ∈ Rr×p is
+an unknown matrix of regression coefficients and x ∼ Px where Px is a distribution on Rp
+such that E(x) = 0 and Cov(x) = Σx. We omit an intercept by assuming E(y) = 0 for easy
+communication.
+The envelope model allows for the possibility that there is a part of the response vector
+that is unaffected by changes in the predictor vector. Specifically, let E ⊆ Rr be a subspace
+such that for all x1 and x2,
+(i) QEy|(x = x1) ∼ QEy|(x = x2) and (ii) PEy ⊥⊥ QEy|x,
+(2)
+where PE is the projection onto E and QE = I − PE. Condition (i) states that the marginal
+distribution of QEy is invariant to changes in x. Condition (ii) says that QEy does not
+affect PEy if x is provided. Conditions together imply that PE includes the relevant depen-
+dency information of y on x (the material part) while QE is the irrelevant information (the
+immaterial part).
+Let B = span(β). The conditions in (2) hold if and only if
+span(β) = B ⊆ E and Σ = PEΣPE + QEΣQE.
+(3)
+5
+
+The definition of an envelope introduced by Cook et al. (2007, 2010) formalizes the smallest
+subspace satisfying the conditions in (2) using the equivalence relation of (2) and (3). The
+envelope is defined as the intersection of all subspaces E satisfying (3) and is denoted by
+EΣ,B, Σ-envelope of B.
+The envelope model arises by parameterizing the multivariate linear model in terms of
+the envelope EΣ,B. The parameterization is as follows. Let u = dim(EΣ,B), Γ ∈ Rr×u be any
+semi-orthogonal basis matrix for EΣ,B, and Γ0 ∈ Rr×(r−u) is any semi-orthogonal basis matrix
+for the orthogonal complement of EΣ,B. Then the multivariate linear model can be written
+as
+y = Γηx + ε, ε ∼ N(0, ΓΩΓT + Γ0Ω0ΓT
+0 ),
+(4)
+where β = Γη with η ∈ Ru×p, and Ω ∈ Rr×r and Ω0 ∈ R(r−u)×(r−u) are symmetric positive
+definite matrices. Model (4) is called the envelope model.
+Envelope estimator The parameters in the envelope model are estimated by maximizing
+the likelihood function from model (4). Assume that p+r < n and u is the dimension u of the
+envelope. SX = n−1XTX, SY = n−1YTY, SY,X = n−1YTX, and SY|X = SY−SY,XS−1
+X SX,Y,
+where Y ∈ Rn×r has rows yT
+i and X ∈ Rn×p has rows xT
+i .
+The envelope estimator of β is determined as
+ˆEΣ,B = span{arg
+min
+G∈Gr(r,u)(log |GTSY|XG| + log |GTS−1
+Y G|)},
+(5)
+where Gr(r, u) = {G ∈ Rr×u : G is a semi-orthogonal matrix}. Define ˆΓ as any semi-
+orthogonal basis matrix for ˆEΣ,B and let ˆΓ0 be any semi-orthogonal basis matrix for the
+orthogonal complement of ˆEΣ,B. The estimator of β is given by
+ˆβ = ˆΓˆΓTSY,XS−1
+X ,
+(6)
+and Σ is estimated by ˆΣ = ˆΓ ˆΩˆΓ + ˆΓT
+0 ˆΩ0ˆΓ0 where
+ˆΩ = ˆΓTSY|XˆΓ,
+ˆΩ0 = ˆΓT
+0 SY ˆΓ0,
+(7)
+2.2
+Envelope regularization
+In this section, we introduce the envelope regularization term that respects the fundamental
+idea in the envelope model by considering the presence of material and immaterial parts,
+6
+
+PEΣ,By and QEΣ,By, in the regression.
+We define the envelope regularization term as
+ρ(η, Ω) = tr(ηTΩ−1η).
+(8)
+The envelope model distinguishes between PEΣ,By and QEΣ,By in the estimation process.
+The log-likelihood function of the envelope model is decomposed into two log-likelihood
+functions. One is the log-likelihood function for the multivariate regression of ΓTy on x,
+ΓTy = ηx+ΓTε where ΓTε ∼ N(0, Ω). The other is the log-likelihood function for the zero-
+mean model of ΓT
+0 y, ΓT
+0 y = ΓT
+0 ε where ΓT
+0 ε ∼ N(0, Ω0). The envelope regularization term
+(8) is the function of η and Ω, the parameters in the likelihood for the material part of the
+envelope model. The envelope regularization term (8) can be seen as imposing the Frobenius
+norm regularization on the coefficient after standardizing the material part of the regression
+to have uncorrelated errors, Ω−1/2ΓTy = Ω−1/2ηx + Ω−1/2ΓTε where Ω−1/2ΓTε ∼ N(0, I).
+We emphasize that the envelope regularization is different from the ridge regularization.
+While the ridge regularization ∥β∥2
+F is the quadratic function of β, the envelope regulariza-
+tion is not because the components of Ω are not fixed values. The envelope regularization is
+the function of both η and Ω, and thus is optimized over η and Ω simultaneously, as shown
+in the next subsection.
+2.3
+The proposed estimator
+We only assume that r ≤ n but p is allowed to be bigger than n. The log-likelihood function
+under the envelope model (4) is
+Lu(η, EΣ,B, Ω, Ω0) = − (nr/2) log(2π) − (n/2) log |ΓΩΓT + Γ0Ω0ΓT
+0 |
+− (1/2)
+n
+�
+i=1
+(yi − Γηxi)T(ΓΩΓT + Γ0Ω0ΓT
+0 )−1(yi − Γηxi).
+By incorporating the envelope regularization term ρ given in the last subsection, we propose
+the following enhanced response envelope estimator via penalized maximum likelihood:
+arg max{Lu(η, EΣ,B, Ω, Ω0) − n
+2λ · ρ(η, Ω)},
+(9)
+where λ > 0 serves as a regularization parameter.
+7
+
+Let SX = n−1XTX, SY = n−1YTY, SY,X = n−1YTX, Sλ
+X = SX + λI and Sλ
+Y|X =
+SY − SY,X(Sλ
+X)−1SX,Y. After some basic calculations, (9) can be expressed as
+ˆEΣ,B(λ) = span{arg
+min
+G∈Gr(r,u)(log |GTSλ
+Y|XG| + log |GTS−1
+Y G|)},
+(10)
+where Gr(r, u) = {G ∈ Rr×u : G is a semi-orthogonal matrix}. Let ˆΓλ be any semi-orthogonal
+basis matrix for ˆEΣ,B(λ) and ˆΓ0,λ be any semi-orthogonal basis matrix for the orthogonal
+complement of ˆEΣ,B(λ). The enhanced envelope estimator of β is given by
+ˆβ(λ) = ˆΓλˆΓT
+λSY,X(Sλ
+X)−1
+(11)
+and Σ is estimated by ˆΣ(λ) = ˆΓλ ˆΩ(λ)ˆΓλ + ˆΓT
+0,λ ˆΩ0(λ)ˆΓ0,λ where
+ˆΩ(λ) = ˆΓT
+λSλ
+Y|XˆΓλ,
+ˆΩ0(λ) = ˆΓT
+0,λSY ˆΓ0,λ,
+(12)
+The enhanced response envelope estimator can naturally handle the case where p ≥ n−r,
+while the original envelope estimator (5) does not. Motivated by the definition of ridgeless
+regression (Hastie et al., 2022), we can consider taking the limit of the enhanced response
+envelope estimator with λ → 0+:
+ˆEΣ,B = span{arg
+min
+G∈Gr(r,u)( lim
+λ→0+ log |GTSλ
+Y|XG| + log |GTS−1
+Y G|)},
+ˆβ = lim
+λ→0+ ˆβ(λ)
+(13)
+We take (13) as the definition of envelope estimator. Obviously, when p < n−r, this extended
+definition recovers the original envelope estimator (5). This definition enables the use of the
+envelope estimator when p ≥ n−r, without altering the definition of the original envelope
+estimator (5) when p < n−r. In practice, we implement (13) by computing the enhanced
+response envelope estimator (10) with a very small value of λ such as 10−8.
+As the enhanced response envelope estimator (9) has flexibility on λ, the enhanced re-
+sponse envelope estimator with an appropriate choice of λ can yield better prediction risk
+compared to the envelope estimator, which is discussed in Section 3. We discuss the Grass-
+mannian manifold optimization required in (10) in the next subsection.
+2.4
+Implementation
+Suppose that the dimension u is specified and λ is given. Our proposed estimator ˆEΣ,B(λ)
+for EΣ(B) requires the optimization over the Grassmannian G(u, r). Such a computation
+8
+
+problem exists for the original envelope model as well. So far, the best-known algorithm for
+solving envelope models is the algorithm introduced by Cook et al. (2016). Thus, we employ
+their algorithm to compute ˆEΣ,B(λ) in (10). Note that we standardize X so that each column
+has a mean of 0 and a standard deviation of 1 before fitting any model.
+In practice, the tuning parameter λ and the dimension u of the envelope are unknown. We
+use the cross-validation method to choose (u, λ). For the original envelope, u can be selected
+by using AIC, BIC, LRT or cross-validation. BIC and LRT may be preferred as shown by
+simulations in Su and Cook (2013). Because the enhanced response envelope model has an
+additional tuning parameter λ, we propose to use cross-validation to find the best tuning
+parameter combination of u and λ.
+We have implemented the enhanced response envelope method in R and the code is
+available upon request.
+3
+Theory
+In this section, we show that the enhanced response envelope can reduce the prediction risk
+over the envelope for any (n, p) pair. We then consider the asymptotic setting when n, p → ∞
+p/n → γ ∈ (0, ∞). This asymptotic regime has been considered in the literature (El Karoui,
+2018; Dobriban and Wager, 2018; Liang and Rakhlin, 2020; Hastie et al., 2022) for analyzing
+the behavior of prediction risk of certain predictive models.
+In our discussion, we consider the case where EΣ(B) is known, which has been assumed
+in the existing envelope papers to understand the core mechanism of envelope methodologies
+(Cook et al., 2013; Cook and Zhang, 2015a,b).
+3.1
+Reduction in prediction risk
+Consider a test point xnew ∼ Px. For an estimator ˆβ, we define the prediction risk as
+R( ˆβ|X) = E[∥ ˆβxnew − βxnew∥2|X].
+Note that this definition has the bias-variance decomposition,
+R( ˆβ|X) = ∥bias(vec( ˆβ)|X)∥2 + tr{Var(vec( ˆβ)|X)}.
+9
+
+Let Γ be a semi-orthogonal basis matrix for EΣ,B. Following the discussion in Section 2.3,
+we take (13) as the definition of the envelope estimator ˆβΓ . The prediction risk of ˆβΓ is
+R( ˆβΓ|X) = vecT(β)[ΠXΣxΠX ⊗ Ir]vec(β)
+�
+��
+�
+bias2
++ tr(Ω)
+n
+tr(S+
+XΣx)
+�
+��
+�
+var
+,
+where ΠX = Ip − S+
+XSX.
+The prediction risk of the enhanced response envelope estimator ˆβΓ(λ) is
+R( ˆβΓ(λ)|X) = E[∥ ˆβΓ(λ)xnew − βxnew∥2|X]
+= λ2vecT(β)[(SX + λI)−1Σx(SX + λI)−1 ⊗ Ir]vec(β)
+�
+��
+�
+bias2
++ tr(Ω)
+n
+tr(ΣxSX(SX + λI)−2)
+�
+��
+�
+var
+.
+(14)
+Theorem 1 shows that using the envelope regularization always improves the prediction
+risk of the envelope model.
+Theorem 1. There always exists a λ > 0 such that R( ˆβΓ(λ)|X) < R( ˆβΓ|X).
+3.2
+Limiting prediction risk and double descent phenomenon
+The asymptotics of the envelope model are well-established in the case where n diverges
+while p is fixed (Cook et al., 2010), while not in a high-dimensional asymptotic setup. In
+this section, we examine the limiting risk of both the enhanced response envelope estimator
+and the envelope estimator in the high-dimensional asymptotic regime where n, p → ∞ with
+p/n → γ ∈ (0, ∞). The number of response variables r is fixed. This kind of asymptotic
+regime has been considered in high-dimensional machine learning theory (El Karoui, 2018;
+Dobriban and Wager, 2018; Liang and Rakhlin, 2020; Hastie et al., 2022) for analyzing the
+behavior of prediction risk of certain predictive models.
+Let x = Σ1/2
+x x∗, where E(x∗) = 0 and Cov(x∗) = Ip. Then the envelope model (4) of y
+on x can be expressed as the envelope model of y on x∗:
+y = Γηx + ε = Γη∗x∗ + ε,
+where η∗ = ηΣ1/2 and ε ∼ N(0, ΓΩΓT + Γ0Ω0ΓT
+0 ). We take advantage of the invariance
+property of the envelope model in the analysis. Considering the envelope on (y, x∗) amounts
+to assuming the covariance of the predictor is Ip.
+10
+
+0
+2
+4
+6
+8
+0
+5
+10
+15
+20
+25
+γ
+Limiting prediction risk
+Envelope
+Enhanced response envelope
+Figure 1: The limiting prediction risks of the enhanced response envelope with λ∗ =
+tr(Ω)γ/c2 (gray solid line) and the envelope (black solid line), illustrating Theorem 2 when
+tr(Ω) = 10 and tr(βTβ) = 10.
+Theorem 2. Assume that x has a bounded 4th moment and that tr(ηTη) = c2 for all n, p.
+Then as n, p → ∞, such that p/n → γ ∈ (0, ∞), almost surely,
+R( ˆβΓ|X) →
+�
+�
+�
+�
+�
+tr(Ω)
+γ
+1−γ
+for γ < 1
+c2(1 − 1
+γ) + tr(Ω)
+1
+γ−1
+for γ > 1,
+and
+R( ˆβΓ(λ∗)|X) → tr(Ω)γm(−λ∗),
+where λ∗ = tr(Ω)γ/c2 and m(z) =
+1−γ−z−√
+(1−γ−z)2−4γz
+(2γz)
+.
+Figure 1 visualizes the limiting prediction risk curves in Theorem 2. It plots the limiting
+risks of envelope (black solid line) and the enhanced response envelope with λ∗ = tr(Ω)γ/c2
+(dark-gray solid line), when tr(Ω) = 10 and tr(ηTη) = 10.
+We have four remarks from Theorem 2. The limiting risk of envelope increases before
+γ = 1 and then decreases after γ = 1. The double descent phenomenon has been observed in
+popular methods such as neural networks, kernel machines and ridgeless regression (Belkin
+et al., 2019; Hastie et al., 2022), but this is the first time that such a result is established
+11
+
+in the envelope literature. Second, the enhanced response envelope estimator always has a
+better asymptotic prediction risk than the envelope estimator (for any c2, tr(Ω), and γ).
+Third, in Theorem 1, we show the existence of a λ that gives a smaller prediction risk of
+the enhanced response envelope than the envelope estimator. In an asymptotic regime, we
+specify such a λ value: λ∗ = tr(Ω)γ/c2. Lastly, the gap between two limiting prediction risks,
+limn,p→∞ R( ˆβΓ|X) and n,p→∞R( ˆβΓ(λ∗)|X), increases as γ increases from 0 to 1. It is easy to
+see as
+1
+1−γ > m(−λ∗), 0 < γ < 1.
+4
+Simulation
+In this section, we use simulations to compare the performance of the enhanced response
+envelope estimator and the envelope estimator in terms of the prediction risk, E[∥ ˆβxnew −
+βxnew∥2|X] = tr[( ˆβ − β)Cov(xnew)( ˆβ − β)T]. In addition, we use simulations to have a
+numeric illustration of the double descent phenomenon to confirm the asymptotic theory.
+We consider a setting where yi ∈ R3 is generated from the model
+yi = βxi + εi, εi ∼ N(0, Σ), i = 1, . . . , n,
+and xi ∈ Rp is generated independently from xi ∼ N(0, Σx(ρ)) where (i, j)th element of
+Σx(ρ) ∈ Rp×p is ρ|i−j|. The covariance matrix Σ is set using three orthonormal vectors and
+has eigenvalues 10, 8 and 2. The columns of Γ are the second and third eigenvectors of Σ.
+Each component of ˜η ∈ R2×p is generated from the standard normal distribution. We then
+set η =
+√
+10 · ˜η/∥˜η∥F. In this setting, tr(ηTη) = 10, tr(Ω) = 10, and tr(Ω0) = 10. We
+assume that dim(EΣ,B) = 2 is known.
+Prediction risk comparison
+In this simulation, we try different combinations of n, p
+and ρ where n ∈ {50, 90, 200, 500}, p/n ∈ {0.1, 0.8, 1.2} and ρ ∈ {0, 0.8}. We compare the
+prediction risk of the enhanced response envelope estimator to three different estimators: the
+envelope estimator, multivariate linear regression, and multivariate ridge regression.
+For the enhanced response envelope and the multivariate ridge regression, we perform
+ten-fold cross-validation on simulated data to select λ among equally spaced 100 candidate
+λ-values in the scale of logarithm base 10. We compute the envelope estimator for data
+12
+
+with n ≤ p−r by taking a very small value of λ = 10−8 in the enhanced response envelope
+estimator. We fit multivariate regression model to n < p data by taking a tiny value of
+λ = 10−8 in the multivariate ridge regression. We then calculate the prediction risk. This
+process is repeated 100 times.
+In Table 1, we report the prediction risk averaged over 100 replications. First, we see that
+the prediction risks from the enhanced response envelope are consistently smaller than the
+envelope, as indicated in Theorem 1. Second, the enhanced response envelope consistently
+gives smaller prediction risks compared to the multivariate ridge regression. When u = r, the
+enhanced response envelope model reduces to the multivariate ridge regression. Therefore,
+the prediction risk of the enhanced envelope model can be smaller than that of multivariate
+ridge regression as long as tr(Ω0) > 0.
+Double descent confirmation
+This simulation is designed to support Theorem 2 and
+to illustrate the double descent phenomenon in the envelope model. We set n ∈ {200, 2000}
+and ρ = 0. p/n varies from 0.1 to 8. We compute the envelope and the enhanced response
+envelope with setting λ∗ = tr(Ω)p/(nc2) = p/n on simulated data. We then calculate the
+prediction risk for each estimator. Again, we fit n ≤ p−r data to the envelope estimator by
+taking a very small value of λ = 10−8 in the enhanced response envelope estimator.
+Figure 2 displays the prediction risks from n = 2000 with various p values. The gray
+rectangle points denote the prediction risk for the enhanced response envelope estimator. The
+black triangle points are the prediction risk for the envelope estimator. We see a fascinating
+double descent prediction risk curve for the envelope model, as discussed in Theorem 2. Also,
+the enhanced response envelope gives a smaller prediction risk across the entire range of p/n.
+Figure 3 plots the prediction risk curves from n = 200. We see that Figure 3 exhibits the
+same messages for the much smaller sample size. Although Theorem 2 is established when
+considering EΣ,B is known, we did not use this information in the actual estimation in the
+simulation study, yet the core message of Theorem 2 is confirmed by the simulation.
+13
+
+n
+p
+Enhanced
+envelope
+Envelope
+Multivariate
+linear reg
+Multivariate
+ridge reg
+Example 1: p/n = 0.1, ρ = 0
+50
+5
+1.31 (0.11)
+1.40 (0.12)
+2.39 (0.17)
+2.04 (0.12)
+90
+9
+1.24 (0.08)
+1.41 (0.10)
+2.33 (0.13)
+1.92 (0.09)
+200
+20
+1.16 (0.04)
+1.26 (0.05)
+2.31 (0.05)
+1.93 (0.04)
+500
+50
+1.06 (0.03)
+1.18 (0.03)
+2.28 (0.04)
+1.85 (0.04)
+Example 2: p/n = 0.8, ρ = 0
+50
+40
+6.73 (0.18)
+60.89 (5.80)
+104.45 (7.09)
+7.16 (0.11)
+90
+72
+6.44 (0.14)
+55.10 (2.93)
+94.81 (3.24)
+7.05 (0.08)
+200
+160
+5.86 (0.10)
+42.50 (0.99)
+81.33 (1.63)
+6.91 (0.06)
+500
+400
+5.67 (0.04)
+40.61 (0.85)
+79.17 (1.11)
+6.89 (0.03)
+Example 3: p/n = 1.2, ρ = 0
+50
+60
+8.02 (0.23)
+33.70 (1.33)
+93.79 (3.83)
+8.08 (0.11)
+90
+108
+7.58 (0.13)
+41.01 (1.36)
+94.38 (3.60)
+7.98 (0.07)
+200
+240
+7.02 (0.07)
+47.91 (1.18)
+99.94 (2.83)
+7.82 (0.04)
+500
+600
+6.78 (0.04)
+50.43 (0.91)
+103.33 (1.55)
+7.75 (0.03)
+Example 4: p/n = 0.1, ρ = 0.8
+50
+5
+1.76 (0.11)
+1.98 (0.19)
+2.39 (0.17)
+1.84 (0.07)
+90
+9
+1.02 (0.05)
+1.40 (0.08)
+2.33 (0.13)
+1.45 (0.06)
+200
+20
+0.90 (0.03)
+1.30 (0.04)
+2.31 (0.05)
+1.31 (0.03)
+500
+50
+0.78 (0.02)
+1.19 (0.03)
+2.28 (0.04)
+1.22 (0.02)
+Example 5: p/n = 0.8, ρ = 0.8
+50
+40
+4.16 (0.17)
+62.50 (6.34)
+104.45 (7.09)
+4.76 (0.12)
+90
+72
+3.78 (0.15)
+55.14 (2.85)
+94.81 (3.24)
+4.63 (0.10)
+200
+160
+3.32 (0.05)
+42.40 (0.99)
+81.33 (1.63)
+4.28 (0.05)
+500
+400
+3.09 (0.03)
+40.69 (0.87)
+79.17 (1.11)
+4.05 (0.03)
+Example 6: p/n = 1.2, ρ = 0.8
+50
+60
+5.24 (0.23)
+36.43 (1.12)
+104.17 (4.37)
+5.80 (0.14)
+90
+108
+4.41 (0.12)
+44.84 (1.68)
+103.01 (4.03)
+5.16 (0.09)
+200
+240
+4.05 (0.07)
+51.46 (1.30)
+109.34 (3.17)
+4.98 (0.06)
+500
+600
+3.86 (0.03)
+54.21 (0.98)
+112.62 (1.71)
+4.82 (0.03)
+Table 1: Prediction risk, averaged over 100 replications. The standard error is given in paren-
+theses. For n ≤ p−r data, we compute the envelope by taking a very small value of λ = 10−8
+in the enhanced response envelope; see the definition of the envelope estimator (13) in Sec-
+tion 2.3. For n < p data, we fit the multivariate regression model by taking a tiny value of
+λ = 10−8 in the multivariate ridge regression.
+14
+
+0
+2
+4
+6
+8
+0
+5
+10
+15
+20
+25
+p/n
+Prediction risk
+Envelope
+Enhanced response envelope
+Figure 2: Prediction risk of the envelope and the enhanced response envelope with λ∗ =
+tr(Ω)p/(nc2), when n = 2000 and p varies. For n ≤ p−r data, we fit the envelope by taking
+a very small value of λ = 10−8 in the enhanced response envelope estimator; see the definition
+of the envelope estimator (13) in Section 2.3.
+5
+Real data
+In this section, we use two real datasets to illustrate the enhanced response envelope esti-
+mator. We use air pollution data in which the number of samples is bigger than the number
+of predictors (n > p) in the next subsection. In Subsection 5.2, we analyze near-infrared
+spectroscopy data in which the number of predictors is much bigger than the number of
+predictors (p ≫ n).
+We compare the prediction performance of the enhanced response envelope estimator to
+the envelope estimator, multivariate regression, and multivariate ridge regression.
+5.1
+Air pollution data
+The air pollution data are available and obtained directly from Table 1.5 of Johnson et al.
+(2002). The response vector y ∈ R5 consists of atmospheric concentrations of CO, NO, NO2,
+O3, and HC, recorded at noon in the Los Angeles area on 42 different days. The two predictors
+15
+
+0
+2
+4
+6
+8
+0
+5
+10
+15
+20
+25
+p/n
+Prediction risk
+Envelope
+Enhanced response envelope
+Figure 3: Prediction risk of the envelope and the enhanced response envelope with λ∗ =
+tr(Ω)p/(nc2), when n = 200 and p varies. For n ≤ p−r data, we fit the envelope by taking a
+very small value of λ = 10−8 in the enhanced response envelope estimator; see the definition
+of the envelope estimator (13) in Section 2.3.
+are wind speed and solar radiation. This data were analyzed in Cook (2018) to illustrate the
+effectiveness of the original envelope model compared to the standard multivariate regression
+model. They showed that the asymptotic standard errors of estimated components of β
+from the envelope model are significantly reduced compared to those from the standard
+multivariate regression model. We use the data to predict atmospheric concentrations from
+wind speed and solar radiation and compare the prediction performance of the enhanced
+response envelope estimator to the envelope estimator, the standard multivariate regression,
+and multivariate ridge regression.
+To compare the prediction performance, we borrow the nested cross validation idea
+(Wang and Zou, 2021; Bates et al., 2021), in which an inner cross-validation is performed
+to tune a model and an outer cross-validation is performed to provide a prediction error of
+the tuned model. We adopt the leave-one-out cross-validation (LOOCV) procedure for the
+outer loop because the LOOCV error is an unbiased estimator of the generalization error
+of the tuned model and is shown to have nice performance compared to other methods for
+16
+
+Enhanced
+envelope
+Envelope
+Multivariate
+linear reg
+Multivariate
+ridge reg
+Error
+8.859
+8.951
+9.192
+9.124
+Table 2: Air pollution data: prediction error of the enhanced response envelope method,
+the original envelope method, the multivariate linear regression, and the multivariate ridge
+regression.
+estimating generalization errors (Wang and Zou, 2021).
+We take the ith observation out from the data and set the remaining n−1 observations
+as the training set to fit and tune models. We standardize X of the training set so that each
+column has a mean of 0 and a standard deviation of 1. We perform ten-fold cross-validation
+to select (u, λ) from a fine grid of u ∈ {0, . . . , 5} and 20 equally spaced candidate λ-values
+in the scale of logarithm base 10 for the enhanced response envelope. For the envelope,
+we perform ten-fold cross-validation to choose u from {0, . . . , 5}. For the multivariate ridge
+model, ten-fold cross-validation is performed to select λ from 20 equally spaced λ-values in
+the scale of logarithm base 10. The ith observation we take out at the beginning is set as
+the test set. We standardize xi of the test set using the mean and standard deviation of the
+training data. We then calculate the squared prediction error, ∥yi − ˆβ(−i)xi∥2
+2/r, where ˆβ(−i)
+is the estimated regression coefficient derived from the training set. We repeat this process
+for i = 1, . . . , n and report �n
+i=1 ∥yi − ˆβ(−i)xi∥2
+2/(nr) in Table 2. We see that the enhanced
+response envelope estimator gives the smallest prediction error among all competitors.
+5.2
+Near-infrared spectroscopy data of fresh cattle manure
+Near-infrared spectroscopy data of cattle manure were collected by Gog´e et al. (2021). The
+data are available in the Data INRAE Repository at https://doi.org/10.15454/JIGO8R. This
+data contain 73 cattle manure samples that were analyzed by near-infrared spectroscopy
+using a NIRFlex device. Near-infrared spectra were recorded every 2 nm from 1100 to 2498
+nm on fresh homogenized samples. In addition, the cattle manure samples were analyzed
+for three chemical properties: the amount of dry matter, magnesium oxide, and potassium
+17
+
+Enhanced
+envelope
+Envelope
+Multivariate
+linear reg
+Multivariate
+ridge reg
+Error
+0.437
+0.460
+0.692
+0.492
+Table 3: Near-infrared spectroscopy data: prediction error from the enhanced response enve-
+lope method, the envelope method, the multivariate linear regression, and the multivariate
+ridge regression. We compute the envelope estimator by taking a very small value of λ = 10−8
+in the enhanced response envelope estimator; see the definition of the envelope estimator
+(13) in Section 2.3. We fit the multivariate regression model by taking a very small value of
+λ = 10−8 in the multivariate ridge regression.
+oxide. We use the data of 62 cattle manure samples which have no missing values. We
+standardize each chemical property to have a sample mean of 0 and a standard deviation of
+1. In our analysis, we consider the multivariate linear model, where xi ∈ R700 is the vector
+of near-infrared spectroscopy measurements and yi ∈ R3 is the vector of three chemical
+measurements to predict the three chemical properties from the absorbance spectra.
+In Table 3, we report the prediction error which is calculated using the same procedure
+described in the previous subsection, except that u is chosen from {0, . . . , 3}. Again, We see
+that the enhanced response envelope estimator has the smallest prediction error among all
+competitors.
+6
+Discussion
+In this paper, we have developed a novel envelope regularization function which is used to
+define the enhanced envelope estimator. We have shown that the enhanced envelope estimator
+is indeed better than the un-regularized envelope estimator in prediction. The asymptotic
+analysis of the risk function of envelope reveals, for the first time in the envelope literature,
+an interesting double descent phenomenon. The numeric examples in this work also suggest
+that the enhanced response envelope estimator is a promising new tool for multivariate
+regression.
+18
+
+Although this paper is focused on the case where the number of responses (r) is less
+than the number of samples and the number of predictors, it is interesting to consider the
+case when r → ∞ in ultrahigh-dimensional problems. Su et al. (2016) studied the response
+envelope for r → ∞ but p is fixed. When both p, r > n and diverge, there are additional
+technical issues to be addressed. For example, we may need another penalty term to handle
+the issues caused by the large r in the model. This direction of research will be investigated
+in a separate paper.
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+Statistics, 48, 161–182.
+A
+Proofs of Theorems
+A.1
+Proof of Theorem 1
+Note that
+R( ˆβΓ(λ)|X) = λ2tr(β(SX + λI)−1Σx(SX + λI)−1βT) + tr(Ω)
+n
+tr(ΣxSX(SX + λI)−2).
+Therefore, we have
+∂
+∂λR( ˆβΓ(λ)|X)
+= 2λ · tr(βSX(SX + λI)−2Σx(SX + λI)−1βT) − 2tr(Ω)
+n
+tr(ΣxSX(SX + λI)−3)
+≤
+p
+�
+i=1
+�
+2λ · σi(βTβ) − 2tr(Ω)
+n
+�
+σi(ΣxSX(SX + λI)−3),
+where σi(M) denotes the i-th largest eigenvalue of M. The inequality above comes from Von
+Neumann’s trace inequality.
+22
+
+Since
+∂
+∂λR( ˆβΓ(λ)|X) < 0 if λ < tr(Ω)/(nσ1
+�
+βTβ)
+�
+, R( ˆβΓ(λ)|X) is a monotonically
+decreasing function if 0 ≤ λ ≤ tr(Ω)/(nσ1
+�
+βTβ)
+�
+. Therefore, we have
+R( ˆβΓ(λ)|X) < tr(Ω)
+n
+tr(ΣxS+
+X),
+when 0 < λ < tr(Ω)/(nσ1
+�
+βTβ)
+�
+. Since
+tr(Ω)
+n
+tr(ΣxS+
+X) ≤ R( ˆβΓ|X),
+we prove the theorem.
+A.2
+Proof of Theorem 2
+Our analyses of limiting prediction risk follow that of Hastie et al. (2022).
+As Σx = I,
+R( ˆβΓ|X) = vecT(β)[ΠX ⊗ Ir]vec(β) + tr(Ω)
+n
+tr(S+
+X),
+R( ˆβΓ(λ)|X) = λ2tr(β(SX + λI)−2βT) + tr(Ω)
+n
+tr(SX(SX + λI)−2),
+where ΠX = Ip − S+
+XSX.
+A.2.1
+Proof for envelope estimator when γ < 1
+Let us consider the case where p/n → γ ∈ (0, 1). From Theorem 1 of Bai and Yin (2008),
+σmin(SX) ≥ (1 − √γ)2/2 and σmax(SX) ≤ 2(1 + √γ)2 almost surely for all sufficiently large
+n. Therefore, in this case, SX is invertible and the bias term of R( ˆβΓ|X) is 0, almost surely.
+The variance term of R( ˆβΓ|X) is
+tr(Ω)
+n
+tr(S+
+X) = p · tr(Ω)
+n
+� 1
+sdFSX(s),
+where FSX(s) is the spectral measure of SX. By the Marchenko-Pastur theorem, which says
+that FSX → Fγ, and the Portmanteau theorem,
+� 2(1+√γ)2/
+(1−√γ)2/2
+1
+sdFSX(s) →
+� 2(1+√γ)2/
+(1−√γ)2/2
+1
+sdFγ(s) =
+� 1
+sdFγ(s).
+23
+
+The equality is because the support of Fγ is [(1 − √γ)2, (1 + √γ)2]. We can also remove the
+upper and lower limits of integration on the left-hand side by Theorem 1 of Bai and Yin
+(2008). Thus, combining above results, we arrive at
+R( ˆβΓ|X) → γ · tr(Ω)
+� 1
+sdFγ(s).
+The Stieltjes transformation of Fγ is given by
+m(z) =
+�
+1
+s − zdFγ(s) = (1 − γ − z) −
+�
+(1 − γ − z)2 − 4γz)
+2γz
+,
+for any real z < 0. By taking the limit z → 0−, the proof is completed.
+A.2.2
+Proof for envelope estimator when γ > 1
+The variance term of R( ˆβΓ|X) is
+tr(Ω)
+n
+tr(S+
+X) = tr(Ω)
+n
+tr((XXT/n)+) = tr(Ω)
+p
+tr((XXT/p)+).
+Considering n/p → τ = 1/γ < 1, by the same arguments from the proof above, we conclude
+that
+tr(Ω)
+n
+tr(S+
+X) → tr(Ω)
+1
+γ − 1.
+Let β = [bT
+1 . . . bT
+r ]. The bias term is
+vecT(β)[ΠX ⊗ Ir]vec(β) =
+r
+�
+i=1
+bT
+i ΠXbi =
+r
+�
+i=1
+lim
+z→0+ zbT
+i (SX + zI)−1bi.
+From Theorem 1 of Bai et al. (2007), we have that
+zbT
+i (SX + zI)−1bi → z
+�
+1
+s + zFγ(s) = z∥bi∥2m(−z) a.s.,
+for any i = 1, . . . , r. We further have that
+r
+�
+i=1
+zbT
+i (SX + zI)−1bi → zc2m(−z) a.s.
+By the Arzela-Ascoli theorem and the Moore-Osgood theorem, we exchange limits and
+arrive at
+lim
+z→0+
+r
+�
+i=1
+zbT
+i (SX + zI)−1bi → c2 lim
+z→0+ zm(−z) = c2(1 − 1/γ) a.s.
+Combining the variance and the bias terms, we complete the proof.
+24
+
+A.2.3
+Proof for enhanced envelope estimator
+We use the similar techniques from the envelope estimator for both variance and bias terms.
+The variance term of R( ˆβΓ(λ)) becomes
+tr(Ω)
+n
+tr(SX(SX + λI)−2) → γtr(Ω)
+�
+s
+(s + λ)2Fγ(s).
+Let gn,λ(η) = λ · tr(β(SX + λ(1 + η)I)−1βT), η ∈ [−1/2, 1/2]. The bias term of R( ˆβΓ(λ))
+is
+λ2tr(β(SX + λI)−2βT) = − ∂
+∂ηgn(λ, 0).
+Because
+gn,λ(η) → λc2m(−λ(1 + η)) = λc2
+�
+1
+s + λ(1 + η)dFγ(s),
+and derivative and limit are exchangeable, we have that
+λ2tr(β(SX + λI)−2βT) → λ2c2
+�
+1
+(s + λ)2dFγ(s).
+We can conclude that,
+R( ˆβΓ(λ)) →
+� λ2c2 + s · γtr(Ω)
+(s + λ)2
+Fγ(s).
+The right-hand side is minimized at λ∗ = γtr(Ω)/c2. In such case, the right-hand side
+becomes γtr(Ω) · m(−λ∗).
+25
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+Bipartite unique-neighbour expanders via Ramanujan graphs
+Ron Asherov and Irit Dinur∗
+Weizmann Institute, Rehovot, Israel
+Abstract
+We construct an infinite family of bounded-degree bipartite unique-neighbour expander graphs with
+arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et
+al., our construction is simpler and may be closer to be implementable in practice due to the smaller
+constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget
+in a way that generalizes the construction of unique neighbour expanders by Alon and Capalbo. For
+the analysis of our construction we prove a strong upper bound on average degrees in small induced
+subgraphs of bipartite Ramanujan graphs. Our bound generalizes Kahale’s average degree bound to
+bipartite Ramanujan graphs, and may be of independent interest. Surprisingly, our bound strongly relies
+on the exact Ramanujan-ness of the graph and is not known to hold for nearly-Ramanujan graphs.
+1
+Introduction
+An infinite family Gn = (Ln ⊔ Rn, En) of (c, d)-biregular graphs with |Ln| + |Rn| → ∞ is called a unique
+neighbour expander family if there exists δ > 0 such that for every n and every set of left side vertices S ⊆ Ln
+of size |S| ≤ δ|Ln| there exists a unique neighbour of S in Gn, namely a vertex in Rn that is connected to
+exactly one vertex in S. We only require that sets of left vertices have unique neighbours, and arbitrarily
+small right side sets may have no unique neighbour.
+Alon and Capalbo [AC02] construct several explicit families of unique neighbour expanders, via an elegant
+composition of a Ramanujan graph and a gadget. They construct three families of general (non-bipartite)
+graphs in which all small sets have unique neighbours, and one family of slightly unbalanced bipartite graphs
+where small sets on the left have unique neighbors on the right. In their construction the left side is 22/21
+times bigger than the right side. The more imbalanced the graph, the harder it is for small left hand side
+sets to expand into the right hand side. Capalbo et. al. [Cap+02] construct arbitrarily unbalanced bipar-
+tite graphs that are lossless expanders, a notion strictly stronger than unique neighbour expansion. Their
+construction is based on a sequence of somewhat involved composition steps using randomness conductors.
+Our main theorem is an efficient construction of an infinite family of bipartite unique neighbour expanders
+for any constant imbalance α, and any sufficiently large left-regularity degrees of a specific form:
+Theorem 1. There is a function ˆq : N × R → N such that for every integer c0 > 5 and real number α > 1,
+if q > ˆq(c0, α) is a prime power and αc0(q + 1) is an integer, then there is a polynomial-time construction of
+an infinite family of (c0(q + 1), αc0(q + 1))-biregular unique neighbour expanders.
+The theorem is proven in Section 6.2, and provides a way to compute ˆq(c0, α). Here are some computed
+values of ˆq(c0, α) for several values of c0, α.
+∗Irit Dinur acknowledges support by ERC grant 772839 and ISF grant 2073/21.
+1
+arXiv:2301.03072v1  [math.CO]  8 Jan 2023
+
+c0
+α
+ˆq(c0, α)
+10
+2
+18907
+35
+2
+1492
+100
+100
+136051
+100
+1.01
+1135
+Notice that ˆq(co, α) increases with α, reflecting the fact that constructions with larger α (namely, more
+imbalanced sides) are harder to come by, and require larger degrees.
+The construction uses an infinite family of bipartite Ramanujan graphs, namely graphs whose non-
+trivial spectrum is contained in the spectrum of the (c, d)-biregular tree (see Preliminaries for details). We
+construct the unique neighbour expander family by taking a family of bipartite Ramanujan graphs and
+combining them with a fixed size graph (“gadget”), with a good unique neighbour property (small sets have
+unique neighbours), whose existence is shown via the probabilistic method (Lemma 11). The combination
+is done as follows. We first place a copy of the gadget for every right side vertex of the Ramanujan graph.
+The vertex is replaced by the right side of the gadget, and its neighbours are identified with the left side of
+the gadget. The gadget is used to route the neighbours of each left side vertex in the Ramanujan graph to
+its neighbours in the product graph.
+Expansion in the product graph comes from unique neighbour expansion of the gadget together with
+low degree vertices in the Ramanujan graph. Sufficiently low degree vertices are guaranteed to exist thanks
+to the following (new) bound on the average degree of induced subgraphs of bipartite Ramanujan graphs,
+which may be of independent interest.
+Theorem 2. Let G = (L ⊔ R, E) be a (c, d)-biregular Ramanujan graph, and let ε > 0. Then there exists
+δ > 0, that depends only on ε, c, d, such that for every S ⊂ L of size |S| ≤ δ|L|, the set N(S) ⊆ R of the
+neighbours of S satisfies
+c|S|
+|N(S)| ≤ 1 + (1 + ε)
+�
+d − 1
+c − 1 .
+The theorem shows that every small set on the left side admits neighbours on the right side with low degree
+in the induced subgraph. The proof involves recursive analysis of non-backtracking paths. Interestingly, the
+recursion has a nice solution only when the graph is Ramanujan. It is unclear whether this method can be
+extended to “nearly-Ramanujan” graphs.
+Combining the average degree upper bound with the gadget, the low-degree right-side vertices in the
+Ramanujan graph imply a small set of left-side vertices in the gadget; this set will have a unique neighbour
+in the gadget, which gives (via Lemma 12) a unique neighbour in the constructed graph.
+Even though Ramanujan graphs are the best spectral expanders one can hope for, an efficient construc-
+tion of Ramanujan graphs (be them bipartite or not) does not immediately imply that we can construct
+unique neighbour expanders. In the d-regular case, Kahale shows ([Kah95, Thm 5.2]) that there are nearly-
+Ramanujan graphs with expansion at most d/2, which is not enough for unique neighbour expansion. In fact,
+recently Kamber and Kaufman [KK22] proved that some Ramanujan graphs strongly fail to have unique
+neighbour expansion, by giving explicit constructions of arbitrarily small sets that do not admit a unique
+neighbour.
+As mentioned, the graph product we define requires a fixed size gadget, whose proof of existence is not
+constructive. In principle, such a gadget could be found by exhaustive search since we are working in a
+constant size search space. The gadget’s size in our construction is at least cubic in q, so exhaustive search
+is impractical for even small values of q. Unfortunately we know of no efficient construction of a gadget with
+the required parameters. It is possible that the graph sampling method present in [AK19] can be used to
+construct fixed size gadgets more efficiently.
+The rest of this work is organized as follows. In Section 2 we survey some of the uses of unique neighbour
+expanders, and mention known constructions of such graphs. Section 3 provides basic definitions and results.
+Our main technical tool, that asserts the low induced degree in bipartite Ramanujan graph, is stated and
+proven in Section 4. We prove the existence of a fixed-size gadget with good unique neighbour expansion
+2
+
+properties in Section 5. In Section 6 we define the way we use the Ramanujan graphs and the gadget to
+construct bipartite unique neighbour expanders, and by that prove Theorem 1.
+2
+Related work
+One of the prominent uses of bipartite expanders in general and bipartite unique neighbour expanders in
+particular, and the motivation for this work, is the construction of error correcting codes. The works of
+Tanner [Tan81] and later Sipser and Spielman [SS96] construct linear error correcting codes C(B, C0) from
+a bipartite graph B and a smaller linear code C0. It is shown that under some assumptions on the code C0
+and the expansion properties of the bipartite graph B, the resulting code has good distance. This gives a
+way to take a family of graphs and transform it into a family of codes. Our work describes a construction
+that, in a sense, goes the other way around: given two bipartite graphs, B and B0, we view B0 as a parity
+check graph1 of the base code C0, and B plays the role of the underlying graph of a Tanner code C(B, C0).
+Our output graph is just the parity check graph of C(B, C0). We give full details of this graph product in
+Section 6.1.
+In [DSW06; BV09] it is shown that codes constructed on top of unique neighbour expanders are weakly
+smooth and can be used to construct robustly testable codes. But the uses of unique neighbour expanders are
+not limited to error correcting codes: for example, such graphs may be used in the context of non-blocking
+networks, where it is required to connect several input-output terminals via paths in a non-intersecting
+fashion. Arora et al. [ALM96] use graphs with expansion beyond the d/2 barrier to establish the existence
+of unique neighbours in the graph, which are useful in finding input-output paths in the online settings.
+Roughly speaking, when routing a set of input-output pairs, the algorithm can use all unique neighbours
+freely since they are guaranteed not to interfere with any other paths.
+Pippenger [Pip93] uses explicit
+constructions of spectral expanders in order to solve a similar problem, in the case where the route planning
+is computed locally. There the spectral expansion of a graph is proven to imply a combinatorial expansion,
+in a similar way to our Theorem 2.
+Another use for unique neigbhour expanders is for load-balancing problems, such as the token distribution
+problem described in [PU89], and the similar pebble distribution problem, briefly discussed in [AC02]. In
+the latter, pebbles are placed arbitrarily on vertices of a graph, and need to be distributed via edges of the
+graph such that no vertex has more than one pebble. Given that the total number of pebbles is small and
+that the graph has the unique neighbour property, we have an efficient parallel algorithm for redistributing
+the pebbles.
+Alon and Capalbo [AC02] construct several families of unique neighbour expanders, one of them is a
+family of bipartite graphs whose left side is 22/21 times bigger than the right side. Similar to the construction
+presented at this work, each graph in the constructed family is a combination of a Ramanujan graph and a
+fixed graph. These graphs are not (bi-)regular but their degrees are bounded by a constant. Becker [Bec16]
+uses a different family of 8-regular Ramanujan graphs in order to construct a family of (non-bipartite) unique
+neighbour expanders, with the additional property that each graph in the family is a Cayley graph.
+A different approach to constructing bipartite graphs uses randomness conductors. Randomness conduc-
+tors are functions that receive a bitstring with some entropy (according to some measure of entropy), and a
+uniformly random bitstring, and output a bitstring, with certain guarantees on its entropy. Some conduc-
+tors can be constructed explicitly via a spectral method, and Capalbo et al. [Cap+02] combine them in a
+zig-zag-like fashion in order to construct an infinite family of bipartite lossless expanders, namely bipartite
+graphs with fixed left-regularity c where small enough sets contained in the left side have at least c(1 − ε)
+neighbours on the right side. These graphs are trivially unique neighbour expanders, since a simple counting
+argument shows that if a set expands by a factor of more than c/2, then it has unique neighbours.
+1This is a bipartite graph whose incidence structure is given by the parity check matrix.
