A Scalable Inter-edge Correlation Modeling in CopulaGNN for Link Sign Prediction
Authors: Jinkyu Sung
Myunggeum Jee
Joonseok Lee
Venue: ICLR 2026 Poster (2026)
Keywords: Signed Graph Representation Learning
Graph Neural Network
Gaussian Copula
Link Sign Prediction
Linear Convergence
Primary area: learning on graphs and other geometries & topologies

Abstract: Link sign prediction on a signed graph is a task to determine whether the relationship represented by an edge is positive or negative. Since the presence of negative edges violates the graph homophily assumption that adjacent nodes are similar, regular graph methods have not been applicable without auxiliary structures to handle them. We aim to directly model the latent statistical dependency among edges with the Gaussian copula and its corresponding correlation matrix, extending CopulaGNN (Ma et al., 2021). However, a naive modeling of edge-edge relations is computationally intractable even for a graph with moderate scale. To address this, we propose to 1) represent the correlation matrix as a Gramian of edge embeddings, significantly reducing the number of parameters, and 2) reformulate the conditional probability distribution to dramatically reduce the inference cost. We theoretically verify scalability of our method by proving its linear convergence. Also, our extensive experiments demonstrate that it achieves significantly faster convergence than baselines, maintaining competitive prediction performance to the state-of-the-art models.

Published as a conference paper at ICLR 2026
A SCALABLE INTER-EDGE CORRELATION MODELING IN COPULAGNN FOR LINK SIGN PREDICTION
Jinkyu Sung, Myunggeum Jee, Joonseok Lee" Seoul National University i finkkusung, go cmaagg, joonseok]esnu.ac.kk
ABSTRACT
Link sign prediction on a signed graph is task to determine whether the rela- tionship represented by an edge is positive or negative. Since the presence of negative edges violates the graph homophily assumption that adjacent nodes are similar, regular graph methods have not been applicable without auxiliary struc- tures handle them. We aim to directly model the latent statistical dependency among edges with the Gaussian copula and its corresponding correlation matrix, extending CopulaGNN (Ma et al., 2021). However, naive modeling of edge-edge relations is computationally intractable even for graph with moderate scale. To address this, we propose to 1) represent the correlation matrix as a Gramian edge embeddings, significantly reducing the number of parameters, and 2) refor- mulate the conditional probability distribution to dramatically reduce the infer- ence cost. We theoretically verify scalability of our method by proving its linear convergence. Also, our extensive experiments demonstrate that it achieves sig- nificantly faster convergence than baselines, maintaining competitive prediction performance to the state-of-the-art models.
INTRODUCTION
A signed graph refers graph where edges can be either positive (+) or negative (-). In the real world, variety of systems containing positive and negative relationship between entities can be modeled as a signed graph; e.g., social networks (friendly/hostile) or content recommendation (like/dislike) (Liao et al., 2018; Kim et al., 2024; Chen et al., 2024; Kim et al., 2025b). On signed graph, we are interested in the link sign prediction task, aiming at predicting the sign of unobserved edges, given the partially observed positive and negative edges in a graph (Agrawal et al., 2013).
In spite of recent advances in general graph neural networks (GNNs) (Kipf & Welling, 2017; Gilmer et al., 2017), they are not directly applicable to the link sign prediction problem, because the pres- ence negative edges violates the traditional homophily assumption, implying that adjacent nodes similar (Zhu al., 2020). As the regular GNN methods cannot be simply adapted to this prob- lem, researchers have proposed signed graph neural networks (SGNNs) to specially handle negative edges, e.g., graph preprocessing inspired by sociological principles like structural balance and status theory (Derr et al., 2018; Huang et al., 2019; 2021) or separate treatment of negative edges (Li et al., 2020; 2023).
However, these additional auxiliary structures often lead to slow convergence, and in some cases, significantly inefficient use of the memory space. contrast adding auxiliary operations, we shift our attention to a more fundamental aspect: statistical dependency among edges. Instead of assum- ing that the adjacent nodes are similar, we assume that adjacent edges connected via a common node are not informationally independent; that is, they are either in accordance with or in the opposite to each other. This relaxation naturally extends the idea of GNNs from an unsigned graph signed one. Note that our assumption is edge-focused since we target the link sign prediction problem.