+3
+
+3
+Preliminaries
+3.1
+Expander graphs
+In this work we deal with undirected graphs, that may contain multiple edges between two vertices, but do
+not contain self-loops. For a graph G and a subset of its vertices S we denote by NG(S) the neighbourhood
+of S, namely all vertices adjacent to some vertex in S. When the graph in discussion is obvious, we may
+omit it and write N(S). We say that v is a unique neighbour of S if there is a unique u ∈ S that is adjacent
+to v.
+Let (Gn) be a series of graphs with the number of vertices growing to infinity. There are several well
+studied notions of expansion in graph families; we note some of them.
+1. Vertex expansion. (Gn) is a (δ, α)-vertex expander if for every n and any subset S ⊆ VGn, if |S| ≤ δ|VGN |
+we have that |NGN (S)| ≥ α|S|.
+2. Edge expansion. (Gn) is a (δ, α)-edge expander if for every n and any subset S ⊆ VGn, if |S| ≤ δ|VGN |
+we have that at least an α-fraction of the edges with one endpoint in S have their other endpoint
+outside of S.
+3. Spectral expansion. Assume that (Gn) are all d-regular, and let An be the adjacency operator associated
+with Gn, so An is indexed by vertices of Gn and (An)uv counts how many edges there are between
+u and v in Gn. Let λ1 ≥ . . . ≥ λVn be its spectrum. It can be seen that λ1 = d. Then (Gn) is a
+λ-spectral expander if for all n and i ̸= 1 we have |λi| ≤ λ.
+4. Unique neighbour expansion. (Gn) is a δ-unique neighbour expander if for every n, any subset S ⊆ VGn
+of size at most δ|VGN | has a unique neighbour.
+These definitions apply to bipartite graphs Gn = (Ln ⊔ Rn, En) as well, with the exception that we usually
+consider sets contained in the left side only, and require that Ln/Rn is a constant, normally greater than
+1. In this case we note that edge expansion is meaningless (since all edges leaving the left side enter the
+right side), and if a bipartite graph is (c, d)-biregular, namely if all left-side vertices have degree c and all
+right-side vertices have degree d, then the largest eigenvalue of the associated adjacency operator is
+√
+cd.
+It can be seen that for d-regular graphs, the best spectral expansion we can hope for is α = 2
+√
+d − 1.
+These graphs are known as Ramanujan graphs.
+3.2
+Bipartite Ramanujan graphs
+Ramanujan graphs have the best spectral gap [Nil91], and their non-trivial eigenvalues are contained in the
+spectrum of the infinite d-regular tree Td. Similarly, in the bipartite case, Biregular Ramanujan graphs are
+defined via their relation to the infinite biregular trees: the infinite (c, d)-biregular tree Tc,d, for d > c, has
+the spectrum
+λ ∈ spec(Tc,d) ⇔ |λ| ∈ {0} ∪
+�√
+d − 1 −
+√
+c − 1,
+√
+d − 1 +
+√
+c − 1
+�
+(see, e.g., [GM88], [LS96].) We therefore say that a finite (c, d)-biregular graph is bipartite Ramanujan if its
+nontrivial eigenvalues lie in this set. That means that every eigenvalue λ of a bipartite Ramanujan graph
+belongs to one of these classes:
+1. Trivial: λ = ±
+√
+cd, with eigenvectors fixed on either sides, or λ = 0;
+2. λ ∈ [
+√
+d − 1 − √c − 1,
+√
+d − 1 + √c − 1] are the nontrivial positive eigenvalues;
+3. λ ∈ [−√c − 1 −
+√
+d − 1, √c − 1 −
+√
+d − 1] are the nontrivial negative eigenvalues. Note that since the
+graph is bipartite, λ is an eigenvalue if and only if −λ is an eigenvalue.
+4
+
+By an extension of the Alon-Boppana bound, given in [FL96], this is the best spectral gap we can hope for,
+at least as far as upper bounds for |λ| are concerned. We note that unlike the d-regular case, we require a
+lower bound to |λ| too, which is essential for our proof.
+While there is a vast literature on the construction of d-regular Ramanujan graph (most prominently
+[LPS88] and [Mar88]), less is known about bipartite Ramanujan graphs. In 2014 Marcus et al. [MSS13]
+proved the existence of biregular graphs with one-sided spectral graphs that resemble the Ramanujan bounds:
+these graphs satisfy the one-sided inequality only, namely |λ| ≤
+√
+d − 1 + √c − 1 for every nontrivial eigen-
+value λ. Gribinski et al. [GM21] showed a polynomial-time construction of such graphs, for every degrees
+(d, kd) for any integers d, k. These graphs do not suffice for our analysis, since we make explicit use of the
+lower bound |λ| ≥
+√
+d − 1 − √c − 1 too.
+In 2021 Brito et al. [BDH22] proved that a random biregular graph is nearly Ramanujan with high
+probability. Interestingly, and unlike other works in this field, our proof strongly relies on the graph to be
+exactly Ramanujan, so we cannot use those constructions either.
+We use an explicit construction of bipartite Ramanujan graphs (with both bounds on non-trivial eigen-
+values) given by Ballantine et al.:
+Theorem 3 ([Bal+15]). For every prime power q, there exists an explicit construction of a (q + 1, q3 + 1)-
+biregular Ramanujan graph.
+4
+Vertex expansion in biregular Ramanujan graphs
+Our main technical tool is the following theorem showing that bipartite Ramanujan graphs exhibit excellent
+left-to-right expansion. We restate the theorem for convenience.
+Theorem 2. Let G = (L ⊔ R, E) be a (c, d)-biregular Ramanujan graph, and let ε > 0. Then there exists
+δ > 0, that depends only on ε, c, d, such that for every S ⊂ L of size |S| ≤ δ|L|, the set N(S) ⊆ R of the
+neighbours of S satisfies
+c|S|
+|N(S)| ≤ 1 + (1 + ε)
+�
+d − 1
+c − 1 .
+We note that the quantity on the left hand side of the inequality can be interpreted as follows. Look
+at the bipartite graph induced by taking the vertices S on the left and N(S) on the right. Since every left
+vertex has c outgoing edges, the total number of edges in the induced subgraph is c|S|. This means that
+the expression on the left hand side of the inequality is exactly the average degree of the right side of the
+induced subgraph. Interestingly, the bound in this theorem is strictly stronger than what we would get from
+just applying the expander mixing lemma which amounts to
+c|S|
+|N(S)| ≤ (1 + ε) ·
+�
+1 + d − 1
+c − 1 + 2
+�
+d − 1
+c − 1
+�
+.
+See Claim 4 for details. The fact that we improve upon the expander mixing lemma is perhaps not surprising
+since our analysis is based on enumerating non-backtracking paths, and not just on magnitude of the second
+largest eigenvalue. We also use lower bounds on the magnitude of all nontrivial eigenvalues, whereas the
+expander mixing lemma uses just upper bounds.
+4.1
+Comparison to known bounds
+As noted above, Theorem 2 is an improvement of the bound that the expander mixing lemma gives in similar
+settings, which only uses the one-sided inequality |λ| ≤
+√
+d − 1 + √c − 1. For reference, we state and prove
+the expander mixing lemma for bipartite Ramanujan graphs.
+5
+
+Claim 4 (Expander mixing lemma for bipartite Ramanujan graphs). Let G = (L⊔R, E) be a (c, d)-biregular
+Ramanujan graph, and let ε > 0. Then there exists δ > 0 such that for every S ⊆ L of size |S| ≤ δ|L|, the
+neighbourhood of S satisfies
+c|S|
+|N(S)| ≤ (1 + ε)
+�
+1 + d − 1
+c − 1 + 2
+√
+d − 1
+√c − 1
+�
+.
+Proof. The expander mixing lemma for biregular graphs says that for every S ⊆ L, T ⊆ R we have
+����
+|e(S, T)|
+|E|
+− |S|
+|L| · |T|
+|R|
+���� ≤
+λ
+√
+cd
+�
+|S|
+|L| · |T|
+|R|
+where λ is the second largest eigenvalue of G (see, e.g., [Hae95]). It is clarified that we consider the spectrum
+of G as an adjacency operator, so the largest eigenvalue is
+√
+cd.
+Picking T = N(S) means all edges coming out from S are in the cut, namely |e(S, T)| = c|S|. Plugging
+that in gives
+����
+c|S|
+c|L| − |S|
+|L| · |N(S)|
+|R|
+���� ≤
+λ
+√
+cd
+�
+|S|
+|L| · |N(S)|
+|R|
+.
+Multiplying both sides by |L|
+|S| gives
+����1 − |N(S)|
+|R|
+���� ≤
+λ
+√
+cd
+�
+|S|
+|L| · |N(S)|
+|R|
+· |L|
+|S| =
+λ
+√
+cd
+�
+|N(S)|
+|R|
+· |L|
+|S| =
+λ
+√
+cd
+�
+|N(S)|
+|S|
+·
+�
+d
+c = λ
+c
+�
+|N(S)|
+|S|
+(1)
+where we also used the fact that |E| = c|L| = d|R|.
+Let us assume that |S| = α|L|. Then we can upper bound |N(S)| by
+|N(S)| ≤ c|S| = αc|L| = αd|R|
+and so we have
+1 − |N(S)|
+|R|
+≥ 1 − dα|R|
+|R|
+= 1 − dα.
+We square (1) and plug in the last inequality to get
+(1 − dα)2 ≤ λ2
+c · |N(S)|
+c|S| .
+Recall that G is bipartite Ramanujan, so |λ| ≤
+√
+d − 1 + √c − 1. Use that and rearrange:
+c|S|
+|N(S)| ≤ λ2
+c (1 − dα)−2
+≤ d − 1 + c − 1 + 2
+√
+d − 1√c − 1
+c
+(1 − dα)−2
+≤ d − 1 + c − 1 + 2
+√
+d − 1√c − 1
+c − 1
+(1 − dα)−2
+=
+�
+1 + d − 1
+c − 1 + 2
+√
+d − 1
+√c − 1
+�
+(1 − dα)−2.
+The claim is proven by noting that there is some δ > 0 such that (1 − dα)−2 ≤ 1 + ε for every α < δ, namely
+whenever |S| ≤ δ|L|.
+6
+
+Kahale proved ([Kah95, Thm 4.2]) that in d-regular Ramanujan graphs (not necessarily bipartite), small
+induced subgraphs have average degree at most 1 +
+√
+d − 1. Interestingly, this result can be deduced almost
+immediately from Theorem 2. This is due to the following lemma, proven in Appendix A, which asserts that
+the edge-vertex incidence graph (see [SS96]) of a d-regular Ramanujan graph is a (2, d)-biregular Ramanujan
+graph:
+Lemma 5. Let G be a d-regular Ramanujan graph, and G′ its edge-vertex incidence graph. Then G′ is a
+(2, d)-biregular Ramanujan graph.
+We state and prove Kahale’s bound, but we will not use it in our construction.
+Corollary 6. Let G = (VG, EG) be a d-regular Ramanujan graph, and let ε > 0. Then there exists δ > 0
+such that for every induced subgraph S with at most δ|VG| vertices, the average degree of S is at most
+dS := 2|ES|
+|VS| ≤ 1 + (1 + ε)
+√
+d − 1.
+Proof. Let G = (VG, EG) be a d-regular Ramanujan graph and ε > 0. We define G′ = (LG′ ⊔ RG′, EG′) as
+the edge-vertex incidence graph, namely LG′ = EG, RG′ = VG, and for every edge e = {u, v} in G we have
+the two edges {e, u} and {e, v} in G′. Since the degree of every vertex in G is d, and since every edge has
+two endpoints, we have that G′ is a (2, d)-biregular graph. Lemma 5 asserts that G′ is Ramanujan in the
+bipartite sense. By Theorem 2, there exists δ > 0 such that if T ⊆ LG′ is of size at most δ|LG′|, then
+2|T|
+|NG′(T)| ≤ 1 + (1 + ε)
+√
+d − 1.
+A subgraph S = (VS, ES) of G satisfies that ES is a subset of left-side vertices in G′, VS is a subset of
+right-side vertices in G′, and VS = NG′(ES) (because if an edge is in the subgraph then both of its endpoints
+are in the subgraph, and we assume that the subgraph does not contain an isolated vertex). Therefore, if ES
+is sufficiently small, namely if |ES| ≤ δ|LG′| = δ|EG|, then by Theorem 2 the average degree of NG′(ES) = VS
+is bounded by 1 + (1 + ε)
+√
+d − 1.
+We add that if we wish to find a bound the number of vertices, we note that |ES| ≤ d
+2|VS|. So every
+induced subgraph with no more than
+2
+dδ|EG| = δ|VG| vertices will satisfy the required average degree
+bound.
+4.2
+Proof of Theorem 2
+Theorem 2 is proven by enumerating non-backtracking paths. A non-backtracking path of length ℓ is a
+sequence of edges ((s(ei), t(ei)))ℓ
+i=1 such that for every i, t(ei) = s(ei+1) and s(ei) ̸= t(ei+1).
+For a bipartite graph G and a subset S of left side vertices we define Mℓ(S) to be the number of all non-
+backtracking paths whose all left-side vertices are in S, and Mℓ(S, G) to be the number of non-backtracking
+paths whose first and last left-side vertices are in S. Clearly Mℓ(S) ≤ Mℓ(S, G), as paths of the latter type
+may leave S ⊔N(S) (before re-entering S at the last step). We use a lower bound on Mℓ(S) due to [Kam19]:
+Lemma 7. For every undirected bipartite graph G = (LG ⊔ RG, EG) and integer l it holds that
+Mℓ(LG) ≥ |EG|
+��
+( ¯dL − 1)( ¯dR − 1)
+�ℓ−1
+where ¯dL, ¯dR are the average degrees of the left and right sides of G respectively.
+We state and prove an upper bound on Mℓ(S, G):
+7
+
+Lemma 8. Let G be a (c, d)-biregular Ramanujan graph with n vertices on the left side, and S a subset of
+the left side. Then for every integer ℓ:
+M2ℓ(S, G) ≤ |S|
+�
+(2 +
+√
+d − 1)ℓ + 2
+�
+(c − 1)ℓ/2(d − 1)ℓ/2
+provided that S is small enough:
+|S|(c − 1)ℓ/2(d − 1)ℓ/2 ≤ n.
+(2)
+Before proving the upper bound, we show how these bounds can be combined to obtain Theorem 2.
+Proof of Theorem 2. Let ℓ be an integer to be determined later, S ⊆ L a sufficiently small subset (where
+sufficiently smalls means (2)). Denote by N(S) ⊆ R the neighbours of S. The subgraph induced on S ∪N(S)
+has c|S| edges, with left degrees all c and average right degree ¯dR =
+c|S|
+|N(S)|.
+Chaining the inequalities in Lemma 7 and Lemma 8, we have
+c|S|
+�
+(c − 1)( ¯dR − 1)
+� 2ℓ−1
+2
+≤ M2ℓ(S) ≤ M2ℓ(S, VG) ≤ |S| ·
+�
+(2 +
+√
+d − 1)ℓ + 2
+�
+· (c − 1)ℓ/2(d − 1)ℓ/2.
+Simplifying, we get,
+c(c − 1)ℓ− 1
+2 ( ¯dR − 1)ℓ− 1
+2 ≤
+�
+(2 +
+√
+d − 1)ℓ + 2
+�
+· (c − 1)ℓ/2 · (d − 1)ℓ/2
+( ¯dR − 1)ℓ− 1
+2 ≤
+�
+(2 +
+√
+d − 1)ℓ + 2
+� √c − 1
+c
+��
+d − 1
+c − 1
+�ℓ
+¯dR − 1 ≤
+�
+�
+�
+�
+�
+(2 +
+√
+d − 1)ℓ + 2
+� √c − 1
+� ¯d − 1
+c
+�
+��
+�
+⋆
+�
+�
+�
+�
+1/ℓ
+·
+�
+d − 1
+c − 1
+Since ¯d ≤ d, we have that ⋆ = O(ℓ), so ⋆1/ℓ = O(1), hence for a fixed ε > 0 there exists a constant ℓ (that
+depends only on ε, c, d) such that ⋆1/ℓ ≤ 1 + ε; this ℓ determines, via inequality (2), a fixed δ such that
+whenever |S| ≤ δn we have
+¯dR ≤ 1 + (1 + ε)
+�
+d − 1
+c − 1 .
+We proceed to prove Lemma 8.
+For a bipartite graph G = (LG ⊔ RG, EG) and an integer ℓ, we define ALL
+ℓ
+, ALR
+ℓ
+, ARL
+ℓ
+, ARR
+ℓ
+as operators
+corresponding to non-backtracking paths of length ℓ, i.e.
+ALL
+ℓ
+: L2(LG) → L2(LG)
+,
+(ALL
+ℓ
+f)(x) =
+�
+(e1,...,eℓ),t(eℓ)=x,s(e1),t(eℓ)∈LG
+f(s(e1))
+with the summation over all non-backtracking paths of length ℓ, and similarly for the other operators.
+Let M be the operator corresponding to a single step from the right side G to the left side of G, namely
+M has |RG| rows and |LG| columns, with Muv counting the number of edges between u ∈ RG and v ∈ LG
+in G. Then the following recursive formulae hold for every integer ℓ > 1:
+M ⊤ALL
+ℓ
+= ARL
+ℓ+1 + (d − 1)ARL
+ℓ−1
+M ⊤ALR
+ℓ
+= ARR
+ℓ+1 + (d − 1)ARR
+ℓ−1
+MARL
+ℓ
+= ALL
+ℓ+1 + (c − 1)ALL
+ℓ−1
+MARR
+ℓ
+= ALR
+ℓ+1 + (c − 1)ALR
+ℓ−1
+8
+
+The first formula is explained as follows.
+Every non-backtracking path from R to L of length ℓ + 1 is
+composed of a non-backtracking path from L to L of length ℓ plus an extra step (that’s the M ⊤ALL
+ℓ
+factor.)
+The opposite is true, except for paths counted in M ⊤ALL
+ℓ
+that do backtrack, namely those made of a non-
+backtracking path of length ℓ − 1, and walking back and forth along the same edge. There are d − 1 ways to
+choose that edge (since it cannot be the one that was last in the path of length ℓ − 1, otherwise it wouldn’t
+be counted in M ⊤ALL
+ℓ
+), so we need to subtract (d − 1)ARL
+ℓ−1. The rest of the equations are explained in an
+analog way.
+Due to symmetry we have:
+(ALL
+ℓ
+)⊤ = ALL
+ℓ
+,
+(ARR
+ℓ
+)⊤ = ARR
+ℓ
+,
+(ALR
+ℓ
+)⊤ = ARL
+ℓ
+And since the graph is bipartite we have:
+ALR
+2ℓ = 0
+,
+ARL
+2ℓ = 0
+ALL
+2ℓ+1 = 0
+,
+ARR
+2ℓ+1 = 0
+These equations yield a recursive formula for ALL
+ℓ
+, with the following initial conditions:
+ALL
+2
+= MM ⊤ − cI
+ALL
+4
+= MM ⊤ALL
+2
+− (c − 1 + d − 1)ALL
+2
+− c(d − 1)I
+MM ⊤ALL
+ℓ
+= ALL
+ℓ+2 + ((c − 1) + (d − 1))ALL
+ℓ
++ (c − 1)(d − 1)ALL
+ℓ−2
+,
+∀ℓ ≥ 4
+(3)
+The following lemma, proven in Appendix A, suggests a way to find a non-recursive formula for ALL
+2ℓ , given
+such linear recursive relations with fixed coefficients.
+Lemma 9. Let (xn) be a series defined via a second order linear recurrence with fixed coefficients A, B ∈ C:
+xn = Axn−1 + Bxn−2
+Assume λ1 ̸= λ2 are (real or complex) roots of the characteristic polynomial λ2 − Aλ − B. Then there are
+α, β ∈ C, that depend on the initial conditions x0, x1, such that
+xn = αλn
+1 + βλn
+2
+for every n ≥ 0.
+If the characteristic polynomial has a single root λ of multiplicity 2, then there are α, β ∈ C such that
+xn = αλn + βnλn
+for every n ≥ 0.
+We use the lemma to bound the eigenvalues of ALL
+2ℓ given bounds on the spectrum of the biregular graph.
+Lemma 10. Let G be a (c, d)-biregular graph. Then there is a sequence of polynomials with integer coeffi-
+cients (pℓ(x)) such that for every eigenpair (λ, v) of G, pℓ(λ2) is an eigenvalue of ALL
+2ℓ , and moreover, for
+every λ ∈ R, if
+|λ| ∈ {0} ∪ [
+√
+d − 1 −
+√
+c − 1,
+√
+d − 1 +
+√
+c − 1]
+(4)
+then
+|pℓ(λ2)| ≤ (2 +
+√
+d − 1)ℓ(c − 1)ℓ/2(d − 1)ℓ/2.
+(5)
+9
+
+Proof. The recursive formulae proven above (3) suggest that there is a series of polynomials pn(x) with
+integer coefficients such that ALL
+2n = pn(MM ⊤). Note that the graph’s adjacency matrix is
+AG =
+� 0
+M
+M ⊤
+0
+�
+And so, if (λ, v) is an eigenpair of G, then (λ2, v) is an eigenpair of
+A2
+G =
+�MM ⊤
+0
+0
+M ⊤M
+�
+.
+This shows that pℓ(λ2) is an eigenvalue of ALL
+2ℓ whenever λ is an eigenvalue of G. The converse is also true.
+The formulae (3) can be transformed so as to convey that pn(x) satisfies these equations:
+p1(x) = x − c
+,
+p2(x) = x2 + (2 − 2c − d)x + c(c − 1)
+xpn(x) = pn+1(x) + (c − 1 + d − 1)pn(x) + (c − 1)(d − 1)pn−1(x)
+for all n > 1. Setting n = 1 gives an equation involving p0(x), p1(x), p2(x). We can solve this equation for
+p0(x) and get a simpler description of the initial conditions:
+p0(x) =
+c
+c − 1
+,
+p1(x) = x − c
+(6)
+xpn(x) = pn+1(x) + (c − 1 + d − 1)pn(x) + (c − 1)(d − 1)pn−1(x)
+(7)
+for all n > 0.
+We fix some t that satisfies (4), namely such that
+|t| ∈ {0} ∪ [
+√
+d − 1 −
+√
+c − 1,
+√
+d − 1 +
+√
+c − 1].
+We first deal with the case where |t| ∈ (
+√
+d − 1 − √c − 1,
+√
+d − 1 + √c − 1), and later we will consider the
+edge cases where t is one of the endpoints of the segment or 0. Let us write x = t2. We have that for this
+fixed x, the series (pn(x))n satisfies a second order linear recurrence with fixed coefficients. Using Lemma 9,
+we conclude that there are functions α(x), λ1(x), β(x), λ2(x) that depend only on x, c and d, such that
+pn(x) = α(x)(λ1(x))n + β(x)(λ2(x))n
+(8)
+for every n.
+In order to find λ1, λ2 we solve for λ the characteristic polynomial, namely the following quadratic
+equation derived from (7):
+xλ = λ2 + (c − 1 + d − 1)λ + (c − 1)(d − 1)
+To obtain
+λ1,2(x) = x − (c − 1) − (d − 1) ±
+�
+∆(x)
+2
+where
+∆(x) = x2 − 2x((c − 1) + (d − 1)) + (c − d)2.
+(9)
+Using the initial values for p0(x), p1(x) from (6), and plugging back into (8) we get the equations
+c
+c − 1 = α(x)(λ1(x))0 + β(x)(λ2(x))0 = α(x) + β(x)
+x − c = α(x)(λ1(x))1 + β(x)(λ2(x))1 = α(x)λ1(x) + β(x)λ2(x)
+10
+
+whose solution is
+α(x) = (c − 1)x − (c − 1)2 − (c − 1) + (c − 1)(d − 1) + (c − 1)
+�
+∆(x) − x + d − 1 +
+�
+∆(x)
+2(c − 1)
+�
+∆(x)
+β(x) =
+c
+c − 1 − α(x).
+We check when ∆(x) = 0 by solving (9) for x:
+x1,2 = 2((c − 1) + (d − 1)) ±
+�
+4(c − 1 + d − 1)2 − 4(c − d)2
+2
+= (c − 1 + d − 1) ±
+�
+(c + d)2 − 4(c + d) + 4 − (c − d)2
+= (c − 1 + d − 1) ±
+�
+c2 + 2cd + d2 − 4c − 4d + 4 − c2 + 2cd − d2
+= (c − 1 + d − 1) ±
+√
+4cd − 4c − 4d + 4
+= (c − 1 + d − 1) ± 2
+√
+c − 1
+√
+d − 1
+= (
+√
+d − 1 ±
+√
+c − 1)2
+We see that ∆(x) is quadratic in x and has roots at (
+√
+d − 1 ± √c − 1)2. This gives a nice factorization of
+∆(x):
+∆(x) = x2 − 2x((c − 1) + (d − 1)) + (c − d)2
+=
+�
+x −
+�√
+d − 1 +
+√
+c − 1
+�2� �
+x −
+�√
+d − 1 −
+√
+c − 1
+�2�
+Recall that for the x we fixed we have √x = t ∈ (
+√
+d − 1 − √c − 1,
+√
+d − 1 + √c − 1), so the first term in the
+product is negative and the second term is positive, so ∆ < 0, and so λ1,2 are complex numbers (conjugate
+to one another), with magnitude
+|λ1,2|2 = (x − (c − 1) − (d − 1))2 − ∆(x)
+4
+= x2 − 2x((c − 1) + (d − 1)) + (c − 1 + d − 1)2 − (x2 − 2x((c − 1) + (d − 1)) + (c − d)2)
+4
+= (c + d − 2)2 − (c − d)2
+4
+= (c − 1)(d − 1)
+(10)
+A very similar calculation shows that α, β are conjugates with magnitude
+|α|2 = |β|2 =
+x(x − cd)
+∆(x) · (c − 1)
+This finishes the proof for all such x’s:
+|pℓ(x)| = |α(x)λℓ
+1 + β(x)λℓ
+2| ≤ |α(x)λℓ
+1| + |β(x)λℓ
+2|
+= |α(x)||λ1|ℓ + |β(x)||λ2|ℓ
+= 2
+�
+x(x − cd)
+∆(x) · (c − 1)(c − 1)ℓ/2(d − 1)ℓ/2
+We keep in mind that x is fixed, so the expression is smaller than (2 +
+√
+d − 1) · ℓ · (c − 1)ℓ/2(d − 1)ℓ/2 for
+large enough ℓ.
+We are left with the cases x = t2 for t = 0,
+√
+d − 1 ± √c − 1:
+11
+
+1. t = 0. We use the same methods and find that the characteristic polynomial is
+λ2 + (c − 1 + d − 1)λ + (c − 1)(d − 1)
+whose roots are
+λ1 = −(c − 1)
+,
+λ2 = −(d − 1).
+Using the initial conditions (p0(0) = c/(c − 1), p1(0) = −c) we obtain
+α(0) =
+c
+c − 1
+,
+β(0) = 0
+and using the fact that c < d we get
+|pℓ(0)| = |α(0)λℓ
+1 + β(0)λℓ
+2|
+=
+c
+c − 1(c − 1)ℓ
+< 2l(c − 1)ℓ/2(c − 1)ℓ/2
+< 2l(c − 1)ℓ/2(d − 1)ℓ/2.
+2. t =
+√
+d − 1 + √c − 1. Then x = t2 = (
+√
+d − 1 + √c − 1)2 = d − 1 + c − 1 + 2
+√
+d − 1√c − 1, and the
+characteristic polynomial has a single root of multiplicity 2, namely
+λ = x − (c − 1) − (d − 1)
+2
+=
+√
+d − 1
+√
+c − 1.
+The solution, therefore, takes the form
+pn(x) = (α(x) + nβ(x))(c − 1)n/2(d − 1)n/2.
+Using the initial values we get
+α(x) =
+c
+c − 1
+,
+β(x) =
+x − c
+√
+d − 1√c − 1 −
+c
+c − 1 = 2 +
+d − 2
+√
+d − 1√c − 1 −
+c
+c − 1.
+1 <
+c
+c−1 ≤ 2 so β(x) ≤
+√
+d − 1 + 1, and in total we get
+|pℓ(x)| = |α(x) + ℓβ(x)|(c − 1)ℓ/2(d − 1)ℓ/2
+≤
+�����
+1
+ℓ ·
+c
+c − 1
+���� + |β(x)|
+�
+ℓ(c − 1)ℓ/2(d − 1)ℓ/2
+≤
+�
+2 +
+√
+d − 1
+�
+ℓ(c − 1)ℓ/2(d − 1)ℓ/2
+For sufficiently large ℓ.
+3. t =
+√
+d − 1 − √c − 1. We get x = t2 = d − 1 + c − 1 − 2
+√
+d − 1√c − 1, and the rest follows the same
+calculations as in the previous case.
+Bounds on the spectrum of ALL
+2ℓ give bounds on the number of non-backtracking paths completely con-
+tained in a small set, hence gives Lemma 8.
+12
+
+Proof of Lemma 8. Recall that M2ℓ(S, G) counts the number of non-backtracking paths of length 2ℓ that
+start and end in S, so by the definition of the ALL
+n
+operatore, we have M2ℓ(S, G) = ⟨ALL
+2ℓ 1S, 1S⟩.
+We note that ALL
+2ℓ 1L = c(c − 1)ℓ−1(d − 1)ℓ1L, because every non-backtracking path starting at a given
+vertex is made of picking the first left-to-right edge (we have c such edges to pick from), and then alternating
+between picking any of the d or c edges adjacent to the current vertex, except for the edge we picked to get
+to it.
+Write 1S =
+|S|
+n 1L + r, with r ⊥ 1L, and ∥r∥2
+2 ≤ ∥1S∥2
+2 = |S|. Since the graph is Ramanujan, the
+nontrivial eigenvalues in its spectral decomposition have their absolute value in the set {0} ∪ [
+√
+d − 1 −
+√c − 1,
+√
+d − 1 + √c − 1].
+We only care about the nontrivial eigenvalues because r ⊥ 1L, hence in the
+writing of r in the orthogonal basis made of eigenvectors, only eigenvectors with nontrivial eigenvalues
+appear. We use Lemma 10 to get
+⟨ALL
+2ℓ r, r⟩ ≤ (2 +
+√
+d − 1)ℓ(c − 1)ℓ/2(d − 1)ℓ/2 · ∥r∥2
+2 .
+Combine everything to get
+M2ℓ(S, G) = ⟨ALL
+2ℓ 1S, 1S⟩ =
+�
+ALL
+2ℓ
+|S|
+n 1L + r, |S|
+n 1L + r
+�
+= |S|2
+n2 ⟨ALL
+2ℓ 1L, 1L⟩ + ⟨ALL
+2ℓ r, r⟩
+= |S|2
+n2 · c(c − 1)ℓ−1(d − 1)ℓ⟨1L, 1L⟩ + ⟨ALL
+2ℓ r, r⟩
+≤ |S|2
+n c(c − 1)ℓ−1(d − 1)ℓ + (2 +
+√
+d − 1)ℓ(c − 1)ℓ/2(d − 1)ℓ/2 ∥r∥2
+2
+≤ |S|
+�|S| · c · (c − 1)ℓ/2(d − 1)ℓ/2
+n(c − 1)
++ (2 +
+√
+d − 1)ℓ
+�
+(c − 1)ℓ/2(d − 1)ℓ/2
+≤ |S|
+�
+c
+c − 1 + (2 +
+√
+d − 1)ℓ
+�
+(c − 1)ℓ/2(d − 1)ℓ/2
+≤ |S|
+�
+(2 +
+√
+d − 1)ℓ + 2
+�
+(c − 1)ℓ/2(d − 1)ℓ/2
+5
+Random gadget
+In this section we prove the existence of bipartite graphs such that every small set of left-side vertices has
+a unique neighbour on the right side. We draw a random biregular graph from a similar distribution as in
+[Pip77], and use techniques similiar to [Vad+12, Thm 4.4].
+Lemma 11. For every integers L, R, c, d with Lc = Rd, L > R, c > 3, if k is an integer that satisfies the
+inequality
+k
+c−3
+2
+≤
+1
+2Le ·
+� R
+3ec
+� c−1
+2
+then there is a (c, d)-biregular graph with sides [L] and [R] such that every set of left vertices of size at most
+k has a unique neighbour.
+We draw a random (c, d)-biregular graph in the following way: fix L vertices on the left side and R
+vertices on the right side (cL = dR), write c copies of each left-side vertex and d copies of each right-side
+vertex, and connect them via a uniformly random matching. That is, pick a uniformly random permutation
+π : L × [c] → R × [d], and for every u ∈ L, v ∈ R, i ∈ [c], j ∈ [d], if π(u, i) = (v, j), then add (u, v) as an edge.
+Note that we allow multiple edges between two vertices (if there are several i, j satisfying π(u, i) = (v, j)).
+13
+
+Let G be a random bipartite graph with L vertices on the left side and R vertices on the right side drawn
+from said distribution. Let A be a subset of left vertices of size k. We note that if A expands by at least
+(c + 1)/2, then, by a simple counting argument, A has a unique neighbour. It is therefore sufficient to find
+the probability that A expands by at least (c + 1)/2.
+Let us fix an arbitary ordering of the ck edges leaving A, and denote it e1, . . . , eck. We say that ei is a
+repeat if it touches a previously covered vertex, that is, if its right endpoint is contained in the set of right
+endpoints of the set e1, . . . , ei−1. We note that if A does not expand by at least (c + 1)/2, then, again by a
+simple counting argument, there are at least (c − 1)k/2 repeats. This is because the number of repeats and
+the size of the set of the neighbours of A add up to the number of edges leaving A, namely ck.
+We note that for every i, ei is a repeat if it touches one of i − 1 or less previously covered vertices. This
+means that Pr[ei is a repeat] ≤ i−1
+R < ck
+R . Moreover, if we condition on the event that some of the first i − 1
+edges are also repeats, then the probability that ei is a repeat may only decrease, since it means that there
+are less “forbidden” endpoints. We conclude that for every set of l edges:
+Pr[ei1, . . . , eil are repeats] =
+l�
+j=1
+Pr[eij is a repeat | ei1, . . . eij−1 are repeats] <
+�ck
+R
+�l
+.
+If A expands too little, then there are many repeats. We can use it to bound the probability that A has
+no unqiue neighbour:
+Pr[A has no unique neighbour] ≤ Pr[A expands by < (c + 1)/2]
+≤ Pr[there are at least (c − 1)k/2 repeats]
+≤
+�
+i1,...,i(c−1)k/2∈(
+ck
+(c−1)k/2)
+Pr[{ei1, . . . , ei(c−1)k/2} are repeats]
+<
+� ck
+c−1
+2 k
+�
+·
+�ck
+R
+� c−1
+2 k
+And by a union bound over the possible choices of A:
+Pr[∃ “bad” A of size k] ≤
+�L
+k
+�
+· Pr[A expands by < (c + 1)/2]
+≤
+�L
+k
+�
+·
+� ck
+c−1
+2 k
+�
+·
+�ck
+R
+� c−1
+2 k
+≤
+�Le
+k
+�k
+·
+� cke
+c−1
+2 k
+� c−1
+2 k
+·
+�ck
+R
+� c−1
+2 k
+=
+�
+Le
+k ·
+� 2ce
+c − 1 · ck
+R
+� c−1
+2 �k
+≤
+�
+Le
+k ·
+�3eck
+R
+� c−1
+2 �k
+(11)
+Where the last inequality follows from assuming that c ≥ 3 so
+2c
+c−1 ≤ 3.
+We are now ready to prove Lemma 11.
+Proof of Lemma 11. Let us draw a (c, d)-biregular graph G = ([L] ⊔ [R], E) from the distribution described
+above. Let k be an integer satsifying (11). Using a union bound and the inequality in (11), we have (where
+14
+
+probability is taken over the choice of G):
+Pr[∃ “bad” A ⊆ [L] of size ≤ k] =
+k
+�
+a=1
+Pr[∃ “bad” A ⊆ [L] of size a]
+≤
+k
+�
+a=1
+�
+Le
+a ·
+�3eca
+R
+� c−1
+2 �a
+<
+∞
+�
+a=1
+�
+Le
+k ·
+�3eck
+R
+� c−1
+2 �a
+=
+∞
+�
+a=1
+�
+k
+c−1
+3
+· Le ·
+�3ec
+R
+� c−1
+2 �a
+≤
+∞
+�
+a=1
+�1
+2
+�a
+< 1.
+We see that with strictly positive probability, a random graph has no “bad” subsets of size ≤ k, hence there
+exists a graph with the desired unique neighbour property.
+6
+Construction
+6.1
+Routed product definition
+Let us begin with a brief coding theory motivation. An error-correcting code is often given via an m × n
+parity check matrix H, so that C = Ker H ⊆ {0, 1}n. The matrix H can be visualized as a bipartite graph,
+called the parity check graph, with n left and m right vertices, and an edge i ∼ j whenever H(j, i) ̸= 0. A
+Tanner code is defined given a bipartite graph B and a base code C0 = Ker H0 [Tan81]. One way to view
+the routed product is through the point of view of codes. Consider the parity check graph B0 of H0 and
+define the routed product of B and B0 to be simply the parity check graph of the Tanner code C(B, C0).
+Here is a more detailed and combinatorial definition of the routed product without mention of codes.
+Let G = (L ⊔ R, E) be a (c, d)-biregular graph and G0 = (L0 ⊔ R0, E0) a (c0, d0)-biregular graph.
+We
+think of G as a big graph (in practice, an infinite family of Ramanujan graphs), and G0 as a fixed size graph
+(gadget). Assume that |L0| = d, and let us think of the edges of G as a function E : R × [d] → L which
+maps a right side vertex v and an index i to the ith neighbour of v in G.
+We can define the routed product graph G′ = G ◦ G0 as the bipartite graph whose left side is L, right
+side is the cartesian product R × R0, and the set of edges is
+E′ = {(E(v, i), (v, j)) : v ∈ R, i ∈ [d], j ∈ [R0], (i, j) ∈ E0}.
+That is, we write R0 copies of each vertex in R, and every right side vertex v in the big graph G and an
+edge (i, j) in the small gadget G0 gives an edge between the ith neighbour of v in G, and the jth vertex of
+the copy of G0 assigned to v in G′. Otherwise put, we use G0 to route every edge of the big graph G to c0
+edges in the product graph G′.
+More precisely, for every v ∈ R, the bipartite subgraph of G′ whose left side is NG(v) and right side
+is (v, ·) is isomorphic to G0. This means that, roughly speaking, unique neighbours are inherited from the
+small graph to the product graph:
+Lemma 12. Let S ⊆ L, v ∈ NG(S). Define S′ = {i : E(v, i) ∈ S} ⊆ [d] as the indexed neighbours of v in
+S. If S′, as a set of vertices in the gadget G0, has a unique neighbour j ∈ R0 in G0, then (v, j) is a unique
+neighbour of S in the product graph G′.
+The proof is immediate while staring at Fig. 1, but for the sake of completion it is given in Appendix A.
+15
+
+Figure 1: An example of a bipartite graph G (dashed, red), a small gadget G0 (dotted, green), and the
+routed product G′ = G ◦ G0 (solid, blue). The set S ⊆ L has a neighbour v ∈ R, and so S is associated with
+a set S′ of left side vertices of the copy of G0 associated with v. Since (i′, j) is the only edge connecting j
+to S′ in G0, we have that (v, j) is a unique neighbour of S in G′.
+16
+
+S6.2
+Proof of Theorem 1
+Let q be a prime power, c0 an integer, and α > 1. Assume that αc0(q + 1) is an integer. We construct an
+infinite family of (c0(q + 1), αc0(q + 1))-biregular graphs with the unique neighbour property under some
+assumptions specified below.
+Denote c = q + 1 and d = q3 + 1. By Theorem 3 there is an efficient construction of an infinite family
+of (c, d)-biregular Ramanujan graphs (Gn). Let G0 = (L0 ⊔ R0, E0) be a gadget: a c0-left-regular bipartite
+graph with |L0| = d = q3 + 1 vertices on the left side and R0 vertices on the right side, such that every
+left-side set of sufficiently small size admits a unique neighbour on the right side, where “sufficiently small”
+here means the bound given in Lemma 11. For the constructed graph to have the left side α times bigger
+than the right side, we set R0 =
+d
+αc =
+q3+1
+α(q+1).
+We define G′
+n = Gn ◦ G0 as the routed product of Gn and G0. For the rest of this (short) proof let us
+suppress n from the notation, for convenience.
+Let ε < 1
+q. By Theorem 2, there exists δ > 0 such that for every S ⊆ L of size at most δ|L|, the “average
+right degree” ¯dS, namely the average of the degrees of vertices in NG(S) in the induced subgraph S ⊔NG(S),
+is bounded:
+¯dS :=
+c|S|
+|NG(S)| ≤ 1 + (1 + ε)
+�
+d − 1
+c − 1 .
+We show that such S has a unique neighbour in G′.
+We note that d−1
+c−1 = q2, so since ε < 1
+q we have a vertex v ∈ R of “degree” at most q + 1 in G, that is, the
+set S′ ⊆ [d] of v’s neighbours in S is of size at most q + 1. By Lemma 12, if S′, as a set of left-side vertices
+in G0, has a unique neighbour j in G0, then our original set S has a unique neighbour (v, j) in G.
+It remains to choose the parameters in a way that all left-side sets of size at most q + 1 have a unique
+neighbour in G0. By Lemma 11, we need to have:
+(q + 1)
+c0−3
+2
+≤
+1
+2(q3 + 1)e ·
+�
+�
+q3+1
+α(q+1)
+3ec0
+�
+�
+c0−1
+2
+.
+(12)
+The LHS is O(q
+c0−3
+2 ) and RHS is Θ(qc0−4), so if c0 > 5 then for sufficiently large q the construction gives a
+unique neighbour expander. That is, there exists some ˆq(c0, α) such that if q > ˆq then (12) holds, hence we
+constructed a bipartite unique neighbour expander as promised in Theorem 1.
+7
+Future work
+The main pitfall of our approach is the non-constructive nature of the gadget. Theoretically since the gadget
+has constant size this is no issue. However, exhaustive search is impractical even for small values of q. This
+is because the gadget’s size is cubic in q so the search space is of size exponential in q3. A natural question
+would be whether it is possible to construct such a gadget in an efficient way, since that would lead to the
+whole unique neighbour expander family to be constructible in practice. For the bipartite Ramanujan family
+chosen in our work (the one by Ballantine et al. [Bal+15]) we ask the following.
+Question 13. For which prime power q and real number α ≥ 1 can one construct efficiently a biregular
+graph with left side q3 + 1, right side
+q3+1
+α(q+1), such that every left side set of size at most q + 1 has a unique
+neighbour?
+We note that the fixed size graph given in [AC02, Lemma 4.3] is a good gadget (for α = 22/21 and the
+edge-vertex incidence graphs of a 44-regular Ramanujan graph family), and indeed these graphs can be used
+to construct bipartite unique neighbour expanders.
+Since we prove that a random gadget is, with non-negligble probability, good for our construction, it
+may be interesting to construct such gadget by simply drawing random gadgets and testing whether they
+are good. Since drawing is simple, we are left with the task of testing. We therefore ask:
+17
+
+Question 14. Given a bipartite graph, can one efficiently find the smallest nonempty set of left-side vertices
+that has no unique neighbours?
+We currently know of no better way than just enumerating all left-side sets, which is exponential in the
+size of the graph, hence impractical. We refer to [AK19] for an interesting approach to testing expansion of
+random graphs.
+The methods presented in this work are not limited to the (q+1, q3 +1)-biregular Ramanujan family. We
+can therefore ask the question the other way around – find a gadget (by sampling or any other way), and see
+whether we can efficiently construct a bipartite Ramanujan family that will make it work, i.e. that would
+allow us to rewrite the proof of Theorem 1. This emphasizes the well-known natural question of constructing
+Ramanujan graphs with arbitrary degrees, specifically in the bipartite and biregular setting,
+Question 15. For which integers c < d can one construct efficiently an infinite family of (c, d)-biregular
+Ramanujan graphs?
+We note that our construction is far from “right-side unique neighbour expansion,” as the complete right
+side of a single gadget is a constant-size set with no unique neighbours on the left. We wonder whether it is
+possible to construct a bipartite graph where all small size sets (be them contained in either sides, or both)
+have unique neighbours.
+18
+
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+20
+
+8
+Appendix A
+We restate and prove the lemmas we used throughout the work.
+Lemma 5. Let G be a d-regular Ramanujan graph, and G′ its edge-vertex incidence graph. Then G′ is a
+(2, d)-biregular Ramanujan graph.
+Proof. Let G = (V, E) a d-regular Ramanujan graph. The adjacency matrix of G′ is A =
+� 0
+M
+M ⊤
+0
+�
+where
+M has |E| rows, each containing two 1’s, and |V | columns, each containing d 1’s. Let v be an eigenvector of
+A with eigenvalue λ; then v is an eigenvector of A2 with eigenvalue λ2. We note that
+A2 =
+�
+MM ⊤
+0
+0
+M ⊤M
+�
+so it suffices to consider the spectrum of M ⊤M, which is essentially the operator corresponding to a walk
+from a vertex of G to an edge that touches it and back to one of its endpoints (possibly the same vertex we
+started at).
+For every v ∈ V , there are d ways to walk from it to an edge and then back to v; all other legal paths
+correspond to picking an edge touching v. We conclude that M ⊤M = dI + A, so every eigenvalue λ of G′
+satisfies λ2 = d + σ where σ is an eigenvalue of G.
+The lemma is proven by noting that |σ| ≤ 2
+√
+d − 1 (since G is Ramanujan), so
+d − 2
+√
+d − 1 ≤ λ2 ≤ d + 2
+√
+d − 1
+The terms on the extreme sides of the inequality can be verified to be (
+√
+d − 1 ± 1)2 so we get |λ| ∈
+[
+√
+d − 1 − 1,
+√
+d − 1 + 1], as needed (recall that in G′ the left-regularity is c = 2 so √c − 1 = 1).
+Lemma 9. Let (xn) be a series defined via a second order linear recurrence with fixed coefficients A, B ∈ C:
+xn = Axn−1 + Bxn−2
+Assume λ1 ̸= λ2 are (real or complex) roots of the characteristic polynomial λ2 − Aλ − B. Then there are
+α, β ∈ C, that depend on the initial conditions x0, x1, such that
+xn = αλn
+1 + βλn
+2
+for every n ≥ 0.
+If the characteristic polynomial has a single root λ of multiplicity 2, then there are α, β ∈ C such that
+xn = αλn + βnλn
+for every n ≥ 0.
+Proof. We note that for every n ≥ 2 we have
+� xn
+xn−1
+�
+=
+�Axn−1 + Bxn−2
+xn−1
+�
+=
+�A
+B
+1
+0
+� �xn−1
+xn−2
+�
+Denote the 2 × 2 matrix by D, so by induction,
+� xn
+xn−1
+�
+= Dn
+�x1
+x0
+�
+Let us diagonalize D. The characteristic polyonmial is
+pD(λ) = det(λI − D) =
+����
+λ − A
+−B
+−1
+λ
+���� = λ(λ − A) − B = λ2 − Aλ − B
+21
+
+If pD(λ) has two distinct roots λ1, λ2, then the matrix is diagonalizable; that means that there exists a 2 × 2
+matrix M such that D = M · diag{λ1, λ2} · M −1. We get:
+� xn
+xn−1
+�
+= M
+�λ1
+0
+0
+λ2
+�n
+M −1
+�x1
+x0
+�
+= M
+�λn
+1
+0
+0
+λn
+2
+�
+M −1
+�x1
+x0
+�
+We can compute M, M −1 explicitly, multiple the matrices and get α, β ∈ C such that xn = αλn
+1 + βλn
+2 as
+required.
+Otherwise, if pD(λ) has a single root λ of multiplicity 2, then we can find its Jordan form, i.e. find M
+such that
+D = M
+�λ
+1
+0
+λ
+�
+M −1
+Dn = M
+�λ
+1
+0
+λ
+�n
+M −1 = M
+�λn
+nλn−1
+0
+λn
+�
+M −1
+Where the last equality follows from a simple induction.
+Similarly, we get
+� xn
+xn−1
+�
+= M
+�λ
+1
+0
+λ
+�n
+M −1
+�x1
+x0
+�
+= M
+�λn
+nλn−1
+0
+λn
+�
+M −1
+�x1
+x0
+�
+And again we can find α, β ∈ C as required.
+For the following lemma we remind that G = (L ⊔ R, E) is a (c, d)-biregular graph, G0 = (L0 ⊔ R0, E0) is
+a (c0, d0)-biregular graph, and G′ = G ◦ G0 is the routed product of G and G0. Recall that the edges of G′
+are (E(v, i), (v, j)) when v ∈ R is a right side vertex of G, i ∈ [d], E(v, i) is the ith neighbour of v according
+to G, and (i, j) ∈ E0.
+Lemma 12. Let S ⊆ L, v ∈ NG(S). Define S′ = {i : E(v, i) ∈ S} ⊆ [d] as the indexed neighbours of v in
+S. If S′, as a set of vertices in the gadget G0, has a unique neighbour j ∈ R0 in G0, then (v, j) is a unique
+neighbour of S in the product graph G′.
+Proof. Assume that i′ ∈ S′ is the unique neighbour of j in G0. By the definition of the routed product
+we have that (E(v, i′), (v, j)) is an edge in G. Since i′ ∈ S′ we have that E(v, i′) ∈ S, so indeed (v, j) is
+a neighbour of S in G′. It is therefore remaining to show that it is unique, i.e. that E(v, i′) is the only
+neighbour of (v, j) in S.
+The neighbours of (v, j) in G are E(v, i) for every i such that (i, j) ∈ E0. If E(v, i) ∈ S, then by the
+definition of S′ we have that i ∈ S′, so i is a neighbour of j in E0. But we know that j is a unique neighbour
+of S′ in E0, so we must have that i = i′, and indeed (v, j) is a unique neighbour of S in G′.
+22
+

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+arXiv:2301.03983v1  [cs.IT]  10 Jan 2023
+On the Performance of Dual RIS-assisted V2I
+Communication under Nakagami-m Fading
+Mohd Hamza Naim Shaikh, Khaled Rabie◦, Xingwang Li#, Theodoros Tsiftsis†, and Galymzhan Nauryzbayev
+School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan City, 010000, Kazakhstan
+◦Department of Engineering, Manchester Metropolitan University, Manchester, M15 6BH, UK
+#School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454000, China
+†Department of Informatics & Telecommunications, University of Thessaly, Greece;
+†School of Intelligent Systems Science and Engineering, Jinan University, China
+Email: {hamza.shaikh, galymzhan.nauryzbayev}@nu.edu.kz, ◦[email protected],
+#[email protected], †[email protected]
+Abstract—Vehicle-to-everything (V2X) connectivity in 5G-and-
+beyond communication networks supports the futuristic intelligent
+transportation system (ITS) by allowing vehicles to intelligently
+connect with everything. The advent of reconfigurable intelligent
+surfaces (RISs) has led to realizing the true potential of V2X
+communication. In this work, we propose a dual RIS-based
+vehicle-to-infrastructure (V2I) communication scheme. Following
+that, the performance of the proposed communication scheme
+is evaluated in terms of deriving the closed-form expressions
+for outage probability, spectral efficiency and energy efficiency.
+Finally, the analytical findings are corroborated with simulations
+which illustrate the superiority of the RIS-assisted vehicular
+networks.
+Keywords— Reconfigurable intelligent surface (RIS), dual RIS,
+energy efficiency, spectral efficiency, vehicular communication.
+I. INTRODUCTION
+As a key enabler for intelligent transportation systems (ITSs),
+vehicle-to-everything (V2X) communication has sparked a re-
+newed interest in the research community. V2X encompasses
+a wide range of wireless technologies such as vehicle-to-
+pedestrian (V2P), vehicle-to-infrastructure (V2I), and vehicle-
+to-vehicle (V2V). Additionally, it also includes the vehicu-
+lar communications with vulnerable road users (VRUs), grid
+(V2G), network (V2N) and cloud (V2C) [1]. The V2X com-
+munications will be a critical component of the futuristic
+connected and self-driving cars, envisioned and enabled by
+the sixth-generation (6G) wireless technologies. Furthermore,
+the V2X communications will also enhance and transform
+the quality-of-service (QoS) in terms of unparalleled user
+experience, ultra-high road safety and air quality improvement.
+In addition, a slew of advanced applications will also be
+supported like platooning, trajectory alignments, exchanging
+sensor data and high precision maps, and so on [2]. Thanks to
+the enhanced capabilities of 6G, vehicles will receive accurate
+safety information, intelligent traffic management support, and
+innovative infotainment features. Thus, the 6G services will be
+used to create a fully automated, autonomous, and ubiquitously
+connected vehicular network [3].
+Recently, reconfigurable intelligent surfaces (RISs) have
+emerged as a breakthrough technology that offers a great deal
+in terms of wireless communication [4]. Inherently, RIS is a
+software-defined artificial structure made up of a large number
+of scattering passive elements, termed as reflecting units (RUs).
+These RUs are capable to adjust the electromagnetic (EM)
+properties of a reflected wave that is incident on them. Thus,
+RIS can use not only this ability to boost the received signal’s
+power, but also the capability to create an additional reflective
+link to mitigate the impact of blockages. With the large number
+of RUs, RISs are particularly known to have large spectral and
+energy efficiency [5]. As a result, RIS may be used to improve
+the quality of vehicular communication through establishing
+a low-cost, highly energy efficient indirect line-of-sight (LoS)
+communications [6].
+In [7], the authors investigated the outage performance for
+RIS-assisted vehicular communication networks. Likewise, the
+secrecy outage performance of RIS-aided vehicular communi-
+cations has been studied in [8]. RISs were also investigated for
+detecting VRUs such as cyclists, pedestrians and wheelchair
+users [9]. Specifically, the authors utilized RISs for enhancing
+the radar visibility for VRUs. Further, in [10], the authors
+provided a optimization framework for resource allocation
+in the RIS-aided vehicular communications. Specifically, they
+jointly optimized the power allocation, RIS reflection coeffi-
+cients and spectrum allocation for different QoS requirements
+of the V2V and V2I communication links. Likewise, in [11],
+the authors discussed a system model where RSU leverages RIS
+to connect the dark zones, i.e., areas blocked due by obstacles.
+Moreover, a comprehensive overview on the recent advances
+in 6G vehicular networks was provided in [12, 13], where the
+authors also described various open challenges and possible
+research directions.
+Motivated by the above, in this work, we investigate the
+performance of a dual RIS-assisted V2I communication net-
+work scenario. Specifically, the proposed scenario considers the
+uplink transmission where the vehicle is communicating with
+the base station. To enhance the communication capabilities, the
+vehicle is supported through two RISs which create a virtual
+line-of-sight (LoS) link, which, otherwise, was inherently non-
+LoS (NLoS). The major contributions can be summarized as
+• Explicitly, we invoked the central limit theorem (CLT) to
+characterize the received signal-to-noise ratio (SNR) for
+
+Vehicle-to-Vehicle (V2V)
+Vehicle-to-Infrastructure (V2I)
+Fig. 1. Schematic for the considered dual RIS-aided V2I communication.
+the proposed dual RIS case. Further, based on this, we
+derived the closed-form expression for outage probability.
+• Further, we derived the closed-form expressions for the
+upper and lower bounds of SE and EE of the proposed
+dual RIS-assisted V2I communication scenario.
+• Finally, as a performance benchmark, the proposed sce-
+nario is compared with the single RIS-assisted V2I com-
+munication and with RIS V2I communication. The results
+show the superiority of the proposed scenario of dual RIS-
+assisted V2I over the single RIS-assisted V2I communi-
+cation case.
+II. SYSTEM MODEL
+As illustrated in Fig. 1, in this work, we consider a V2I
+communication model, wherein the vehicular user (V) tries to
+communicate with a nearby base station (BS). Apart from the
+direct cellular link, a reflected path through RISs is considered
+to support this uplink transmission. In particular, we consider
+a dual RIS-assisted uplink V2I transmission with two RISs,
+one each placed near V and BS both, respectively. For the two
+RISs, the number of RUs is assumed to be M1 and M2 for
+RIS-1 and RIS-2, respectively, while keeping the total number
+of RUs unchanged, i.e., M1+M2 = N, where N is the number
+of RUs in large RIS for the single RIS scenario, which is the
+benchmark for comparison. Thus, based on RIS, the following
+scenarios are considered in this work
+• Dual RIS-assisted Transmission (DRAT): In DRAT, the
+transmission takes place only through the two RISs and
+the reflected link, as shown in Fig. 1.
+• Single RIS-assisted Transmission (SRAT): In SRAT, the
+transmission takes place through single large RIS which
+is placed near to BS.
+• Direct Cellular Transmission (DCT): In DCT, V commu-
+nicates with BS directly without utilizing RISs. Thus, the
+transmission is inherently NLoS and experiences a higher
+pathloss exponent. This would also serve as the baseline
+scheme for the performance comparison of the above two
+cases.
+A. Channel Model
+The channels between V-to-RIS-1 and RIS-2-to-BS can
+be modeled as deterministic LOS channels as the distances
+are small and the probability of having LoS is very high.
+However, the distance between RIS-1 and RIS-2 is large and
+thus the small scale fading for the channel between the ith
+element of RIS-1 and the jth element of RIS-2, denoted by
+hRR
+ij , is modeled through Nakagami-m fading. Hence, for
+i = {1, 2, . . ., M1} and j = {1, 2, . . ., M2}. Further, the
+distances related to the V-to-RIS-1, RIS-1-to-RIS-2 and RIS-2-
+to-BS links are represented by d1, dRR and d2, respectively.
+B. Received Signal Model
+The received base-band signal at BS, denoted by r, for the
+dual RIS-aided transmission case can be expressed as
+r =
+�
+B Pt
+��M1
+i=1
+�M2
+j=1 ejφ(1)
+i hRR
+ij ejφ(2)
+j
+�
+s + No,
+(1)
+where Pt is the transmit power constraint at V, B is the distance-
+dependent pathloss, s ∼ CN (0, 1) is the transmitted symbol,
+and No ∼ CN
+�
+0, σ2�
+is the additive white Gaussian noise
+(AWGN). Further, φ1 and φ2 are the phase of the V-to-RIS1
+and RIS2-to-BS channels. Further, for a link distance d, B at
+the carrier frequency of 3 GHz can be given by [14]
+B(d) [dB] =
+�
+−37.5 − 22 log10(d/1 m)
+if LOS,
+−35.1 − 36.7 log10(d/1 m)
+if NLOS.
+(2)
+Likewise, instantaneous SNR at BS can be formulated as
+γ =
+����
+�M1
+i=1
+�M2
+j=1 δije
+j
+�
+φ(1)
+i
++φ(2)
+j
+−ϕij
+�����
+2
+B Pt
+σ2
+,
+(3)
+where δij and ϕij denote the amplitude and phase of hRR
+ij .
+1) RIS Reflection Parameters: Now, SNR at BS can be
+maximized through adjusting the phase at RISs to cancel
+the resultant phase, i.e., φ(1)
+i
++ φ(2)
+j
+− ϕij = 0, for i =
+{1, 2, . . ., M1} and j = {1, 2, . . ., M2}. Thus, by substituting
+ϕij = φ(1)
+i
++ φ(2)
+j , ∀i, j, the received signal power at BS can
+be maximized. Consequently, maximized SNR corresponding
+to the optimal phase can be given as
+γmax =
+����M1
+i=1
+�M2
+j=1 δij
+���
+2
+B Pt
+σ2
+= A2B Pt
+σ2
+= A2 B ¯γ,
+(4)
+where A2 =
+���
+�M1
+i=1
+�M2
+j=1 δij
+���
+2
+is the cascaded channel gain
+provided by RISs, and ¯γ = Pt/σ2 is transmit SNR.
+Likewise, proceeding in the similar way, for the SRAT
+scenario, maximized SNR at BS can be given as1
+ˆγmax =
+��N
+i=1 βi
+�2
+¯�� = B2¯γ,
+(5)
+where βi is the amplitude of a channel between RIS and
+V, denoted by hRU
+i
+, i.e., hRU
+i
+= βie−jϕi, and B2 is the
+corresponding channel gain provided by single RIS.
+1For the SRAT scenario, the analysis is similar. Thus, the detailed description
+is omitted for the sake of brevity. In particular, for SRAT, large RIS with N
+RUs is present near BS, where N = M1 + M2. Likewise, the RIS-to-BS link
+is also modeled as Nakagami-m fading with the rest of the parameters being
+the same, as in DRAT, like transmit power constraint at V, etc.
+
+III. PERFORMANCE ANALYSIS
+This section initially evaluates SNR for the dual RIS-aided
+V2I scenario. Utilizing the SNR expressions formulated earlier,
+the outage probability, SE and EE are derived.
+A. Statistical Characterization of the Dual RIS Channel Gain
+Now utilizing CLT for M
+≫ 1, A = �M1
+i=1
+�M2
+j=1 δij
+can be approximated through a Gaussian distribution, i.e.,
+A ∼ N(µy, σ2
+y) [15], with a probability density function (PDF)
+given by
+fA(y) =
+
+
+
+1
+√
+2πσ2
+A exp
+�
+−(y−µA)2
+2σ2
+A
+�
+,
+if y > 0,
+0,
+if y = 0,
+(6)
+where µA = �M1
+i=1
+�M2
+j=1 µij, σ2
+A = �M1
+i=1
+�M2
+j=1 σ2
+ij. Here,
+µij and σ2
+ij are the mean and variance of the random variable
+δij, which follows the Nakagami-m distribution. Hence, µij =
+Γ(m1+ 1
+2 )
+Γ(m1)
+�� Ωm1
+m1
+�
+and σ2
+ij = Ωm1
+�
+1 −
+1
+m1
+�
+Γ(m1+ 1
+2 )
+Γ(m1)
+�2�
+,
+for all i = {1, . . . , M1} and j = {1, . . ., M2}.
+Likewise the cumulative distribution function (CDF) of A
+can be derived from its PDF as
+FA(y)=
+� y
+−∞
+fA(y)dy =
+�
+1−Q
+�
+y−µA
+σ2
+A
+�
+,
+if y > 0,
+0,
+if y = 0.
+(7)
+B. Outage Probability
+The normalized instantaneous rate, denoted by Rin, for the
+DRAT scenario can be formulated from (4) and expressed as
+Rin = log2 (1 + γmax) = log2
+�
+1 + A2¯γ
+�
+.
+(8)
+Now, the end-to-end outage from V to BS via RIS, denoted by
+Pout, can be defined in terms of a rate threshold, Rth, as
+Pout = Pr [Rin < Rth] = Pr
+�
+log2
+�
+1 + A2¯γ
+�
+< Rth
+�
+= Pr
+
+A <
+�
+2Rth − 1
+¯γ
+
+ = Pr [A < Υth] ,
+(9)
+where Υth =
+�
+2Rth −1
+¯γ
+. Thus, the closed-form expression of
+the outage probability DRAT can be evaluated as
+Pout =
+� Υth
+0
+fA(y)dy,
+=FA (Υth) = 1 − Q
+�Υth − µA
+σ2
+A
+�
+.
+(10)
+C. Spectral Efficiency
+SE for the DRAT scenario can be defined from (8) as
+SE =E [Rin] = E
+�
+log2
+�
+1 + A2 B ¯γ
+��
+,
+=
+� ∞
+0
+log2
+�
+1 + y2 B ¯γ
+�
+fA(y)dy.
+(11)
+The exact derivation of the integral in (11) is mathematically
+intractable, and thus a closed-form expression may not be
+derived. Hence, we resort to approximate SE with tight upper
+and lower bounds by invoking Jensen’s inequality.
+1) Upper Bound: Applying Jensen’s inequality, we define
+the upper bound for SE as SEu, where SE ≤ SEu. Now,
+SEu can be evaluated from (11) as
+SEu = log2
+�
+1 + ¯γ B E
+�
+A2��
+,
+(12)
+and expressed as
+SEu = log2 [1 + ¯γ B M1M2 Ωm1
+×
+�
+1 + (M1 M2 − 1)
+m1
+�Γ(m1 + 1
+2)
+Γ(m1)
+�2��
+.
+(13)
+Evaluation of Upper Bound: In (12), E
+�
+A2�
+can be evaluated
+utilizing the relation Var [X] = E
+�
+X2�
+− (E [X])2 as
+E
+�
+A2�
+=Var [A] + (E [A])2 = σ2
+A + µ2
+A.
+(14)
+After substituting the values of µ2
+A and σ2
+A in (12), the upper
+bound for DRAT-based SE can be evaluated.
+2) Lower Bound: Likewise, we define the lower bound for
+SE as SEl, where SE ≥ SEl. Now, SEl can again be be
+defined from (11) as
+SEl = log2
+�
+1 +
+¯γ B
+E
+� 1
+A2
+�
+�
+,
+(15)
+and expressed as given in (16), on the top of next page.
+Evaluation of Lower Bound: In (15), the expectation
+E
+�
+1/A2�
+can be solved utilizing the Taylor series expansion
+and approximated as [15]
+E
+� 1
+A2
+�
+≈
+1
+E [A2] + Var
+�
+A2�
+[E [A2]]3 .
+(17)
+Since the statistical characteristics of A is known to be Gaussian
+distributed (as discussed earlier in subsection A), A2 will
+follow a non-central chi-square distribution. Thus, the mean
+and variance of A2 can be expressed as
+Var
+�
+A2�
+= 2 σ2
+A
+�
+σ2
+A + 2 µ2
+A
+�
+,
+(18)
+E
+�
+A2�
+= σ2
+A + µ2
+A,
+(19)
+respectively. Thus, utilizing these expressions and substituting
+the values of µ2
+A and σ2
+A, the lower bound for SE of the DRAT
+scenario can be evaluated.
+3) Approximation for Large M: We define SE as approx-
+imate SE (ASE) for large M1 and M2. Now, with the upper
+and lower bounds of SE of the DRAT scenario, exact SE lies
+in-between and can be expressed as
+SEl ≤ SE ≤ SEu.
+(20)
+However, for larger M1 and M2, i.e., M1, M2 ≫ 1, both SEl
+and SEu converge to SE. Thus, ASE can be given as
+SE = log2
+�
+1 + ¯γ B M 2
+1 M 2
+2 Ωm1
+m1
+�Γ(m1 + 1
+2)
+Γ(m1)
+�2�
+.
+(21)
+It can be noted from (21) that, through utilizing dual RIS,
+the fourth order channel gain can be realized, i.e., M 2
+1 M 2
+2 ,
+whereas, for single RIS, the maximum channel gain is of the
+second order, i.e., N 2.
+
+SEl = log2
+
+1 + ¯γ B
+M1M2 Ωm1
+�
+1 + (M1 M2−1)
+m1
+� Γ(m1+ 1
+2 )
+Γ(m1)
+�2�3
+2
+�
+1 + (2M1 M2−1)
+m1
+� Γ(m1+ 1
+2 )
+Γ(m1)
+�2� �
+1 −
+1
+m1
+� Γ(m1+ 1
+2 )
+Γ(m1)
+�2�
++
+�
+1 + (M1 M2−1)
+m1
+� Γ(m1+ 1
+2 )
+Γ(m1)
+�2�2
+
+
+(16)
+TABLE I
+SIMULATION PARAMETERS
+Parameter
+Simulation Values
+Circuit Power
+PBS=10 dBm, PU=10 dBm [5]
+Fading Parameter for DRAT
+m1= 10
+Fading Parameter for Direct Links
+m3= 1
+RIS Power Consumption
+PRE = 10 dBm [5]
+HPA Power Consumption Factor
+α = 1.2
+Noise Floor
+σ2 = -120 dBm
+D. Energy Efficiency
+Now, EE of the dual RIS-aided system is defined as the
+ratio of SE over the total power consumed and can be ex-
+pressed as EE =
+SE
+Ptot , where Ptot denotes the total power
+consumed, which consists of the transmit power, the circuit
+power consumption at BS and V, and the power consumed at
+RIS. Considering all the power consumed, the EE in can be
+expressed
+EE =
+SE
+(1 + ξ)Pt + P c
+V + (M1 + M2)P c
+RIS + P c
+BS
+,
+(22)
+where P c
+RIS denotes the power utilized by each RU, ξ =
+1
+ω
+and ω is the drain efficiency of HPA. Likewise, P c
+V , i.e., the
+power consumed in other circuit components excluding HPA at
+V and P c
+BS is the circuit power consumption at BS.
+This completes the analytical derivation of the outage, SE,
+and EE for DRAT of the uplink of V2I communication.
+IV. SIMULATION RESULTS
+This section discusses and presents the simulation results for
+the performance of the dual RIS-assisted V2I communication.
+Further, the results for the SRAT and DCT scenarios are
+presented for the sake of comparison. The distances between
+V-to-RIS1, RIS1-to-RIS2 and RIS2-to-BS are assumed to be
+5, 100 and 5 meters, respectively. Similarly for the simulation
+purpose, M = M1 = M2 and N is taken as to be N = 2 M,
+in order to maintain the fairness in the comparison. The rest of
+the simulation parameters are summarized in Table I.
+Fig. 2 shows the SE performance for the DRAT scenario,
+where the solid lines without marker points show the exact
+(simulation) performance of DRAT, whereas the markers show
+the analytically derived upper and lower bounds on SE. Ad-
+ditionally, ASE for large M is also plotted. The simulation
+verifies that the derived upper and lower bounds are quite
+tight as the analytically derived bounds are remarkably close
+to the actual performance. Further, it can also be noted that
+the difference between exact SE and ASE (as shown in (21))
+diminishes as M increases. For instance, at M = 10 and
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR (dB)
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+SE (bps/Hz)
+Sim
+UB
+LB
+Approx
+M = 10, 20, 50, 100
+Fig. 2. SE with respect to ¯γ for different M of the proposed DRAT scenario.
+0
+500
+1000
+1500
+2000
+2500
+M 
+0
+5
+10
+15
+20
+25
+SE (bps/Hz)
+DRAT
+SRAT
+DCT
+Fig. 3. SE with respect to M for the proposed DRAT scenario.
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+SNR (dB)
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Outage
+Rth=5
+Rth=7.5
+Rth=10
+Fig. 4. Outage with respect to Pt for different rate thresholds for DRAT.
+¯γ = 30 dB, SE is 1.5037 bps/Hz whereas SE is 1.5034
+bps/Hz; however, at M = 50 and ¯γ = 15 dB, SE is 5.2201
+whereas SE is 5.2200 bps/Hz. Thus, it shows that the bounds
+are quite accurate and near to the exact simulation value.
+Fig. 3 shows the SE results for the DRAT scenario, and
+compares them with the SRAT and DCT scenarios. Specifically,
+it shows SE for a varying number of RUs. The following
+observations can be easily inferred from this plot: 1) Apart
+from smaller M, SE of DRAT is always better than SE of the
+SRAT scheme due to the fourth order gain provided by dual
+RIS. This can also be inferred from the analytical evaluation in
+(31). 2) Due to the multiplicative pathloss, for less number of
+RUs, i.e., smaller M, the DCT scenario may provide better SE
+performance than the RIS-reflected link for both DRAT and
+
+0
+500
+1000
+1500
+2000
+2500
+M 
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+EE (bits/Hz/Joule)
+DRAT
+SRAT
+DCT
+(a) EE with respect to M, here Pt = 10 dBm.
+0
+5
+10
+15
+20
+25
+30
+SNR (dB)
+0
+0.5
+1
+1.5
+EE (bits/Hz/Joule)
+DRAT
+SRAT
+DCT
+(b) EE with respect to SNR, here M = 1000.
+Fig. 5. EE comparison for DRAT with respect to the SRAT and DCT scenarios.
+SRAT scenarios. However, as the number of RUs increases,
+the RIS-based scenarios outperform DCT. 3) Similar to single
+RIS, dual RIS-based DRAT also suffers from the multiplicative
+effect of pathloss. Thus, for smaller RUs, the SRAT scenario
+shows better SE than the DRAT one. 4) As the number of RUs
+increases sufficiently, DRAT outperforms SRAT significantly.
+Fig. 4 shows the outage probability of the DRAT scenario for
+three different rate thresholds, i.e., Rth = {5, 7.5, 10} bps/Hz.
+As evident from the result, the outage can be improved either
+through increasing the transmit power or the number of RUs.
+Since, the transmit power at BS is usually constrained, RIS
+provides an alternate to improve the outage through increasing
+RUs, instead of increasing the transmit power. Thus, to circum-
+vent the power constraint, the number of RUs at RIS can be
+scaled accordingly.
+Fig. 5 shows the EE results of the DRAT scenario, the EE
+plots of the SRAT and DCT scenarios are also plotted here
+for comparison. Specifically, in Fig. 5(a), the performance is
+with respect to M, while in Fig. 5(b), the EE curve is plotted
+against SNR. It can be observed that, for large M, the DRAT
+scenario is the most energy-efficient. Although, for smaller M,
+single RIS provides better EE; this is due to the fact that the
+received signal of the dual RIS-reflected link suffers from the
+multiplicative pathloss that can be mitigated by by large M.
+From the above results on SE and EE, it can be easily inferred
+that the proposed DRAT scheme outperforms SRAT in terms of
+both SE as well as EE. Similarly, the above results also show
+that, for a fixed rate requirement, DRAT requires lower transmit
+power and hence is more energy efficient.
+V. CONCLUSION
+V2X has opened up a slew of novel possibilities in the
+wireless vehicular communication arena, but its potential for
+enabling true ITS has yet to be explored completely, despite
+its significant importance in the safety of autonomous driving.
+In this work, we have envisioned the integration of RIS into
+vehicular networks to realize the true potential in enhancing the
+performance of the V2I communication. Specifically, we have
+evaluated the performance of a dual-RIS assisted V2I uplink
+communication scenario in terms of the outage probability, SE
+and EE. Novel closed-form expressions are derived and verified
+through the extensive numerical simulations. The results show
+a significant gain in the performance can be achieved through
+the proposed RIS scenario.
+VI. ACKNOWLEDGEMENT
+This work was supported by the Nazarbayev University CRP
+Grant no. 11022021CRP1513.
+REFERENCES
+[1] J. Wang, K. Zhu, and E. Hossain, “Green internet of vehicles (IoV) in the
+6G era: Toward sustainable vehicular communications and networking,”
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+[2] Y. Cao, S. Xu, J. Liu, and N. Kato, “Toward smart and secure V2X
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+

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+page_content=' Khaled Rabie◦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Xingwang Li#,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Theodoros Tsiftsis†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' and Galymzhan Nauryzbayev  School of Engineering and Digital Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Nazarbayev University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Nur-Sultan City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
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+page_content=' Manchester Metropolitan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
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+page_content=' Henan Polytechnic University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Jiaozuo 454000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
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+page_content=' Greece;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  †School of Intelligent Systems Science and Engineering, Jinan University, China  Email: {hamza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='shaikh, galymzhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
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+page_content='kz, ◦k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
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+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='uk,  #lixingwang@hpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='cn, †tsiftsis@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='org  Abstract—Vehicle-to-everything (V2X) connectivity in 5G-and-  beyond communication networks supports the futuristic intelligent  transportation system (ITS) by allowing vehicles to intelligently  connect with everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The advent of reconfigurable intelligent  surfaces (RISs) has led to realizing the true potential of V2X  communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' In this work, we propose a dual RIS-based  vehicle-to-infrastructure (V2I) communication scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Following  that, the performance of the proposed communication scheme  is evaluated in terms of deriving the closed-form expressions  for outage probability, spectral efficiency and energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Finally, the analytical findings are corroborated with simulations  which illustrate the superiority of the RIS-assisted vehicular  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Keywords— Reconfigurable intelligent surface (RIS), dual RIS,  energy efficiency, spectral efficiency, vehicular communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' INTRODUCTION  As a key enabler for intelligent transportation systems (ITSs),  vehicle-to-everything (V2X) communication has sparked a re-  newed interest in the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' V2X encompasses  a wide range of wireless technologies such as vehicle-to-  pedestrian (V2P), vehicle-to-infrastructure (V2I), and vehicle-  to-vehicle (V2V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Additionally, it also includes the vehicu-  lar communications with vulnerable road users (VRUs), grid  (V2G), network (V2N) and cloud (V2C) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The V2X com-  munications will be a critical component of the futuristic  connected and self-driving cars, envisioned and enabled by  the sixth-generation (6G) wireless technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Furthermore,  the V2X communications will also enhance and transform  the quality-of-service (QoS) in terms of unparalleled user  experience, ultra-high road safety and air quality improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  In addition, a slew of advanced applications will also be  supported like platooning, trajectory alignments, exchanging  sensor data and high precision maps, and so on [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thanks to  the enhanced capabilities of 6G, vehicles will receive accurate  safety information, intelligent traffic management support, and  innovative infotainment features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the 6G services will be  used to create a fully automated, autonomous, and ubiquitously  connected vehicular network [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Recently, reconfigurable intelligent surfaces (RISs) have  emerged as a breakthrough technology that offers a great deal  in terms of wireless communication [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Inherently, RIS is a  software-defined artificial structure made up of a large number  of scattering passive elements, termed as reflecting units (RUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  These RUs are capable to adjust the electromagnetic (EM)  properties of a reflected wave that is incident on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus,  RIS can use not only this ability to boost the received signal’s  power, but also the capability to create an additional reflective  link to mitigate the impact of blockages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' With the large number  of RUs, RISs are particularly known to have large spectral and  energy efficiency [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' As a result, RIS may be used to improve  the quality of vehicular communication through establishing  a low-cost, highly energy efficient indirect line-of-sight (LoS)  communications [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  In [7], the authors investigated the outage performance for  RIS-assisted vehicular communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Likewise, the  secrecy outage performance of RIS-aided vehicular communi-  cations has been studied in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' RISs were also investigated for  detecting VRUs such as cyclists, pedestrians and wheelchair  users [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, the authors utilized RISs for enhancing  the radar visibility for VRUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, in [10], the authors  provided a optimization framework for resource allocation  in the RIS-aided vehicular communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, they  jointly optimized the power allocation, RIS reflection coeffi-  cients and spectrum allocation for different QoS requirements  of the V2V and V2I communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Likewise, in [11],  the authors discussed a system model where RSU leverages RIS  to connect the dark zones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', areas blocked due by obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Moreover, a comprehensive overview on the recent advances  in 6G vehicular networks was provided in [12, 13], where the  authors also described various open challenges and possible  research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Motivated by the above, in this work, we investigate the  performance of a dual RIS-assisted V2I communication net-  work scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, the proposed scenario considers the  uplink transmission where the vehicle is communicating with  the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' To enhance the communication capabilities, the  vehicle is supported through two RISs which create a virtual  line-of-sight (LoS) link, which, otherwise, was inherently non-  LoS (NLoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The major contributions can be summarized as  Explicitly, we invoked the central limit theorem (CLT) to  characterize the received signal-to-noise ratio (SNR) for  Vehicle-to-Vehicle (V2V)  Vehicle-to-Infrastructure (V2I)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Schematic for the considered dual RIS-aided V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  the proposed dual RIS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, based on this, we  derived the closed-form expression for outage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Further, we derived the closed-form expressions for the  upper and lower bounds of SE and EE of the proposed  dual RIS-assisted V2I communication scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Finally, as a performance benchmark, the proposed sce-  nario is compared with the single RIS-assisted V2I com-  munication and with RIS V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The results  show the superiority of the proposed scenario of dual RIS-  assisted V2I over the single RIS-assisted V2I communi-  cation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' SYSTEM MODEL  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 1, in this work, we consider a V2I  communication model, wherein the vehicular user (V) tries to  communicate with a nearby base station (BS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Apart from the  direct cellular link, a reflected path through RISs is considered  to support this uplink transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' In particular, we consider  a dual RIS-assisted uplink V2I transmission with two RISs,  one each placed near V and BS both, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' For the two  RISs, the number of RUs is assumed to be M1 and M2 for  RIS-1 and RIS-2, respectively, while keeping the total number  of RUs unchanged, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M1+M2 = N, where N is the number  of RUs in large RIS for the single RIS scenario, which is the  benchmark for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, based on RIS, the following  scenarios are considered in this work  Dual RIS-assisted Transmission (DRAT): In DRAT, the  transmission takes place only through the two RISs and  the reflected link, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Single RIS-assisted Transmission (SRAT): In SRAT, the  transmission takes place through single large RIS which  is placed near to BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Direct Cellular Transmission (DCT): In DCT, V commu-  nicates with BS directly without utilizing RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the  transmission is inherently NLoS and experiences a higher  pathloss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' This would also serve as the baseline  scheme for the performance comparison of the above two  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Channel Model  The channels between V-to-RIS-1 and RIS-2-to-BS can  be modeled as deterministic LOS channels as the distances  are small and the probability of having LoS is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  However, the distance between RIS-1 and RIS-2 is large and  thus the small scale fading for the channel between the ith  element of RIS-1 and the jth element of RIS-2, denoted by  hRR  ij , is modeled through Nakagami-m fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Hence, for  i = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M1} and j = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, the  distances related to the V-to-RIS-1, RIS-1-to-RIS-2 and RIS-2-  to-BS links are represented by d1, dRR and d2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Received Signal Model  The received base-band signal at BS, denoted by r, for the  dual RIS-aided transmission case can be expressed as  r =  �  B Pt  ��M1  i=1  �M2  j=1 ejφ(1)  i hRR  ij ejφ(2)  j  �  s + No,  (1)  where Pt is the transmit power constraint at V, B is the distance-  dependent pathloss, s ∼ CN (0, 1) is the transmitted symbol,  and No ∼ CN  �  0, σ2�  is the additive white Gaussian noise  (AWGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, φ1 and φ2 are the phase of the V-to-RIS1  and RIS2-to-BS channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, for a link distance d, B at  the carrier frequency of 3 GHz can be given by [14]  B(d) [dB] =  �  −37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5 − 22 log10(d/1 m)  if LOS,  −35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='1 − 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='7 log10(d/1 m)  if NLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (2)  Likewise, instantaneous SNR at BS can be formulated as  γ =  ����  �M1  i=1  �M2  j=1 δije  j  �  φ(1)  i  +φ(2)  j  −ϕij  �����  2  B Pt  σ2  ,  (3)  where δij and ϕij denote the amplitude and phase of hRR  ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  1) RIS Reflection Parameters: Now, SNR at BS can be  maximized through adjusting the phase at RISs to cancel  the resultant phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', φ(1)  i  + φ(2)  j  − ϕij = 0, for i =  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M1} and j = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, by substituting  ϕij = φ(1)  i  + φ(2)  j , ∀i, j, the received signal power at BS can  be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Consequently, maximized SNR corresponding  to the optimal phase can be given as  γmax =  ����M1  i=1  �M2  j=1 δij  ���  2  B Pt  σ2  = A2B Pt  σ2  = A2 B ¯γ,  (4)  where A2 =  ���  �M1  i=1  �M2  j=1 δij  ���  2  is the cascaded channel gain  provided by RISs, and ¯γ = Pt/σ2 is transmit SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Likewise, proceeding in the similar way, for the SRAT  scenario, maximized SNR at BS can be given as1  ˆγmax =  ��N  i=1 βi  �2  ¯γ = B2¯γ,  (5)  where βi is the amplitude of a channel between RIS and  V, denoted by hRU  i  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', hRU  i  = βie−jϕi, and B2 is the  corresponding channel gain provided by single RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  1For the SRAT scenario, the analysis is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the detailed description  is omitted for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' In particular, for SRAT, large RIS with N  RUs is present near BS, where N = M1 + M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Likewise, the RIS-to-BS link  is also modeled as Nakagami-m fading with the rest of the parameters being  the same, as in DRAT, like transmit power constraint at V, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' PERFORMANCE ANALYSIS  This section initially evaluates SNR for the dual RIS-aided  V2I scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Utilizing the SNR expressions formulated earlier,  the outage probability, SE and EE are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Statistical Characterization of the Dual RIS Channel Gain  Now utilizing CLT for M  ≫ 1, A = �M1  i=1  �M2  j=1 δij  can be approximated through a Gaussian distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=',  A ∼ N(µy, σ2  y) [15], with a probability density function (PDF)  given by  fA(y) =  \uf8f1  \uf8f2  \uf8f3  1  √  2πσ2  A exp  �  −(y−µA)2  2σ2  A  �  ,  if y > 0,  0,  if y = 0,  (6)  where µA = �M1  i=1  �M2  j=1 µij, σ2  A = �M1  i=1  �M2  j=1 σ2  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Here,  µij and σ2  ij are the mean and variance of the random variable  δij, which follows the Nakagami-m distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Hence, µij =  Γ(m1+ 1  2 )  Γ(m1)  �� Ωm1  m1  �  and σ2  ij = Ωm1  �  1 −  1  m1  �  Γ(m1+ 1  2 )  Γ(m1)  �2�  ,  for all i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' , M1} and j = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Likewise the cumulative distribution function (CDF) of A  can be derived from its PDF as  FA(y)=  � y  −∞  fA(y)dy =  �  1−Q  �  y−µA  σ2  A  �  ,  if y > 0,  0,  if y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (7)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Outage Probability  The normalized instantaneous rate, denoted by Rin, for the  DRAT scenario can be formulated from (4) and expressed as  Rin = log2 (1 + γmax) = log2  �  1 + A2¯γ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (8)  Now, the end-to-end outage from V to BS via RIS, denoted by  Pout, can be defined in terms of a rate threshold, Rth, as  Pout = Pr [Rin < Rth] = Pr  �  log2  �  1 + A2¯γ  �  < Rth  �  = Pr  \uf8ee  \uf8f0A <  �  2Rth − 1  ¯γ  \uf8f9  \uf8fb = Pr [A < Υth] ,  (9)  where Υth =  �  2Rth −1  ¯γ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the closed-form expression of  the outage probability DRAT can be evaluated as  Pout =  � Υth  0  fA(y)dy,  =FA (Υth) = 1 − Q  �Υth − µA  σ2  A  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (10)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Spectral Efficiency  SE for the DRAT scenario can be defined from (8) as  SE =E [Rin] = E  �  log2  �  1 + A2 B ¯γ  ��  ,  =  � ∞  0  log2  �  1 + y2 B ¯γ  �  fA(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (11)  The exact derivation of the integral in (11) is mathematically  intractable, and thus a closed-form expression may not be  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Hence, we resort to approximate SE with tight upper  and lower bounds by invoking Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  1) Upper Bound: Applying Jensen’s inequality, we define  the upper bound for SE as SEu, where SE ≤ SEu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Now,  SEu can be evaluated from (11) as  SEu = log2  �  1 + ¯γ B E  �  A2��  ,  (12)  and expressed as  SEu = log2 [1 + ¯γ B M1M2 Ωm1  ×  �  1 + (M1 M2 − 1)  m1  �Γ(m1 + 1  2)  Γ(m1)  �2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (13)  Evaluation of Upper Bound: In (12), E  �  A2�  can be evaluated  utilizing the relation Var [X] = E  �  X2�  − (E [X])2 as  E  �  A2�  =Var [A] + (E [A])2 = σ2  A + µ2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (14)  After substituting the values of µ2  A and σ2  A in (12), the upper  bound for DRAT-based SE can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  2) Lower Bound: Likewise, we define the lower bound for  SE as SEl, where SE ≥ SEl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Now, SEl can again be be  defined from (11) as  SEl = log2  �  1 +  ¯γ B  E  � 1  A2  �  �  ,  (15)  and expressed as given in (16), on the top of next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Evaluation of Lower Bound: In (15), the expectation  E  �  1/A2�  can be solved utilizing the Taylor series expansion  and approximated as [15]  E  � 1  A2  �  ≈  1  E [A2] + Var  �  A2�  [E [A2]]3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (17)  Since the statistical characteristics of A is known to be Gaussian  distributed (as discussed earlier in subsection A), A2 will  follow a non-central chi-square distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, the mean  and variance of A2 can be expressed as  Var  �  A2�  = 2 σ2  A  �  σ2  A + 2 µ2  A  �  ,  (18)  E  �  A2�  = σ2  A + µ2  A,  (19)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, utilizing these expressions and substituting  the values of µ2  A and σ2  A, the lower bound for SE of the DRAT  scenario can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  3) Approximation for Large M: We define SE as approx-  imate SE (ASE) for large M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Now, with the upper  and lower bounds of SE of the DRAT scenario, exact SE lies  in-between and can be expressed as  SEl ≤ SE ≤ SEu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (20)  However, for larger M1 and M2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M1, M2 ≫ 1, both SEl  and SEu converge to SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, ASE can be given as  SE = log2  �  1 + ¯γ B M 2  1 M 2  2 Ωm1  m1  �Γ(m1 + 1  2)  Γ(m1)  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  (21)  It can be noted from (21) that, through utilizing dual RIS,  the fourth order channel gain can be realized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', M 2  1 M 2  2 ,  whereas, for single RIS, the maximum channel gain is of the  second order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  SEl = log2  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f01 + ¯γ B  M1M2 Ωm1  �  1 + (M1 M2−1)  m1  � Γ(m1+ 1  2 )  Γ(m1)  �2�3  2  �  1 + (2M1 M2−1)  m1  � Γ(m1+ 1  2 )  Γ(m1)  �2� �  1 −  1  m1  � Γ(m1+ 1  2 )  Γ(m1)  �2�  +  �  1 + (M1 M2−1)  m1  � Γ(m1+ 1  2 )  Γ(m1)  �2�2  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  (16)  TABLE I  SIMULATION PARAMETERS  Parameter  Simulation Values  Circuit Power  PBS=10 dBm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' PU=10 dBm [5]  Fading Parameter for DRAT  m1= 10  Fading Parameter for Direct Links  m3= 1  RIS Power Consumption  PRE = 10 dBm [5]  HPA Power Consumption Factor  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='2  Noise Floor  σ2 = -120 dBm  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Energy Efficiency  Now, EE of the dual RIS-aided system is defined as the  ratio of SE over the total power consumed and can be ex-  pressed as EE =  SE  Ptot , where Ptot denotes the total power  consumed, which consists of the transmit power, the circuit  power consumption at BS and V, and the power consumed at  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Considering all the power consumed, the EE in can be  expressed  EE =  SE  (1 + ξ)Pt + P c  V + (M1 + M2)P c  RIS + P c  BS  ,  (22)  where P c  RIS denotes the power utilized by each RU, ξ =  1  ω  and ω is the drain efficiency of HPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Likewise, P c  V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', the  power consumed in other circuit components excluding HPA at  V and P c  BS is the circuit power consumption at BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  This completes the analytical derivation of the outage, SE,  and EE for DRAT of the uplink of V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' SIMULATION RESULTS  This section discusses and presents the simulation results for  the performance of the dual RIS-assisted V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Further, the results for the SRAT and DCT scenarios are  presented for the sake of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The distances between  V-to-RIS1, RIS1-to-RIS2 and RIS2-to-BS are assumed to be  5, 100 and 5 meters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Similarly for the simulation  purpose, M = M1 = M2 and N is taken as to be N = 2 M,  in order to maintain the fairness in the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The rest of  the simulation parameters are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 2 shows the SE performance for the DRAT scenario,  where the solid lines without marker points show the exact  (simulation) performance of DRAT, whereas the markers show  the analytically derived upper and lower bounds on SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Ad-  ditionally, ASE for large M is also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The simulation  verifies that the derived upper and lower bounds are quite  tight as the analytically derived bounds are remarkably close  to the actual performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Further, it can also be noted that  the difference between exact SE and ASE (as shown in (21))  diminishes as M increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' For instance, at M = 10 and  0  5  10  15  20  25  30  35  40  45  50  SNR (dB)  0  2  4  6  8  10  12  14  16  18  20  SE (bps/Hz)  Sim  UB  LB  Approx  M = 10, 20, 50, 100  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' SE with respect to ¯γ for different M of the proposed DRAT scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  0  500  1000  1500  2000  2500  M  0  5  10  15  20  25  SE (bps/Hz)  DRAT  SRAT  DCT  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' SE with respect to M for the proposed DRAT scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  0  2  4  6  8  10  12  14  16  18  20  SNR (dB)  10-6  10-5  10-4  10-3  10-2  10-1  100  Outage  Rth=5  Rth=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5  Rth=10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Outage with respect to Pt for different rate thresholds for DRAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  ¯γ = 30 dB, SE is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5037 bps/Hz whereas SE is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5034  bps/Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' however, at M = 50 and ¯γ = 15 dB, SE is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='2201  whereas SE is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='2200 bps/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, it shows that the bounds  are quite accurate and near to the exact simulation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 3 shows the SE results for the DRAT scenario, and  compares them with the SRAT and DCT scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically,  it shows SE for a varying number of RUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The following  observations can be easily inferred from this plot: 1) Apart  from smaller M, SE of DRAT is always better than SE of the  SRAT scheme due to the fourth order gain provided by dual  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' This can also be inferred from the analytical evaluation in  (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 2) Due to the multiplicative pathloss, for less number of  RUs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', smaller M, the DCT scenario may provide better SE  performance than the RIS-reflected link for both DRAT and  0  500  1000  1500  2000  2500  M  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='7  EE (bits/Hz/Joule)  DRAT  SRAT  DCT  (a) EE with respect to M, here Pt = 10 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  0  5  10  15  20  25  30  SNR (dB)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5  EE (bits/Hz/Joule)  DRAT  SRAT  DCT  (b) EE with respect to SNR, here M = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' EE comparison for DRAT with respect to the SRAT and DCT scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  SRAT scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' However, as the number of RUs increases,  the RIS-based scenarios outperform DCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 3) Similar to single  RIS, dual RIS-based DRAT also suffers from the multiplicative  effect of pathloss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, for smaller RUs, the SRAT scenario  shows better SE than the DRAT one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 4) As the number of RUs  increases sufficiently, DRAT outperforms SRAT significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 4 shows the outage probability of the DRAT scenario for  three different rate thresholds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=', Rth = {5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='5, 10} bps/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  As evident from the result, the outage can be improved either  through increasing the transmit power or the number of RUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Since, the transmit power at BS is usually constrained, RIS  provides an alternate to improve the outage through increasing  RUs, instead of increasing the transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Thus, to circum-  vent the power constraint, the number of RUs at RIS can be  scaled accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 5 shows the EE results of the DRAT scenario, the EE  plots of the SRAT and DCT scenarios are also plotted here  for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 5(a), the performance is  with respect to M, while in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' 5(b), the EE curve is plotted  against SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' It can be observed that, for large M, the DRAT  scenario is the most energy-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Although, for smaller M,  single RIS provides better EE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' this is due to the fact that the  received signal of the dual RIS-reflected link suffers from the  multiplicative pathloss that can be mitigated by by large M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  From the above results on SE and EE, it can be easily inferred  that the proposed DRAT scheme outperforms SRAT in terms of  both SE as well as EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Similarly, the above results also show  that, for a fixed rate requirement, DRAT requires lower transmit  power and hence is more energy efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' CONCLUSION  V2X has opened up a slew of novel possibilities in the  wireless vehicular communication arena, but its potential for  enabling true ITS has yet to be explored completely, despite  its significant importance in the safety of autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  In this work, we have envisioned the integration of RIS into  vehicular networks to realize the true potential in enhancing the  performance of the V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Specifically, we have  evaluated the performance of a dual-RIS assisted V2I uplink  communication scenario in terms of the outage probability, SE  and EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' Novel closed-form expressions are derived and verified  through the extensive numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' The results show  a significant gain in the performance can be achieved through  the proposed RIS scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
+page_content=' ACKNOWLEDGEMENT  This work was supported by the Nazarbayev University CRP  Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfkwf2/content/2301.03983v1.pdf'}
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+AmbieGen: A Search-based Framework for Autonomous
+Systems Testing
+Dmytro Humeniuk, Foutse Khomh, Giuliano Antoniol
+Polytechnique Montr´eal, 2500 Chemin de Polytechnique, QC H3T 1J4, Montr´eal,
+Canada
+Abstract
+Thorough testing of safety-critical autonomous systems, such as self-driving
+cars, autonomous robots, and drones, is essential for detecting potential fail-
+ures before deployment. One crucial testing stage is model-in-the-loop test-
+ing, where the system model is evaluated by executing various scenarios in
+a simulator. However, the search space of possible parameters defining these
+test scenarios is vast, and simulating all combinations is computationally in-
+feasible. To address this challenge, we introduce AmbieGen, a search-based
+test case generation framework for autonomous systems.