similar perspective has been developed by CopulaGNN (Ma et al., 2021) for node regression tasks on unsigned graphs. By learning Gaussian copula (Stute, 1986; Schweizer, 1991) and its corre- sponding correlation matrix, CopulaGNN aims at discovering the underlying dependency structure
"Corresponding author

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Published as a conference paper at ICLR 2026
among nodes in the graph. In this paper, extend the node-centric CopulaGNN framework to the edge-centric task of link sign prediction. This extension, however, is not straightforward due to the following challenges. At training, unlike an unsigned graph where its inter-node correlations can be simply parameterized with a Graph Laplacian based on edge connectivity, a signed graph requires direct modeling of correlations between edges. However, since the correlation matrix quadratically scales with the number of edges, the memory consumption prohibitively grows up to O(1V)*) where is the set of nodes in the graph. Unless the target graph sufficiently sparse, it impractical to directly model this correlation matrix. At inference, CopulaGNN estimates and samples from the conditional probability distribution of the unobserved data given the observed one. This process includes inverting the huge correlation matrix, incurring prohibitively high computational cost.
To overcome these challenges, we propose CopulaI.SP (CopulaGNN for Link Sign Prediction), novel framework that enables efficient and scalable modeling of inter-edge correlations for link sign prediction. Our framework introduces two key components training, we construct the correlation matrix Gramian of edge embeddings. This novel strategy significantly reduces the number of learnable parameters, while still allowing the direct modeling of inter-edge dependencies. At inference, we reformulate the conditional probability distribution using the Woodbury matrix identity (Woodbury, 1950), which simplifies the inversion of matrix with the Gramian structure, transforming the problem into the inversion of considerably smaller matrix. Our proposed reformulation adapts the CopulaGNN to edge-centric tasks, resolving the prohibitive memory and computation costs that make the naive approach intractable.
achieve this scalability. At
Through extensive experiments, we demonstrate that our proposed method converges significantly faster than the baselines. We subsequently provide a theoretical foundation for this empirical ob- servation, proving that our approach achieves linear convergence. This theoretical finding validates that the explicit and scalable modeling of inter-edge correlation is the key driver behind the dramatic acceleration in convergence speed.
2 PRELIMINARIES
Problem Formulation. 9 (V, ] be signed graph with a set of nodes V and edges E. The objective of the semi-supervised link sign prediction task is to infer the missing signs for set of unobserved edges. The set of edges is partitioned into the observed Eohs and the missing Emiss- Each edge e Eobs has its known sign label y E {-1,+11, while it is unknown for the edges in Emiss. We denote the total number of edges by n := -E and the number of observed edges by m := |otsl- For convenience,
assume all n edges (C1, C2,..., come first, followed by the unobserved edges cm++ == (em+1, cm+ ..........,. c.))
ordered such that the observed ones C1am
Sklar's Theorem. According to Sklar's theorem (Sklar, 1959), any multivariate joint cumulative distribution function (CDF) H of n univariate contimuous random variables XI.. can be decomposed into its marginal CDFs Fx(x)) = P(X; S x) and the copula function C [0, 1 [0, 1] by:
H( = (2()))
F2(x2), .... Fn(a.)).
(1)
The copula function isolates and models the underlying dependence structure across the random variables, separating it from their individual marginal distributions.
Moreover, by the probability integral transform (David & Johnson, 1948), the transformed random variable U, : (x,),) follows the standard uniform distribution on [O, 1]. Then, the copula function C can be expressed in the following explicit form, not as composite function of F;:
C(ui..) = H(FFF" (u,) F. (u2), Fr+((u)))
(2)
where u, = Fx)))) If H is differentiable, its probability density function (PDF) H'' is defined as:
0"H
H' Hl(z))) = 0x, 0x2 ... dz. = =((1))))
II [xx,). (=1
(3)
where ft is the PDF of X,, and c(uim)) ==
Ouy S00..... Ou, is the copula density.
Gaussian Copula. We extend CopulaGNN (Ma et al., 2021) for the link sign prediction task, framed as binary classification of edge labels. While various copula functions can model the dependencies between variables, we specifically choose the Gaussian copula C. [0, 1]" + [0. 1] defined as: (u,) C(uia; R) = 4, ( (o-(a(((v -w(1), Φ ) Φ $-((wn); 0, R),
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learnable positive edge negative edge
Parameter Structure Embedding Copula
Joint Label Distribution
Gaussian Copula e
unseen edge
Linear Projection (Eq. 10)
Correlation R
SNEA encoder
Element
Product
Edge Embedding Q
C
Marginals
(n 6
A)
only abserved
Node Embedding
Signed Graph
Gramian (Eq. 8)
(Eq. 9)
Model Training
Model Inference
Maximum Likellhood Estimation
Conditional Gaussian Sampling
Woodbury Reformulation (Eq. 20)
H
Emiss Mobs
Step1. sample 3mm1 for unobserved
(Eq. 9) H. ==
StepZ. predicted Sin+1
Joint Label Distribution (Model)
8m+1 Free1 (()(4)))))
Step3. predict label c
(Eq. 13) [ tm lo Negative Log. likelihood Loss
positive (5m=11
[0,
R
negative (fmes 0.5)
Figure 1: CopulaLSP (our proposed model) architecture and its training, inference process.