+AmbieGen uses
+evolutionary search to identify the most critical scenarios for a given system,
+and has a modular architecture that allows for the addition of new systems
+under test, algorithms, and search operators. Currently, AmbieGen supports
+test case generation for autonomous robots and autonomous car lane keep-
+ing assist systems. In this paper, we provide a high-level overview of the
+framework’s architecture and demonstrate its practical use cases.
+Keywords:
+evolutionary search, autonomous systems, self driving cars,
+autonomous robots, neural network testing
+Metadata
+The project metadata is presented in Table 1.
+1. Motivation and significance
+Autonomous systems, including autonomous vehicles, robots, or drones
+can provide a number of benefits such as driving assistance, high-risk zone
+Preprint submitted to Science of Computer Programming
+January 4, 2023
+arXiv:2301.01234v1  [cs.RO]  1 Jan 2023
+
+Nr.
+Code metadata description
+Please fill in this column
+C1
+Current code version
+v0.1.0
+C2
+Permanent link to code/repository
+used for this code version
+For
+example:
+https://github.
+com/swat-lab-optimization/
+AmbieGen-tool
+C3
+Permanent
+link
+to
+Reproducible
+Capsule
+https://codeocean.com/
+capsule/1741442/tree
+C4
+Legal Code License
+MIT license (MIT)
+C5
+Code versioning system used
+git
+C6
+Software code languages, tools, and
+services used
+python
+C7
+Compilation requirements, operat-
+ing environments and dependencies
+indicated in requirements.txt
+C8
+If available, link to developer docu-
+mentation/manual
+https://github.com/
+swat-lab-optimization/
+AmbieGen-tool/blob/master/
+README.md
+C9
+Support email for questions
+[email protected]
+Table 1: Code metadata (mandatory)
+exploration, and aid in rescue operations. At the same time, these are safety-
+critical systems and it is very important to ensure they are robust to unseen
+environments and conditions. This can be done by thorough testing prior
+to their deployment. Typically, at the initial development stages model-in-
+the-loop testing is performed [1], where the system is tested in a simulation
+environment. Given the complexity of autonomous systems, the number of
+potential test scenarios is vast and exhaustive execution is not feasible. For
+example, an autonomous vehicle scenario could involve a variety of param-
+eters such as road topology, the movement and behavior of other vehicles
+and pedestrians, traffic signs, weather conditions, etc. We surmise that in
+order to identify the most critical scenarios for a given system, application
+of search algorithms is necessary.
+In this work, we propose AmbieGen, a search based framework for gen-
+erating adversarial test scenarios for autonomous systems.
+By leveraging
+evolutionary search AmbieGen allows to find challenging and diverse test
+scenarios.
+2
+
+The problem of identifying critical scenarios for a system has been ad-
+dressed in several previous works on falsifying temporal logic requirements
+of cyber-physical systems, such as S-Taliro [2], Breach [3], and ARIsTEO [4].
+These works typically consider falsifying a model of the system that takes a
+set of input signals and produces a set of output signals.
+In our work, we focus on testing autonomous systems for which the input
+signals are complex and may include data from various sensors and cameras.
+Generating a valid combination of falsifying input signals (such as lidar read-
+ings and RGB camera readings) directly would be challenging. Therefore,
+we propose a method for generating test cases that specify a virtual environ-
+ment for the autonomous system, rather than the input signals. The input
+signals are generated in the virtual environment during simulation based on
+the actions of the autonomous agent.
+Several approaches have been proposed for generating virtual environ-
+ments for testing autonomous driving and robotics systems, including As-
+Fault [5], Frenetic [6], DeepJanus [7], DeepHyperion [8] and others presented
+at the SBST 2021 [9] and SBST 2022 [10] tool competitions.
+The tool we present in this paper, AmbieGen, is the winner of SBST 2022
+tool competition. It could produce the biggest number of diverse fault reveal-
+ing scenarios for an autonomous vehicle lane keeping assist system (LKAS)
+given a limited time budget. More details about the search algorithm im-
+plementation can be found in our research paper [11]. In our work we have
+shown that the simplified model of the system can be effective in guiding the
+search for producing the test scenarios for the full, simulator based, model
+of the system.
+Our framework can be used for further research in the search algorithms,
+search operator and fitness function design for autonomous systems adver-
+sarial testing. We built the framework to be modular, and thus easily cus-
+tomizable. By referring to project documentation as well as the example
+implementations we provide, researchers can specify their own test scenario
+generation problems, fitness functions, crossover and mutation operators.
+This tool is developed in Python and can be easily run as a python package.
+More instructions and examples are provided in the AmbieGen repository.
+2. Software description
+In this work, we present AmbieGen, an open-source Python framework
+that utilizes evolutionary search for the generation of test scenarios for au-
+3
+
+tonomous systems. Currently, AmbieGen supports the creation of test sce-
+narios for lane keeping assist systems (LKAS) in autonomous vehicles and
+for autonomous robots navigating a closed room with obstacles.
+The test scenarios for LKAS in vehicles are designed to challenge the
+system with various road topologies, while the scenarios for autonomous
+robots involve navigating a closed room with obstacles.
+Examples of the
+generated scenarios can be seen in Figure 1.
+Figure 1: An example of the test case for LKAS system (a) and an autonomous robot (b).
+The x-axis represents the map length in meters, and the y-axis represents the map width
+in meters.
+2.1. Software architecture
+This subsection provides a detailed description of the software imple-
+mentation of AmbieGen. The key components of AmbieGen are illustrated
+in Figure 2, which are common components for implementing evolutionary
+search. We use the Pymoo framework [12] to implement the search algo-
+rithms. The most important modules and classes are outlined below:
+• Solution - this is one of the most important classes, which contains all
+the necessary attributes and functions needed to represent the candi-
+date solution of the algorithm. It should contain a scenario attribute
+with the list of test case parameters, function for fitness evaluation,
+novelty calculation, as well as, optionally, image generation.
+4
+
+200
+a
+40
+b
+口
+Ci
+■
+35-
+■
+■
+■
+30
+■
+25
+Robotpath
+20-
+Walls
+V
+国
+15 -
+■
+■
+口
+■
+10
+5
+■
+■
+fo
+0
+0
+5
+10
+15
+20
+25
+200
+30
+35
+40Figure 2: AmbieGen architecture
+• Sampling - this is the class for initial population generation. At the
+output it provides N instances of the Solution class, with the initial-
+ized scenario attribute, defining the test scenario. Typically the test
+scenario is represented by a two dimensional array, randomly initial-
+ized based on the minimum and maximum values of the test case pa-
+rameters, defined in the configuration file. Each column of the array
+corresponds to some part of the environment. More information about
+the representation of the test scenarios that we used can be found in
+the repository page as well as in our research article.
+• Problem - in this class, we define the logic for evaluating the fitness
+of each solution. For single-objective search (using GA), we specify
+the fitness function for evaluating the scenario fault revealing power.
+For two-objective search (using NSGA-II), we define two objectives:
+fault revealing power and novelty calculation. The novelty objective
+is calculated as the average novelty of a given test scenario relative to
+the 5 solutions with the highest fault revealing power fitness. If the
+problem has any constraints, such as a minimum required fitness value,
+they should also be specified in this class.
+• TC to environment - this is a function to transform the test case (TC)
+encoded as a 2D array of parameters, to the input format suitable for
+the system model. For example, for the LKAS problem, the model
+input is a list of the 2D coordinates of points, defining the road topol-
+ogy. The test case itself is represented as a sequence of transformations
+5
+
+Pymoo
+post_processing()
+Sampling
+Solution object 1
+TC to environment ()
++gen_randomscenario()
+Solution object 2
+Problem
+fitness evaluation ()
+Solution object 3
+Solution
+Crossover
++map_size
+Solution object 4
+configuration file
++scenario
+Mutation
++fitness eval()
+Population size
+Solution object N
+Numberofgenerations
++novelty eval()
+Crossover/mutationrate
++build image()
+TCparameterranges
+Folderto save resultsneeded to perform to obtain the points. For the autonomous robot the
+test scenario is represented as a sequence of parameters describing the
+2D map with obstacles. The TC to environment module is used to
+create a 2D bitmap from the given parameters. The bitmap is given
+as the input to the autonomous robot model, which runs a planning
+algorithm to find the shortest path between the start and goal location.
+• fitness evaluation - a function to calculate the fitness i.e fault revealing
+power of the scenario. It takes the output of the TC to environment
+function as the input and execute the system model. It collects the
+data about the model behaviour during execution and computes the
+fitness score. For the LKAS system, the fitness is defined by the biggest
+deviation from the lane center and for the autonomous robot - by the
+length of the path to reach the goal.
+• Crossover - in this class the crossover operator is defined. Currently
+we are using a one point crossover, which can be applied to fixed and
+variable length solutions.
+• Mutation - in this class the mutation operator is implemented. We
+have 2 types of mutations: exchange and change of variable. In ex-
+change mutation, two randomly selected columns of the test case are
+exchanged. In the case of the road topology, it would correspond to
+exchanging the positions of two random road segments. In change of
+variable mutation, a randomly selected parameter value in the test case
+matrix is changed. In the road topology example it could correspond
+to the change of the length of one of the straight road segments.
+• post processing - The post-processing module of our framework includes
+several functions for handling the test suite and its metadata.
+The
+function get test suite() retrieves the test suite, get stats() retrieves
+metadata such as fitness and novelty scores, and save tcs images()
+saves the images of the test cases. The size of the test suite, denoted
+as T, can be specified in the configuration file. In our experiments, T
+was typically set to 30, representing the best solutions found by the
+algorithm.
+Metadata for the test suite includes the fitness of the top T solutions,
+their novelty (calculated as the average novelty between all pairs of
+scenarios in the test suite), and the convergence (best solution fitness
+6
+
+found at each epoch).
+The post-processing module also includes a
+compare.py script for comparing the results of different algorithms,
+using the collected metadata to generate convergence plots and fitness
+and diversity boxplots.
+• configuration file - finally we have a configuration file, where the pa-
+rameters of the algorithm, such as: the population size, the number
+of generations, crossover/mutation rate, and the test suite size are de-
+fined. Users should also specify the allowable ranges for the test case
+parameters and the paths for saving the resulting test suite and its
+metadata.
+Currently, when adding a new problem, one should provide the implemen-
+tation of each of the modules as well as the TC to environment and fitness
+evaluation functions. We are working on reducing the number of additional
+implementations needed. Our framework includes the implementation of all
+the modules for the LKAS and autonomous robot test case generation prob-
+lems.
+2.2. Software functionalities
+AmbieGen public version 0.1.0 as presented in this paper offers the fol-
+lowing major functionalities:
+• Autonomous vehicle LKAS system testing: generating scenarios, rep-
+resented as a list of 2D coordinates defining the road topology.
+• Autonomous robot testing: generating scenarios, represented as the 2D
+bitmap, defining obstacle locations in a fixed sized map.
+• Search-based generation: our framework provides options for search-
+based test suite generation, including random search, single-objective
+genetic algorithm (GA), and two-objective genetic algorithm (NSGA-
+II). The search algorithms are implemented using the Pymoo frame-
+work [12], and can be easily extended to support additional algorithms
+supported by Pymoo with minor modifications.
+The single-objective GA optimizes the test suite for scenario fault re-
+vealing power, while the two-objective NSGA-II optimizes for both
+fault revealing power and diversity.
+As demonstrated in our previ-
+ous work [11], the two-objective algorithm allows to produce a more
+diverse set of test scenarios compared to the single-objective search.
+7
+
+• Experiment data tracking: AmbieGen tracks the results of each ex-
+periment and saves them in a user-defined location. The saved data
+includes the T (as determined by the user) best test scenarios identified
+based on their fitness or crowding distance, as well as their associated
+metadata such as fitness, average diversity, and visualizations. This
+allows for easy analysis and comparison of the results of different ex-
+periments.
+2.3. Use cases of the software
+In this subsection we provide an illustrative example of how to use Am-
+bieGen to generate test cases for an autonomous robot planning algorithm
+testing. Suppose we want to perform 30 runs of the NSGA-II algorithm with
+150 individuals and 200 generations to evaluate this configuration. We want
+to save the generated test cases, their illustrations as well as their metadata,
+such as fitness and diversity. Below you can see the configuration file entries
+with the parameters we chose for the genetic algorithm and well as the path
+to save the experiment results:
+ga = {" pop_size ": 150, "n_gen ": 200, "mut_rate ": 0.4, "cross_rate ": 0.9,
+" test_suite_size ": 30 }
+files = {" stats_path ": "stats", "tcs_path ": "tcs", "images_path ": images "}
+Now we are ready to start the test case generation. We can launch Am-
+bieGen with the following command and parameters:
+python
+optimize.py --problem =" robot" --algo =" nsga2" --runs =30 \\
+--save_results=True
+The search will start and you could see some printouts, such as in Fig. 3 with
+the current number of generation (n gen), number of evaluations (n eval),
+constraint violation (cv min), number of non-dominant solution for NSGA-
+II algorithm (n nds) and the best solution found (f opt) for GA algorithm.
+More details about the printed information can be found on the Pymoo page
+(https://pymoo.org/interface/display.html).
+After a successful run, you will see the confirmation about the run exe-
+cution time, saved test cases, their metadata and the images, as in Fig. 4
+In Fig. 5 you can see examples of the metadata saved, such as the algo-
+rithm convergence 5a (the best fitness value at each generation in the format
+”evaluation number”: best fitness found), the fitness of the test cases in the
+test suite as well as their average diversity i.e., novelty 5b. Novelty is cal-
+culated as the average diversity of all of the pairs of the test cases in the
+8
+
+Figure 3: Printouts during the search
+Figure 4: Successful run confirmation
+test suite. In Fig. 6 we show an example of the test case images saved for a
+particular run.
+(a) Scenario fitness convergence
+(b) Final test suite fitness and diversity
+Figure 5: Metadata for the generated scenarios
+Finally, let us suppose we also want to run a random search with the same
+evaluation budget to be able to compare the performance of our configuration
+of NSGA-II algorithm to some baseline. We can run the random search by
+9
+
+01:0602.320 INFO
+started test generation,writing logs to file: logs.txt
+-12-9101:0602,320INFO
+Running the optimization
+2-12-9801:0602,321INFO
+Problem: robot,Algorithm:nsga2,Runs number:3e,Saving the results:True
+2-12-9101:06.02,343INFO
+Executing run o:
+2-12-9101:06:02344INFO
+Using random seed:1753925990
+n_gen
+n_eval
+innds
+cvmin
+cv_avg
+eps
+indicator
+1
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+1
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+2
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+1
+4.684567E+01
+7.742613E+01
+3
+450
+1
+4.436039E+01
+7.231653E+01
+4
+600
+工
+3.167410E+01
+6.692135E+01
+5
+750
+1
+7.8751083190
+6.161694E+0103:21:13,072INFO
+Execution time,6909.677314 sec
+,088INFO
+Test suite of 3o test scenarios generated
+$,103INFO
+Thehighest fitnessfound:224.994949
+3:211L3,103INFO
+Average diversity:0.720751
+03:21:25,148INFO
+Stats savedas stats nsga231-12-2022-stats.json
+03:21:25,157INFO
+Stats saved asstats nsga231-12-2022-conv.json
+3:21:25,361INFO
+Test cases saved as tcs nsga2l31-12-2022-tcs.json
+03:21:53,871INFO
+Images saved in tc images nsga2
+21:53,871INFO
+Images saved in tcimagesnsga2rung":f
+"158"：97.81219330881972
+200"：99.59797974644661
+"250"：99.59797974644661,
+"300"：186.18376618487352,
+"350"：186.18376618407352,
+"480"：106.18376618407352,
+"450：107.25483399593897,
+"500"：107.25483399593897,
+"550":130.56854249492375,
+"600"：130.56854249492375,
+"650"：130.56854249492375,
+"700"：130.56854249492375,
+"888"130.56854249492375
+"850":
+130.56854249492375,
+"900":rune":f
+"fitness":[
+198.7106781186548
+171.8538238691624,
+192.02438661763966,
+194.46803743153552,
+190.36753236814718,
+211.88225099390866,
+209.88225099390866,
+194.85281374238588,
+168.71067811865476,
+191.8822509939086,
+181.15432893255073,
+183.39696961967007
+novelty":0.23096571372433472
+runi"Figure 6: Images of the generated scenarios
+executing the following command:
+python
+optimize.py --problem =" robot" --algo =" random" --runs =30 \\
+--save_results=True
+The random search will be run and the metadata saved, as in the previous
+case. Now we can compare the results produced by the two different search
+algorithms via executing the following command:
+python
+compare.py --stats_path =" stats_nsga2" " stats_random" \\
+--stats_names "NSGA -II" "Random"
+In the stats path argument we specify the paths of the metadata for the
+runs we wish to compare and in the stats names the names we assign for the
+runs.
+In Fig.7 and Fig. 8 we can see examples of the outputs produced by the
+compare.py script. Fig. 7a shows the fitness and Fig. 7b the diversity of the
+scenarios in the test suites produced over the specified number of runs. Fig.
+8 shows the best values found by the compared search algorithms over the
+generations.
+3. Illustrative examples
+In this section, we present the summarized results of several test genera-
+tion case studies using the AmbieGen tool. The full results can be found in
+our research paper [11] and the SBST 2022 competition report [10].
+We conducted a case study on an autonomous robot with an obstacle
+avoidance algorithm based on nearness diagrams [13]. The robot model was
+a Pioneer 3-AT equipped with a SICK LMS200 laser with a sensing range
+of 10 meters. The simulations were run in the Player/Stage simulator [14].
+10
+
+2022-10-15-images_fin_rob
+Vruno
+o.png
+1.png
+2.png
+Test case fitenss 198.7106781186548
+3.png
+Robotpathm
+质4.png
+Walls
+35
+5.png
+6.png
+30
+7.png
+25 
+8.png
+9.png
+U
+20
+10.png
+U
+15 
+11.png
+U
+10
+12.png
+U
+13.png
+U
+5
+14.png
+U
+15.png
+U
+0
+5
+10
+15
+20
+30
+35
+40(a) Scenario fitness
+(b) Scenario diversity
+Figure 7: Evaluating the NSGA-II algorithm for autonomous robot test case generation
+Figure 8: Comparing the convergence of NSGA-II and random search for autonomous
+robot case study
+You can see an illustration of the simulation environment in Fig. 9a. We
+used AmbieGen to generate diverse maps with obstacles to test the robot’s
+performance. We identified several scenarios in which the robot became stuck
+and failed to reach its goal location. An example of such a scenario can be
+found in the following video: Video.
+To evaluate the effectiveness of our tool, we allocated a two-hour budget
+for AmbieGen to generate test scenarios. The generated scenarios were then
+passed to the simulator and executed. We repeated the experiment 30 times,
+using both the NSGA-II and random search configurations of AmbieGen.
+The average number of failures detected is shown in Fig. 9b. On average,
+11
+
+220
+NSGA-II
+Random
+200
+180
+Fitness
+160
+140
+120
+100
+80
+0
+20
+40
+60
+80
+100
+120
+140
+Numberofgenerations250
+200
+itness
+.150
+左 100
+50
+0
+Random
+NSGA-II
+Algorithm0.8
+0.6
+Novelty
+0.4
+0.2
+0.0
+Random
+NSGA-II
+AlgorithmAmbieGen detected 9 failures in two hours, compared to 2 failures for random
+search
+(a) Executing autonomous robot scenario in the Play-
+er/Stage simulator
+(b) The number of failures revealed by AmbieGen for
+the robot case study
+Figure 9: Using AmbieGen for testing autonomous robot navigation algorithm
+In the second case study, we evaluated the performance of our test gener-
+ation tool on an autonomous vehicle lane keeping assist system (LKAS) using
+the BeamNg simulator [15]. We used the AmbieGen tool to generate diverse,
+fault-revealing road topologies, which were then simulated in the BeamNg
+environment (shown in Fig. 10a). During the simulations, we identified a
+number of scenarios in which the vehicle left its lane (an example of which
+can be seen in the video at Video).
+We ran our tool for a time budget of 2 hours, using the SBST22 compe-
+tition code pipeline. The failure criterion for the LKAS system was defined
+as more than 85% of the car’s area leaving the lane. The driving agent had
+a maximum speed of 70 Km/h. We compared the results of AmbieGen’s
+NSGA-II configuration, Random Search configuration, and the Frenetic tool
+[6], which was also given a 2-hour time budget for test generation.
+As shown in Fig. 10b, out of 30 runs, AmbieGen and Frenetic on average
+produced almost the same number of failures (14), while Random Search
+produced an average of 9 failures.
+The obtained results suggest that AmbieGen could effectively identify
+failures in the autonomous systems under test.
+4. Impact
+Autonomous systems testing is an important area of research, and finding
+test scenarios that reveal a diverse range of system failures within a limited
+12
+
+25
+faults
+20
+15
+Revealed
+10
+5
+0
+AmbieGen
+RandomSearch
+Generationmethod(a) Executing the LKAS scenario in the BeamNg sim-
+ulator
+(b) The number of failures revealed by AmbieGen for
+the LKAS case study
+Figure 10: Using AmbieGen to test autonomous vehicle LKAS model
+time and evaluation budget is a significant challenge [16]. One of the common
+solutions is to use evolutionary search to guide the sampling towards more
+challenging scenarios [5, 7]. These search based techniques allow to identify
+potential failures and improve the overall reliability of the system.
+AmbieGen is a test generation tool that uses evolutionary search to gen-
+erate test scenarios for autonomous systems. Its modular design allows for
+customization of the initial population generation function, fitness evaluation
+function, search operators (such as crossover and mutation), and the search
+algorithm itself. Out of the box, AmbieGen supports testing of autonomous
+robots and vehicle LKAS systems, and additional systems can be added using
+the provided implementations as examples.
+AmbieGen is a valuable resource for research on search-based test case
+generation for autonomous systems. Its built-in modules enable easy com-
+parison of different search algorithms and their modifications, based on the
+quality and diversity of the generated solutions, as well as the convergence
+of the algorithm over time.
+AmbieGen can help answer research questions that are not frequently
+discussed in the literature, such as:
+• To what extent the diversity preservation technique A helps improve
+the diversity of the test suite? The importance of the diversity in test
+case generation is extensively discussed in the work of Klikovits et al.
+[17].
+• To what extent does the search operator A helps improve the conver-
+gence over the operator B? To what extent the algorithm A outperforms
+13
+
+30
+25
+ults
+20
+led
+15
+veal
+Rev
+10
+5
+0
+AmbieGen
+Frenetic
+RandomSearch
+Generationmethodthe algorithm B for the test case generation? Improvements to the base-
+line genetic algorithms implementations can lead to better results, as
+discussed by Abdessalem et al. [18], where multi-objective population-
+based search algorithms and decision tree classification were combined.
+• What fitness criteria are more relevant for guiding the system towards
+fault revealing scenarios? This question includes the comparison of the
+single, multi-objective based search as well surrogate model assisted
+search.
+AmbieGen can also be useful in the pursuit of actively studied research ques-
+tions, where the fault revealing test case generation is required, such as:
+transferability of failures from simulation to the real world [19], autonomous
+system failure prediction [20], test case prioritization [21] and others.
+AmbieGen has proven its effectiveness in fault revealing by winning this
+year’s edition of the SBST 2022 cyber-physical testing tool competition. Our
+submission is described in the following article [22] and is available at the fol-
+lowing link https://github.com/dgumenyuk/tool-competition-av.
+We
+have always kept our tool open sourced and we expect more people to start
+using it. We welcome all the contributions for expanding our framework.
+5. Conclusions
+In this paper, we present the AmbieGen framework for search based test
+case generation for autonomous systems, in its public version 0.1.0.
+We
+briefly outline the motivation for developing this framework, its workflow and
+main functionalities. We also provide illustrative examples for using the tool
+for autonomous vehicle lane keeping assist system testing and autonomous
+robot obstacle avoiding algorithm testing.
+The main features of our tool
+include:
+• modular architecture, which allows researchers to easily modify the
+existing modules, such as initial population generation, crossover, mu-
+tation, fitness function as well as introduce new problems and run ex-
+periments;
+• we provide implementations of test case generation for two systems
+under test: autonomous vehicle LKAS system and autonomous robot;
+this implementation includes three search algorithms: random search,
+14
+
+single objective genetic algorithm and a two-objective NSGA-II genetic
+algorithm;
+• our framework is built to be compatible with Pymoo framework [12],
+allowing to fully benefit from the Pymoo framework features, such as
+high number of implemented algorithms in Pymoo.
+6. Future Plans
+Our framework currently includes the implementation of two test case
+generation problems, as well as three algorithms (random search, GA, NSGA-
+II) for generating test cases. The fitness function is calculated based on a
+simplified model of the system, and test scenarios are represented as 2D
+arrays, with each column describing a discrete aspect of the scenario. In the
+future, we plan to expand the capabilities of our framework to include:
+• new algorithms, especially the ones based on the quality-diversity search
+[23]
+• new test case generation problems, for instance more complex test sce-
+narios that include moving pedestrians, other vehicles and traffic signs;
+• new fitness functions e.g based on surrogate models of the system under
+test, as in the work of Ramakrishna et al. [24], functions based on
+neuron coverage [25] and surprise adequacy [26] dedicated to testing
+systems containing neural networks;
+• add new problem representations, supporting popular scenario specifi-
+cation languages such as SCENIC [27];
+• add an integration with popular simulators, for instance CARLA [28]
+or LGSVL [29]. This will allow to directly evaluate the system model
+with the generated scenarios. Also the feedback from the simulators
+could be incorporated in fitness functions for guiding the test scenario
+sampling.
+Acknowledgements
+This work is partly funded by the by the Fonds de Recherche du Qu´ebec
+(FRQ), the Natural Sciences and Engineering Research Council of Canada
+(NSERC), and the Canadian Institute for Advanced Research (CIFAR).
+15
+
+References
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+2020, pp. 1–6.
+19
+

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@@ -0,0 +1,419 @@
+Simulating the radio emission of dark matter for new
+high-resolution observations with MeerKAT
+M Sarkis and G Beck
+School of Physics, University of the Witwatersrand, Private Bag 3, WITS-2050, Johannesburg,
+South Africa
+E-mail: [email protected]
+Abstract.
+Recent work has shown that searches for diffuse radio emission by MeerKAT - and
+eventually the SKA - are well suited to provide some of the strongest constraints yet on dark
+matter annihilations. To make full use of the observations by these facilities, accurate simulations
+of the expected dark matter abundance and diffusion mechanisms in these astrophysical objects
+are required. However, because of the computational costs involved, various mathematical and
+numerical techniques have been developed to perform the calculations in a feasible manner.
+Here we provide the first quantitative comparison between methods that are commonly used in
+the literature, and outline the applicability of each one in various simulation scenarios. These
+considerations are becoming ever more important as the hunt for dark matter continues into a
+new era of precision radio observations.
+1. Introduction
+Despite decades of work, indirect Dark Matter (DM) searches – those that look for emission from
+the annihilation and decay products of DM particles – are yet to find a signal that can be solely
+attributed to DM. Until such a detection is made, and as our observing capabilities improve
+with newer and more sophisticated telescopes, we continue to methodically move through the
+parameter spaces of candidate DM models and eliminate those that conflict with the data. The
+recent public release of the MeerKAT Galaxy Cluster Legacy Survey data [1], together with recent
+studies that show the competitiveness of using DM radio emission for indirect detection [2, 3, 4],
+provides strong motivation for a renewed and continued effort in radio DM searches. In this work
+we take a brief but detailed look at the various theoretical aspects involved in the modelling
+of the radio emission from DM, and comment on how the choice of model will likely play an
+important role in indirect searches with high-resolution instruments.
+Our analysis includes simulations of the DM host environments for two source targets, the
+Coma galaxy cluster and the M31 galaxy, and a calculation of the synchrotron emission resulting
+from the annihilation of Weakly Interacting Massive Particles (WIMPs) therein. We model our
+DM halos with a set of reasonable source parameters and find the emission after solving the
+electron propagation equation in each environment. The methods of solving this equation are
+a major focus point of this work, as the choice of technique used can lead to a non-negligible
+change in the observed emission, particularly in smaller source targets where diffusion effects are
+significant. With < 10 arcsecond resolution capabilities, observations with MeerKAT (and soon
+the SKA) are for the first time able to probe the inner regions of these targets, which is where
+the strongest constraints on DM can be found. Therefore, accurate spatial modelling of these
+targets is essential for us to make full use of the new data.
+arXiv:2301.03326v1  [astro-ph.CO]  9 Jan 2023
+
+2. Modelling
+The two source targets in this work, the Coma galaxy cluster and the M31 galaxy, were chosen
+for their well-characterised properties in the literature. Of particular importance are the profiles
+of their magnetic fields and thermal gas densities; as these quantities appear in the modelling
+process (but are often underspecified), the uncertainty of the final solution depends strongly
+on the treatment of these factors [5]. However, since the simulation of the halo environment
+is not the central focus of this work (and for the sake of brevity), we refer the reader to the
+following sources for details regarding the parameters in the Coma cluster [6, 7] and in the M31
+galaxy [8, 9].
+In each halo environment, the emission of synchrotron radiation will be determined by the
+spatial and energy equilibrium distribution of charged annihilation products, ψ(x, E). In this
+work the products considered are electrons and positrons. The evolution of these distributions
+over time is then given by the following propagation equation, which includes the dominant
+effects of energy losses and spatial diffusion:
+∂ψ(x, E)
+∂t
+= ∇ ·
+�
+D(x, E)∇ψ(x, E)
+�
++ ∂
+∂E
+�
+b(x, E)ψ(x, E)
+�
++ Q(x, E).
+(1)
+Here D, b and Q are the diffusion, energy-loss and DM annihilation source functions respectively,
+and the determination of the exact forms of these functions follows the methods laid out in [5].
+2.1. Solving the propagation equation
+We determine the equilibrium electron distribution ψ using two independent techniques. The
+first, referred to here as the ‘Green’s Function (GF) method’ [2, 10], uses a Green’s function
+with simplified forms of D and b to solve Eq. 1 semi-analytically. The second, referred to as the
+‘Alternating Direction Implicit (ADI) method’ [11, 12], uses a numerical approach to solve Eq. 1
+iteratively. In both methods we consider the halo environment to be spherically symmetric, so
+that x may be replaced by r in Eq. 1. We also note here that we have assumed a simplified
+form of D, which would be a tensor in a more general case. As our methodology closely follows
+the above-mentioned literature, we only summarise these methods and point out any major
+differences in the following sections.
+GF method
+If the forms of the diffusion and energy-loss functions are simplified so that they have
+no spatial dependence, a solution to Eq. 1 can be found directly with the use of Green’s functions
+and image charges. However, these simplifications often have an impact on the calculated
+emission (for a review on this topic, see [5]). In this work we use non-weighted averages for the
+magnetic field and thermal gas densities, found using an averaging scale radius that matches
+the scale radius of the DM halo. This choice encapsulates the region in the halo that contains
+the majority of WIMP annihilations – and thus best represents the spatial structure of the
+halo – while allowing us to forgo any explicit spatial dependence in Eq. 1. Now, the equilibrium
+distribution of electrons in the halo can be calculated using
+ψ(r, E) =
+1
+b(E)
+� mχ
+E
+dE′G(r, ∆v)Q(r, E′) ,
+(2)
+with mχ as the WIMP mass and the Green’s function (G) given by
+G(r, ∆v) =
+1
+√
+4π∆v
+∞
+�
+n=−∞
+(−1)n
+� rmax
+0
+dr′ r′
+rn
+�
+�exp
+�
+−(r′ − rn)2
+4∆v
+�
+− exp
+�
+−(r′ + rn)2
+4∆v
+��
+� Q(r′)
+Q(r) .
+(3)
+
+Here rmax is the maximum radius for any diffusion processes and rn = (−1)nr + 2nrmax is the
+location of the nth image charge. The quantity ∆v is calculated as
+∆v = v(E) − v(E′) ,
+(4)
+where
+v(E) =
+� mχ
+E
+dx D(x)
+b(x) .
+(5)
+ADI method
+In this method, we discretise Eq. 1 and solve for the equilibrium distribution
+iteratively. Since the ADI method retains the radial dependence in the diffusion and energy loss
+functions (where the GF method does not), the problem becomes 2-dimensional in energy and
+space. Using a traditional finite-difference technique in this scenario could be computationally
+expensive, which is why we opt for a method that uses so-called ‘operator splitting’ to treat
+each dimension separately and divide the problem into smaller, more manageable pieces. Thus,
+during each step of the method, we use a general form of the 1-dimensional Crank-Nicolson
+(CN), scheme (see, for instance, [13]) which is a finite-differencing technique that includes the
+average of second-order implicit and explicit terms in the updating equation, thereby leveraging
+the unconditional stability of a fully implicit method while maintaining second-order accuracy in
+space and time. This scheme is relatively easy to solve, as the updating equation turns out to be
+a set of linear equations with tridiagonal coefficient matrices. We write this, as in [11, 12], as
+− α1
+2 ψn+1
+x−1 +
+�
+1 + α2
+2
+�
+ψn+1
+x
+− α3
+2 ψn+1
+x+1 = α1
+2 ψn
+x−1 +
+�
+1 − α2
+2
+�
+ψn
+x + α3
+2 ψn
+x+1 + Qx∆t.
+(6)
+Here n is the temporal grid index (with the spacing between indices given by ∆t) and x represents
+either the energy or spatial grid index. The forms of the α coefficients, which encapsulate the
+diffusion and energy loss effects, need to be found by discretising the relevant operators from
+Eq. 1. The scheme we have used for this is as follows:
+1
+r2
+∂
+∂r
+�
+r2D∂ψ
+∂r
+�
+−−−−−−−−→
+discretisation C−2
+˜r
+�
+�ψi+1 − ψi−1
+2∆˜r
+�
+log(10)D + ∂D
+∂˜r
+������
+˜r=˜ri
++ ψi+1 − 2ψi + ψi−1
+∆˜r2
+D|˜r=˜ri
+�
+(7)
+for the radial operator and
+∂
+∂E (bψ) −−−−−−−−→
+discretisation C−1
+˜E
+�b| ˜E= ˜Ej(ψj+1 − ψj)
+∆ ˜E
+�
+(8)
+for energy operator, where C˜r = (r0 log(10)10˜ri), C ˜E = (E0 log(10)10 ˜Ej) and ∆˜r, ∆ ˜E represent
+the radial and energy grid spacings, respectively. We use i and j to denote positions in the
+radial and energy grids, and have omitted the temporal indices as these forms will apply to both
+implicit and explicit terms in the same way. The vertical bars denote that the functions which
+they are attached to are evaluated at the given grid index. We have also made the variable
+transformations ˜r = log10(r/r0) and ˜E = log10(E/E0) (similarly to [12], except with base 10
+instead of e), which allows us to more accurately track the electron distribution in our grids
+when the involved processes operate over a wide range of physical scales. Finally, note that in
+the case of energy losses, we only consider upstream differencing.
+
+The α values can now be found by taking Eqs. 7 and 8 and equating coefficients with Eq. 6;
+once these are found, the updating equation can be solved with some matrix solution algorithm.
+If we represent the discretisation schemes shown above by the symbol Ψ, the overall iterative
+solution can be summarised with the steps
+ψn+1/2 = Ψ ˜E(ψn)
+ψn+1 = Ψ˜r(ψn+1/2) ,
+(9)
+which are repeatedly solved (using Eq. 6) until the value of ψ has converged to the equilibrium
+value. The other minutiae of this method, including initial and boundary conditions, convergence
+criteria and stability considerations, can be found in [11, 12].
+2.2. Synchrotron emission
+Once found via the GF or ADI methods, the equilibrium distribution is used to calculate the
+radio emissivity, given by
+jsync(ν, r) =
+� mχ
+0
+dE ψe±(E, r)Psync(ν, E, r) ,
+(10)
+where ν is the synchrotron frequency, ψe± is the sum of electron and positron equilibrium
+distributions and Psync is the power emitted by an electron with an energy of E (this is calculated
+as in [2]). The emissivity is then used to calculate the two main results in this work. Firstly, the
+azimuthally averaged surface brightness curves,
+Isync(ν, r, Θ, ∆Ω) =
+�
+∆Ω
+dΩ
+�
+l.o.s.
+dl jsync(ν, l)
+4π
+,
+(11)
+where l.o.s. is the line-of-sight to a point in the halo at radius r, which makes an angle of Θ with
+the centre of the halo, and ∆Ω is the solid angle over which the surface brightness is calculated.
+In this work we show results for a single representative frequency of ν = 0.5 GHz. Secondly, we
+calculate the integrated flux density by
+Ssync(ν, R) =
+� R
+0
+d3r′ jsync(ν, r′)
+4πd2
+L
+,
+(12)
+where the emissivity is integrated over the region enclosed by R and dL is the luminosity distance
+to the target. For the results shown in this work we consider R to be the virial radius of the halo.
+3. Results
+Here we provide the details of the simulations we have performed, and show the results for two
+observables: the radio surface brightness (Eq. 11) and integrated flux (Eq. 12). We use a set of
+reasonable source parameters for the halo environments that respect observational constraints,
+and aim to use WIMP parameter values that are representative of the many viable candidates.
+We thus consider a large range of particle masses, from 10 to 1000 GeV, and use a set of four
+annihilation channels, {bb, e+e−, µ+µ−, τ +τ −}. Since the focus of this work is on a comparison
+between the two solution methods, particurly in the way that they differ with various source
+targets, we show the results side-by-side and in the same manner for both targets. In Fig. 1 we
+show the surface brightness curves for the Coma cluster (left-hand panels) and M31 (right-hand
+panels), and Fig. 2 shows the integrated fluxes from the same targets for a range of frequencies.
+
+10−8
+10−4
+100
+b¯b
+e+e−
+10−2
+10−1
+100
+101
+10−8
+10−4
+100
+µ+µ−
+10−2
+10−1
+100
+101
+τ +τ −
+Angular radius from halo centre (arcmin)
+Surface Brightness (Jy/arcmin2)
+10−7
+10−4
+10−1
+b¯b
+e+e−
+100
+102
+10−7
+10−4
+10−1
+µ+µ−
+100
+102
+τ +τ −
+Angular radius from halo centre (arcmin)
+χ Mass
+10 GeV
+1000 GeV
+Method
+ADI
+GF
+Figure 1. Surface brightness curves for the Coma galaxy cluster (left-hand panels) and the M31
+galaxy (right-hand panels). Each of the four panels show different annihilation channels, given
+by the label in the top right of each plot. The ADI and GF methods are represented by the
+red and blue colours respectively, and the region in which the results overlap are given by the
+combination of these (the purple colour). These shaded regions represent the full mass range of
+the WIMPs (from 10 to 1000 GeV), and the domain of each panel runs up until the halo’s virial
+radius R (in angular units).
+600
+800
+1000
+1200
+1400
+Frequency (MHz)
+10−3
+10−2
+Flux (Jy)
+Channels
+bb
+µ+µ−
+τ +τ −
+e+e−
+Methods
+ADI
+Greens
+600
+800
+1000
+1200
+1400
+Frequency (MHz)
+10−3
+10−2
+10−1
+100
+Channels
+bb
+µ+µ−
+τ +τ −
+e+e−
+Methods
+ADI
+Greens
+Figure 2. The integrated fluxes, calculated using Eq. 12, for the Coma galaxy cluster (left) and
+the M31 galaxy (right). The different linestyles represent the two solution methods presented in
+Sec. 2.1, and each colour indicates the use of a different annihilation channel.
+4. Discussion and conclusions
+In Figs. 1, we see generally good agreement between the GF and ADI methods, which can be
+inferred from the significant amount of overlap between the curves in each panel. Noticeably
+however, we see more disagreement (less overlap) between the methods in the M31 galaxy than
+we do for the Coma galaxy cluster. Our explanation for this lies in the mathematical techniques
+employed by each method, and how they each treat the spatial dependence of the diffusion
+function in particular. In the galaxy cluster environment of Coma, diffusion effects are negligible
+on sufficiently large scales [10, 5], whereas in the physically smaller galaxy, diffusion effects
+start to influence the surface brightness distribution at all relevant scales. Since the GF and
+ADI methods leverage a spatially independent and dependent diffusion function (respectively),
+the resulting equilibrium distributions will tend to differ in the environments where the length
+
+scales in question do not greatly exceed the diffusion length, as is the case for M31. This trend
+is also seen in the fluxes from Fig. 2, which show a clear disagreement in all channels for the
+M31 galaxy, and relative agreement in all channels in the Coma cluster. Based on these results
+and the comparison of target environments presented in [5], we also expect that smaller target
+environments (like the dwarf spheroidal satellite galaxies of the Milky Way) would show further
+disagreement between the solution methods, as diffusion effects would be even more significant
+in these environments.
+The other notable result we see from these simulations is that the methods differ on small
+scales, even in the large Coma cluster. This is significant, as the inner regions of the DM halos
+are where we would observe the strongest emission. With high-resolution radio interferometers
+allowing us to resolve these smaller scales, our models could be tested against the strongest
+possible DM emission, allowing us to find more stringent constraints on DM properties than
+previously possible. In this regard, the surface brightness curves displayed here would be especially
+valuable results when determining new observational limits, as their emission profiles are highly
+dependent on the spatial structure of the DM halo.
+With the impressive spatial resolution of telescopes like MeerKAT and the SKA, we are now
+able to probe the inner regions of these DM halos – regions which have formerly been hidden
+from our view. The need for accurate modelling techniques is thus more necessary than ever
+before, and the considerations presented in this work should help inform the modelling choices
+made in future radio searches for DM.
+Acknowledgments
+This work is based on the research that was supported by the National Research Foundation
+of South Africa (Bursary No. 112332). G.B. acknowledges support from a National Research
+Foundation of South Africa Thuthuka grant no. 117969.
+References
+[1] Knowles K, Cotton W D, Rudnick L et al. 2022 Astronomy & Astrophysics 657 A56
+[2] Beck G 2019 Galaxies 7 16
+[3] Regis M, Reynoso-Cordova J, Filipovi´c M D et al. 2021 Journal of Cosmology and Astroparticle Physics 2021
+046
+[4] Chan M 2021 Galaxies 9 11
+[5] Sarkis M and Beck G 2022 The Proceedings of SAIP2021 ed Prinsloo A (UJ) pp 316–322 ISBN 978-0-620-
+97693-0
+[6] Bonafede A, Feretti L, Murgia M et al. 2010 Astronomy and Astrophysics 513 A30
+[7] �Lokas E L and Mamon G A 2003 Monthly Notices of the Royal Astronomical Society 343 401–412
+[8] Ruiz-Granados B, Rubi˜no-Mart´ın J A, Florido E et al. 2010 The Astrophysical Journal 723 L44–L48
+[9] Tamm A, Tempel E, Tenjes P et al. 2012 Astronomy & Astrophysics 546 A4
+[10] Colafrancesco S, Profumo S and Ullio P 2006 Astronomy & Astrophysics 455 21–43
+[11] Strong A W and Moskalenko I V 1998 The Astrophysical Journal 509 212–228
+[12] Regis M, Richter L, Colafrancesco S et al. 2015 Monthly Notices of the Royal Astronomical Society 448
+3747–3765
+[13] Press W H, Teukolsky S A and Vetterling W T 2007 Numerical Recipes: The Art of Scientific Computing 3rd
+ed (Cambridge University Press) ISBN 978-0-521-88068-8
+

AdE1T4oBgHgl3EQfpAU_/content/tmp_files/load_file.txt

@@ -0,0 +1,139 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf,len=138
+page_content='Simulating the radio emission of dark matter for new  high-resolution observations with MeerKAT  M Sarkis and G Beck  School of Physics, University of the Witwatersrand, Private Bag 3, WITS-2050, Johannesburg,  South Africa  E-mail: michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='sarkis@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='wits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='za  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  Recent work has shown that searches for diffuse radio emission by MeerKAT - and  eventually the SKA - are well suited to provide some of the strongest constraints yet on dark  matter annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' To make full use of the observations by these facilities, accurate simulations  of the expected dark matter abundance and diffusion mechanisms in these astrophysical objects  are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' However, because of the computational costs involved, various mathematical and  numerical techniques have been developed to perform the calculations in a feasible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  Here we provide the first quantitative comparison between methods that are commonly used in  the literature, and outline the applicability of each one in various simulation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' These  considerations are becoming ever more important as the hunt for dark matter continues into a  new era of precision radio observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Introduction  Despite decades of work, indirect Dark Matter (DM) searches – those that look for emission from  the annihilation and decay products of DM particles – are yet to find a signal that can be solely  attributed to DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Until such a detection is made, and as our observing capabilities improve  with newer and more sophisticated telescopes, we continue to methodically move through the  parameter spaces of candidate DM models and eliminate those that conflict with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The  recent public release of the MeerKAT Galaxy Cluster Legacy Survey data [1], together with recent  studies that show the competitiveness of using DM radio emission for indirect detection [2, 3, 4],  provides strong motivation for a renewed and continued effort in radio DM searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In this work  we take a brief but detailed look at the various theoretical aspects involved in the modelling  of the radio emission from DM, and comment on how the choice of model will likely play an  important role in indirect searches with high-resolution instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  Our analysis includes simulations of the DM host environments for two source targets, the  Coma galaxy cluster and the M31 galaxy, and a calculation of the synchrotron emission resulting  from the annihilation of Weakly Interacting Massive Particles (WIMPs) therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We model our  DM halos with a set of reasonable source parameters and find the emission after solving the  electron propagation equation in each environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The methods of solving this equation are  a major focus point of this work, as the choice of technique used can lead to a non-negligible  change in the observed emission, particularly in smaller source targets where diffusion effects are  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' With < 10 arcsecond resolution capabilities, observations with MeerKAT (and soon  the SKA) are for the first time able to probe the inner regions of these targets, which is where  the strongest constraints on DM can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Therefore, accurate spatial modelling of these  targets is essential for us to make full use of the new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='03326v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='CO]  9 Jan 2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Modelling  The two source targets in this work, the Coma galaxy cluster and the M31 galaxy, were chosen  for their well-characterised properties in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Of particular importance are the profiles  of their magnetic fields and thermal gas densities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' as these quantities appear in the modelling  process (but are often underspecified), the uncertainty of the final solution depends strongly  on the treatment of these factors [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' However, since the simulation of the halo environment  is not the central focus of this work (and for the sake of brevity), we refer the reader to the  following sources for details regarding the parameters in the Coma cluster [6, 7] and in the M31  galaxy [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  In each halo environment, the emission of synchrotron radiation will be determined by the  spatial and energy equilibrium distribution of charged annihilation products, ψ(x, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In this  work the products considered are electrons and positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The evolution of these distributions  over time is then given by the following propagation equation, which includes the dominant  effects of energy losses and spatial diffusion:  ∂ψ(x, E)  ∂t  = ∇ ·  �  D(x, E)∇ψ(x, E)  �  + ∂  ∂E  �  b(x, E)ψ(x, E)  �  + Q(x, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  (1)  Here D, b and Q are the diffusion, energy-loss and DM annihilation source functions respectively,  and the determination of the exact forms of these functions follows the methods laid out in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Solving the propagation equation  We determine the equilibrium electron distribution ψ using two independent techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The  first, referred to here as the ‘Green’s Function (GF) method’ [2, 10], uses a Green’s function  with simplified forms of D and b to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1 semi-analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The second, referred to as the  ‘Alternating Direction Implicit (ADI) method’ [11, 12], uses a numerical approach to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1  iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In both methods we consider the halo environment to be spherically symmetric, so  that x may be replaced by r in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We also note here that we have assumed a simplified  form of D, which would be a tensor in a more general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' As our methodology closely follows  the above-mentioned literature, we only summarise these methods and point out any major  differences in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  GF method  If the forms of the diffusion and energy-loss functions are simplified so that they have  no spatial dependence, a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1 can be found directly with the use of Green’s functions  and image charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' However, these simplifications often have an impact on the calculated  emission (for a review on this topic, see [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In this work we use non-weighted averages for the  magnetic field and thermal gas densities, found using an averaging scale radius that matches  the scale radius of the DM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' This choice encapsulates the region in the halo that contains  the majority of WIMP annihilations – and thus best represents the spatial structure of the  halo – while allowing us to forgo any explicit spatial dependence in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Now, the equilibrium  distribution of electrons in the halo can be calculated using  ψ(r, E) =  1  b(E)  � mχ  E  dE′G(r, ∆v)Q(r, E′) ,  (2)  with mχ as the WIMP mass and the Green’s function (G) given by  G(r, ∆v) =  1  √  4π∆v  ∞  �  n=−∞  (−1)n  � rmax  0  dr′ r′  rn  �  �exp  �  −(r′ − rn)2  4∆v  �  − exp  �  −(r′ + rn)2  4∆v  ��  � Q(r′)  Q(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  (3)  Here rmax is the maximum radius for any diffusion processes and rn = (−1)nr + 2nrmax is the  location of the nth image charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The quantity ∆v is calculated as  ∆v = v(E) − v(E′) ,  (4)  where  v(E) =  � mχ  E  dx D(x)  b(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  (5)  ADI method  In this method, we discretise Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1 and solve for the equilibrium distribution  iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Since the ADI method retains the radial dependence in the diffusion and energy loss  functions (where the GF method does not), the problem becomes 2-dimensional in energy and  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Using a traditional finite-difference technique in this scenario could be computationally  expensive, which is why we opt for a method that uses so-called ‘operator splitting’ to treat  each dimension separately and divide the problem into smaller, more manageable pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Thus,  during each step of the method, we use a general form of the 1-dimensional Crank-Nicolson  (CN), scheme (see, for instance, [13]) which is a finite-differencing technique that includes the  average of second-order implicit and explicit terms in the updating equation, thereby leveraging  the unconditional stability of a fully implicit method while maintaining second-order accuracy in  space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' This scheme is relatively easy to solve, as the updating equation turns out to be  a set of linear equations with tridiagonal coefficient matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We write this, as in [11, 12], as  − α1  2 ψn+1  x−1 +  �  1 + α2  2  �  ψn+1  x  − α3  2 ψn+1  x+1 = α1  2 ψn  x−1 +  �  1 − α2  2  �  ψn  x + α3  2 ψn  x+1 + Qx∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  (6)  Here n is the temporal grid index (with the spacing between indices given by ∆t) and x represents  either the energy or spatial grid index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The forms of the α coefficients, which encapsulate the  diffusion and energy loss effects, need to be found by discretising the relevant operators from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The scheme we have used for this is as follows:  1  r2  ∂  ∂r  �  r2D∂ψ  ∂r  �  −−−−−−−−→  discretisation C−2  ˜r  �  �ψi+1 − ψi−1  2∆˜r  �  log(10)D + ∂D  ∂˜r  ������  ˜r=˜ri  + ψi+1 − 2ψi + ψi−1  ∆˜r2  D|˜r=˜ri  �  (7)  for the radial operator and  ∂  ∂E (bψ) −−−−−−−−→  discretisation C−1  ˜E  �b| ˜E= ˜Ej(ψj+1 − ψj)  ∆ ˜E  �  (8)  for energy operator, where C˜r = (r0 log(10)10˜ri), C ˜E = (E0 log(10)10 ˜Ej) and ∆˜r, ∆ ˜E represent  the radial and energy grid spacings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We use i and j to denote positions in the  radial and energy grids, and have omitted the temporal indices as these forms will apply to both  implicit and explicit terms in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The vertical bars denote that the functions which  they are attached to are evaluated at the given grid index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We have also made the variable  transformations ˜r = log10(r/r0) and ˜E = log10(E/E0) (similarly to [12], except with base 10  instead of e), which allows us to more accurately track the electron distribution in our grids  when the involved processes operate over a wide range of physical scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Finally, note that in  the case of energy losses, we only consider upstream differencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  The α values can now be found by taking Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 7 and 8 and equating coefficients with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  once these are found, the updating equation can be solved with some matrix solution algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  If we represent the discretisation schemes shown above by the symbol Ψ, the overall iterative  solution can be summarised with the steps  ψn+1/2 = Ψ ˜E(ψn)  ψn+1 = Ψ˜r(ψn+1/2) ,  (9)  which are repeatedly solved (using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 6) until the value of ψ has converged to the equilibrium  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The other minutiae of this method, including initial and boundary conditions, convergence  criteria and stability considerations, can be found in [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Synchrotron emission  Once found via the GF or ADI methods, the equilibrium distribution is used to calculate the  radio emissivity, given by  jsync(ν, r) =  � mχ  0  dE ψe±(E, r)Psync(ν, E, r) ,  (10)  where ν is the synchrotron frequency, ψe± is the sum of electron and positron equilibrium  distributions and Psync is the power emitted by an electron with an energy of E (this is calculated  as in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The emissivity is then used to calculate the two main results in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Firstly, the  azimuthally averaged surface brightness curves,  Isync(ν, r, Θ, ∆Ω) =  �  ∆Ω  dΩ  �  l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  dl jsync(ν, l)  4π  ,  (11)  where l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' is the line-of-sight to a point in the halo at radius r, which makes an angle of Θ with  the centre of the halo, and ∆Ω is the solid angle over which the surface brightness is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  In this work we show results for a single representative frequency of ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Secondly, we  calculate the integrated flux density by  Ssync(ν, R) =  � R  0  d3r′ jsync(ν, r′)  4πd2  L  ,  (12)  where the emissivity is integrated over the region enclosed by R and dL is the luminosity distance  to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' For the results shown in this work we consider R to be the virial radius of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Results  Here we provide the details of the simulations we have performed, and show the results for two  observables: the radio surface brightness (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 11) and integrated flux (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' We use a set of  reasonable source parameters for the halo environments that respect observational constraints,  and aim to use WIMP parameter values that are representative of the many viable candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  We thus consider a large range of particle masses, from 10 to 1000 GeV, and use a set of four  annihilation channels, {bb, e+e−, µ+µ−, τ +τ −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Since the focus of this work is on a comparison  between the two solution methods, particurly in the way that they differ with various source  targets, we show the results side-by-side and in the same manner for both targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1 we  show the surface brightness curves for the Coma cluster (left-hand panels) and M31 (right-hand  panels), and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2 shows the integrated fluxes from the same targets for a range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  10−8  10−4  100  b¯b  e+e−  10−2  10−1  100  101  10−8  10−4  100  µ+µ−  10−2  10−1  100  101  τ +τ −  Angular radius from halo centre (arcmin)  Surface Brightness (Jy/arcmin2)  10−7  10−4  10−1  b¯b  e+e−  100  102  10−7  10−4  10−1  µ+µ−  100  102  τ +τ −  Angular radius from halo centre (arcmin)  χ Mass  10 GeV  1000 GeV  Method  ADI  GF  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Surface brightness curves for the Coma galaxy cluster (left-hand panels) and the M31  galaxy (right-hand panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Each of the four panels show different annihilation channels, given  by the label in the top right of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The ADI and GF methods are represented by the  red and blue colours respectively, and the region in which the results overlap are given by the  combination of these (the purple colour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' These shaded regions represent the full mass range of  the WIMPs (from 10 to 1000 GeV), and the domain of each panel runs up until the halo’s virial  radius R (in angular units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  600  800  1000  1200  1400  Frequency (MHz)  10−3  10−2  Flux (Jy)  Channels  bb  µ+µ−  τ +τ −  e+e−  Methods  ADI  Greens  600  800  1000  1200  1400  Frequency (MHz)  10−3  10−2  10−1  100  Channels  bb  µ+µ−  τ +τ −  e+e−  Methods  ADI  Greens  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The integrated fluxes, calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 12, for the Coma galaxy cluster (left) and  the M31 galaxy (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The different linestyles represent the two solution methods presented in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='1, and each colour indicates the use of a different annihilation channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Discussion and conclusions  In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 1, we see generally good agreement between the GF and ADI methods, which can be  inferred from the significant amount of overlap between the curves in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Noticeably  however, we see more disagreement (less overlap) between the methods in the M31 galaxy than  we do for the Coma galaxy cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Our explanation for this lies in the mathematical techniques  employed by each method, and how they each treat the spatial dependence of the diffusion  function in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In the galaxy cluster environment of Coma, diffusion effects are negligible  on sufficiently large scales [10, 5], whereas in the physically smaller galaxy, diffusion effects  start to influence the surface brightness distribution at all relevant scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Since the GF and  ADI methods leverage a spatially independent and dependent diffusion function (respectively),  the resulting equilibrium distributions will tend to differ in the environments where the length  scales in question do not greatly exceed the diffusion length, as is the case for M31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' This trend  is also seen in the fluxes from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2, which show a clear disagreement in all channels for the  M31 galaxy, and relative agreement in all channels in the Coma cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' Based on these results  and the comparison of target environments presented in [5], we also expect that smaller target  environments (like the dwarf spheroidal satellite galaxies of the Milky Way) would show further  disagreement between the solution methods, as diffusion effects would be even more significant  in these environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  The other notable result we see from these simulations is that the methods differ on small  scales, even in the large Coma cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' This is significant, as the inner regions of the DM halos  are where we would observe the strongest emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' With high-resolution radio interferometers  allowing us to resolve these smaller scales, our models could be tested against the strongest  possible DM emission, allowing us to find more stringent constraints on DM properties than  previously possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' In this regard, the surface brightness curves displayed here would be especially  valuable results when determining new observational limits, as their emission profiles are highly  dependent on the spatial structure of the DM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  With the impressive spatial resolution of telescopes like MeerKAT and the SKA, we are now  able to probe the inner regions of these DM halos – regions which have formerly been hidden  from our view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' The need for accurate modelling techniques is thus more necessary than ever  before, and the considerations presented in this work should help inform the modelling choices  made in future radio searches for DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='  Acknowledgments  This work is based on the research that was supported by the National Research Foundation  of South Africa (Bursary No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 112332).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' acknowledges support from a National Research  Foundation of South Africa Thuthuka grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 117969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
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+page_content=' 2010 The Astrophysical Journal 723 L44–L48  [9] Tamm A, Tempel E, Tenjes P et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2012 Astronomy & Astrophysics 546 A4  [10] Colafrancesco S, Profumo S and Ullio P 2006 Astronomy & Astrophysics 455 21–43  [11] Strong A W and Moskalenko I V 1998 The Astrophysical Journal 509 212–228  [12] Regis M, Richter L, Colafrancesco S et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}
+page_content=' 2015 Monthly Notices of the Royal Astronomical Society 448  3747–3765  [13] Press W H, Teukolsky S A and Vetterling W T 2007 Numerical Recipes: The Art of Scientific Computing 3rd  ed (Cambridge University Press) ISBN 978-0-521-88068-8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQfpAU_/content/2301.03326v1.pdf'}

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+Adversarial Attacks on Neural Models of Code via
+Code Difference Reduction
+Zhao Tian
+College of Intelligence and
+Computing, Tianjin University
+Tianjin, China
+[email protected]
+Junjie Chen†
+College of Intelligence and
+Computing, Tianjin University
+Tianjin, China
+[email protected]
+Zhi Jin
+Key Lab of High Confidence Software
+Technologies, Peking University
+Beijing, China
+[email protected]
+Abstract—Deep learning has been widely used to solve various
+code-based tasks by building deep code models based on a
+large number of code snippets. However, deep code models
+are still vulnerable to adversarial attacks. As source code is
+discrete and has to strictly stick to the grammar and semantics
+constraints, the adversarial attack techniques in other domains
+are not applicable. Moreover, the attack techniques specific to
+deep code models suffer from the effectiveness issue due to
+the enormous attack space. In this work, we propose a novel
+adversarial attack technique (i.e., CODA). Its key idea is to
+use the code differences between the target input and reference
+inputs (that have small code differences but different prediction
+results with the target one) to guide the generation of adversarial
+examples. It considers both structure differences and identifier
+differences to preserve the original semantics. Hence, the attack
+space can be largely reduced as the one constituted by the two
+kinds of code differences, and thus the attack process can be
+largely improved by designing corresponding equivalent structure
+transformations and identifier renaming transformations. Our
+experiments on 10 deep code models (i.e., two pre-trained models
+with five code-based tasks) demonstrate the effectiveness and
+efficiency of CODA, the naturalness of its generated examples,
+and its capability of defending against attacks after adversarial
+fine-tuning. For example, CODA improves the state-of-the-art
+techniques (i.e., CARROT and ALERT) by 79.25% and 72.20%
+on average in terms of the attack success rate, respectively.
+I. INTRODUCTION
+In recent years, deep learning (DL) has been widely used
+to solve code-based software engineering tasks, such as code
+clone detection [1], vulnerability prediction [2], and code com-
+pletion [3], by building DL models based on a large amount of
+training code snippets (also called deep code models). Indeed,
+deep code models have achieved notable performance and
+largely promoted the process of software development and
+maintenance [4]–[7]. In particular, some industrial products on
+deep code models have been released and received extensive
+attention, such as AlphaCode [8] and Codex [9].
+Like DL models in other areas (e.g., image processing [10]
+and speech recognition [11]), the robustness of deep code
+models is also critical [12]. However, the existing adversarial
+attack techniques proposed in other areas are not applicable to
+deep code models. This is because these techniques perturb an
+input in a continuous space for altering the model prediction
+result, while the inputs (i.e., source code) for deep code models
+†Junjie Chen is the corresponding author.
+are discrete. Moreover, source code has to strictly stick to
+the grammar and semantics constraints, i.e., the adversarial
+example generated from an original input should have no
+grammar errors and preserve the original semantics.
+Indeed, some adversarial attack techniques specific to deep
+code models have been proposed recently, such as MHM [13],
+CARROT [12], and ALERT [14]. In general, they share two
+main steps: (1) designing a series of semantic-preserving code
+transformation rules (e.g., identifier renaming or dead code
+insertion), and (2) searching ingredients from the space defined
+by the rules (e.g., a valid identifier name is an ingredient for
+the rule of identifier renaming) for transforming an input to a
+semantic-preserving adversarial example. For example, CAR-
+ROT designs two semantic-preserving code transformation
+rules (i.e., identifier renaming and dead code insertion), and
+uses the hill-climbing algorithm to search for the ingredients
+from the entire space with the guidance of gradients and
+changes of model prediction results. ALERT considers the rule
+of identifier renaming, and uses the naturalness (i.e., natural
+semantics of code) and changes of model prediction results to
+guide the ingredient search process from the entire space.
+Although some of them have been demonstrated to be
+effective to some degree, these existing techniques still suffer
+from major limitations:
+• The ingredient space defined by the code transformation
+rules is enormous. For example, for the rule of identifier
+renaming, all valid identifier names could be the ingredi-
+ents for renaming the target identifier. Hence, searching
+for the ingredients that can help attack the target model
+successfully is challenging. The existing techniques tend
+to utilize the changes of model prediction results after
+performing semantic-preserving transformations on the
+target input to guide the search process, which is very
+likely to fall into local optimum in the enormous space
+and thus limits their attack effectiveness.
+• Frequently invoking the target model can negatively af-
+fect the efficiency of adversarial attack techniques, as
+model invocation is the most costly part during the
+attack process [12]. Also, when the model is deployed
+remotely, frequent model invocations could be identified
+as malicious attacks and thus lead to blocking access to
+the model. However, the existing techniques often involve
+1
+arXiv:2301.02412v1  [cs.CR]  6 Jan 2023
+
+frequent model invocations due to calculating gradients
+or guiding the search direction via model prediction.
+• Developers care about the natural semantics of code since
+it is helpful to assist human comprehension [15]. Hence,
+guaranteeing the naturalness of generated adversarial ex-
+amples (i.e., source code in our task) is important. How-
+ever, all the existing techniques (except ALERT [14]) do
+not consider this factor. For example, CARROT designs
+the rule of dead code insertion, but it may largely damage
+the naturalness of the generated examples (especially
+when a large amount of dead code is inserted).
+Overall, a more effective adversarial attack technique spe-
+cific to deep code models should enhance the attack effective-
+ness by improving the ingredient search process, and guarantee
+the naturalness of generated adversarial examples as much as
+possible and the times of model invocations as few as possible.
+Our work does propose such a technique, called CODA (COde
+Difference guided Attacking).
+To improve the attack effectiveness, the key idea of CODA
+is to use the inputs, which have small code differences with
+the target input but have different prediction results, to largely
+reduce the ingredient space. For ease of presentation, we call
+such inputs reference inputs. Actually, reference inputs can
+be regarded as invalid successfully-attacking adversarial ex-
+amples generated from the target input, where “invalid” refers
+to altering the original semantics and “successfully-attacking”
+refers to producing different prediction results. Over the target
+input, the code differences brought by reference inputs con-
+tribute to the invalid but successful attack to a large extent.
+Hence, if we extract the ingredients from the code differences
+to support semantic-preserving transformations on the target
+input, their code differences can be gradually reduced without
+altering the original semantics, and thus a valid successfully-
+attacking adversarial example is likely to be generated. In
+this way, the ingredient space is effectively reduced as the
+one constituted by only code differences between reference
+inputs and the target input, and thus the search process
+can be largely improved. Please note that taking reference
+inputs (especially code differences brought by them) as the
+guidance for generating adversarial examples is an innovative
+perspective, which closely utilizes the unique characteristics
+of deep code models (e.g., source code is discrete).
+To preserve the semantics of the target input during the
+attack process, CODA considers code structure differences
+and identifier differences, and thus extracts the ingredients to
+support equivalent structure transformations and identifier re-
+naming transformations. Equivalent structure transformations
+(e.g., transforming a for loop to an equivalent while loop)
+do not affect the naturalness of generated examples, and thus
+CODA first applies this kind of transformations to reduce
+code differences for generating adversarial examples. Then,
+identifier renaming transformations are applied to further
+reduce code differences to improve the attack effectiveness.
+To ensure the naturalness of generated examples by this
+kind of transformations, CODA measures semantic similarity
+between identifiers for guiding iterative transformations. In
+particular, CODA just involves necessary model invocations
+to check whether the generated example attacks successfully,
+without extra gradient calculation and a large amount of model
+prediction for guiding the search process.
+We conducted an extensive study to evaluate CODA based
+on two popular pre-trained models (i.e., CodeBERT [6] and
+GraphCodeBERT [7]) and five code-based tasks (i.e., vul-
+nerability prediction [2], clone detection [16], authorship
+attribution [17], functionality classification [18], and defect
+prediction [12]). In total, we used 10 subjects. Our results
+demonstrate the effectiveness and efficiency of CODA. For
+example, on average across the ten subjects, CODA improves
+the two state-of-the-art adversarial attack techniques specific
+to deep code models (i.e., CARROT [12] and ALERT [14])
+by 79.25% and 72.20% respectively in terms of the attack
+success rate. The time spent by CODA on completing the
+attack process for the ten subjects is just 39.59 hours, while
+those by CARROT and ALERT are 159.19 hours and 198.89
+hours, respectively. Also, we investigated the value of the
+generated adversarial examples by using them to improve
+the robustness of the target model via an adversarial fine-
+tuning strategy. The results show that the models after fine-
+tuning with the examples generated by CODA can successfully
+defend against attacks from 63.64%, 66.96%, and 76.68%
+of adversarial examples generated by CARROT, ALERT, and
+CODA on average, respectively. Besides, we conducted a user
+study to confirm the naturalness of the generated examples by
+CODA and an ablation experiment to confirm the contribution
+of each main component in CODA.
+To sum up, our work makes the four major contributions:
+• Novel Perspective. We propose a novel perspective of
+utilizing code differences between reference inputs and
+the target input to guide the adversarial attack process
+for deep code models.
+• Technique Implementation. We implement CODA fol-
+lowing the novel perspective by measuring code structure
+and identifier differences and designing the corresponding
+semantic-preserving code transformation rules.
+• Performance Evaluation. We conducted an extensive
+study on two popular pre-trained models and five code-
+based tasks, demonstrating the effectiveness and effi-
+ciency of CODA over two state-of-the-art techniques.
+• Public Artifact. We released all the experimental data
+and our source code at the project homepage [19] for
+experiment replication, future research, and practical use.
+II. BACKGROUND AND MOTIVATION
+In this section, we first introduce the background of deep
+code models (Section II-A), define our problem (Section II-B),
+and motivate our key idea with an example (Section II-C).
+A. Deep Code Models
+In the area of software engineering, DL has been widely-
+used to process source code [2], [3], [16], [20]. In particular,
+some popular pre-trained DL models have been constructed
+based on a large number of code snippets, among which
+2
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+void f1(int a[], int n){
+  int i; int j; int k;
+  for (i = 0; i < n; i++) {
+    for (j = 0; j < ((n - i) - 1); j++) {
+      if (a[j] > a[j + 1]){
+        k = a[j];
+        a[j] = a[j + 1];
+        a[j + 1] = k;
+      }
+    }
+  }
+}
+int f2(int t[], int len){
+  int i; int j;
+  i = 0; j = 0;
+  while (len != 0) {
+    t[i] = len % 10;
+    len /= 10;
+    i = i + 1;
+  } 
+  while (j < i){
+    if (t[j] != t[(i - j) - 1]) return 0;
+    j = j + 1;
+  }
+  return 1;
+}
+void f3(int t[], int len){
+  int i; int j; int k;
+  i = 0;
+  while (i < len) {
+    j = 0;
+    while (j < ((len - i) - 1)) {
+      if (t[j] > t[j + 1]){
+        k = t[j];
+        t[j] = t[j + 1];
+        t[j + 1] = k;
+      } j = j + 1;
+    } i = i + 1;
+  }
+}
+Ground-truth Label: sort
+Prediction Result: sort (96.52%)
+Ground-truth Label: palindrome
+Prediction Result: palindrome (99.98%)
+Ground-truth Label: sort
+Prediction Result: palindrome (90.88%)
+Fig. 1. An illustrating example (the target input f1, a reference input f2, and a successfully-attacking adversarial example f3 generated from f1)
+CodeBERT [6] and GraphCodeBERT [7] are two state-of-the-
+art pre-trained models. CodeBERT learns features from bi-
+modal data in the form of programming languages and natural
+languages, while GraphCodeBERT takes into consideration the
+code structure and data flow information. Same as the existing
+work [14], we used them in our evaluation (Section IV).
+These
+pre-trained
+models
+have
+brought
+breakthrough
+changes to many downstream code-based tasks [21], including
+both classification tasks and generation tasks, by fine-tuning
+them on the datasets of the corresponding tasks. The former
+makes classification based on the given code snippets (e.g.,
+clone detection [16] and vulnerability prediction [2]), while the
+latter produces a sequence of information based on code snip-
+pets or natural language descriptions (e.g., code completion [3]
+and code summarization [22]). Following most of the existing
+work on attacking deep code models [12]–[14], our work also
+focuses on the classification tasks and takes the generation
+tasks as our future work. In particular, in our study, we adopted
+all the tasks used in the studies of evaluating the state-of-the-
+art attack techniques (i.e., CARROT [12] and ALERT [14]),
+i.e., five classification tasks including vulnerability prediction,
+clone detection, authorship attribution, functionality classifica-
+tion, and defect prediction.
+B. Problem Definition
+Given a code snippet x that is processed as the required
+format by the target deep code model M (e.g., abstract syntax
+trees required by code2seq [4], control-flow graphs required
+by DGCNN [23], or data-flow graphs required by GraphCode-
+BERT [7]), M can predict a probability vector for x, each
+element in which represents the probability classifying x to
+the corresponding class. The class with the largest probability
+is the final prediction result of M for x. If the prediction result
+is different from the ground-truth label (denoted as y) of x, it
+means that M makes a wrong prediction on x; otherwise, M
+makes a correct prediction.
+Although deep code models can achieve great performance
+on the given test sets, they may be vulnerable to adversarial
+examples [12]–[14]. The goal of our work is to generate
+successfully-attacking adversarial examples as much as pos-
+sible, so as to improve the model robustness. As source code
+is discrete and has to stick to the grammar and semantics con-
+straints, the existing adversarial example generation techniques
+proposed in other domains are not applicable.
+The existing attack techniques specific to deep code models
+always generate adversarial examples from a target input by
+performing a series of semantic-preserving code transforma-
+tions [12]–[14], which is also followed by our work. For ease
+of understanding, we formally define our target problem is to
+find {x′|x′ ∈ ϵ ∧ y = M(x) ̸= M(x′)} from a target input x
+for the target model M. Here, ϵ refers to the universal set of
+code snippets that satisfy the grammar constraints and preserve
+the semantics of x. y = M(x) means that we just regard the
+test inputs on which M makes correct predictions as target
+inputs, where M(x) refers to the prediction result of M on
+x. M(x) ̸= M(x′) means that x′ successfully attacks M, that
+is, it is a successfully-attacking adversarial example generated
+from x. Besides, an effective attack technique should be also
+efficient to find x′ and ensure the naturalness of x′ (i.e., natural
+to human comprehension [14]), which are indeed carefully
+considered by the proposed technique in our work.
+C. Motivating Example
+We then use a real-world example (simplified for ease of
+illustration) to help motivate our key idea: utilizing the code
+differences between reference inputs and the target input to
+guide the generation of adversarial examples. In Figure 1, the
+first code snippet f1 is the target input from the test set of
+the functionality classification task [18], and the two state-
+of-the-art techniques (i.e., CARROT [12] and ALERT [14])
+do not generate successfully-attacking adversarial examples
+from it since they can fall into local optimum in the enormous
+ingredient space. In this figure, the second code snippet f2 is
+a reference input from the training set of this task, which has
+the different label with f1.
+In fact, f2 can be regarded as an invalid successfully-
+attacking example from f1, as they are semantically inconsis-
+tent but have different prediction results. The code differences
+between f1 and f2 mainly contribute to this phenomenon. From
+this perspective, to generate a valid successfully-attacking
+adversarial example (denoted as f3) from f1, we should per-
+form semantic-preserving code transformations on f1, and the
+transformations should reduce the code differences between f1
+and f2 in order to alter the prediction result of the target model
+on f1. That is, the ingredients supporting these transformations
+3
+
+Initial
+Snippet
+Adversarial
+Snippet
+Input
+Output
+Training 
+Data
+Input
+Target
+Model
+Test
+Test
+Report
+Attack
+Success ?
+Fig. 2. Overview of CODA.
+should be extracted from the code differences brought by
+f2. With this intuition, by performing equivalent structure
+transformations on f1 (i.e., transforming for loops to while
+loops, where while loops are the used loop structure in f2)
+and identifier remaining transformations (i.e., renaming a and
+n to t and len respectively, where t and len are the used
+identifier names in f2), f3 is generated as shown in the third
+code snippet in Figure 1 and indeed attacks successfully,
+i.e., making a wrong prediction (palindrome) with a high
+confidence (90.88%).
+Based on the code differences between the target input and
+the reference input, the ingredient space is largely reduced. For
+example, the ingredient space defined by identifier renaming
+transformations is reduced from all valid identifier names
+(i.e., almost infinite) to the identifier names occurring in the
+reference input but not in the target input (i.e., only two
+identifier names in this simplified example). Hence, it can help
+improve the ingredient search process and thus improve the
+attack effectiveness. On the other hand, too small ingredient
+space could also lose too many ingredients useful to successful
+attacks, and thus we will select a set of reference inputs (rather
+than only one reference input) for guiding the attack process
+in order to balance the size of the ingredient space and the
+number of useful ingredients.
+III. APPROACH
+A. Overview
+In this work, we propose a novel perspective to attack
+deep code models more effectively and more efficiently, which
+utilizes the code differences between reference inputs and the
+target input to guide the generation of adversarial examples.
+From this perspective, we design an effective attack technique,
+called CODA. Specifically, the code differences brought by
+reference inputs provide effective ingredients for altering the
+prediction result of the target input by transforming it with
+these ingredients, which can contribute to successful attacks in
+CODA. However, as the semantics of reference inputs and the
+target input are different, the ingredients from some kinds of
+code differences can alter the original semantics, which is not
+allowed by the adversarial attack for deep code models. Hence,
+in CODA, we consider the structure differences and identifier
+differences for measuring code differences between them,
+which can preserve the original semantics during the attack
+process. In this way, the ingredient space can be effectively
+reduced as the one constituted by the two kinds of code
+differences between reference inputs and the target input, and
+thus the ingredient search process (for generating adversarial
+examples) can be largely improved.
+In fact, not all the inputs that have different prediction
+results with the target one, can be regarded as effective
+reference inputs for improving the adversarial attack process.
+In other words, different inputs could have different degrees
+of capabilities for reducing the ingredient search space and
+providing effective ingredients for altering the prediction result
+of the target input. Therefore, the first step in CODA is
+to select effective reference inputs for the target input in
+order to improve the attack effectiveness as much as possible
+(to be presented in Section III-B). Based on the selected
+reference inputs, CODA then measures the structure differ-
+ences and identifier differences over the target input, which
+support extracting the ingredients for two corresponding kinds
+of semantic-preserving code transformations (i.e., equivalent
+structure transformations and identifier renaming transforma-
+tions). With the guidance of reducing their code differences
+based on the two kinds of transformations, the target input
+could be effectively transformed to a successfully-attacking
+adversarial example. As equivalent structure transformations
+do not affect the naturalness of generated examples, CODA
+first applies this kind of transformations to reduce the code
+differences for improving the attack effectiveness (to be pre-
+sented in Section III-C). Then, we apply identifier renaming
+transformations to further reduce the code differences for
+improving the generation of successfully-attacking adversarial
+examples (to be presented in Section III-D). In particular,
+CODA measures the semantic similarity between identifiers
+to guarantee the naturalness of generated examples.
+Figure 2 shows the overview of CODA. In a nutshell, by
+successively applying equivalent structure transformations and
+identifier renaming transformations to the target input with
+the ingredient space defined by the code differences between
+the selected reference inputs and the target one, adversarial
+examples can be generated towards the direction of reducing
+the code differences without altering the original semantics. In
+this way, the prediction result of the target input is more likely
+to be changed, leading to a successfully-attacking adversarial
+example. Due to the smaller ingredient search space (but
+including effective ingredients) and the clearer attack direction,
+the attack effectiveness could be largely improved by CODA.
+B. Reference Inputs Selection
+The goal of reference inputs is to largely reduce the in-
+gredient space. Also, the reduced space should include the
+ingredients that are effective to transform the target input
+to a successfully-attacking adversarial example. In this way,
+the adversarial attack process can be largely improved by
+searching for effective ingredients more efficiently.
+4
+
+TABLE I
+DESCRIPTIONS OF EQUIVALENT STRUCTURE TRANSFORMATIONS
+Transformation
+Description
+Example Before Transformation
+Example After Transformation
+R1-loop
+equivalent transformation among for structure
+for ( i=0; i<9; i++ ) {
+i=0; while ( i<9 ) {
+and while structure
+Body; }
+Body; i++; }
+R2-branch
+equivalent transformation between if-else(-if)
+if ( A ) { BodyA; }
+if ( A ) { BodyA; }
+structure and if-if structure
+else if ( B ) { BodyB; }
+if ( !A && B ) { BodyB; }
+R3-calculation
+equivalent numerical calculation transformation, e.g.,
+i += 1;
+i = i + 1;
+++, --, +=, -=, *=, /=, %=, <<=, >>=, &=, |= , ˆ =
+R4-constant
+equivalent transformation between a constant and
+println("Hello, World!");
+String i = "Hello, World!";
+a variable assigned by the same constant
+println(i);
+Although all the inputs that have different prediction results
+with the target one can provide ingredients for altering the pre-
+diction result of the target one after transformations, their capa-
+bilities for successful attacks could be different. To transform
+the target input to a successfully-attacking adversarial example
+with fewer perturbations, CODA should select the reference
+inputs, which can provide the ingredients that are more likely
+to conduct successful attacks for the target input. Similar to the
+existing work [24]–[27], we assume that the prediction result
+of the target input is more likely to be changed from its original
+class denoted as ci (with the largest probability predicted
+by the target model) to the class with the second largest
+probability (denoted as cj). Hence, the ingredients in the inputs
+belonging to cj are more likely to attack successfully on the
+target input, and thus CODA selects the inputs belonging to
+cj as the initial set of reference inputs. Please note that all
+the reference inputs are selected from the training set to avoid
+introducing the contents beyond the cognitive scope of the
+target model. Meanwhile, we only consider the training inputs
+whose prediction results are consistent with their ground-truth
+labels in order to avoid introducing noise.
+However, the number of inputs belonging to the same class
+(i.e., cj as above) could be large, and thus the ingredient space
+constituted by code differences between them and the target
+input could be also large. Hence, to further reduce the ingre-
+dient space for more effective adversarial example generation,
+CODA selects a subset of inputs with high similarity to the
+target input from the initial set of reference inputs, as the
+final set of reference inputs used by CODA. This is because
+smaller code differences can effectively limit the number of
+ingredients, leading to smaller ingredient space. CODA does
+not select only one reference input, as too small ingredient
+space could incur a high risk of missing too many ingredients
+contributing to successful attacks. That is, CODA selects a
+small set of reference inputs following the above two steps of
+selection to balance the ingredient space size and the amount
+of ingredients contributing to successful attacks in the space.
+We further introduce how to measure the similarity be-
+tween the target input (denoted as t) and a reference input
+(denoted as r) for the second step of selection in CODA. In
+general, we can adopt some pre-trained models to represent
+the code as a vector and then measure code similarity by
+calculating the vector distance, like many existing studies [5],
+[6], [28]. However, as presented in Section III-A, CODA
+first applies equivalent structure transformations (rather than
+identifier renaming transformations) to reduce code differences
+for adversarial attacks, as this kind of transformations does not
+affect the naturalness of generated examples. Moreover, the
+identifiers used in different code snippets are usually different
+due to the enormous identifier space, which may lead to the
+low similarity between various code snippets. Hence, when
+measuring code similarity, CODA eliminates the influence
+of identifiers by replacing them with the placeholder <unk>.
+Specifically, CODA first represents t and r after placeholder
+replacement as vectors respectively based on CodeBERT [6]
+(one of the most widely-used pre-trained models [29]–[31]),
+and then calculates the cosine similarity between the two
+vectors. As the descending order of the calculated similarity,
+CODA selects Top-N reference inputs for the follow-up ad-
+versarial attack process. Please note that to make the selection
+process efficient, we randomly sampled U inputs from the
+initial set for the second step of selection. We will investigate
+the influence of both U and N on CODA in Section VI.
+C. Equivalent Structure Transformation
+Based on the small set of selected reference inputs, CODA
+then extracts ingredients from the space defined by their
+brought code differences over the target input. That is, CODA
+transforms the target input to an adversarial example towards
+the direction of reducing code differences. CODA first reduces
+structure differences by applying equivalent structure transfor-
+mations to the target input as they do not affect the naturalness
+of generated examples.
+To preserve the semantics of the target input, we design four
+categories of equivalent structure transformations in CODA
+inspired by the existing work in metamorphic testing and
+code refactoring [32], [33]. In particular, we systematically
+consider all common kinds of code structures, i.e., loop struc-
+tures, branch structures, and sequential structures (including
+numerical calculation and constant usage). We explain the four
+categories in detail in Table I, each of which is also illustrated
+with an example. For each category of transformations, it
+may include several specific rules. For example, the rules
+of transformation on += and transformation on -= belong to
+the category of R3-calculation, and the rules of transforming
+for loop to while loop and transforming while loop to for
+loop belong to the category of R1-loop. In total, CODA has
+20 specific rules for the four categories of transformations.
+Please note that not all the rules are applicable to the code
+5
+
+programmed by any programming language. For example, ++
+and -- in R3-calculation are not supported by Python. Also,
+in R4-constant, the newly-defined variable cannot be the same
+as the existing variables in the code; otherwise, it may incur
+grammar errors and alter the original semantics. Due to the
+space limit, we put more details about all these specific rules
+at our project homepage [19].
+Then, we illustrate how to apply each rule for reducing code
+differences. Each rule involves two structures, i.e., the one
+before transformation (sb) and the one after transformation
+(sa). CODA first counts the occurring times of sb and sa in
+the set of selected reference inputs (denoted as nb and na),
+and then calculates their occurring distribution, i.e.,
+nb
+nb+na
+and
+na
+nb+na . Further, CODA applies each rule in a probabilistic
+way to reduce the occurring distribution differences in terms
+of sb and sa between reference inputs and the target input. In
+this way, the structure differences in terms of sb and sa can
+be reduced effectively. More specifically, for each occurrence
+of sb in the target input, CODA applies this rule with the
+probability of
+na
+nb+na , also indicating that it can be retained
+with the probability of
+nb
+nb+na .
+In this step, CODA obtains M inputs from the target input,
+each of which is generated by applying all the applicable rules
+in the above probabilistic way, and then selects the input with
+the highest average similarity (also measured by the method
+described in Section III-B) to the selected reference inputs as
+the one for the follow-up adversarial attack process.
+D. Identifier Renaming Transformation
+To facilitate the generation of successfully-attacking ad-
+versarial examples, CODA then applies identifier renaming
+transformations to further reduce code differences. Inspired by
+the existing work [12]–[14], identifier renaming transformation
+in CODA refers to replacing the name of an identifier in the
+target input with the name of an identifier in the selected
+reference inputs. For ease of presentation, we denote the set of
+identifiers in the target input as Vt and the set of identifiers in
+the selected reference inputs as Vr. To preserve the semantics
+of the target input and guarantee the grammatical correctness
+of the generated example, CODA ensures that the identifier
+used for replacement does not exist in the target input.
+Then, we illustrate how to apply this kind of transformations
+to the input obtained from the last step (i.e., equivalent
+structure transformations). As demonstrated by the existing
+work [12]–[14], renaming identifiers is effective to generate
+successfully-attacking adversarial examples, but can negatively
+affect the naturalness of generated examples. To ensure the
+naturalness of generated examples, CODA considers the se-
+mantic similarity between identifiers and designs an iterative
+transformation process like ALERT [14]. Specifically, CODA
+measures the semantic similarity between each identifier in
+Vt and each identifier in Vr by representing each identifier as
+a vector via word embedding. Here, CODA builds the pre-
+trained language model with the FastText algorithm [34] and
+calculates the cosine similarity between vectors to measure
+their semantic similarity. Then, CODA prioritizes each pair
+TABLE II
+STATISTICS OF OUR USED SUBJECTS
+Task
+Train/Validate/Test
+Class
+Language
+Model
+Acc.
+Vulnerability
+21,854/2,732/2,732
+2
+C
+CB
+63.76%
+Prediction
+GCB
+63.65%
+Clone
+90,102/4,000/4,000
+2
+Java
+CB
+96.97%
+Detection
+GCB
+97.36%
+Authorship
+528/–/132
+66
+Python
+CB
+90.35%
+Attribution
+GCB
+89.48%
+Functionality
+41,581/–/10,395
+104
+C
+CB
+98.18%
+Classification
+GCB
+98.66%
+Defect
+27,058/–/6,764
+4
+C/C++
+CB
+84.37%
+Prediction
+GCB
+83.98%
+* CB is short for CodeBERT and GCB is short for GraphCodeBERT.
+of identifiers as the descending order of their semantic sim-
+ilarity, and iteratively applies this transformation based on
+each pair of identifiers in the ranking list, which ensures
+that more natural transformations can be first performed.
+After each iteration, CODA invokes the target model to
+check whether a successfully-attacking adversarial example
+is generated. The iterative attack process terminates until a
+successfully-attacking adversarial example is generated or all
+the pairs are used by this transformation. Please note that
+CODA ensures that the pair of identifiers will not introduce
+repetitive identifiers in the generated example in each iteration;
+otherwise, this pair will be discarded.
+Overall, CODA only invokes the target model when check-
+ing if a successfully-attacking adversarial example is gen-
+erated. They are necessary model invocations for this task.
+Hence, CODA can largely reduce the number of model
+invocations compared with the existing techniques (e.g.,
+ALERT [14]), which is confirmed by our study (Section V-A).
+IV. EVALUATION DESIGN
+In the study, we address four research questions (RQs):
+• RQ1: How does CODA perform in terms of effectiveness
+and efficiency compared with state-of-the-art techniques?
+• RQ2: Are the adversarial examples generated by CODA
+natural for humans?
+• RQ3: Are the adversarial examples generated by CODA
+useful to improve the robustness of deep code models?
+• RQ4: Does each main component contribute to the over-
+all effectiveness of CODA?
+A. Subjects
+1) Datasets and Tasks: To sufficiently evaluate CODA,
+we consider all the five code-based tasks in the studies of
+evaluating state-of-the-art techniques (i.e., CARROT [12] and
+ALERT [14]). The statistics of datasets is shown at the first
+four columns in Table II, each of which represents the task, the
+number of inputs in the training/validation/test set, the number
+of classes for the classification task, and the programming
+language for the inputs.
+The task of vulnerability prediction aims to predict whether
+a given code snippet has vulnerabilities. Its used dataset is ex-
+tracted from two C projects (i.e., FFmpeg [35] and Qemu [36])
+by Zhou et al. [2] and has been integrated as part of the
+CodeXGLUE benchmark [30]. The task of clone detection
+6
+
+aims to detect whether two given code snippets are equivalent
+in semantics. Its used dataset is from BigCloneBench [37],
+the most widely-used dataset for clone detection. The existing
+work [14] randomly sampled 90,102/4,000/4,000 inputs from
+the benchmark for training/validation/testing, to make the
+experiment at a computationally friendly scale. In our study,
+we used the same dataset. The task of authorship attribution
+aims to identify the author of a given code snippet. Its used
+dataset is the Google Code Jam (GCJ) dataset [17]. The task
+of functionality classification aims to classify the functionality
+of a given code snippet. If code snippets solve the same
+problem, they are regarded to have the same functionality [18].