where (.; 0, R.) is the multivariate Gaussian CDF with zero mean and its covariance is equivalent the correlation matrix R. and (-) is the inverse CDF of the univariate standard Gaussian distribution. The Gaussian copula density function c(:; R) is naturally defined as:
dcc
c(uini R) = Juy ... dun det exp
= 2
(R-1 (R
I
(5)
where L, := diag(I,
_ax" and == (Φ ( (un), +b - (u2). (
2- (u,,) R..
This choice motivated by its ability to naturally model the underlying dependency structure be- tween edges through the correlation matrix, well-established by related studies (Jia & Benson, 2020; Ma et al., 2021).
3 METHODOLOGY
This section details our approach to effectively model the correlations between edges. We begin by defining the marginal probability distribution for each edge sign, followed by the introduction of our proposed correlation structure. As illustrated in Fig. 1, the marginal distributions and the correlation structure then integrated using the Gaussian copula form the joint probability distribution over all edge signs. Subsequently, we describe the training and inference procedures of our approach.
3.1 JOINT EDGE LABEL DISTRIBUTION AS LINK SIGN PREDICTION MODEL
Edge Label Distribution. To model inter-edge correlation with the Gaussian copula, we first define marginal distribution F for the sign of each edge. Recalling that model the link sign prediction as a binary classification task, the edge signs can be intuitively modeled as a Bernoulli distribution. For the tractability of the joint PDF, we employ the continuous relaxation (Jang al., 2017: Mad- dison et al., 2017) to convert the discrete Bernoulli distribution into continuous one. The resulting PDF of the relaxed Bernoulli distribution given by:
[(x; a, == atx -t-1-11pp 1-1 (1 ---
(ax-1 + (1 xx) -1)2 for x E (0, 1),
(6)
where a and & are the location and temperature parameters, respectively. our model, (0, oo) determines the sign of an edge, depending on whether it is greater or less than The other parameter t E (0. 1) represents the confidence of the sign. Fig. 2 illustrates that when the confidence high (top), the relaxed Bernoulli PDF shows higher density to the direction of the label (here, positive, to the right). On the other hand, when the confidence is low (bottom), the PDF exhibits neg-nggli density not just for the label direction (negative, left), but for the opposite (positive, right).
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Shape of Relaxed Bernoulli PDF f(r;a, and its Semantic
Log-scale Contour Map of Joint PDF
Positive Edge
5.00
a > positive sign
high
strong
f = 0.99
[ closed to 1 strong confidence in pasitive
Gaussian Copula
(corre nottonation 0.5)
Negative Edge
weak
ag 0.70
a < = negative sign
Negatiee
12 0.70
weak confidence negative
Positive Edge
Figure 2: Examples of relaxed Bernuuli marginals and joint PDF using the Gaussian copula
In order to employ the Gaussian copula, we need the closed-form CDF F(;a,t) and its inverse, which are derived as follows (see Appendix A):
x²
1//t
F(x; a) a(1 - 2) rt F (x;a,t) a a-1/t11- - x) / x
for [0,
(7)
Gramian of Edge Embedding as Edge Label Correlation. To represent the dependency structure across the marginals, we require a correlation matrix R E [-1, 1|mxx. However, directly learning Moreover, R must be positive definite, a constraint
this n matrix R is memory-prohibitin required by the det R term in the Gaussian copula density (Eq. (5)). To satisfy these two conditions, construct the correlation matrix R from the Gramian of an edge embedding matrix Q E Randd
R (E) DD- such that E QQ + cIn,
where is the size of edge embeddings, e is a hyperparameter in (0, oc), and v(-) normalizes a covariance matrix into correlation matrix with diagonal matrix D;; := V Recalling that a Gramian is only positive semi-definite, we ensure covariance Σ becomes positive definite by adding a scaled identity term, <I,, Consequently, by Sylvester's 's law of inertia (Sylvester, 1852), R is also positive definite, with its minimum eigenvalue being positively lower-bounded.