+Its used dataset is the Open Judge (OJ) benchmark [38],
+which has been also integrated as part of the CodeXGLUE
+benchmark [30]. The task of defect prediction aims to predict
+whether a given code snippet is defective and its defect type.
+Its used dataset is the CodeChef dataset [39], which is labeled
+by the execution results on the CodeChef platform (i.e., four
+defect types: no defect, wrong answer, timeout, runtime error).
+2) Models: Following the existing work [14], we adopted
+two state-of-the-art pre-trained models for code-based tasks,
+i.e., CodeBERT [6] and GraphCodeBERT [7], and then fine-
+tuned them on the five tasks based on the corresponding
+datasets, respectively. In total, we obtained 10 deep code
+models as the subjects. The last two columns in Table II show
+the used pre-trained model and the accuracy of the deep code
+model after fine-tuning, respectively. When fine-tuning Code-
+BERT and GraphCodeBERT on these tasks (except Graph-
+CodeBERT on functionality classification and defect predic-
+tion), we used the same hyper-parameter settings provided by
+the existing work [12], [14]. As there is no instruction on the
+hyper-parameter settings for fine-tuning GraphCodeBERT on
+functionality classification and defect prediction, we used the
+same settings as the one used by authorship attribution (they
+are all multi-class classification tasks). Indeed, the achieved
+model performance outperforms that achieved by the models
+(e.g., TBCNN [38] and CodeBERT [6]) used in the existing
+work [12] on the same datasets [12], indicating that the
+transferred hyper-parameter settings are reasonable.
+Overall, the subjects used in our study are indeed diverse,
+involving different tasks, different pre-trained models, different
+numbers of classes, different programming languages, etc. It is
+very helpful to sufficiently evaluate the performance of CODA.
+B. Compared Techniques
+In the study, we compared CODA with two state-of-the-art
+techniques attacking deep code models, i.e., CARROT [12]
+and ALERT [14], which have been introduced in Section I (the
+third paragraph). We adopted their implementations and the
+recommended parameter settings provided by the correspond-
+ing papers [12], [14]. As the original version of CARROT can
+only support to attack C/C++ code, we extended it to attack
+Python and Java code for sufficient comparison.
+C. Implementations
+We implemented CODA in Python and adopted tree-
+sitter [40] to extract identifiers from code following the exist-
+ing work [14]. We set the parameters in CODA by conducting
+a preliminary experiment, i.e., U = 256, N = 64, and
+M = 64. We discuss the influence of the settings of main
+parameters on CODA in Section VI. We released our code
+and experimental data at our project homepage [19]. All the
+experiments were conducted on a server with an Ubuntu 20.04
+system with Intel(R) Xeon(R) Silver 4214 @ 2.20GHz CPU,
+256GB memory, and NVIDIA GeForce RTX 2080 Ti GPU.
+V. RESULTS AND ANALYSIS
+A. RQ1: Effectiveness and Efficiency
+1) Setup: For each deep code model, we applied CODA,
+CARROT, ALERT to generate adversarial examples from each
+target input in the test set, respectively. We measured their
+effectiveness and efficiency based on the following metrics.
+To reduce the influence of randomness, we repeated all the
+experiments (including those for other RQs) 10 times, and
+reported the average results.
+Following the existing work [14], we adopted the at-
+tack success rate (ASR) to measure the effectiveness of
+each technique. ASR is the percentage of the target inputs
+from which an attack technique can generate a successfully-
+attacking adversarial example. Larger ASR values mean better
+attack effectiveness. Also, it is important to measure whether
+the prediction confidence (i.e., the probability of being the
+ground-truth class of the target input) is decreased by the gen-
+erated examples (although there is no successfully-attacking
+adversarial example generated from a target input). Hence, we
+also calculated prediction confidence decrement (PCD) to
+measure the effectiveness of each technique. PCD is calculated
+by the prediction confidence of the target input minus the min-
+imum prediction confidence of the set of generated examples
+from the target input. If the former is smaller than the latter,
+we regard PCD to be 0, indicating that the generated examples
+cannot decrease the prediction confidence of the target input.
+Larger PCD values mean better attack effectiveness.
+In addition, following the existing work [12], [14], we used
+the time spent on the overall attack process (i.e., completing
+the adversarial example generation process from all the target
+inputs) and the average number of model invocations for
+generating examples from one target input, to measure
+the efficiency of each technique. Less time and fewer model
+invocations mean higher efficiency.
+2) Results: Table III shows the comparison results among
+CARROT, ALERT, and CODA in terms of ASR. From this
+table, CODA always outperforms CARROT and ALERT on
+all the tasks based on both CodeBERT and GraphCodeBERT,
+demonstrating the stable attack effectiveness of CODA. On
+average, CODA improves 70.11% and 89.83% higher ASR
+than CARROT and ALERT across all the five tasks on
+CodeBERT, and improves 89.34% and 57.67% higher ASR
+on GraphCodeBERT, respectively.
+Figures 3(a) and 3(b) show the comparison results among
+the three techniques in terms of PCD on CodeBERT and
+GraphCodeBERT, respectively. From these figures, the upper
+quartile, median, and lower quartile of CODA are always
+7
+
+TABLE III
+EFFECTIVENESS COMPARISON IN TERMS OF ASR
+Task
+CodeBERT
+GraphCodeBERT
+CARROT
+ALERT
+CODA
+CARROT
+ALERT
+CODA
+Vulnerability Prediction
+33.72%
+53.62%
+89.58%
+37.40%
+76.95%
+94.72%
+Clone Detection
+20.78%
+27.79%
+44.65%
+3.50%
+7.96%
+27.37%
+Authorship Attribution
+44.44%
+35.78%
+79.05%
+31.68%
+61.47%
+92.00%
+Functionality Classification
+44.15%
+10.04%
+56.74%
+42.76%
+11.22%
+57.44%
+Defect Prediction
+71.59%
+65.15%
+95.18%
+79.08%
+75.87%
+96.58%
+Average
+42.94%
+38.48%
+73.04%
+38.88%
+46.69%
+73.62%
+(a) PCD on attacking CodeBERT
+(b) PCD on attacking GraphCodeBERT
+Fig. 3. Comparison in terms of prediction confidence decrement
+(a) Model invocation times on attacking CodeBERT
+(b) Model invocation times on attacking GraphCodeBERT
+Fig. 4. Comparison in terms of model invocation times (y-axis refers to the normalized values following the existing work [14])
+larger than (or equal to) those of both CARROT and ALERT
+regardless of the tasks and the pre-trained models, demon-
+strating that CODA produces more significant attacks for
+decreasing prediction confidence of target inputs. For example,
+on CodeBERT, the average improvements of CODA over
+CARROT and ALERT are 101.88% and 520.65% across all
+the tasks in terms of average PCD, respectively. Similarly, on
+GraphCodeBERT, the average improvements of CODA over
+CARROT and ALERT are 76.35% and 560.15%, respectively.
+Besides, CARROT, ALERT, and CODA take 159.19 hours,
+198.89 hours, and 39.59 hours to complete the entire attack
+process on all five tasks, respectively. Further, we measured
+the number of model invocations for each target input during
+the attack process, whose results are shown in Figure 4(a)
+and 4(b). From these figures, CODA performs fewer model
+invocations than both CARROT and ALERT regardless of the
+tasks and pre-trained models. On average, CODA performs
+65.73% and 78.58% fewer model invocations than CARROT
+and ALERT across all the tasks on CodeBERT, and 34.07%
+and 75.31% fewer model invocations on GraphCodeBERT,
+respectively. The results demonstrate that CODA has the
+significantly highest efficiency among the three techniques.
+Answer to RQ1: CODA spends less time with fewer
+model invocations on completing the entire attack
+process, but generates more successfully-attacking ex-
+amples with more significant prediction confidence
+decrement on all the subjects, than the state-of-the-art
+techniques (i.e., CARROT and ALERT).
+B. RQ2: Naturalness of Adversarial Examples
+1) Setup: It is important to check whether the generated
+adversarial examples are natural to human judges [14], [41].
+Here, we conducted a user study to compare the naturalness
+of examples generated by CODA, CARROT, and ALERT, and
+our user study shares the same design as the one conducted
+by the existing work [14]:
+Data Preparation. For each subject, we randomly sampled
+10 target inputs, and then for each technique we randomly
+sampled an adversarial example from the set of examples
+generated from each sampled target input. That is, for each
+sampled target input, we construct three pairs of code snippets,
+each of which contains the target input and an adversarial
+example generated by CODA, CARROT, or ALERT. In total,
+we obtained 300 pairs of code snippets for the user study due
+to 10 subjects × 3 techniques.
+Participants. Same as the existing work [14], the user study
+also involves four non-author participants, each of whom has
+8
+
+Fig. 5. Average score to evaluate naturalness of examples per participant
+a Bachelor/Master degree in Computer Science with at least
+five years of programming experience.
+Process. For objective evaluation, we did not tell partici-
+pants which technique generates the adversarial example in a
+pair of code snippets. Also, we highlighted the changes in each
+pair of code snippets for facilitating manual evaluation. Then,
+each participant individually evaluated each pair by evaluating
+to what extent the changes are natural to the code context
+and the changed identifiers preserve the original semantics,
+following the existing work [14]. Specifically, participants
+gave a score for each pair based on a 5-point Likert scale [42]
+(1 means strongly disagree and 5 means strongly agree)
+following the existing work [14], [43].
+2) Results: Figure 5 shows the average score of the ad-
+versarial examples generated by each technique for each
+participant. From this figure, the conclusions from different
+participants are consistent: the naturalness of the adversarial
+examples generated by CODA and ALERT is closely high
+(round 4.50 on average), and significantly higher than that by
+CARROT (just 2.89 on average). ALERT is a naturalness-
+aware technique, whose core contribution is to ensure the
+naturalness of generated examples, but CODA achieves similar
+naturalness scores to it, demonstrating that CODA can also
+generate highly natural adversarial examples.
+Answer to RQ2: The adversarial examples gener-
+ated by CODA are natural closely to the state-of-the-
+art naturalness-aware attack technique (i.e., ALERT),
+which is consistently confirmed by participants.
+C. RQ3: Model Robustness Improvement
+1) Setup: We studied the value of generated adversarial
+examples by using them to improve the robustness of the target
+model via an adversarial fine-tuning strategy. For each subject,
+we divided the test set into two equal parts (S1 and S2),
+so as to avoid data leakage between the adversarial training
+set and the evaluation set constructed by the same technique.
+Specifically, we applied each technique to generate examples
+from S1, and selected one generated adversarial example
+for each target input, i.e., the one that successfully attacks
+the model or achieves the largest decrement on prediction
+confidence (if no successfully-attack example is generated).
+The selected examples were integrated with the training set to
+form the adversarial training set, which is used for fine-tuning
+the model. Thus, for a given subject, the size of the adversarial
+training set constructed by each technique is the same.
+After obtaining a fine-tuned model for each subject with
+each technique, we evaluated it on the evaluation set of
+the successfully-attacking examples generated from S2 by
+CODA, CARROT, ALERT, respectively. Then, we measured
+the accuracy of the fine-tuned model on the three evaluation
+sets to measure its ability of defending against attacks.
+2) Results: Table IV shows the effectiveness of improving
+the model robustness with the generated examples by the
+studied techniques, respectively. The first row represents the
+evaluation set constructed by the corresponding technique,
+while the second row represents the adversarial training set
+constructed by the corresponding technique. The value in each
+cell represents the ratio of the adversarial examples in the
+evaluation set that can be defended by the fine-tuned model
+based on the adversarial training set. We found on most sub-
+jects, CODA improves the model robustness to defend against
+attacks from the largest ratio of adversarial examples generated
+by CODA, CARROT, ALERT, respectively. On average, the
+models fine-tuned by CODA can defend against attacks from
+63.64%, 66.96%, 76.68% of successfully-attacking examples
+generated by CARROT, ALERT, CODA respectively, with
+the improvement of 6.35%, 25.69%, 42.67% over those by
+CARROT and 32.65%, 11.70%, 25.99% over those by ALERT
+respectively. Besides, the results of attack defense between
+different techniques indicate that the examples generated by
+CODA could subsume those by CARROT and ALERT to
+a large extent. We also applied each fine-tuned model to
+the corresponding test set, and found its accuracy is almost
+consistent with the original accuracy (all the absolute accuracy
+differences are less than 1%). The results demonstrate that
+CODA is more helpful to improve the model robustness than
+CARROT and ALERT without damaging the original model
+performance. In four evaluation sets (constructed by ALERT or
+CARROT), CODA performs worse than ALERT or CARROT,
+as the adversarial training set and the evaluation set generated
+by the same technique could be more similar.
+Answer to RQ3: CODA helps improve the model ro-
+bustness more effectively than CARROT and ALERT,
+in terms of defending against attacks from the adver-
+sarial examples generated by itself as well as the adver-
+sarial examples generated by the other two techniques.
+D. RQ4: Contribution of Each Main Component
+1) Setup: We studied the contribution of each main compo-
+nent in CODA, i.e., reference inputs selection (RIS), equivalent
+structure transformations (EST), and identifier renaming trans-
+formations (IRT). We constructed three variants of CODA:
+• w/o RIS: we replaced RIS with the method that randomly
+selects N inputs from training data as reference inputs.
+• w/o EST: we removed EST from CODA, i.e., it directly
+performs identifier renaming transformations after select-
+ing reference inputs.
+• w/o IRT: we removed IRT from CODA, i.e., it directly
+checks whether a successfully-attacking example is gen-
+erated after equivalent structure transformations.
+9
+
+TABLE IV
+ROBUSTNESS IMPROVEMENT OF THE TARGET MODELS AFTER ADVERSARIAL FINE-TUNING
+Task
+Model
+CARROT
+ALERT
+CODA
+CARROT
+ALERT
+CODA
+CARROT
+ALERT
+CODA
+CARROT
+ALERT
+CODA
+Vulnerability
+CodeBERT
+29.14%
+21.11%
+29.69%
+23.43%
+26.27%
+34.44%
+32.16%
+31.73%
+38.82%
+Prediction
+GraphCodeBERT
+12.37%
+19.59%
+21.65%
+16.33%
+17.35%
+23.71%
+25.77%
+24.74%
+34.02%
+Clone
+CodeBERT
+83.15%
+42.31%
+94.44%
+52.65%
+72.46%
+75.32%
+38.51%
+71.45%
+89.78%
+Detection
+GraphCodeBERT
+75.00%
+66.67%
+77.50%
+79.17%
+84.29%
+92.31%
+35.71%
+57.69%
+92.97%
+Authorship
+CodeBERT
+45.06%
+40.67%
+41.03%
+51.25%
+56.25%
+58.82%
+45.67%
+43.33%
+76.47%
+Attribution
+GraphCodeBERT
+81.75%
+67.08%
+72.40%
+79.41%
+78.67%
+100.00%
+45.59%
+80.39%
+84.75%
+Functionality
+CodeBERT
+83.46%
+72.80%
+81.51%
+70.83%
+71.75%
+79.41%
+78.92%
+71.18%
+95.43%
+Classification
+GraphCodeBERT
+67.53%
+75.19%
+77.27%
+32.04%
+52.62%
+62.98%
+91.22%
+90.81%
+93.08%
+Defect
+CodeBERT
+52.73%
+25.81%
+66.03%
+74.88%
+75.87%
+83.12%
+76.86%
+68.66%
+85.36%
+Prediction
+GraphCodeBERT
+68.20%
+48.54%
+74.88%
+52.73%
+63.91%
+59.45%
+67.08%
+68.66%
+76.14%
+Average
+59.84%
+47.98%
+63.64%
+53.27%
+59.94%
+66.96%
+53.75%
+60.86%
+76.68%
+TABLE V
+ABLATION TEST FOR CODA IN TERMS OF AVERAGE ASR
+Model
+w/o RIS
+w/o EST
+w/o IRT
+CODA
+CodeBERT
+30.83%
+62.73%
+35.14%
+73.04%
+GraphCodeBERT
+29.49%
+62.41%
+26.24%
+73.62%
+TABLE VI
+INFLUENCE OF HYPER-PARAMETER U.
+U
+64
+128
+256
+512
+1024
+CodeBERT
+60.14%
+67.90%
+73.04%
+75.27%
+75.83%
+GraphCodeBERT
+61.92%
+70.16%
+73.62%
+74.98%
+75.69%
+2) Results: Table V shows the average ASR values of each
+technique across all the tasks on CodeBERT and GraphCode-
+BERT, respectively. The results on each task can be found at
+our project homepage [19] due to the space limit. We found
+that CODA outperforms all three variants in terms of average
+ASR with improvements of 17.20%∼143.14%, demonstrating
+the contribution of each main component in CODA. Also, ref-
+erence inputs selection and identifier renaming transformations
+contribute more than equivalent structure transformations. The
+possible reason is that not all the rules of equivalent structure
+transformations can be applicable to all the target inputs,
+but identifier renaming transformations are applicable to all
+the inputs. We can enrich the rules of equivalent structure
+transformations in the future to further improve the attack
+effectiveness.
+Answer to RQ4: All the components of reference in-
+put selection, equivalent structure transformations, and
+identifier renaming transformations make contributions
+to the overall effectiveness of CODA, demonstrating
+the necessity of each of them in CODA.
+VI. THREATS TO VALIDITY
+The main threat to validity lies in the settings of param-
+eters in CODA. Here, we investigated the influence of two
+important parameters in CODA (i.e., U and N introduced in
+Section III-B). They affect the selection of reference inputs.
+Tables VI and VII show the influence of U and N in
+terms of average ASR across all the tasks. As U increases,
+CODA performs better, as incorporating more inputs for the
+second step of selection can increase the possibility of finding
+TABLE VII
+INFLUENCE OF HYPER-PARAMETER N
+N
+1
+4
+16
+32
+64
+128
+CodeBERT
+28.08%
+46.33%
+61.07%
+67.12%
+73.04%
+76.38%
+GraphCodeBERT
+31.84%
+46.46%
+60.40%
+66.12%
+73.62%
+74.93%
+more effective reference inputs. Similarly, as N increases
+within our studied range, more effective ingredients could
+be included, leading to better effectiveness. However, the
+amount of increase in terms of average ASR becomes smaller
+with U and N increasing, and meanwhile incorporating more
+inputs can incur more costs in similarity calculation or code
+transformations. Hence, by balancing the effectiveness and
+efficiency of CODA, we set U to 256 and N to 64 as the
+default settings in CODA for practical use.
+VII. RELATED WORK
+Besides the state-of-the-art techniques compared in our
+study (i.e., CARROT [12] and ALERT [14]), there are some
+other adversarial example generation techniques for deep code
+models. For example, Yefet et al. [44] proposed DAMP, which
+changes variables in the target input by gradient computation.
+It only works for the models using one-hot encoding to process
+code, and thus cannot attack the models based on state-
+of-the-art CodeBERT [6] and GraphCodeBERT [7] due to
+different encoding methods. Zhang et al. [13] proposed MHM,
+which iteratively performs identifier renaming transformations
+to generate adversarial examples based on the Metropolis-
+Hastings [45]–[47] algorithm. MHM underperforms CARROT
+and ALERT as presented by the existing studies [12], [14].
+Pour et al. [48] proposed a search-based technique with an
+iterative refactoring-based process. It does not ensure the
+naturalness of generated examples, especially with the rule
+of dead code insertion. These techniques still search for
+effective ingredients in the enormous space, limiting their
+effectiveness. Different from them, our work designs the first
+code-difference-guided attack technique, which can largely
+reduce ingredient space for improving the attack effectiveness.
+There are also many adversarial attack techniques in other
+domains, such as FGSM [49], JSMA [50], and BIM [10] in
+image processing. However, they are not applicable to attack
+deep code models as source code is discrete and has to strictly
+stick to the grammar and semantics constraints.
+10
+
+VIII. CONCLUSION
+To improve the attack effectiveness to deep code models, we
+propose a novel perspective by exploiting the code differences
+between reference inputs and the target input to guide the
+generation of adversarial examples. From this perspective, we
+design CODA, which reduces the ingredient space as the one
+constituted by structure and identifier differences and designs
+equivalent structure transformations and identifier renaming
+transformations to preserve original semantics. We conducted
+an extensive study on two popular pre-trained models with
+five tasks. The results demonstrate that CODA performs more
+successful attacks with less time than the state-of-the-art
+techniques (i.e., CARROT and ALERT), and confirm the
+naturalness of its generated examples as well as the capability
+of improving the model robustness.
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+

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+MLMC techniques for discontinuous functions
+Michael B. Giles
+Abstract The Multilevel Monte Carlo (MLMC) approach usually works well when
+estimating the expected value of a quantity which is a Lipschitz function of inter-
+mediate quantities, but if it is a discontinuous function it can lead to a much slower
+decay in the variance of the MLMC correction. This article reviews the literature
+on techniques which can be used to overcome this challenge in a variety of different
+contexts, and discusses recent developments using either a branching diffusion or
+adaptive sampling.
+1 Introduction
+The Multilevel Monte Carlo (MLMC) method is based on the telescoping sum
+E[ �𝑃𝐿] = E[ �𝑃0] +
+𝐿
+∑︁
+ℓ=1
+E[ �𝑃ℓ−�𝑃ℓ−1]
+where �𝑃ℓ represents an approximation to an output quantity of interest 𝑃 on level ℓ,
+with the weak error
+���E[ �𝑃ℓ−𝑃]
+��� and MLMC variance V[ �𝑃ℓ−�𝑃ℓ−1], both decreasing
+as the level ℓ increases, but with the corresponding computational cost per sample
+increasing.
+If �𝑌ℓ has expected value E[ �𝑃ℓ−�𝑃ℓ−1], with variance 𝑉ℓ and cost 𝐶ℓ, then for a
+given target RMS error 𝜀, the number of levels 𝐿 and the number of independent
+samples on each level can be optimised [13, 14] to give an overall cost which is
+approximately equal to 𝜀−2
+� 𝐿
+∑︁
+ℓ=0
+√︁
+𝐶ℓ𝑉ℓ
+�2
+.
+Michael B. Giles
+University of Oxford Mathematical Institute, Woodstock Rd, Oxford OX2 6GG, UK
+e-mail: [email protected]
+1
+arXiv:2301.02882v1  [math.NA]  7 Jan 2023
+
+2
+Michael B. Giles
+If 𝐶ℓ𝑉ℓ → 0 as ℓ → ∞, then the cost is dominated by the first term from level 0,
+and the cost is approximately 𝜀−2𝐶0𝑉0, so proportional to 𝜀−2.
+If 𝐶ℓ𝑉ℓ → const as ℓ → ∞, then the contributions to the cost are spread almost
+equally across all levels and the cost is approximately 𝜀−2𝐿2𝐶𝐿𝑉𝐿, proportional to
+𝜀−2| log 𝜀|2 if E[ �𝑃ℓ−𝑃] decays exponentially with ℓ.
+Even worse, if 𝐶ℓ𝑉ℓ → ∞ then the cost is dominated by the contribution from the
+finest level and so is approximately 𝜀−2𝐶𝐿𝑉𝐿 which is 𝑂(𝜀−2−(𝛾−𝛽)/𝛼) if E[ �𝑃ℓ−𝑃] ∼
+2−𝛼ℓ, 𝑉ℓ ∼ 2−𝛽ℓ and 𝐶ℓ ∼ 2𝛾ℓ.
+In most MLMC applications, 𝑃 is a smooth function of some intermediate solution
+quantities, such as the solution of an SDE, a PDE with stochastic coefficients or
+initial/boundary data, or an estimate of an inner conditional expectation. Under
+these circumstances we usually have 𝛽 ≥ 𝛾 and so the MLMC complexity is 𝑂(𝜀−2)
+or 𝑂(𝜀−2| log 𝜀|2).
+This article is concerned with the small but important class of applications where
+𝑃 is a discontinuous function of the intermediate quantities, and because of this the
+MLMC variance 𝑉ℓ can decay much more slowly, leading to the complexity falling
+into the third category of being 𝑂(𝜀−2−(𝛾−𝛽)/𝛼).
+The good news is that there has been considerable research within the MLMC
+community to address this challenge. This article surveys the variety of methods
+which have been developed in the hope that this can aid researchers meeting similar
+challenges in future applications.
+To illustrate things, we begin by detailing two specific application challenges
+which have motivated much of this research. We then discuss the many approaches
+which have been developed, several of which have borrowed ideas from the literature
+on computing sensitivities (the “greeks” in mathematical finance literature) of the
+form
+𝜕
+𝜕𝛼E [ 𝑓 (𝜔, 𝛼)]
+using the pathwise sensitivity approach [24] (also known as Infinitesimal Perturba-
+tion Analysis, IPA for short [30]) or Likelihood Ratio Method [31], or from methods
+from improving integrand smoothness to improve the rate of convergence for QMC
+integration [1, 6].
+1.1 Challenge 1: nested expectation
+Suppose 𝑓 is a scalar function and we want to estimate the nested expectation
+E [ 𝑓 (E[𝑍|𝑋]) ], where the outer expectation is with respect to a random variable
+𝑋 and we will assume that the inner conditional expectation E[𝑍|𝑋] has a bounded
+density near zero.
+A very simple MLMC treatment 1 uses 𝑀ℓ = 2ℓ𝑀0 inner samples on level ℓ, so
+estimators on level 0 and the higher levels are simply
+1 Note that if 𝑓 is smooth, or at least Lipschitz, then it is better to use an “antithetic” estimator
+[8, 14, 15, 18], but this does not give a better order of convergence when 𝑓 is discontinuous.
+
+MLMC techniques for discontinuous functions
+3
+�𝑌0 = 𝑓 (𝑍
+(0,𝑀0)),
+�𝑌ℓ = 𝑓 (𝑍
+(ℓ,𝑀ℓ)) − 𝑓 (𝑍
+(ℓ,𝑀ℓ−1)),
+where 𝑍
+(ℓ,𝑀ℓ) and 𝑍
+(ℓ,𝑀ℓ−1) represent independent averages of 𝑀ℓ and 𝑀ℓ−1 inde-
+pendent samples of 𝑍, all conditional on the same value of 𝑋 [14, 15].
+If V[𝑍|𝑋] is finite, and 𝑓 is Lipschitz with constant 𝐿 𝑓 , then
+E
+��
+𝑓 (𝑍
+(ℓ,𝑀ℓ)) − 𝑓 (E[𝑍|𝑋])
+�2
+| 𝑋
+�
+≤ 𝐿2
+𝑓 E
+��
+𝑍
+(ℓ,𝑀ℓ) − E[𝑍|𝑋]
+�2
+| 𝑋
+�
+= 𝐿2
+𝑓 𝑀−1
+ℓ V[𝑍|𝑋],
+and hence E[�𝑌2
+ℓ |𝑋] ≤ 4 𝐿2
+𝑓 (𝑀−1
+ℓ
++ 𝑀−1
+ℓ−1)V[𝑍|𝑋] for ℓ>0. If V[𝑍|𝑋] is uniformly
+bounded it follows that 𝑉ℓ = 𝑂(𝑀−1
+ℓ ). If the cost of each conditional sample of 𝑍 is
+𝑂(1) then 𝐶ℓ = 𝑂(𝑀ℓ) and hence the complexity is 𝑂(𝜀−2| log 𝜀|2).
+Unfortunately, the situation is significantly poorer when 𝑓 is the Heaviside step
+function 𝐻 defined by 𝐻(𝑥)=0 if 𝑥<0, and 𝐻(𝑥)=1 if 𝑥≥0. This occurs in many
+applications because P [E[𝑍|𝑋] > 𝐾] = E [𝐻(E[𝑍|𝑋] − 𝐾)] , so it corresponds to
+the probability of a conditional expectation exceeding some threshold 𝐾, which is a
+very important quantity in risk calculations.
+If 𝐾=0 and 𝐸[𝑍|𝑋] has a bounded density near zero then there is an 𝑂(𝑀−1/2
+ℓ
+)
+probability that |𝐸[𝑍|𝑋] | = 𝑂(𝑀−1/2
+ℓ
+), which is the circumstance under which there
+is an 𝑂(1) probability that �𝑌ℓ = ±1 due to 𝑍
+(ℓ,𝑀ℓ) being positive and 𝑍
+(ℓ,𝑀ℓ−1) being
+negative, or vice versa. Hence 𝑉ℓ ≈ 𝑂(𝑀−1/2
+ℓ
+) and the complexity is approximately
+𝑂(𝜀−5/2) [19].
+This challenge is the primary motivation for Section 7, and also arises in the
+context of Section 3.
+1.2 Challenge 2: discontinuous payoff function
+In the case of a scalar SDE
+d𝑆𝑡 = 𝑎(𝑆𝑡) d𝑡 + 𝑏(𝑆𝑡) d𝑊𝑡,
+(1)
+with an output quantity of interest 𝑃 ≡ 𝑓 (𝑆𝑇 ), the standard estimator is
+�𝑌ℓ = �𝑃ℓ − �𝑃ℓ−1
+where the same Brownian motion 𝑊𝑡 is used to calculate both �𝑃ℓ and �𝑃ℓ−1, but with
+different uniform timesteps ℎℓ and ℎℓ−1.
+If 𝑓 is Lipschitz with constant 𝐿 𝑓 , then
+𝑉ℓ ≤ E
+�
+( �𝑃ℓ − �𝑃ℓ−1)2�
+≤ 𝐿2
+𝑓 E
+�
+(�𝑆ℓ − �𝑆ℓ−1)2�
+
+4
+Michael B. Giles
+where �𝑆ℓ is the level ℓ numerical approximation to 𝑆𝑇 . Hence, based on standard
+strong convergence results [28] we have 𝑉ℓ = 𝑂(ℎℓ) for an Euler-Maruyama dis-
+cretisation of the SDE, and 𝑉ℓ = 𝑂(ℎ2
+ℓ) for the first order Milstein discretisation.
+The cost 𝐶ℓ is 𝑂(ℎ−1
+ℓ ) in both cases, giving MLMC complexities of 𝑂(𝜀−2| log 𝜀|2)
+and 𝑂(𝜀−2), respectively,
+In mathematical finance, a digital call option payoff is 0 or 1, depending on
+whether 𝑆𝑇 is below or above the strike 𝐾, so the payoff function can be written
+as 𝑓 (𝑆𝑇 ) = 𝐻(𝑆𝑇 −𝐾). The MLMC problem is that a small difference between the
+coarse and fine paths can give a payoff difference of ±1 if the two paths straddle the
+strike, i.e. are on different sides of the strike.
+When using the Euler-Maruyama approximation of the SDE, �𝑆ℓ−�𝑆ℓ−1 = 𝑂(ℎ1/2
+ℓ ).
+Speaking loosely (see [4, 21] for the rigorous analysis) an 𝑂(ℎ1/2
+ℓ ) fraction of
+fine/coarse pairs straddle the strike, so 𝑉ℓ = 𝑂(ℎ1/2
+ℓ ) and hence the complexity is
+𝑂(𝜀−5/2).
+Similarly, using the Milstein approximation gives �𝑆ℓ−�𝑆ℓ−1 = 𝑂(ℎℓ) so 𝑉ℓ =
+𝑂(ℎℓ). This is clearly better, and gives a complexity which is 𝑂(𝜀−2| log 𝜀|2), but
+there is still the problem that most MLMC samples 𝑌ℓ are zero on the finer levels, so
+the kurtosis is 𝑂(ℎ−1
+ℓ ) which causes problems in practice in estimating 𝑉ℓ accurately
+to determine the number of samples 𝑁ℓ to use on level ℓ. In addition, there is the
+difficulty that the Milstein discretisation of multi-dimensional SDEs often requires
+the simulation of Lévy areas, though this problem can be addressed through the use
+of an antithetic estimator [23].
+This challenge is the primary motivation for Sections 2, 4, 5 and 6, also also arises
+in Sections 3 and 7.
+2 Explicit smoothing
+The pathwise sensitivity analysis (or IPA) approach to compute the parameter sen-
+sitivities known as Greeks in mathematical finance [24] requires that the payoff
+function 𝑓 is continuous and piecewise smooth. This is clearly a problem with digi-
+tal options, and one standard approach is to smooth the payoff function by replacing
+the Heaviside step function 𝐻 with a smoothed approximation 𝐻𝛿(𝑥) ≡ 𝑔(𝑥/𝛿),
+with 𝑔(𝑥) → 0 as 𝑥 → −∞ and 𝑔(𝑥) → 1 as 𝑥 → +∞, so the discontinuity is
+smoothed over a width of 𝑂(𝛿).
+For financial reasons, the preference is often to use a one-sided smoothing, such
+as the piecewise linear approximation shown in yellow in Figure 1. This one-sided
+approximation introduces a weak error, or bias, which is 𝑂(𝛿). If it is used for
+MLMC, then 𝐻′
+𝛿(𝑆𝑇 ) = 𝛿−1 for the 𝑂(𝛿) fraction of the paths which end up in the
+ramp region, and therefore 𝑉ℓ = 𝑂(𝛿 × (𝛿−1)2) = 𝑂(𝛿−1). Hence the optimal choice
+of 𝛿 involves a tradeoff between bias and variance.
+The bias can be reduced by making the smoothing anti-symmetric about 𝑥 = 0 so
+that 𝐻𝛿(𝑥) − 𝐻(𝑥) = −(𝐻𝛿(−𝑥) − 𝐻(−𝑥)), for example by choosing 𝑔(𝑥) ≡ Φ(𝑥)
+
+MLMC techniques for discontinuous functions
+5
+as illustrated in orange in Figure 1. If 𝑆𝑇 has the smooth probability density 𝜌(𝑆)
+then the weak error is
+∫ ∞
+−∞
+(𝐻𝛿(𝑆−𝐾) − 𝐻(𝑆−𝐾)) 𝜌(𝑆) d𝑆 = 𝛿
+∫ ∞
+−∞
+(𝑔(𝑥) − 𝐻(𝑥)) 𝜌(𝐾+𝑥𝛿) d𝑥
+and a Taylor series expansion of 𝜌(𝐾+𝑥𝛿) results in the asymptotic error expansion
+𝑎1𝜌(𝐾) 𝛿 + 𝑎2𝜌′(𝐾) 𝛿2 + 𝑎3𝜌′′(𝐾) 𝛿3 + 𝑎4𝜌′′′(𝐾) 𝛿4 + 𝑂(𝛿5)
+where
+𝑎𝑘 =
+∫ ∞
+−∞
+𝑥𝑘−1 (𝑔(𝑥) − 𝐻(𝑥)) d𝑥.
+If 𝑔(𝑥) − 𝐻(𝑥) = − (𝑔(−𝑥) − 𝐻(−𝑥)) then 𝑎1 = 𝑎3 = 0, and
+𝑎2 = 2
+∫ ∞
+0
+𝑥(𝑔(𝑥) − 1) d𝑥,
+𝑎4 = 2
+∫ ∞
+0
+𝑥3(𝑔(𝑥) − 1) d𝑥.
+If 𝑔(𝑥) is monotonic, then 𝑎2 ≠ 0, but by considering non-monotonic functions
+such as 𝑔(𝑥) = (4/3) Φ(𝑥) − (1/3) Φ(2𝑥) it is possible to set 𝑎2 = 0 making the
+weak error 𝑂(𝛿4). Hence, to achieve 𝑂(𝜀) accuracy overall we need 𝛿=𝑂(𝜀1/4), and
+then on the coarsest levels 𝑉ℓ = 𝑂(𝛿−1) = 𝑂(𝜀−1/4) so the overall complexity is
+approximately 𝑂(𝜀−2−1/4) in the best cases where the overall cost is dominated by
+the cost on the coarsest levels.
+Giles, Nagapetyan & Ritter [22] used explicit smoothing for estimating cumulative
+distribution functions (CDFs). For a scalar random variable 𝑋, to estimate 𝐶(𝑥) =
+P(𝑋 < 𝑥) = E[𝐻(𝑥−𝑋)], the approach they adopted was to use MLMC to estimate
+𝐶(𝑥 𝑗) for a set of spline points 𝑥 𝑗 and then interpolate these values with a cubic
+spline. Overall, their method balanced three weak errors, the SDE discretisation error
+on the finest level, the smoothing error due to 𝐻𝛿, and the cubic spline interpolation
+error, in addition to the MLMC sampling error.
+0.5
+1
+1.5
+ST
+0
+0.2
+0.4
+0.6
+0.8
+1
+f(S T)
+Fig. 1 Two explicitly smoothed versions of the Heaviside step function for a digital call option
+
+6
+Michael B. Giles
+3 Integration/differentiation and Malliavin calculus
+Krumscheid & Nobile [29] used a slightly different approach for estimating CDFs,
+particularly in the context of risk estimation. Starting from the identity
+d
+d𝑥 E [ max(0, 𝑥−𝑆𝑇 ) ] = E[ 𝐻(𝑥−𝑆𝑇 ) ]
+they used MLMC to estimate E[ max(0, 𝑥 𝑗−𝑆𝑇 ) ] for a set of spline points 𝑥 𝑗,
+interpolated these with a cubic spline, and and then differentiated the spline to
+obtain an approximation to the CDF 𝐶(𝑥). This avoids the extra weak error due to
+smoothing the Heaviside function, but differentiating the cubic spline amplifies the
+noise in the spline data.
+On a similar note, Altmayer & Neuenkirch [2] used Malliavin calculus integration
+by parts to treat discontinuous payoffs based on solutions of the Heston stochastic
+volatility SDE. They observed that asymptotically this improves the MLMC vari-
+ance on the finer levels, but it increases the variance on coarse levels. To address
+this, they split the payoff into a smooth part which they treated with the standard
+MLMC approach, and a compact-support discontinuous part for which they used the
+Malliavin MLMC.
+Malliavin calculus was originally developed for computing sensitivities, so this
+is another example of the literature on sensitivity calculations being exploited to
+develop improved MLMC algorithms.
+4 Conditional expectation
+When using the first order Milstein discretisation for an SDE, one way to improve
+the MLMC variance for digital options is to switch to the Euler-Maruyama approxi-
+mation for the final timestep, and then take the conditional expectation with respect
+to the final fine path Brownian increment Δ𝑊 [12, 17].
+For the fine path approximation of the scalar SDE (1) with 𝑁 timesteps of size
+ℎℓ, the path value 𝑆𝑇 at the final time 𝑇 is given by
+�𝑆 𝑓
+𝑇 = �𝑆 𝑓
+𝑇 −ℎℓ + 𝑎(�𝑆 𝑓
+𝑇 −ℎℓ) ℎℓ + 𝑏(�𝑆 𝑓
+𝑇 −ℎℓ) Δ𝑊𝑁 ,
+and therefore the conditional expected value for the digital call option is
+�𝑃 𝑓
+ℓ
+= E
+�
+𝐻(�𝑆 𝑓
+𝑇 −𝐾) | �𝑆 𝑓
+𝑇 −ℎℓ
+�
+= Φ ��
+�
+�𝑆 𝑓
+𝑇 −ℎℓ + 𝑎(�𝑆 𝑓
+𝑇 −ℎℓ) ℎℓ − 𝐾
+𝑏(�𝑆 𝑓
+𝑇 −ℎℓ) √ℎℓ
+��
+�
+.
+Similarly, for the coarse path with coarse timestep ℎℓ−1 = 2 ℎℓ, the Brownian incre-
+ment for the final coarse timestep is the sum of the last two Brownian increments for
+the fine path, Δ𝑊𝑁 −1+Δ𝑊𝑁 , and therefore
+
+MLMC techniques for discontinuous functions
+7
+�𝑆𝑐
+𝑇 = �𝑆𝑐
+𝑇 −ℎℓ−1 + 𝑎(�𝑆𝑐
+𝑇 −ℎℓ−1) ℎℓ−1 + 𝑏(�𝑆𝑐
+𝑇 −ℎℓ−1) (Δ𝑊𝑁 −1+Δ𝑊𝑁 ) ,
+from which we obtain
+�𝑃𝑐
+ℓ−1 = E
+�
+𝐻(�𝑆𝑐
+𝑇 −𝐾) | �𝑆𝑐
+𝑇 −ℎℓ−1, Δ𝑊𝑁 −1
+�
+= Φ
+� �𝑆𝑐
+𝑇 −ℎℓ−1 + 𝑎(�𝑆𝑐
+𝑇 −ℎℓ−1) ℎℓ−1 + 𝑏(�𝑆𝑐
+𝑇 −ℎℓ−1) Δ𝑊𝑁 −1 − 𝐾
+𝑏(�𝑆𝑐
+𝑇 −ℎℓ−1) √ℎℓ
+�
+.
+With �𝑌ℓ ≡ �𝑃ℓ−�𝑃ℓ−1, numerical analysis [17] proves that 𝑉ℓ ≈ 𝑂(ℎ3/2
+ℓ ) so the
+MLMC complexity is 𝑂(𝜀−2). Heuristically, this is because there is an 𝑂(ℎ1/2
+ℓ )
+probability of paths being within 𝑂(ℎ1/2
+ℓ ) of the strike 𝐾, and for these
+�𝑆 𝑓
+𝑇 −ℎℓ−1 − �𝑆𝑐
+𝑇 −ℎℓ−1 = 𝑂(ℎℓ),
+𝜕 �𝑃
+𝜕�𝑆
+= 𝑂(ℎ−1/2
+ℓ
+)
+=⇒
+�𝑃ℓ − �𝑃ℓ−1 = 𝑂(ℎ1/2
+ℓ ),
+so 𝑉ℓ ≈ 𝑂(ℎ1/2
+ℓ
+× (ℎ1/2
+ℓ )2) = 𝑂(ℎ3/2
+ℓ ). In addition, the kurtosis is improved to
+𝑂(ℎ−1/2
+ℓ
+).
+Unfortunately, the conditional expectation approach does not help when the Euler-
+Maruyama discretisation is used for the entire path since �𝑆 𝑓
+𝑇 −ℎℓ−1−�𝑆𝑐
+𝑇 −ℎℓ−1 = 𝑂(ℎ1/2
+ℓ )
+and so �𝑃ℓ − �𝑃ℓ−1 = 𝑂(1)
+The use of this kind of conditional expectation is a standard technique for smooth-
+ing the payoff to enable IPA/pathwise sensitivity calculations [24]. Another example
+is a down-and-out barrier option, where the option is knocked out if the path drops
+below a certain value. In this case the payoff can be smoothed by computing the
+probability of this happening, conditional on the computed path approximations at
+discrete timesteps [24]. Again, this works well for MLMC when using the first or-
+der Milstein discretisation [12, 17], but it does not help with the Euler-Maruyama
+discretisation.
+A different kind of conditional expectation smoothing was introduced by Achtsis,
+Cools & Nuyens [1] and Bayer, Siebenmorgen & Tempone [6] to improve the
+convergence of QMC computations, and then used by Bayer, Ben Hammouda &
+Tempone [5] to improve the MLMC variance for digital options.
+In its simplest form, they split the random inputs for the numerical simulation into
+a scalar 𝑍 and the remainder 𝑍𝑟, and express the desired MLMC level ℓ expectation
+as
+E[ �𝑃ℓ−�𝑃ℓ−1] = E
+�
+E[ �𝑃ℓ−�𝑃ℓ−1 | 𝑍𝑟]
+�
+and observe that in many financial applications it is possible to perform this split
+in a way such that the conditional expectations E[ �𝑃ℓ | 𝑍𝑟], E[ �𝑃ℓ−1 | 𝑍𝑟] are smooth
+functions of 𝑍𝑟, and can be evaluated analytically or very accurately by 1D numerical
+quadrature when there is just a single discontinuity with respect to changes in 𝑍.
+
+8
+Michael B. Giles
+For a scalar SDE, 𝑍 could be the terminal value of the driving Brownian motion,
+in which case 𝑍𝑟 would represent the other Normal random variables required for a
+Brownian Bridge construction of the Brownian increments.
+5 Change of measure
+Another approach to treating digital options using the Milstein discretisation is to
+use a change of measure [9, 14], which has connections to the Likelihood Ratio
+Method (LRM) that is used for sensitivity analysis [31].
+For both the fine and coarse paths, we have conditional Gaussian distributions for
+�𝑆𝑇 , with slightly different means and variances, as illustrated in Figure 2. We can
+therefore perform a change of measure to the same Gaussian distribution with mean
+𝜇 and variance 𝜎2, also illustrated in Figure 2, and then pick the same sample �𝑆𝑇
+for both paths from this common Gaussian distribution.
+The resulting MLMC estimator is
+�𝑌ℓ = 𝑓 (�𝑆𝑇 ) (𝑅ℓ − 𝑅ℓ−1)
+where 𝑅ℓ, 𝑅ℓ−1 are the respective Radon-Nikodym derivatives for the fine and coarse
+paths. For the scalar SDE (1), 𝑅ℓ is
+𝑅ℓ =
+𝜎
+𝑏(�𝑆 𝑓
+𝑇 −ℎℓ)√ℎℓ
+exp
+�
+− (�𝑆𝑇 − �𝑆 𝑓
+𝑇 −ℎℓ − 𝑎(�𝑆 𝑓
+𝑇 −ℎℓ) ℎℓ)2 / (2 𝑏2(�𝑆 𝑓
+𝑇 −ℎℓ) ℎℓ)
+�
+exp
+�
+− (�𝑆𝑇 −𝜇)2 / (2𝜎2)
+�
+and 𝑅ℓ−1 is defined similarly. It can be shown that the difference 𝑅ℓ − 𝑅ℓ−1 is
+approximately 𝑂(ℎ1/2
+ℓ ), which implies that 𝑉ℓ ≈ 𝑂(ℎℓ). To improve the variance we
+note that the conditional expected value of Radon-Nikodym derivatives is always 1,
+0.6
+0.8
+1
+1.2
+1.4
+ST
+0
+0.1
+0.2
+0.3
+0.4
+p(ST)
+Fig. 2 Coarse and fine path conditional Gaussian distributions, plus third common distribution
+
+MLMC techniques for discontinuous functions
+9
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+t
+1
+1.5
+2
+S
+Fig. 3 Illustration of coarse and fine jump-diffusion paths with jumps before and after 𝑇 = 1.
+i.e. E[𝑅ℓ | �𝑆 𝑓
+𝑇 −ℎℓ] = E[𝑅ℓ−1 | �𝑆𝑐
+𝑇 −ℎℓ−1, Δ𝑊𝑁 −1] = 1, and therefore we can change
+the definition of �𝑌ℓ to
+�𝑌ℓ =
+�
+𝑓 (�𝑆𝑇 ) − 𝑓 (𝜇)
+�
+(𝑅ℓ − 𝑅ℓ−1)
+without changing its expected value. This estimator is now non-zero only when �𝑆𝑇
+and 𝜇 are on opposite sides of the strike 𝐾, which occurs for an 𝑂(ℎ1/2
+ℓ ) fraction of
+coarse/fine paths. Hence the new MLMC variance 𝑉ℓ is approximately 𝑂(ℎ3/2
+ℓ ), as
+with the use of the analytic conditional expectation.
+The benefit of this approach is that it works well in multiple dimensions when it is
+often not possible to evaluate the analytic conditional expectation [9, 14]. However,
+again it does not help with the full path Euler-Maruyama discretisation because that
+gives 𝑅ℓ − 𝑅ℓ−1 = 𝑂(1).
+An earlier use of a change of measure in an MLMC computation was by Xia
+[33, 34] for a Merton-style jump-diffusion SDE with a path-dependent jump rate
+𝜆(𝑆, 𝑡). The challenge in this application, as illustrated in Figure 3, is that the
+coarse and fine paths will jump at different times, and one might jump just before
+the final time 𝑇, and the other just after, leading to a large jump in the computed
+value of 𝑓 (𝑆𝑇 ). The path-dependent jump rate was treated by using the thinning
+technique of Glasserman & Merener [25], over-sampling possible jump times using
+a uniform rate 𝜆𝑠𝑢𝑝 > 𝜆(𝑆, 𝑡) and then using an acceptance/rejection step to select
+the real jump times. Xia modified this with a change of measure to ensure the same
+acceptance/rejection decision for both the fine and coarse paths so that they both
+jump at the same time. This leads to an estimator of the form
+�𝑌ℓ = �𝑃ℓ 𝑅ℓ − �𝑃ℓ−1 𝑅ℓ−1.
+
+10
+Michael B. Giles
+When combined with a first order Milstein discretisation of the SDE between the
+jump times, this gives 𝑉ℓ = 𝑂(ℎ2
+ℓ) for Lipschitz payoff functions such as a standard
+put or call option [33, 34].
+6 Splitting
+Returning to the challenge of digital options arising from the solution of an SDE,
+a third approach is to use path-splitting to generate an unbiased estimate of the
+conditional expectation introduced in Section 4 [14].
+This is a variant of the general splitting technique [3]. As illustrated in Figure 4,
+it involves performing a standard fine path simulation up until one timestep before
+the final time 𝑇, and then performing multiple independent simulations of the final
+timestep, averaging the payoff for each of these to get an approximation of the
+conditional expectation. The same is done for the coarse path except that each of the
+splits uses the same Δ𝑊𝑁 −1 that was used for the second to last fine path timestep.
+Since the computational cost of the path up to the splitting time is 𝑂(ℎ−1
+ℓ ), it means
+that up to 𝑂(ℎ−1
+ℓ ) splits can be used without increasing the path cost significantly. If
+𝑀ℓ splits are used, then the standard splitting variance analysis gives
+V[�𝑌ℓ] = V
+�
+E[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]
+�
++ 𝑀−1
+ℓ E
+�
+V[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]
+�
+.
+As discussed previously V
+�
+E[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]
+�
+= 𝑂(ℎ3/2
+ℓ ), and similarly it
+can be argued that E
+�
+V[ �𝑃ℓ−�𝑃ℓ−1 | {Δ𝑊𝑛}𝑛<𝑁 ]
+�
+= 𝑂(ℎℓ). Therefore choosing 𝑀ℓ
+to lie between 𝑂(ℎ−1
+ℓ ) and 𝑂(ℎ−1/2
+ℓ
+) ensures the benefits of the splitting are obtained
+without significantly increasing the computational cost per sample.
+0
+0.2
+0.4
+0.6
+0.8
+1
+t
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+St
+Fig. 4 Path splitting in final timestep to estimate conditional expectation
+
+MLMC techniques for discontinuous functions
+11
+As an additional bonus, one can use the more accurate Milstein discretisation
+for the final timestep, instead of switching to the Euler-Maruyama discretisation.
+Burgos [9, 10] gives more details of the analysis, and also used the same approach
+for pathwise sensitivity analysis for a variety of financial options.
+Giles & Bernal [16] also used splitting for Feynman-Kac functionals arising for
+stopped diffusions, SDE simulations which terminate when the solution path leaves
+a prescribed domain. The issue here is that when a fine path exits, there is an 𝑂(ℎ1/2
+ℓ )
+probability that the corresponding coarse path does not leave until much later. This
+is addressed by estimating a conditional expectation by splitting the coarse path into
+𝑂(ℎ−1/2
+ℓ
+) independent sub-simulations which continue until each of them leaves the
+domain. 𝑉ℓ is improved from 𝑂(ℎ1/2
+ℓ ) to approximately 𝑂(ℎℓ) without a significant
+increase in the cost per sample, and finally the MLMC complexity achieved is
+𝑂(𝜀−2| log 𝜀|3).
+None of the three methods introduced so far (conditional expectation, change of
+measure, splitting) helps when using the Euler-Maruyama discretisation. For this, a
+new method has recently been developed by Giles & Haji-Ali [20].
+It again uses splitting, but inspired by the simulation of branching diffusions, it
+considers splits at multiple deterministic times, as illustrated in Figure 5 which shows
+the logical structure of a set of split paths. Here we are considering a simulation
+on the unit time interval. A single pair of fine/coarse paths is calculated up to time
+𝑡 = 1/2, with the number of fine timesteps being 1
+2 ℎ−1
+ℓ . This simulation is then
+split into two separate independent simulations up to time 𝑡 = 3/4, with the two
+simulations between them accounting for an additional 1
+2 ℎ−1
+ℓ
+fine timesteps. There
+are further splits at 𝑡 = 3/4, then at 𝑡 = 7/8, and so on, with the final split when there
+is just one coarse timestep left.
+The total number of fine timesteps simulated is 𝑂(ℎ−1
+ℓ | log ℎℓ|) so the computa-
+tional cost is only slightly increased compared to the original method with a single
+pair of fine/coarse paths. �𝑌ℓ is defined to be the average of the values �𝑃ℓ−�𝑃ℓ−1 for
+each of the final paths, and it can be proved that its variance is 𝑂(ℎℓ), the same
+0
+0.2
+0.4
+0.6
+0.8
+1
+t
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+Fig. 5 Repeated path splitting to estimate conditional expectation
+
+12
+Michael B. Giles
+asymptotic order of convergence as for Lipschitz payoff functions [20]. The kurtosis
+is also improved, so this technique fully addresses the challenge of using MLMC
+with the Euler-Maruyama discretisation to estimate digital option values.
+7 Adaptive sampling
+We return now to the challenge of estimating the nested expectation E [ 𝐻 (E[𝑍|𝑋]) ]
+and we note that we only need an accurate estimate of the inner conditional expecta-
+tion E[𝑍|𝑋] when it is near zero. This observation is the basis for the development
+of adaptive sampling by Broadie, Du & Moallemi [7] within a standard Monte Carlo
+procedure. This was then extended to adaptive sampling combined with MLMC by
+Giles & Haji-Ali [19] by defining the number of inner samples 𝑀ℓ on level ℓ to be
+•
+𝑀ℓ = 2ℓ𝑀0 inner samples when |E[𝑍|𝑋]| ≫
+√︁
+V[𝑍|𝑋]/(2ℓ𝑀0)
+This is the smallest number of samples used on level ℓ.
+√︁
+V[𝑍|𝑋]/(2ℓ𝑀0) is
+the standard deviation of the Monte Carlo estimate for E[𝑍|𝑋], so the inequality
+means that this number of samples is sufficient to be very sure that the estimate
+has the correct sign.
+•
+𝑀ℓ = 4ℓ𝑀0 inner samples when |E[𝑍|𝑋]| = 𝑂(
+√︁
+V[𝑍|𝑋]/(4ℓ𝑀0))
+This is the maximum number of samples used on level ℓ. In this case, the estimate
+of E[𝑍|𝑋] may have the incorrect sign, but this will only happen when |E[𝑍|𝑋]| =
+𝑂(2−ℓ) which occurs with probability 𝑂(2−ℓ). Likewise, the total cost of the
+higher number of samples in this region is 𝑂(2−ℓ × 4ℓ) = 𝑂(2ℓ), so it does not
+significantly increase the overall average cost.
+•
+2ℓ𝑀0 < 𝑀ℓ < 4ℓ𝑀0 for intermediate values
+In this region the number of samples is chosen to be very sure that the estimate
+of E[𝑍|𝑋] has the correct sign, and at the same time the total cost is 𝑂(2ℓ).
+Overall, this adaptive sampling approach leads to 𝐶ℓ ∼ 2ℓ, 𝑉ℓ ∼ 2−ℓ and hence
+a complexity of roughly 𝑂(𝜀−2) [19]. However, the kurtosis is 𝑂(2ℓ) since only an
+𝑂(2−ℓ) fraction of the outer samples give non-zero values for �𝑌ℓ.
+Haji-Ali, Spence & Teckentrup [27] have further extended this to estimate quan-
+tities of the form
+P[𝐺 ∈ Ω] ≡ E[1𝐺∈Ω]
+where 𝐺 is a 𝑑-dimensional random variable which cannot be sampled directly.
+In their paper they consider in particular the two challenges in this article. In the
+context of the digital option with the Euler-Maruyama discretisation on the unit time
+interval, the adaptive sampling varies the timestep used on level ℓ so that
+•
+ℎℓ = 2−ℓ when |�𝑆ℓ − 𝐾| is large compared to the strong error in the path approx-
+imation
+•
+ℎℓ = 4−ℓ when |�𝑆ℓ − 𝐾| is of the same order as the strong error
+
+MLMC techniques for discontinuous functions
+13
+•
+2−ℓ < ℎℓ < 4−ℓ for intermediate values
+A Brownian bridge construction is used when the timestep needs to be refined as
+part of the adaptation procedure from its initial value ℎℓ = 2−ℓ. The adaptation again
+leads to 𝐶ℓ ∼ 2ℓ, 𝑉ℓ ∼ 2−ℓ and hence a complexity of roughly 𝑂(𝜀−2), but there is
+again a high kurtosis [27].
+In earlier research, Elfverson, Hellman & Malqvist [11] considered estimation of
+E[𝐻(𝑋)] where 𝑋 cannot be sampled exactly but there is a sequence of approxi-
+mations 𝑋′
+0, 𝑋′
+1, 𝑋′
+2, . . . 𝑋 of increasing accuracy and increasing cost. Motivated by
+PDE applications with a well-behaved truncation error so that there are uniform
+geometric bounds on |𝑋′
+𝑗 − 𝑋|, level ℓ in their method uses
+�𝑋ℓ = 𝑋′
+𝑗,
+𝑗 = min{ℓ, min 𝑗 : | �𝑋′
+𝑗 − 𝑋| < |𝑋|}
+and achieves similarly good MLMC benefits. This idea is essentially the same as in
+the work of Haji-Ali et al but requiring a uniform bound on |𝑋′
+𝑗−𝑋| is significantly
+more restrictive than the bounds on E[ |𝑋′
+𝑗−𝑋|𝑞] for some 𝑞>2 required by Haji-Ali
+et al.
+A final comment is that the analysis of Haji-Ali, Spence & Teckentrup can be gen-
+eralised to a product of an indicator function and a Lipschitz function, E[1𝐺∈Ω 𝑓 (𝑆)],
+and so can handle barrier options. Furthermore, Haji-Ali & Spence have extended
+the adaptive sampling methodology to an extremely challenging triply-nested expec-
+tation which arises in mathematical finance [26]. By incorporating the randomised
+MLMC treatment of Rhee & Glynn [32] to handle the time discretisation of the
+underlying SDEs as well as the sampling for the inner conditional expectations, they
+achieve an overall complexity of approximately 𝑂(𝜀−2) which is very impressive for
+such a difficult application.
+-10
+-5
+0
+5
+10
+E[Z|X]
+0
+0.5
+1
+1.5
+2
+2.5
+Fig. 6 Error distributions for two conditional expectations with i) few samples being needed to
+ensure the correct sign (left), and ii) many samples being insufficient to ensure the correct sign
+(centre). The blue line represents the Heaviside step function.
+
+14
+Michael B. Giles
+8 Conclusions
+It is worth repeating that in most MLMC applications the output quantity of interest
+is a Lipschitz function of the intermediate simulation quantities, so good strong
+convergence for the intermediate quantities leads automatically to a good rate of
+convergence of the MLMC variance 𝑉ℓ.
+For those applications in which the function is discontinuous, this article shows
+there is an extensive literature with a variety of different approaches to improve the
+MLMC variance and try to recover the optimal 𝑂(𝜀−2) complexity. It is notable that
+many of these methods have adapted ideas from Monte Carlo sensitivity analysis
+which also has problems with discontinuous functionals. It is hoped that this survey
+will assist future researchers facing similar challenges in other new application areas.
+Acknowledgements This paper is based on research with many students, postdocs and other
+collaborators and I am grateful to all of them. Funding for the research has been provided by the UK
+Engineering and Physical Sciences Research Council through grants EP/E031455/1, EP/H05183X/1
+and EP/P020720/2 as well as the Hong Kong Innovation and Technology Commission (InnoHK
+Project CIMDA). The paper was written while visiting the Oden Institute at UT Austin, and I thank
+my hosts for their warm hospitality.
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+

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+arXiv:2301.04457v1  [gr-qc]  11 Jan 2023
+Ghost and Laplacian Instabilities in Teleparallel Horndeski Gravity
+Salvatore Capozziello,1, 2, 3, ∗ Maria Caruana,4, 5, † Jackson Levi Said,4, 5, ‡ and Joseph Sultana6, §
+1Dipartimento di Fisica ”E. Pancini”, Universit`a degli Studi di Napoli,
+”Federico II”, Complesso Universitario Monte S. Angelo,
+Via Cinthia 9 Edificio G, 80126 Napoli, Italy
+2Istituto Nazionale di Fisica Nucleare (INFN),
+Sezione di Napoli Complesso Universitario Monte S. Angelo,
+Via Cinthia 9 Edificio G, 80126 Napoli, Italy
+3Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy
+4Institute of Space Sciences and Astronomy, University of Malta, Msida, Malta
+5Department of Physics, University of Malta, Msida, Malta
+6Department of Mathematics, University of Malta, Msida, Malta
+Teleparallel geometry offers a platform on which to build up theories of gravity where torsion
+rather than curvature mediates gravitational interaction. The teleparallel analogue of Horndeski
+gravity is an approach to teleparallel geometry where scalar-tensor theories are considered in this
+torsional framework. Being teleparallel gravity of lower order in dynamics, this turns out to be more
+general than metric Horndeski gravity. In other words, the class of teleparallel Horndeski gravity
+models is much broader than the standard metric one. In this work, we explore constraints on this
+wide range of models coming from ghost and Laplacian instabilities. The aim is to limit pathological
+branches of the theory by fundamental considerations. It is possible to conclude that a very large
+class of models results physically viable.
+I.
+INTRODUCTION
+General relativity (GR) has been the gravitational foundation of cosmology for over a century. Its latest
+embodiment appears in a picture wherein the evolutionary processes of the Universe are described in the
+framework of the so-called ΛCDM model [1, 2]. Supported by overwhelming observational evidences, this
+model describes a Universe started with a big bang, then driven through an inflationary phase and other well
+known epochs to eventually produce an accelerating Universe at late-times [3, 4]. In the ΛCDM model, the
+late time acceleration expansion is driven by the cosmological constant or some form of dark energy. Despite
+a lot of foundational works, internal consistency issues persist in this respect [5–7]. Recently, observational
+challenges to the standard model has arisen in the form of cosmological tensions with statistically significant
+differences between predictions of expansion from early time data [8–10] and measurements from late time
+[11, 12]. These tensions continue to increase with new survey data [13–15], and may permeate into other
+sectors of cosmology besides expansion [16–18]. This situation leads to take into account potential alternatives
+to the standard cosmological ΛCDM model in the context of possible modifications to the gravitational sector.
+A solution to the problem of cosmic tensions, as well as of other longstanding issues, can be offered by
+alternative theories of gravity, in particular by scalar-tensor gravity. Scalar fields improve GR in view of Mach
+principle and could naturally address several issues in cosmology and astrophysics like inflation, dark matter
+and dark energy [19]. The most general scalar-tensor theory of gravity, where scalar fields are minimally and
+non-minimally coupled to curvature, is the so-called Horndeski gravity [20]. It is the most general theory
+of gravity producing second order field equations. This is advantageous because nature appears to produce,
+generally, second order equations of motion. Furthermore, higher order theories can produce Ostrogradsky
+instabilities making second order theories more attractive from a fundamental perspective [21–23]. Horndeski
+gravity is formulated by four free functions of the scalar field and its kinetic term [24, 25]. This offers a
+rich phenomenology on which to produce dark energy models but also dark matter and inflation. However,
+recent multimessenger observations by the LIGO collaboration, in the gravitational event GW170817 [26], and
+measurements of the companion electromagnetic counterpart, namely GRB170817A [27], has placed stringent
+constraints on the speed of propagation of gravitational waves (GW) to within deviations of at most one part
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
+
+2
+in 1015. Considering these constraints, many branches of Horndeski gravity have been disqualified in regular
+curvature-based gravity [28–30]. However, Horndeski gravity has a number of interesting generalizations
+[31] where higher order terms, avoiding the Ostrogradsky instabilities, can be incorporated into the theory.
+Another possibility is to reconsider the geometric foundations of curvature on which Horndeski gravity is
+built. Curvature can be expressed through the Levi-Civita connection
+◦Γ
+λ
+µν obtained from the metric [1].
+Here an over-circle represents quantities determined by the Levi-Civita connection.
+In teleparallel gravity (TG), the teleparallel connection (Γλ
+µν) replaces the Levi-Civita connection and so
+dynamics of gravity is replaced from curvature to torsion [32–35]. The teleparallel connection is curvatureless
+and satisfies metricity. For this reason, the teleparallel Ricci scalar turns out to identically vanish, i.e. R = 0
+(this is not to say that the standard Ricci scalar is zero, being, in general,
+◦R ̸= 0). On the other hand, TG
+naturally produces a torsion scalar (T ) [33], which turns out to be equal to the curvature-based Ricci scalar
+up to a boundary term (B). For this reason, the linear torsion scalar produces the Teleparallel equivalent of
+General Relativity (TEGR) which is dynamically equivalent to GR in the classical regime. This distinction
+between the torsion scalar, producing second order terms in the equations of motion, and the boundary
+term, producing fourth order terms in the equations of motion (when the boundary term is nonlinear in the
+action), gives rise to a much richer landscape of theories on which to build cosmological models, if compared
+with the standard Einstein-Hilbert action.
+Another way to conceptualize this feature is through the teleparallel analogue of the Lovelock theorem
+[36–38] which produces a much wider range of actions with second order field equations. Equivalences and
+differences of these representations are discussed in Ref. [39].
+Similar to GR, TEGR features many modifications such as f(T ) gravity [34, 40–46] which is generically
+second order in nature, as well as f(T, B) gravity [47–55] which now features fourth order contributions
+similar to f(
+◦R) gravity [19, 56–62]. There have also been various formulations of scalar–tensor gravity. See,
+e.g. Refs.[63–65].
+Equipped with the TG formalism, one can consider the teleparallel analogue of Horndeski gravity [38] where
+the generically lower order nature of TG turns out to produce a much richer structure to be developed. This
+feature directly comes from the teleparallel analogue of the Lovelock theorem. This larger framework of
+models means that there are many more classes of theories that now satisfy the speed of GWs constraint
+[66, 67] as required from the above mentioned multimessenger analysis [68].
+This class of gravitational
+models also satisfies the parameterized post-Newtonian observational constraints [69] for a large number of
+models. The teleparallel analogue of Horndeski gravity also produces a number of interesting cosmologies
+such as those which are well-tempered [70, 71], as well as those that are derived from Noether symmetry
+considerations [72–76]. In this way, all models that were previously disqualified in regular Horndeski may be
+revived in this new formalism based on TG.
+In this work, we perform a stability analysis of the teleparallel analogue of Horndeski gravity through
+considerations on a Minkowski spacetime background. The Minkowski background is an ideal arena to probe
+the fundamental behavior of theories with large classes of models since any astrophysical or cosmological
+setting must first be stable on a Minkowski spacetime. In our analysis, we consider both background and
+leading order perturbations in the context of a scalar-vector-tensor decomposition. This is important to fully
+decompose any potentially problematic part of the theory in detail. In the TG setting, this is more intricate
+since the metric tensor (gµν) is replaced as the fundamental dynamical variable of the field equations with
+the tetrad (eA
+µ) and the spin connection (ωA
+Bµ), which are discussed in detail later on. In all cases, we
+focus our study on the ghost and gradient instabilities, which can wreak havoc on gravitational models since
+this may, respectively, produce scalar field kinetic terms with the wrong sign and unbounded propagation
+speeds of particular perturbations, which can lead to unphysical models.
+The structure of the manuscript is the following: in Sec. II, we summarize TG and its teleparallel analogue
+of Horndeski gravity; this then leads to the Minkowski background equations of motion in Sec. III. Scalar–
+vector–tensor perturbations are discussed in Sec. IV. From these perturbations, we are able to explore
+potential instabilities in Sec. V where our main results are reported. Discussion and conclusions are drawn
+in Sec. VI. In this work, we use geometric units and the signature (−, +, +, +).
+
+3
+II.
+TELEPARALLEL HORNDESKI GRAVITY
+GR, a curvature-based theory, is constructed on torsionless Levi-Civita connection
+◦Γλ
+µν, where, as said
+above, the overhead circle (◦) represents quantities built with this connection.
+This leads to numerous
+theories of gravity beyond GR as the gravitational field can be expressed through the Riemann tensor and
+its contractions such as the Ricci tensor and the Riemann scalar [59], the latter produces the Einstein-
+Hilbert action [1]. On the other hand, TG incorporates the teleparallel connection, dubbed as teleparallel
+connection Γλ
+µν, leading to a theory satisfying the curvature-less and metricity conditions [32–35], resulting
+in a torsionful theory. Later in this section, it will be shown that even though the Ricci scalar R vanishes,
+due to the symmetry of the connection, the curvature-ful connection gives a non-zero value for the Riemann
+tensor. Hence, the Ricci scalar
+◦R can be defined through teleparallel quantities.
+The fundamental dynamical object of GR is the metric tensor gµν, but within TG, the metric is expressed
+through the tetrad eA
+µ, which acts as a transformation between the local and general manifold spaces. The
+tetrad and the inertial spin connection ωB
+Cν become the fundamental objects [33] while creating a link
+between the general manifold denoted through Greek indices to the local Minkowski manifold denoted by
+Latin indices. Thus, the relationship between Minkowski and general spacetimes is given by [77]
+gµν = ηAB eA
+µ eB
+ν ,
+ηAB = gµν E
+µ
+A E
+ν
+B ,
+(1)
+where E
+µ
+A
+is the inverse tetrad which satisfies orthogonality conditions
+eA
+µ E
+µ
+B
+= δA
+B ,
+eA
+µ E
+ν
+A
+= δν
+µ .
+(2)
+The teleparallel connection is defined through the TG variables as [34, 35]
+Γλ
+µν = E
+λ
+A
+�
+∂νeA
+µ + ωA
+BνeB
+µ
+�
+.
+(3)
+Note, the quantities without an overhead circle will correspond to those objects that are related to teleparallel
+gravity or calculated on its connection. The condition
+∂[µωA
+|B|ν] + ωA
+C[µωC
+|B|ν] ≡ 0 ,
+(4)
+results in a flat spin connection [32], where square brackets denote the antisymmetric operator, and can
+equivalently be used to determine the components of the spin connection. The local Lorentz transformation
+(LLT), wherein ΛA
+B represents Lorentz boosts and rotations, is used to define the spin connection ωA
+Bµ =
+ΛA
+C ∂µΛ
+C
+B
+[33]. It plays a role in the field equations since an infinite number of tetrad choices could satisfy
+Eq. (1) for a particular metric, thus the spin connection is used to counterbalance inertial effects. This
+ensures the theory, in this case TG, remains covariant [78]. Moreover, there exists a Lorentz frame such that
+the spin connection is set to zero. It is referred to as the Weitzenb¨ock gauge [35, 79]. Hence, from this point
+onwards, the spin connection will be dropped following the application of the aforementioned gauge.
+Similar to how the Riemann tensor, constructed on the Levi-Civita connection, is associated to the cur-
+vature property of GR, the analogy of this for TG is the torsion tensor defined by the teleparallel connec-
+tion [35, 80]
+T A
+µν = ΓA
+νµ − ΓA
+µν .
+(5)
+While curvature in GR is defined as a deformation of spacetime, the torsion tensor in TG represents the
+field strength of gravitation that transforms covariantly under LLTs and diffeomorphisms [78]. Another
+important aspect of TG is that the torsion tensor can be decomposed in irreducible parts [81–83], namely
+aµ = 1
+6ǫµνλρ T νλρ ,
+(6)
+vµ = T λ
+λν ,
+(7)
+tλµν = 1
+2(Tλµν + Tµλν) + 1
+6(gνλvµ + gνµvλ) − 1
+3gλµvν ,
+(8)
+
+4
+giving the axial, vectorial and tensorial pieces, respectively, and ǫABCD being the four-dimensional Levi-
+Civita tensor. Hence, the respective scalar invariants are given by [47, 82]
+Tax = aµaµ = − 1
+18Tλµν(T λµν − 2T µλν) ,
+(9)
+Tvec = vµvµ = T λ
+λµT
+ρµ
+ρ
+,
+(10)
+Tten = tλµνtλµν = 1
+2Tλµν(T λµν + T µλν) − 1
+2T λ
+λµT ρµ
+ρ
+,
+(11)
+which, under parity transformation, are invariant scalars. On the other hand, terms such as P1 = vµaµ and
+P2 = ǫµνσρtλµνtλ
+ρσ are excluded due to parity-violation [38, 81]. The combination of the scalar invariants
+leads to the torsion scalar
+T = 3
+2Tax + 2
+3Tten − 2
+3Tvec = 1
+2
+�
+E
+λ
+A gρµE
+ν
+B
+− 2E
+ρ
+B gλµE
+ν
+A + 1
+2ηABgµρgνλ
+�
+T A
+µνT B
+ρλ .
+(12)
+An identical result can be obtained through contraction between the torsion tensor and the superpotential
+S
+µν
+A
+which represents the potential relation of the gravitational energy-momentum tensor [80, 84, 85]:
+S
+µν
+A
+= 1
+2
+�
+Kµν
+A − E
+ν
+A T αµ
+α + E
+µ
+A T αν
+α
+�
+,
+(13)
+where
+Kλ
+µν = Γλ
+µν −
+◦Γλ
+µν = 1
+2
+�
+T
+λ
+µ ν + T
+λ
+ν µ − T λ
+µν
+�
+(14)
+is the contorsion tensor relating TG and GR. These quantities can be used to form the torsion scalar [34],
+written as
+T = S
+µν
+A
+T A
+µν ,
+(15)
+which can be shown to be equivalent to the result obtained in Eq. (12). The Ricci scalar dependent on
+the teleparallel connection, which vanishes, can be related, using the contorsion tensor, to the regular Ricci
+scalar. This results in a relationship between the Levi-Civita and the teleparallel connections defined Ricci
+scalar [47, 77]
+R =
+◦R + T − B = 0 ,
+(16)
+where B = 2
+e∂µ(e T λ µ
+λ ) = 2
+◦∇µT λ µ
+λ
+is a boundary term and e = det(eA
+µ) = ���−g is the tetrad determinant.
+Hence, the curvature-ful Ricci scalar is non-vanishing being
+◦R = −T + B .
+(17)
+The total divergence term found in B accounts for the fourth order derivative contributions to the field
+equations in modified theories of gravity [47, 54]. This is embodied within the Ricci scalar in GR [57, 59].
+Thus, the Teleparallel Equivalent of General Relativity (TEGR) [86, 87] is defined as the linear appearance
+of the torsion scalar since both Lagrangians are equal up to a boundary term.
+The equivalence principle in GR allows one to raise local Lorentz frames from a Minkowski metric to
+a general metric tensor, and additionally, partial derivatives are raised to covariant derivatives defined by
+the Levi-Civita connection [1]. This procedure, referred to as the minimal coupling prescription, if applied
+within TG, is preserved for additional fields. Minkowski tetrads are raised to arbitrary ones and tangent
+space partial derivatives are exchanged for covariant derivatives based on Levi-Civita connection [33, 88]
+∂µ →
+◦∇µ .
+(18)
+Thus, by having both gravitational and scalar fields well developed, it is possible to look at the teleparallel
+analogue of Horndeski gravity, referred to at times as Bahamonde-Dialektopoulos-Levi Said (BDLS) the-
+ory [38, 66, 69]. The construction of BDLS theory depends on the following criteria: (1) field equations are
+
+5
+at most second order with respect to the tetrad and scalar; (2) as previously mentioned, scalar invariants do
+not violate parity; (3) contractions of the torsion tensor are at most quadratic [38]. All of these requirements
+allow for an adequate extension of the standard metric Horndeski gravity. Note that Lovelock’s theorem
+states that Einstein fields equations are the only second-order field equations from a Lagrangian density
+constructed through a four-dimensional metric. TG allows for the weakening of Lovelock [36, 37] theory as
+an additional scalar field φ is introduced, giving rise to further terms in the gravitational action.
+Starting off with the non-minimal coupling of scalar and torsion field, the linear contraction is given by [38]
+I2 = vµ ◦∇µφ ,
+(19)
+being the scalar I1 = tλµν
+◦∇λφ
+◦∇µφ
+◦∇µφ = 0, due to the complete symmetry of the tensor decomposition,
+Furthermore I3 = aλ
+◦∇λφ violates the parity condition due to an odd number of axial parts. Moreover, the
+quadratic contractions of this nature are given by [38]
+J1 = aµaν ◦∇µφ
+◦∇νφ ,
+(20)
+J3 = vλtλµν ◦∇µφ
+◦∇νφ ,
+(21)
+J5 = tλµνt α
+λ ν
+◦∇µφ
+◦∇αφ ,
+(22)
+J6 = tλµνt αβ
+λ
+◦∇µφ
+◦∇νφ
+◦∇αφ
+◦∇βφ ,
+(23)
+J8 = tλµνt
+α
+λµ
+◦∇νφ
+◦∇αφ ,
+(24)
+J10 = ǫµ
+νλρaνtαρλ ◦∇µφ
+◦∇αφ ,
+(25)
+while other possibilities are eliminated as they have already been included:
+J2 = vµvν ◦∇µφ
+◦∇νφ = I2
+2 ,
+J4 = vµtλµν ◦∇λφ
+◦∇νφ = J3 ,
+J7 = tλµνtαβ
+λ
+◦∇µφ
+◦∇νφ
+◦∇αφ
+◦∇βφ = −2J6 ,
+(26)
+while J9 = tλµνtαβγ
+◦∇λφ
+◦∇µφ
+◦∇νφ
+◦∇αφ
+◦∇βφ
+◦∇γφ = 0 due to the total symmetry of the tensor irreducible part.
+Hence, BDLS action is described by [38]
+SBDLS =
+1
+2κ2
+�
+d4x e LTele +
+1
+2κ2
+5
+�
+i=2
+�
+d4x e Li +
+�
+d4x e Lm ,
+(27)
+where
+LTele = GTele(φ, X, T, Tax, Tvec, I2, J1, J3.J5, J6, J8, J10) .
+(28)
+Here GTele is an arbitrary function, X := − 1
+2∂µφ∂µφ and
+L2 := G2(φ, X) ,
+(29)
+L3 := −G3(φ, X)
+◦□φ ,
+(30)
+L4 := G4(φ, X)(−T + B) + G4,X(φ, X)[(
+◦□φ)2 −
+◦∇µ
+◦∇νφ
+◦∇µ ◦∇νφ] ,
+(31)
+L5 := G5(φ, X)
+◦Gµν
+◦∇µ
+◦∇νφ
+− 1
+6G5,X(φ, X)[(
+◦□φ)3 + 2
+◦∇µ
+◦∇νφ
+◦∇ν
+◦∇αφ
+◦∇α
+◦∇µφ − 3
+◦∇µ
+◦∇νφ
+◦∇µ ◦∇νφ
+◦□φ] ,
+(32)
+which are identical to the standard metric Horndeski Lagrangians [20] but calculated using the tetrad.
+Finally, Lm is the matter Lagrangian in Jordan conformal frame,
+◦Gµν is the Einstein tensor, and κ2 = 8πG
+where G is the gravitational constant.
+III.
+MINKOWSKI BACKGROUND EQUATIONS IN TELEPARALLEL HORNDESKI GRAVITY
+Let us explore now perturbations on a Minkowski background within the gravitational theory of teleparallel
+analogue of Horndeski. Firstly, we tackle the background equations for a general flat Friedman–Lemaˆıtre–Robertson–Walker
+
+6
+(FLRW) metric to obtain the necessary constraints. Then, this is followed by the analysis for perturbation
+theory up to second order which is needed for our approach since we consider the Euler-Lagrange equations
+to get the cosmological equations of motion.
+The flat FLRW metric in Cartesian coordinates is given by
+ds2 = −N(t)2dt2 + a(t)2(dx2 + dy2 + dx2) ,
+(33)
+where N(t) is the lapse function and a(t) is the scale factor. As previously mentioned, this follows the metric
+signature that is mostly positive. The tetrad can be written in diagonal form as [35]
+eA
+µ = diag(N(t), a(t), a(t), a(t)) ,
+(34)
+which is compatible with the Weitzenb¨ock gauge giving a vanishing spin connection. Hence, the action in
+Eq. (27) can be re-expressed in terms of these background quantities. Friedman equations are then obtained
+by varying with respect to the lapse function and scale factor. Additionally, the scalar field Klein-Gordon
+equation can be obtained by varying with respect to the scalar field [38].
+Since we will be working in
+Minkowski background, the limits N(t) → 1 and a(t) → 1 are taken. Hence, the constraints obtained from
+the Friedman equation and scalar field variation in Minkowski background are given by
+0 = −G2 − GTele + 2XG2,X − 2XG3,φ + 2XGTele,X ,
+(35)
+0 = G2 + GTele − 2XG3,φ + 4XG4,φφ + 2¨φG4,φ − 2X ¨φG3,X + 4X ¨φG4,φX − d
+dt( ˙φGTele,I2) ,
+(36)
+0 = G2,φ + GTele,φ − 2XG2,φX + 2XG3,φφ − 2XGTele,φX − G2,X ¨φ + 2G3,φ ¨φ − GTele,X ¨φ
+− 2X ¨φGTele,X − 2X ¨φG2,XX + 2X ¨φG3,φX − 2X ¨φGTele,XX ,
+(37)
+where all scalar invariants are background quantities and comma ( , ) denotes partial derivative. Moreover,
+the scalar field equation can also be expressed in terms of a Klein-Gordon equation as shown in Ref. [38].
+IV.
+SECOND ORDER PERTURBED ACTION WITH SCALAR-VECTOR-TENSOR
+DECOMPOSITION
+Here, we calculate the perturbed action up to second order terms which are necessary for the Euler-
+Lagrange method to find the equations of motion of the system. By taking perturbations about a Minkowski
+background for both tetrad and scalar field, we can build the action up to second order. The tetrad and
+scalar perturbation are respectively given by
+eA
+µ → eA
+µ + ǫ δeA
+µ = δA
+µ + ǫ δeA
+µ ,
+(38)
+φ → φ + ǫ δφ ,
+(39)
+where ǫ is the perturbation parameter representing the perturbation order of background Minkowski tetrad.
+It is given by a four-dimensional identity matrix given by the Kronecker delta δA
+µ . It is sufficient to expand
+the scalar field up to first order since higher order terms do not contribute. Additionally, it should be noted
+that the background scalar field can be taken to be as a function of time and apply the unitary gauge such
+that
+φ → φ(t) .
+(40)
+Moreover, the perturbations of arbitrary functions present in the Lagrangian are obtained by performing a
+Taylor expansion up to second order such that
+Gi(φ, X) = Gi + ǫ Gi,XX(1) + ǫ2 �
+1
+2Gi,XX(X(1))2 + Gi,XX(2)�
+,
+(41)
+GTele(φ, X, T, Tax, Tvec, I2, J1, J3, J5, J6, J8, J10)
+= GTele + ǫ(GTele,XX(1) + GTele,I2I(1)
+2 ) + 1
+2ǫ2(GTele,XX(X(1))2 + GTele,I2I2(I(1)
+2 )2)
++ ǫ2�
+GTele,XX(2) + GTele,T T + GTele,TaxTax + GTele,TvecTvec + GTele,I2I(2)
+2
+�
+,
+(42)
+
+7
+where, for i = {2, 3, 4, 5}, j = {X, XX} and k = {X, T, Tax, Tvec, XX, I2I2}, Gi,j and GTele,k are background
+functions, such that XX and I2I2 are the second order derivatives with respect to X and I2. The numbered
+superscripts represent the order of perturbation of the scalar invariant.
+In this section, we consider a scalar-vector-tensor (SVT) decomposition of the tetrad based on the formal-
+ism applied in Ref. [89] for first order perturbations:
+δeA
+µ :=
+
+
+ϕ
+−(∂iβ + βi)
+δI
+i (∂ib + bi)
+δIi �
+−ψδij + ∂i∂jh + ∂ihj + ∂jhi + 1
+2hij + ǫijk(∂kσ + σk)
+�
+
+ ,
+(43)
+where {ϕ, β, b, ψ, h} are scalars and σ is a pseudoscalar of 1 degree of freedom (DoF) each, {βi, bi, hi} are
+vectors and σi is a pseudovector of 1 DoFs each, and hij is the tensor mode of 2 DoFs, for a total of of 16
+DoFs. The tensor modes are symmetric hij = h(ij), traceless δijhij = 0 and divergenceless ∂ihij = 0. See
+also Ref. [90, 91]. The divergenceless property also applies for vectors and pseudovectors such that ∂iαi = 0
+where α = {β, b, h, σ}. Unlike Ref. [92, 93], the pseudoscalar and pseudovector are included to account for the
+anti-symmetry of the tetrad. The mid-range Latin indices are spacial coordinates: {I, J, K, . . .} for spacial
+inner bundle and {i, j, k, . . .} for spacial spacetime manifold. Note, δij is the spacial Minkowski metric such
+that δij = −ηij. In turn, the first order metric perturbation from Eq. (1) is given by
+δgµν =
+
+
+2ϕ
+∂iB + Bi
+∂iB + Bi
+2
+�
+−ψδij + ∂i∂jh + ∂ihj + ∂jhi + 1
+2hij
+�
+
+ ,
+(44)
+where B = −β + b and Bi = −βi + bi. The off-diagonals are identical hence verifying the symmetry of the
+metric. The pseudoscalar and pseudovectors no longer play a role, and thus, the number of DoFs reduces to
+10.
+The gauge transformation through the coordinate change [94–96] is
+˜xµ → xµ + ξµ ,
+(45)
+where ξµ is a vector field applied to study the gauge transformation of perturbative quantities. Such a
+coordinate transformation can be extended to include orders higher than second one, but it is sufficient to
+consider up to this point. Thus, the transformations for first tetrad are given by [97]
+δ˜eA
+µ = δeA
+µ + Lξ(1)eA (0)
+µ
+,
+(46)
+where Lξ is the Lie derivative along ξµ wherein ξµ = (ξ0, ξi + δij∂jξ) further splits and once again obeys
+divergencelessness as ∂iξi = 0. The analysis of each combination of temporal and spatial parts of the tetrad
+indicates that ψ, σ, βi and hij are gauge invariant in Minkowski background while the rest have the following
+gauge transformations
+˜ϕ = ϕ − ˙ξ0 ,
+β = β − ξ0 ,
+˜b = b − ˙ξ ,
+˜h = h − ξ ,
+˜bi = bi + ˙ξi ,
+˜hi = hi + 1
+2ξi ,
+˜σi = σi − 1
+2ǫijk∂jξk .
+(47)
+This shows that since the pseudoscalar σ is gauge-invariant, it can be treated separately than the rest of
+the scalar modes, but the same cannot be said for the pseudovector. In fact, the pseudovector σi can be
+expressed in terms of hi (and its derivative in terms of bi). This implies that the vector and pseudovector
+cannot be decomposed [98].
+Next, we construct groups of non-gauge invariant quantities: {ϕ, β}, {b, h} and {bi, hi, σi}. For gauge
+choice, a quantity from each group is set to zero [98]. In particular, we will choose β = 0, h = 0 and σi = 0.
+V.
+GHOST AND LAPLACIAN INSTABILITIES IN MINKOWSKI BACKGROUND
+Ghost instabilities stem from a negative kinetic term in the action associated to a propagating degree of
+freedom. A ghost mode can be determined by expanding the action up to second order perturbations about
+
+8
+a background. Therefore, for a Lagrangian of the form L = ˙⃗χtA ˙⃗χ + . . ., we impose that the eigenvalues of
+A should be positive to eliminate ghost modes. The procedure is applied in what follows. The second order
+perturbation of the action is obtained by separately applying the scalar, vector and tensor field perturbations
+given by Eq. (43). The dynamical fields in the action are identified, while a system of equations is obtained
+by varying the action with respect to the auxiliary fields. This system of equations is substituted back into
+the action to eliminate the non-dynamic fields [99, 100].
+In order to obtain a gauge invariant action, the action is varied with respect to the temporal and spatial
+part of the vector field associated with the coordinate transformation, and imposing that this vanishes. Then,
+a diagonalized kinetic matrix is constructed and a constraint is generated for each entry. In general, these
+constraints could be time dependent: this feature arises from the background spacetime and the background
+quantities of fields. When looking at the propagating speeds of these modes, a positive definite value should
+be applied in order to ensure that the perturbation does not lead to an exponential growth [101] i.e. c2 > 0,
+referred to as gradient or Laplacian instability. Hence, both ghost and gradient instabilities are checked for
+each mode in order to obtain the constraints which lead to a stable model.
+A.
+Tensor Perturbations
+The tensor mode perturbations, presented in Eq. (43), can be extended as
+eA
+µ →
+
+1
+0
+0 δij + 1
+2ǫhij − 1
+8ǫ2hikhk
+j
+
+ ,
+(48)
+analogous to the Arnowitt-Deser-Misner (ADM) for tensor modes [100]. The action can be extended to
+second order perturbations such that
+S(2)
+T
+= 1
+2
+�
+d4x
+�
+MT ˙hij ˙hij − NT ∂mhij∂mhij + Phijhij�
+(49)
+where
+MT = G4 − 2XG4,X + XG5,φ − GTele,T + 1
+2XGTele,J5 + 2XGTele,J8 ,
+(50)
+NT = G4 − X(G5,φ + ¨φG5,X) − GTele,T ,
+(51)
+PT = − 1
+2
+d
+dt( ˙φGTele,I2) .
+(52)
+For the sake of simplicity, we switch to Fourier space such that
+S(2)
+T
+= 1
+2
+�
+dt d3k
+(2π)3
+�
+MT ˙hij ˙hij +
+�
+−k2NT + PT
+�
+hijhij .
+�
+(53)
+This result is obtained after applying integration by parts, removing the surface terms and applying the
+background Eqs. (35-37). By imposing MT > 0, the theory is ghost free in tensor modes. When considering
+a constant background scalar φ, this implies that G4 − GTele,T > 0 which corresponds to the condition
+G4 −GTele,T ̸= 0 imposed in Ref. [102] in order to ensure that there are tensor mode DoFs propagating. The
+propagation speed of tensor modes is given by
+c2
+T = NT
+MT
+=
+G4 − X(G5,φ + ¨φG5,X) − GTele,T
+G4 − 2XG4,X + XG5,φ − GTele,T + 1
+2XGTele,J5 + 2XGTele,J8
+,
+(54)
+for which c2
+T > 0 is required to ensure gradient stability. Hence, MT > 0 and NT > 0 result in ghost
+and gradient stability, respectively. The same result would by obtained when eliminating I2 contribution
+from the GTele function. Following the observations of gravitational wave signal GW170817 [103] and its
+electromagnetic counterpart GRB170817A [27], it is interesting to calculate the deviation from speed of light
+propagation such that the excess speed is given by
+αT = c2
+T − 1 =
+X(2G4,X − 2G5,φ − ¨φG5,X − 1
+2GTele,J5 − 2GTele,J8)
+G4 − 2XG4,X + XG5,φ − GTele,T + 1
+2XGTele,J5 + 2XGTele,J8
+,
+(55)
+
+9
+which corresponds to the result obtained through the GW propagation equation [66], a value which is highly
+constrained such that a graviton mass would be minute.
+Additionally, from the general result of teleparallel Horndeski analogue, given by Eq. (48), one may obtain
+the results of well-studied theories from literature, as summarized in Table I. See also Ref.[73] for the
+metric cases derived from Noether symmetries. As an extension analogous to standard Horndeski theory,
+the stability conditions can be obtained by setting GTele = 0. The ghost modes are excluded for when
+the constant of the kinetic term is G4 − 2XG4,X + XG5,X > 0 while gradient stability is obtained for
+G4 − X(G5,φ + ¨φG5,X) > 0, in agreement with Refs. [101, 104–106] when taking the appropriate limits
+to Minkowski spacetime. An example of a subclass of Horndeski gravity is the Brans-Dicke theory, where
+G2 = 2wBDX
+φ
+and G4 = φ for which wBD is the Dicke coupling constant [107], other constants are set to
+vanish and ghost instabilities are avoided for φ > 0. By considering the generalized Brans Dicke theory,
+ghost instabilities are avoided for a positive value of a function of the scalar field, F(φ) > 0 [108]. Another
+example is f(
+◦R) where G2 = f(φ) − φf ′(φ), G4 = f ′(φ), the rest of the constants are zero, implying that
+ghost stability is achieved for f ′(φ) > 0. A final subcase of Horndeski gravity considered here is GR. In this
+case, the only non-vanishing constant is G4 = 1 for which there are no ghost modes since MT = 1 > 0. All
+of these subcases do not result in any gradient instabilities as cT = 1 > 0.
+Next, we look at cases that arise due to the inclusion of teleparallel terms. For a purely teleparallel theory
+such as f(T ) gravity, all terms are set to vanish except for GTele,T = f(T ). This implies in −f ′(T ) > 0 to
+avoid ghost modes similar to the result obtained in Ref. [98]. An equivalent result is obtained for scalar-
+tensor theory with a Lagrangian of the form L = f(φ, T ) + XP(φ), an extension of f(T ) [100, 109]. Once
+again, gradient instabilities are not an issue. Finally, the case where only I2 contributions are present is
+considered. In this case, all constants are going to vanish while GTele,I2 ̸= 0 leads to a non-dynamical degree
+of freedom.
+Theory
+Case
+MT
+NT
+Horndeski
+GTele = 0
+G4 − 2XG4,X + ¨φXG5,X G4 − X(G5,φ + ¨φG5,X)
+Generalized
+Brans-Dicke
+GTele = G5 = 0, G2 = B(φ)X
+G3 = 2ξ(φ)X, G4 = 1
+2F(φ)
+1
+2F(φ)
+1
+2F(φ)
+Brans-Dicke
+GTele = G3 = G5 = 0
+G2 = 2wBDX
+φ
+, G4 = φ
+φ
+φ
+f(
+◦R)
+GTele = G3 = G5 = 0
+G2 = f(φ) − φf ′(φ), G4 = f ′(φ)
+f ′(φ)
+f ′(φ)
+General Relativity
+GTele = G2 = G3 = G5 = 0
+G4 = 1
+1
+1
+Teleparallel
+or f(T )
+G2 = G3 = G4 = G5 = 0,
+GTele = f(T )
+−f ′(T )
+−f ′(T )
+f(φ, T ) + XP(φ)
+G3 = G4 = G5 = 0
+G2 = XP(φ), GTele = f(φ, T )
+−f,T (φ, T )
+−f,T (φ, T )
+GTele,I2 only
+G2 = G3 = G4 = G5 = 0
+GTele,T = 0
+no propagating mode
+TABLE I. List of literature models with the respective ghost MT and gradient stability NT conditions are positive
+definite, while the propagation speed is cT = NT/MT for tensor modes. The models include Horndeski theory [20,
+101, 104, 105], Generalized Brans-Dicke [108] and Brans-Dicke [107], f(˚
+R) theory, General Relativity, f(T ) theory [98],
+f(φ, T ) theory [100] and the case where the action is dependent on I2 only.
+
+10
+B.
+Vector Perturbations
+Next, we consider the vector perturbation of the tetrad (43) with the application of gauge fixing which
+eliminates the pseudovector and any coupling with it such that
+eA
+µ →
+
+
+1
+−ǫβi
+ǫδI
+i bi δIi(δij + ǫ(∂ihj + ∂jhi))
+
+ .
+(56)
+This result is a case of only vector modes in this portion of decomposition. Extending the action to second
+order vector perturbations, we obtain
+S(2)
+V
+=
+�
+dt d3k
+(2π)3
+�
+4k4Ahihi + E ˙βi ˙βi
++ k2(4B ˙hi( ˙hi − bi) + Cbibi + Dβiβi + 4Fhiβi + 4Ghi ˙βi − 2Hβibi)
+�
+,
+(57)
+where
+A = GTele,Tvec + 1
+9X
+�
+−GTele,J8 + XGTele,J6 − 5
+2GTele,J5 − 3GTele,J3
+�
+,
+B = G4 − 2XG4,X + XG5,φ − GTele,T + X
+�
+2GTele,J8 + 1
+2GTele,J5
+�
+,
+C = B + 1
+2XGTele,J5 + 2
+3XGTele,J10 + 2
+9GTele,Tax ,
+D = C + XGTele,J5 − 2XGTele,J8 − 2XGTele,J10 ,
+E = 4A − 3GTele,Tvec + 2XGTele,J3 ,
+F = 1
+2 ˙φGTele,I2 − d
+dt (G4 − 2XG4,X + XG5,φ) .
+G = 2A − B − 3GTele,Tvec + 1
+3XGTele,J3 − 2XGTele,J8 + 5
+2XGTele,J5 ,
+H = C − 2XGTele,J8 − 4
+9GTele,Tax ,
+(58)
+While the decomposition along unitary gauge ensures that there are no higher order time derivatives, one
+can see that the action contains higher order spatial derivatives [110]. Here, we will set the coefficients of
+these derivatives to vanish, but it should be noted that a general analysis for each constraint should be
+carried out before solving the next one and leading to a very complicate solution for the action. By imposing
+the conditions A = 0 and B = 0, the latter condition accounting for the higher order mix of temporal and
+spatial derivatives [111], the action reduces to
+S(2)
+V
+=
+�
+dt d3k
+(2π)3
+�
+E ˙βi ˙βi + k2(Cbibi + Dβiβi + 4Fhiβi + 4Ghi ˙βi − 2Hβibi)
+�
+,
+(59)
+where bi and hi are auxiliary vector modes. By varying with respect to each of these non-dynamical modes,
+the respective field equations are given by
+2k2(Cbi − Hβi) = 0 ,
+and
+4k2Fβi = 0 .
+(60)
+By solving for each auxiliary field and plugging it back in into Eq. (59), we get
+S(2)
+V
+=
+�
+dt d3k
+(2π)3
+�
+E ˙βi ˙βi − k2 �
+H2
+C − D
+�
+βiβi
+�
+,
+(61)
+wherein the action is expressed in terms of the dynamical non-gauge invariant mode βi. Thus the ghost and
+Laplacian conditions are given respectively by
+MV = E > 0 ,
+and NV = H2
+C − D > 0 .
+(62)
+suggesting that there is a propagating vector mode with speed
+cV = H2 − CD
+CE
+.
+(63)
+This propagating mode generally steps from the introduction of the Ji terms in the BDLS action (27)
+representing the quadratic contractions of the scalar and torsion fields.
+
+11
+a.
+Vanishing Ji terms:
+When considering the case were J1 = J3 = J5 = J6 = J10 = 0, the action (61)
+reduces to
+Svanishing J terms
+V
+=
+�
+dt d3k
+(2π)3 k2
+�
+C − H2
+C
+�
+βiβi ,
+(64)
+exhibiting no dynamical modes i.e. no propagating vector mode. In the case of only teleparallel function,
+the same conditions apply with vanishing Horndeski terms in each constant.
+The subcases of standard
+Horndeski [31, 104], f(
+◦R) [99, 112], Generalized Brans-Dicke[108] and f(T ) [98] fall under this category of
+non-viable propagating vector mode.
+b.
+C = 0
+: The Laplacian constraint, given in Eq. (62), is undefined for C = 0. By considering this
+particular case, the action results in
+SC=0
+V
+=
+�
+dt d3k
+(2π)3 E ˙βi ˙βi ,
+(65)
+to which the Laplacian condition is violated since it is null. On the other hand, imposing only E = 0, the
+ghost instability is generated since this value should be a positive definite value.
+c.
+Ji terms only:
+On the other hand, when considering only the contribution of the quadratic contrac-
+tions, the only surviving term is that with coefficient E leading to a ghost stability condition but leading
+to Laplacian instability, similar to the C = 0 case. This implies the importance of having a combination of
+the Ji terms along with the rest of the teleparallel scalar invariants to maintain stability conditions and a
+possible propagating vector mode.
+d.
+Constant Background Scalar Field:
+With regards to the scalar field, the unitary gauge is applied to
+the perturbative part of the scalar, while, at background level, it is a function of time only. Alternatively,
+if one considers a constant scalar field φ = c, then the entire action is vanishing such that no vector modes
+propagate, thus it corresponds to the results in Ref. [67] for the DoF and polarisation modes in the vector
+portion of the decomposition.
+C.
+Scalar Perturbations
+Let us now consider the scalar modes. Since there is no mixing between the scalar and pseudoscalar modes,
+we will treat them separately. The tetrad perturbation using scalar modes is given by
+eA
+µ →
+
+1 + ϕ
+0
+δI
+i ∂ib (1 − ψδIiδij .
+
+
+(66)
+As already stated, for the gauge invariant quantities, given by Eq. (47), the gauge choice is β = h = 0. The
+pseudoscalar will be analyzed separately. It should be noted that this situation is identical to the Arnowitt-
+Deser-Misner (ADM) decomposition used in torsion-based theories [100]. It leads to the same metric as that
+applied to curvature-based theories [101]. On the other hand, the choice β = b = 0 results in the Newtonian
+(longitudinal) gauge. Substituting the tetrad perturbation to the action, followed by integration by parts
+and application of the background equations (35-37), the second order perturbation of the action is given by
+S(2)
+S
+=
+�
+dt d3k
+(2π)3
+�
+¯
+Aϕ2 + 6 ¯Dψ2 − 6 ¯F ˙ψ2 − 6 ¯Gϕ ˙ψ
+− k2(− ¯Bϕ2 − 2 ¯Eψ2 + 4 ¯Hϕψ − 2 ¯Gϕb − 4 ¯F ˙ψb) + k4 ¯Cb2�
+(67)
+
+12
+where
+¯
+A = XG2,X + 2X2G2,XX − 2XG3,φ − 2X2G3,φX + XGTele,X + 2X2GTele,XX ,
+¯B = GTele,Tvec + 2
+9X (2GTele,J8 + 2XGTele,J6 − 5GTele,J5 + 3GTele,J3) ,
+¯C = −GTele,Tvec + XGTele,I2I2 + 1
+3X(4GTele,J8 + GTele,J5) ,
+¯D = d
+dt( ˙φGTele,I2) ,
+¯E = G4 − X(G5,φ − ¨φG5,X) − GTele,T + 2GTele,Tvec + 1
+9X (−2GTele,J8 + 2XGTele,J6 − 5GTele,J5 − 6GTele,J3) ,
+¯F = G4 − 2XG4,X + XG5,φ − GTele,T + 3
+2(GTele,Tvec − XGTele,I2I2) ,
+¯G = − ˙φXG3,X + ˙φG4,φ + 2 ˙φXG4,φX + 1
+2 ˙φGTele,I2 + ˙φXGTele,XI2 ,
+¯H = G4 − 2XG4,X + XG5,φ − GTele,T + GTele,Tvec + 1
+9X(2GTele,J8 − 2XGTele,J6 + 5GTele,J5 + 3
+2GTele,J3) .
+(68)
+Here k is the co-vector. As shown in coefficient of ¯C in Eq. (67), there is a higher derivative contribution,
+hence it is set to vanish [110]. It is clear that variables ϕ and b are non-dynamical with respect to the time
+component. By varying with respect to each of these auxiliary fields, one obtains a set of equations
+0 = ( ¯
+A + k2 ¯B)ϕ − 3 ¯G ˙ψ + k2(2 ¯Hψ − ¯Gb) ,
+(69)
+0 = ¯Gϕ + 2 ¯F ˙ψ .
+(70)
+By solving these equations of motion for the modes ϕ and b, it leads to an action expressed in terms of the
+dynamical mode only, that is
+S(2)
+S
+=
+�
+dt d3k
+(2π)3
+�
+4k2 ¯B ¯F
+2
+¯G
+2
+˙ψ2 +
+�
+4 ¯
+A ¯F
+2
+¯G
+2
++ 6 ¯F
+�
+˙ψ2 − k2
+�
+−2 ¯E + 4 d
+dt
+� ¯F ¯H
+¯G
+��
+ψ2 + 6 ¯Dψ2
+�
+.
+(71)
+Taking into account the mix of spatial and temporal higher-order derivatives, the first term of the action
+vanishes [111]. This implies that either ¯F = 0 (which renders the whole action without a dynamical mode),
+or ¯B = 0. By applying the latter condition, the action can be expressed in terms of a gauge invariant quantity
+within a Minkowski background and no higher-order terms. It is
+S(2)
+S
+=
+�
+dt d3k
+(2π)3
+��
+4 ¯
+A ¯F
+2
+¯G
+2
++ 6 ¯F
+�
+˙ψ2 − k2
+�
+−2 ¯E + 4 d
+dt
+� ¯F ¯H
+¯G
+��
+ψ2 + 6 ¯Dψ2
+�
+.
+(72)
+In the case of teleparallel analogue of Horndeski theory, ghost stability and gradient stability are obtained,
+respectively, when
+MS = 4 ¯
+A ¯F
+2
+¯G
+2
++ 6 ¯F > 0 ,
+(73)
+¯
+N S = −2 ¯F + 4 d
+dt
+� ¯F ¯H
+¯G
+�
+> 0 ,
+(74)
+and the propagating speed is
+cS = NS
+MS
+.
+(75)
+Different subcases are reported in Table II. Through the substitution of Eqs (73) and (74), conditions to
+avoid ghost and Laplacian instabilities are realised.
+a.
+Horndeski:
+The standard Horndeski case has identical expressions for ghost and gradient stability
+conditions as those for teleparallel Horndeski (73-74) with the difference that each constant does not have
+teleparallel contributions.
+
+13
+b.
+Generalized Brans-Dicke:
+Considering Generalised Brans-Dicke (GBD) theory means that G2 =
+B(φ)X, G3 = 2Xξ(φ), G4 = 1
+2F(φ) and all other terms are set to zero. When we take the background
+equations
+B − 4Xξ′ = 0 ,
+and
+˙φ2F ′′ + ¨φ(F ′ − 2 ˙φ2ξ) = 0 ,
+(76)
+for ghost stability, we have
+MGBD
+S
+= F[3F ′2 − 4 ˙φ2(Fξ′ + 3ξF ′) + 12 ˙φ4ξ2]
+[F ′ − 2 ˙φ2ξ]2
+> 0 ,
+(77)
+where ′ denotes a derivative with respect to the respective variable of the function. In this case, it is φ, while
+the gradient condition is
+N GBD
+S
+= F[3F ′3 − 2 ˙φ2(5F ′2ξ − 4ξFF ′′ − 2Fξ′F ′) + 4 ˙φ4(F ′ξ2 − 2Fξ′ξ) + 8 ˙φ6ξ3]
+[F ′ − 2 ˙φ2ξ]3
+> 0 .
+(78)
+With these conditions, the speed of propagation is positive. It is
+(cGBD
+S
+)2 = 3F ′3 − 2 ˙φ2(5F ′2ξ − 4ξFF ′′ − 2Fξ′F ′) + 4 ˙φ4(F ′ξ2 − 2Fξ′ξ) + 8 ˙φ6ξ3
+[3F ′2 − 4 ˙φ2(Fξ′ + 3ξF ′) + 12 ˙φ4ξ2][F ′ − 2 ˙φ2ξ]
+,
+(79)
+which correlates with results obtained in Ref. [108] for a Minkowski background.
+c.
+Brans-Dicke:
+As for the Brans-Dicke theory in Ref. [107] with G2 = 2wBDX
+φ
+and G4 = φ, ghosts
+instabilities are avoided if 2φ(3 + 2wBD) > 0.
+Provided that φ > 0, this implies that the Brans-Dicke
+constant is wBD > − 3
+2 [99]. The gradient stability condition is given by 2φ(3 − 2φ¨φ/ ˙φ2) > 0 to ensure
+a positive propagating speed. Moreover, when applying the second Friedman Eq. (36), gradient condition
+reduces to 2φ(3 + wBD) such that wBD > −3 and a unitary propagating speed [113], analogous to ξ = 0,
+reported in Ref. [108], is obtained. The first Friedman Eq. (35) shows that the only non-trivial solution is
+that for wBD = 0, which results in identical ghost and gradient expressions.
+d.
+f(˚
+R):
+The scalar-tensor theory equivalent to f(R) [114] is given by setting G2 = f(φ) − φf ′(φ) and
+G4 = f ′(φ) while the rest of the functions are set to vanish. See also Ref. [73]. In order to have ghost
+stability, it has to be 6f ′(φ) > 0, i.e. f ′(φ) > 0. Once again, upon applying the background Eqs. (35-36),
+the gradient condition reduces to f ′(φ) > 0. In fact, as a subcase of GBD theory, the case ξ = 0 always
+results in a unitary speed propagating mode.
+e.
+GR, f(T ), f(φ, T ):
+When considering the cases of GR, f(T ) and f(φ, T ), the substitution in Eqs˜(73-
+74) results in an undefined expression. In these cases, although taking the appropriate limits of each coeffi-
+cient does lead to a positive ghost condition value, the propagating speed is negative, thus scalar modes are
+not viable. This is expected in GR. Additionally, Ref. [98] shows that the only propagating scalar field is that
+dependent on the perturbation of scalar field φ, while, in this paper, we are applying the unitary gauge. As
+in Ref. [100], f(φ, T ) theory gives rise to a propagating mode when considering cosmological perturbations.
+The same applies for GR with an additional canonical scalar field such that G2 = X − V (φ), G4 = M 2
+Pl/2,
+G3 = G5 = GTele = 0 [101].
+f.
+Teleparallel:
+If one considers only the teleparallel portions of the action, all terms from Eq. (71) are
+retained, with all standard Horndeski contributions set to vanish. This implies that the ghost and Laplacian
+instabilities coincide with the form of Eqs. (73) and (51), respectively, provided that the background equations
+are satisfied.
+D.
+Pseudoscalar Perturbations
+The gauge invariant pseudoscalar σ can be treated separately as
+eA
+µ →
+
+1
+0
+0 δIi(δij + ǫ εijk∂kσ)
+
+ ,
+(80)
+
+14
+Theory
+Case
+MS
+NS
+Horndeski
+GTele = 0
+6 ¯
+F + 4 ¯
+A ¯
+F2
+¯
+G2
+−2 ¯
+E + 4 d
+dt ( ¯
+F2
+¯
+G )
+Generalized
+Brans-Dicke
+GTele = G5 = 0,
+G2 = B(φ)X, G3 = 2ξ(φ)X,
+G4 = 1
+2 F (φ)
+F [3F ′2−4 ˙φ2(F ξ′+3ξF′)+12 ˙φ4ξ2]
+[F ′−2 ˙φ2ξ]2
+F [3F ′3−2 ˙φ2(5F ′2ξ−4ξFF ′′−2F ξ′F ′)+4 ˙φ4(F ′ξ2−2F ξ′ξ)+8 ˙φ6ξ3]
+[F ′−2 ˙φ2ξ]3
+Brans-Dicke
+GTele = G3 = G5 = 0
+G2 = 2wBDX
+φ
+, G4 = φ
+2φ(3 + 2wBD)
+2(3φ −
+¨
+φ
+X ) → 2φ(3 + 2wBD)
+f( ◦
+R)
+GTele = G3 = G5 = 0
+G2 = f(φ) − φf′(φ),
+G4 = f′(φ)
+6f′(φ)
+−2f′(φ) + 4 d
+dt ( f′(φ)2
+˙φf′′(φ) ) → 6f′(φ)
+GR
+GTele = G2 = G3 = G5 = 0
+G4 = 1
+no propagating mode
+f(T )
+G2 = G3 = G4 = G5 = 0,
+GTele = f(T )
+no propagating mode
+f(φ, T ) + XP (φ)
+G3 = G4 = G5 = 0,
+G2 = XP (φ),
+GTele = f(φ, T )
+no propagating mode
+Teleparallel
+G2 = G3 = G4 = G5 = 0
+4 ¯
+A ¯
+F2
+¯
+G2
++ 6 ¯
+F
+−2 ¯
+F + 4 d
+dt
+� ¯
+F ¯
+H
+¯
+G
+�
+TABLE II. List of literature models with the respective ghost MS and gradient stability NS conditions are positive
+definite, and propagation speed cS = NS/MS for scalar modes. The models include Horndeski theory [20, 101, 104,
+105], Generalized Brans-Dicke [108] and Brans-Dicke [107], f(˚
+R) theory , General Relativity, f(T ) theory [98], f(φ, T )
+theory [100]. and an action with only teleparallel terms.
+The action in Fourier space is expanded up to second order
+S(2)
+PS = −
+�
+dt d3k
+(2π)3
+��
+k2 d
+dt( ˙φGTele,I2) + 4
+9k4 (GTele,Tax − 2XGTele,J1)
+�
+σ2
++k2
+�4
+9GTele,Tax + X
+�
+GTele,J5 + 4
+3GTele,J10
+��
+˙σ2
+�
+,
+(81)
+where the first term exhibits higher-order spatial derivatives and the second term exhibits higher order mixed
+derivatives. By imposing that GTele,Tax = 2XGTele,J1 = −X
+� 9
+4GTele,J5 + 3GTele,J10
+�
+, the action reduces to
+S(2)
+PS =
+�
+dt d3k
+(2π)3
+�
+− k2 d
+dt( ˙φGTele,I2)σ2�
+,
+(82)
+where σ is a non-dynamical mode and thus it is a non-propagating mode. This shows that in Minkowski
+background, the additional scalar invariants, provided by teleparallel analogue of Horndeski theory, do not
+contribute to any additional propagating DoF. See also Ref. [98].
+VI.
+DISCUSSION AND CONCLUSIONS
+We have investigated constraints emerging by considering the stability of perturbations about a Minkowski
+background in teleparallel analogue of Horndeski gravity. The approach is performed by exploring the ghost
+instabilities through looking at potentially negative expressions of kinetic term associated with propagating
+DoFs, as well as Laplacian instabilities emerging when propagation speeds of perturbations are not positive.
+This leads to possible exponential growth rates. These considerations are fundamental for the construction
+of robust and self-consistent cosmological models related to the scalar-tensor gravity.
+Specifically, we have discussed teleparallel analogue of Horndeski gravity where the most general scalar-
+tensor action is considered, provided that the Lagrangian terms are not parity violating and that scalars
+are produced by no more than quadratic contractions of torsion scalar. The naturally lower order nature
+of teleparallel gravity induces a much richer landscape of functional models when compared with standard
+metric Horndeski gravity, based on Levi-Civita connection. In other words, the replacement of torsion with
+curvature to lead dynamics turns out to manifest as a generalization of the standard Horndeski theory. Im-
+portantly, this produces a generalization of tensor mode propagation speed which means that the restrictions
+appearing in standard Horndeski gravity, due to the constraints on the gravitational waves speed, do not
+have the same effect here.
+
+15
+Our calculations have proceeded by first deriving the background equations of motion (35–37) which are
+obtained by considering a flat FLRW background in which we take the Minkowski limit. We find this to
+be a convenient approach to determine the system of equations. These are useful equations for reducing
+the expressions of perturbations that ensue. The procedure is described in Sec. IV. We then proceeded to
+directly determine constraints to prevent ghost or Laplacian instabilities starting with tensor modes which
+results in the action in Eq. (53) and tensor propagation speed in Eq. (54). We collected the constraints for
+popular subclasses of BDLS theory in Table I where the known results are obtained for standard Horndeski
+gravity as well as f(
+◦R) and Brans-Dicke gravity, but new conditions are found in other cases. As for the
+vector perturbations, Sec. V B shows that vector modes are indeed propagating for some cases of GTele
+contribution, namely when Ji scalars appear in this function. However, these contributions may be small
+and within observational constraints.
+The scalar perturbations contain the most diverse range of DoFs. This rich structure produces the intricate
+action in Eq. (72), together with the propagation speed in Eq. (75). As in the case of tensor modes, constraints
+for each subclass of popular models is reported in Table II where results for the Minkowski perturbations are
+recovered for standard metric Horndeski gravity together with f(
+◦R) and Brans-Dicke gravity. Interestingly,
+either no constraints are set on some of the purely teleparallel models or only lightly limited cases are set.
+Importantly, these constraints are consistent with the tensor mode constraints on the functional models.
+These results are promising in terms of the viability of BDLS theory. As future work, it is intriguing to
+explore constraints from tachyonic instabilities. These constraints have been connected to Jeans instability
+where exponential growth of perturbations is slowed by the expansion of the Universe and where the matter
+dispersion relation has a negative mass term that renders it to vanish. This could lead to a more general
+consideration of perturbations about a flat FLRW background spacetime which would then be directly linked
+to observations such as those related to the growth of large scale structure. In forthcoming studies, possible
+observational constraints will be scrutinized.
+ACKNOWLEDGMENTS
+This paper is based upon work from COST Action CA21136 Addressing observational tensions in cosmology
+with systematics and fundamental physics (CosmoVerse) supported by COST (European Cooperation in
+Science and Technology). SC acknowledges the Istituto Nazionale di Fisica Nucleare (iniziative specifiche
+QGSKY and MOONLIGHT2). MC acknowledges funding by the Tertiary Education Scholarship Scheme
+(TESS, Malta).
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+Dynamic Behavior of a Railway Track Under a Moving Wheel
+Load Modelled as a Sinusoidal Pulse
+Mohammed Touati1, Nouzha Lamdouar1 and Chakir Tajani2,∗
+1Civil Department, Mohammadia School of Engineers -Mohammed V University - Morocco
+[email protected], [email protected]
+2Department of Mathematics, Polydisciplinary faculty of Larache,
+Abdelmalek Essadi University, Morocco
+chakir [email protected]
+Abstract
+The aim of this paper is to evaluate the train/track induced loads on the substructure
+by modelling the wheel, at each instant, as a moving sinusoidal pulse applied in a very short
+period of time. This assumption has the advantage of being more realistic as it reduces the
+impact of time on the load definition. To that end, mass, stiffness, and dumping matrices
+of an elementary section of track will be determined. As a result, the equations of motion
+of a section of track subjected to a sinusoidal pulse and a rectangular pulse respectively is
+concluded. Two numerical methods of resolution of that equation, depending on the nature
+of the dumping matrix, will be presented. The computation results will be compared in
+order to conclude about the relevance of that load model. This approach is used in order to
+assess the nature and the value of the loads received by the substructure.
+2020 MSC: Primary 74S05, 37M05, 74-10 Secondary 37N30
+Key Words and Phrases: Dynamic properties, Finite elements modelling, Railway
+track dynamics, Sinusoidal pulse load.
+1
+Introduction
+Various theoretical and experimental researches have been performed in order to assess train/track
+induced loads on the substructure. Mohammed Touati and al. [1] determined the loads induced
+by a non-linear 3D multi-body modelled train on the track with taking into account wheel/rail
+contact properties and track irregularities. Yang Xinwen and al. [2] concluded, through a vehicle-
+track-subgrade coupling dynamic theory and finite element method, about the train/track in-
+duced loads on each layer of the substructure. As an experimental study, Al Shaer and al. [3]
+presented the dynamic behavior of a portion of ballasted railway track subjected to cyclic loads
+in substitution of a moving wheelset. In conclusion, the dynamics behavior of the substructure
+is widely studied in the literature [4-8] based on the train/track coupling model.
+Actually, even if modelling a wheel load as a rectangular pulse is a common assumption, real
+measurements don’t show the same shape. In fact, ONCF (Moroccan railway network manager)
+has many tools that record wheel pulse like GOTCHA. This system shows that the shape of
+the load has never been rectangular, but it’s more likely compared to a sinusoidal pulse. Then,
+this paper deals with evaluating train/track induced loads on the substructure by proposing a
+new approach when it comes to modelling the shape of the wheel impact. Indeed, it’s common
+to consider a moving load as a rectangular impulse applied on the nodes of a mesh structure in
+each period of time depending on signal sampling. This paper shows that assuming the wheel
+1
+arXiv:2301.01524v1  [math.NA]  4 Jan 2023
+
+Figure 1:
+Elementary track modelling
+load as a sinusoidal pulse may reduce the impact of the period of time of its application and,
+consequently, minimize the loads induced on the substructure oversized by the common assump-
+tion. In that matter, a finite element model of the track will be presented and the numerical
+results will be compared.
+2
+Track elementary section modeling
+2.1
+Determination of mass, stiffness et dumping matrices
+Let’s assume a portion of ballasted track composed of two elements of rail considered as a contin-
+uous Euler-Bernoulli beam, fixed to two sleepers by a couples of springs/dampers representing
+the railpads. The ballast is modelled as a couples of springs/dampers under each sleeper (Figure
+1).
+The displacement vector is written as:
+U = [u1, θ1, u2, θ2, u3, θ3, uT1, uT2]
+The effective mass and the stiffness matrices of an element of rail [9], are given by:
+Mr = (ρrArL/420)
+�
+�������
+156
+22L
+54
+−13L
+0
+0
+22L
+4L2
+13L
+−3L2
+0
+0
+54
+13L
+312
+0
+54
+−13L
+−13L
+−3L2
+0
+8L2
+13L
+−3L2
+0
+0
+54
+13L
+156
+−22L
+0
+0
+−13L
+−3L2
+−22L
+4L2
+�
+�������
+Kr =
+�
+ErIr/L3�
+�
+�������
+12
+6L
+−12
+6L
+0
+0
+6L
+4L2
+−6L
+2L2
+0
+0
+−12
+−6L
+24
+0
+−12
+6L
+6L
+2L2
+0
+8L2
+−6L
+2L2
+0
+0
+−12
+−6L
+12
+−6L
+0
+0
+6L
+2L2
+−6L
+4L2
+�
+�������
+2
+
+element1
+blement 2
+ua
+2
+kn2
+c
+"
+ur2
+Masse (m-)
+Masse (m-)where ρr is the density of the rail, Ar is the surface of the rail section, Er is Young modulus,
+and Ir is the rail moment of inertia. The dumping matrix of the rail is obtained as a linear
+combination of mass and stiffness matrices by assuming that the displacements u1 and u3 are
+completely dumped by the effect of railpads.
+Therefore, the dumping matrix is written as:
+C∗
+r = a0 · M∗
+r + a1 · K∗
+r
+where,
+M∗
+r = (ρrArL/420)
+�
+���
+4L2
+13L
+−3L2
+0
+13L
+312
+0
+−13L
+−3L2
+0
+8L2
+−3L2
+0
+13L
+−3L2
+4L2
+�
+���
+K∗
+r =
+�
+ErIr/L3�
+�
+���
+4L2
+−6L
+2L2
+0
+−6L
+24
+0
+6L
+2L2
+0
+8L2
+2L2
+0
+6L
+2L2
+4L2
+�
+���
+a0 and a1 are concluded from the equation:
+� a0
+a1
+�
+=
+�
+2ω1ω2/
+�
+ω2
+2 − ω2
+1
+�� �
+ω2
+−ω1
+−1/ω2
+1/ω1
+� � ζ1
+ζ2
+�
+where ωi2, (i = 1, 2) are the eigenvalues associated to the vibration of the rail described by
+the matrices Mr∗ and Kr∗, and ζi, (i = 1, 2) are the dumping ratios according to the first and
+second modes.
+In one hand, the equation of motion of the rail is written as:
+Mr ¨U ∗ + Cr ˙U ∗ + KrU ∗ = F
+(1)
+where Cr is the transformation of the matrix C∗
+r in the base U ∗, and U ∗ is defined by:
+U ∗ = [u1, θ1, u2, θ2, u3, θ3]
+F is given by:
+F =
+�
+�������
+−ks (u1 − uT1) − cs ( ˙u1 − ˙uT1)
+0
+0
+0
+−ks (u3 − uT3) − cs ( ˙u3 − ˙uT3)
+0
+�
+�������
+In the other hand, the equations of motion of the sleepers are written as:
+� mT ¨uT1 = ks (u1 − uT1) + cs ( ˙u1 − ˙uT1) − kbuT1 − cb ˙uT1
+mT ¨uT2 = ks (u3 − uT2) + cs ( ˙u3 − ˙uT2) − kbuT2 − cb ˙uT2
+(2)
+From (1) and (2), we may conclude about the equation of motion of the track elementary
+section as it’s modelled. It’s written as:
+M ¨U + C ˙U + KU = 0
+where M, C and K are the mass, dumping, and the stiffness of the track elementary section
+respectively.
+3
+
+Symbol
+Quantity
+Value
+ρr
+Rail density (kg/m3)
+7850
+Ar
+Rail section surface (cm²)
+76.70
+Er
+Young modulus of the rail (GPa)
+210
+Ir
+Rail moment of inertia (cm4)
+3038.6
+mT
+Sleeper mass (kg)
+90.84
+ks
+Railpad stiffness (MN/m)
+90
+cs
+Railpad damping (kN.s/m)
+30
+kb
+Ballast stiffness (MN/m)
+25.5
+cb
+Ballast damping (kN.s/m)
+40
+ζ
+Rail dumping ratio
+5%
+Table 1: Track properties
+2.2
+Numerical application
+Let’s assume a track elementary section characterized by the data given in table 1 (we can refer
+to ([10], [11], [12]).
+The figure 2 illustrates the evolution of natural frequencies according to vibration modes. It
+shows that:
+• The frequencies of the 1st and 2nd modes correspond to a movement in phase between rail
+and sleepers. It’s equal to 81.62 Hz;
+• The frequency of the 3rd mode corresponds to a movement in opposition of phase between
+rail and sleepers. It’s equal to 381.1 Hz.
+3
+Track response to a rectangular and a sinusoidal pulses
+3.1
+Description of the studied track
+Let’s assume a section of track composed of N track elementary sections subjected to an external
+load F as it’s shown in figure 3.
+The number of degrees of freedom is given by:
+Ndof = 8N − 3(N − 1)
+The displacement vector is written as:
+U =
+�
+���
+...
+uj,k
+...
+�
+���
+where,
+� uj,k = u∗
+j,k
+where
+k ∈ [1, 8]
+if
+j = 1
+uj,k = u∗
+j,k
+where
+k ∈ [3, 4, 5, 6, 8]
+if
+j ̸= 1
+and,
+U ∗
+j = [uj,1, θj,1, uj,2, θj,2, uj,3, θj,3, uj,T1, uj,T2]
+4
+
+Figure 2:
+Natural frequencies of an elementary track section
+Figure 3:
+Track section modelling
+5
+
+4.5
+X104
+3.5
+(ZH)
+3
+2.5
+2
+1.5
+1
+0.5
+0
+0
+0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+6
+10
+NumerodumodeFigure 4:
+Sinusoidal and rectangular pulses over a period of td
+j refers to the element’s number.
+The mass, stiffness and dumping matrices in the base U are obtained by assembling those
+of a track elementary section determined earlier.
+The vector of loads is defined by:
+F =
+�
+���
+...
+fj
+...
+�
+���
+where,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+N is even
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+N = 2
+� fj = P
+if
+j = 5
+fj = 0
+else
+N ̸= 2
+� fj = P
+if
+j = (5N/2) + 1
+fj = 0
+else
+N is uneven
+� fj = P
+if j = (5(N + 1)/2) − 1
+fj = 0
+else
+P is a rectangular or a sinusoidal load given as:
+• Sinusoidal pulse:
+� P = P0 sin ωt
+if
+t ≤ td
+P = 0
+else
+• Rectangular pulse:
+� P = P0
+if
+t ≤ td
+P = 0
+else
+Its shape is shown in the figure 4.
+6
+
+sinusoidal pulse
+rectangular pulse
+PO
+P
+x=0
+x=td
+X = 2td
+→3.2
+Description of the methods of resolution
+The dynamic behavior of the section of track may be analyzed by modal superposition if the
+dumping matrix verifies orthogonality properties.
+That method is used in particular for an
+undumped system. In that case, the equation of motion is reduced to:
+M ¨U + KU = F
+Let’s assume that ω2
+i are the eigenvalues associated to the track vibration. We note {φi} the
+normalized eigenvectors related to ω2
+i . Therefore, the equation of motion is written as:
+¨Z + diag(ω2
+i )Z = φT F
+(3)
+where diag(ω2
+i ) is a diagonal matrix of the eigenvalues and:
+U = Φ.Z
+The system of equations (3) is uncoupled where each equation is written as:
+¨zi + ω2
+i zi = Φj,iP(t)
+The resolution of that equation is given by DUHAMEL integral:
+zi(t) = (1/ωi)
+� t
+0
+Φj,iP(τ) sin ωi(t − τ)dτ
+Therefore, the solution for a sinusoidal pulse load is given as:
+zi(t) =
+� (Φj,iP0/ω2
+i ).(1/(1 − β2))(sin ωt − β sin ωit)
+if
+t ≤ td
+( ˙zi(td)/ωi) sin ωi(t − td) + zi(td) cos ωi(t − td)
+if
+t ≥ td
+where,
+β = ω/ωi
+and the solution for a rectangular pulse load is given as:
+zi(t) =
+� (Φj,iP0/ω2
+i )(1 − cos ωit)
+if
+t ≤ td
+(Φj,iP0/ω2
+i )(cos ωi(t − td) − cos ωit
+if
+t ≥ td
+The figure 5 shows the response z(t) to a sinusoidal and a rectangular pulse. It’s obvious that
+in the forced phase, the maximum rectangular response is higher than the maximum sinusoidal
+response.
+In general, the dumping matrix doesn’t verify the orthogonality characteristics. Therefore,
+the modal superposition method is substituted by the following method.
+The equation of motion can be written as:
+¨Z + φT Cφ. ˙Z + diag(ω2
+i ).Z = φT F
+(4)
+Where diag(ωi2) and φ are defined earlier. Knowing that:
+˙Z − ˙Z = 0
+(5)
+(4) and (5) could be written as:
+˙Y = D.Y + F ∗
+(6)
+7
+
+Figure 5:
+z(t) response to a rectangular and sinusoidal pulse td/T = 0.75
+where,
+Y =
+� Z
+˙Z
+�
+,
+D = A−1B,
+F ∗ = A−1
+� φT F
+0
+�
+and,
+A =
+� φT Cφ
+I
+I
+0
+�
+,
+B =
+� diag(ω2
+i )
+0
+0
+−I
+�
+Let’s assume that {λi} are the eigenvalues associated to the matrix D. We note {ψi} the
+normalized eigenvectors related to {ωi2}. We define X(t) as:
+Z = ψ.X
+The equation (6) is written as:
+˙X = diag(λi).X + ψ−1F ∗
+(7)
+The system of equations (7) is uncoupled where each equation is written as:
+˙xi(t) = ai.xi(t) + bi.P
+(8)
+where,
+ai = λi
+and
+bi = χi
+and,
+χ = ψ−1
+� φT
+0
+0
+0
+�
+The resolution of the equation (8) gives:
+8
+
+sinusoidal pulse
+rectangularpulse
+2
+-2
+-3
+-4
+X=0
+PI = X
+x = 2td
+PI =X
+X = 4td
+t• Sinusoidal pulse:
+xi(t) =
+�
+biPω
+a2
+i +ω2 eait − aibiP
+a2
+i +ω2 sin ωt −
+biPω
+a2
+i +ω2 cos ωt
+if
+t ≤ td
+xi(td)eai(t−td)
+if
+t ≥ td
+• Rectangular pulse:
+xi(t) =
+� (bP/a)(eait − 1)
+if
+t ≤ td
+xi(td)eai(t−td)
+if
+t ≥ td
+3.3
+Results and discussion
+The figures presented in this section show the numerical resolution of the system of equations
+of a dumped track section subjected to a rectangular and sinusoidal loads. The properties of
+the track are defined in table 1. In figure 6 and figure 7, the sinusoidal pulse is presented in red;
+however, the rectangular pulse is presented in black.
+1. Displacements and rotations of the rail
+2. Displacements of the sleepers
+It’s clear that the maximum values of rail and sleepers movement under rectangular pulse
+are higher than those reached under a sinusoidal pulse. The figure 8 shows the maximum loads
+induced in the substructure. The table 2 shows the repartition of the loads under the sleepers.
+These results have many consequences in the railway field. Actually, we may optimize railway
+infrastructure components for example (like ballast height). Moreover, the study is made by
+considering a static load (10 T). This load is mainly amplified by rail/wheel interaction and
+train speed [1].
+Sleeper
+number
+% of load
+(undumped -
+sinusoidal load)
+% of load
+(dumped -
+sinusoidal load)
+% of load
+(undumped -
+rectangular
+load)
+% of load
+(dumped -
+rectangular
+load)
+13
+4.49%
+1.73%
+5.50%
+-
+14
+11.52%
+6.12%
+13.66%
+6.63%
+15
+23.24%
+15.29%
+26.00%
+16.41%
+16
+30.78%
+22.43%
+34.35%
+23.55%
+17
+23.24%
+15.29%
+26.00%
+16.41%
+18
+11.52%
+6.12%
+13.66%
+6.63%
+19
+4.49%
+1.73%
+5.50%
+-
+Table 2:
+Repartition of the loads under the sleepers (N = 30, td = 0.01s and P = 10 T)
+4
+Conclusion
+Based on the results of the model analysis studied in order to determine the loads induced on
+the substructure, the following conclusions can be drawn:
+9
+
+Figure 6:
+Rail response under sinusoidal and rectangular pulse (N = 4, td = 0.01s and P = 10
+T)
+10
+
+10-
+X10-
+X104
+X10-
+w)
+0
+0.02
+0.04
+0.06
+0.02
+0.04
+0.06
+0
+0.02
+0.04
+0.06
+0
+0.02
+0.04
+0.06
+t (s)
+t (s)
+t (s)
+t (s)
+a) Displacements and rotations of the 1st node
+f) Displacements and rotations of the 6th node
+×10~4
+X104
+(pe))
+X104
+PO'O
+0.06
+0.02
+0.04
+0.06
+t (s)
+t (s)
+b) Displacements and rotations of the 2nd node
+0.02
+0.04
+0.06
+0.02
+0.04
+0.06
+t (s)
+t (s)
+104
+g) Displacements and rotations of the 7th node
+(e)
+X104
+0.02
+0.06
+0
+0.02
+0.04
+0.06
+t (s)
+t (s)
+c) Displacements and rotations of the 3rd node
+0.02
+0.04
+90'0
+0
+0.02
+0.06
+t (s)
+t (s)
+×10+
+X104
+h) Displacements and rotations of the 8th node
+10-4
+10
+5-10
+0.02
+0.04
+0.06
+0.02
+0.04
+0.06
+t (s)
+t (s)
+d) Displacements and rotations of the 4th node
+0.020.04
+0.06
+t (s)
+0.02
+0.04
+0.06
+t (s)
+i) Displacements and rotations of the 9th node
+×10
+X10-19
+-
+0.02
+PO'O
+0.06
+2
+0.02
+0.04
+0.06
+t (s)
+t (s)
+e) Displacements and rotations of the 5th nodeFigure 7:
+Sleeper response under sinusoidal and rectangular pulse (N = 4, td = 0.01s and P =
+10 T)
+Figure 8:
+Loads induced in the substructure (N = 30, td = 0.01s and P = 10 T)
+11
+
+10-4
+×10
+[w)
+-2
+-2
+0
+0.02
+0.04
+0.06
+0
+0.02
+0.04
+0.06
+t (s)
+t (s)
+a) Sleeper 1
+b) Sleeper 2
+×104
+cement (m)
+X104
+-2
+-5
+10
+0
+0.02
+0.04
+0.06
+0
+0.02
+0.04
+0.06
+(s) 1
+t (s)
+c) Sleeper 3
+d) Sleeper 4
+2
+0
+0.02
+0.04
+0.06
+t (s)
+e) Sleeper 50.5
+X104
+-0.5
+-1
+(N) 
+Force
+-1.5
+-2
+-2.5
+Undumped strcture-Sinusoidal pulse
+Dumped strcture -Sinusoidalpulse
+-3
+Undumped strcture - Rectangular pulse
+Dumpedstrcture-Rectangularpulse
+-3.5
+1
+2
+4
+5
+6
+7
+8
+6
+101112
+13141516171819202122232425262728293031
+Sleepernumber• The common modelling of the load applied on the track due to a moving wheel as a
+rectangular pulse acting in the time sample of a force signal generates a higher rate of
+movement in the track and over sizes the loads induced in the substructure than a sinusoidal
+pulse model;
+• Dumping matrix has a major influence on reducing the loads induced in the substructure.
+Therefore, it’s necessary to preserve the quality of the track components while maintaining
+it.
+As an application, we may evaluate the track behavior according to different characteristics
+of the track elements that degrade because of maintenance operations. Indeed, the ballast is
+considered as the most affected element because of operations of damping required for track
+geometry corrections.
+References
+[1] M. Touati, N. Lamdouar and A. Bouyahyaoui: Railway vehicle response under random
+irregularities on a tangent track – nonlinear 3d multi-body modelling. Inter. J. of Mech.
+Eng. and Tech. 9 (2018) 944–956.
+[2] Y. Xiwen, G. Shaojie, Z. Shunhua, S. Yao, M. Xiaoyun: Vertical Vibration Analysis of
+Vehicle-Track-Subgrade Coupled System in High Speed Railway with Dynamic Flexibility
+Method. Transportation Research Procedia 25 (2017) 291–300.
+[3] A. Al Shaer, D. Duhamel, K. Sab, G. Foret, L. Schmitt: Experimental settlement and
+dynamic behavior of a portion of ballasted railway track under high speed trains. J. Sound
+Vib. 316(1-5) (2008) 211–233.
+[4] A. Kaynia, C. Madshus, P. Zackrisson: Ground Vibration from High-speed Trains: Predic-
+tion and Countermeasure, J. of Geotech. and Geo-env. Eng.. 126 (2000) 531–537.
+[5] H. Takemiya: Simulation of Track–ground Vibrations due to High-speed Trains, Proceedings
+of the Eighth International Congress on Sound and Vibration. Hong Kong, China (2000)
+2875–2882.
+[6] C. Madshus, A. Kaynia: High-speed Railway Lines on Soft Ground: Dynamic Behaviour at
+Critical Train Speed, Journal of Sound and Vibration. 231(3) (2000) 689-701.
+[7] S. Qian, C. Ying: A Spatial Time-Varying Coupling Model for Dynamic Analysis of High
+Speed Railway Subgrade, J. of South. Jiao. Univ. (2001) 36(5) 509-513.
+[8] H. Chebli, D. Clouteau and L. Schmitt: Dynamic response of high-speed ballasted rail-
+way tracks: 3D periodic model and in situ measurements. Soil Dynamics and Earthquake
+Engineering, 28(2) (2008) 118-131.
+[9] P. Mario and L. William, Structural dynamics: Theory and Computation, Springer US,
+(2004).
+[10] EN 13481 : Railway applications. Track. Performance requirements for fastening systems.
+Definitions.
+[11] G. Kouroussis, O. Verlinden and C. Conti: Prediction of vibratory nuisances of rail transport
+vehicles. In 6th National Congress of Theoretical and Applied Mechanics, Ghent (Belgium),
+(2003) National Committee for Theoretical and Applied Mechanics.
+12
+
+[12] W. Zhai and X. Sun: A detailed model for investigating vertical interaction between railway
+vehicle and track. Vehicle System Dynamics 23(sup1) (1994) 603-615.
+13
+