We construct the edge embedding between two nodes using the element wise product of the two node embeddings. The node embeddings are obtained from the signed graph using a signed graph encoder, e.g., Derr et al. (2018); Li et al. (2020); Huang et al. (2021); Li et al. (2023).
Our approach is highly memory-efficient for two reasons. First, each edge is represented by vector of a small size (i.e., < n). Second, the model learns only the parameters of the signed graph encoder, rather than the edge embeddings themselves. Additionally, demonstrate that our approach has sufficient representational power to capture inter-edge correlation (see Appendix B).
Overall Architecture. We now have the Gaussian copula with relaxed Bernoulli marginals for edges and the learnable correlation matrix to model dependencies between them. Putting them together, model is given by the following joint distribution:
H(x H(I1m ii Oiin 1im, R) = C(Fi (21ja1,41) (2 a2, 12), .. (Zn Ons bn);; R),
where Fi(:; az t.) is the relaxed Bernoulli CDF of edge e, E E, and C(:; R) is the Gaussian copula.
is noteworthy that while the correlation matrix R [-1, 1| is usually huge, its structure can be represented as the Gramian form thanks to the properties of the normalization function v(-). For this reason, the Gaussian copula C can be implemented with a low-rank multivariate Gaussian distribution without imposing memory bottleneck. (see Appendix C).
The parameters of the marginal distributions, a; and t,, are obtained from the edge embeddings Q through two different learnable linear projections W1, W2 E Rd as follows:
Oin == exp((ww) o0), tin = sigmoid sigmoid(Qw,,) E (0,
(10)
We use exp(-) and sigmoid(-) to satisfy the mathematical domains a, E (0, xc) and t, E (0, respectively. The sigmoid(.) for I is selected as it also performed best empirically.
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3.2 MODEL TRAINING
Notation. Without loss of generality, we assume that the observed edges are indexed before unob- served ones. Based this, we denote each partition of the correlation matrix R follows:
R
Roo
Roi R11
such that Roo E [-11 1)mx and R111 E [-1, ((==m)((-)) (11)
R10
where the subseripts and denote the sets of observed Eabs and unobserved edges Emiss, respec- tively. Thus, the upper-left submatrix Roo corresponds to the training set, where the true signs are known. Also, order to match the domain between our marginal distributions defined on the interval (0, 1) and the observed labels in {-1,+1}. remap the negative labels (-1) to zeros.
Label Smoothing. Eqs. (6) and (7) illustrate the marginal PDF is undefined at the boundary points O and 1, and the CDF evaluates to constants (F; (0;;,,4,) 4,) = and F, 4,) = 1), regardless of its parameters ag and t4. Consequently, this provides no informative gradient for training the parameters of the marginal distributions. To resolve these issues and enable effective training, apply the label smoothing (Müller et al., 2019). We map the original hard labels in {-1, (-1,+1} to the smooth labels, denoted by Ui:
if -1
74
(12)
17 if y/4 +1
where E (0,0.5) is hyperparameter. By preventing evaluation at the boundaries, this method guarantees an informative gradient for the marginal parameters. Similarly, we denote the smooth uniform variable, which serves as the input to the Gaussian copula, as ū, = (iss; (4.
Loss Function. We train our model H. via maximum likelihood estimation by minimizing the following negative log-likelihood loss, formulated with the smooth labels Zy:
C = log H' (J aims 1ims Roo) log
c((1mm Roo) II (i(Bsa,ttt) E=1
2 log det Roo + 22obe (Rro)
Mohs
log fi ai, 4),
(13)
where Zobs == (4 (ū,), $- (u2). + (um)) 1 and Roo [-1, 1mmm is the observed portion of the correlation matrix R as in Eq. (11).
Generalization to Unobserved Edges. Although we train our model using only the observed edges in the graph, the signed graph encoder (implemented as SNEA (Li et al., 2020) in Fig. 1) gen- cralizes this learned knowledge to all nodes. Crucially, we adhere to the standard setting of link sign prediction. Under this standard setting, the objective is to determine the sign of relationship given the premise that the edge structurally exists. Therefore, an wnobserved edge under our context refers to existing edge with its sign unknown, distinct from the regular link prediction task. This formulation allows us to directly compute not only their edge embeddings but also their marginal distributions via the learned linear projections. Consequently, model H is applicable the entire R as in Eq. (9), not just to its observed portion Roo-
3.3 EFFICIENT INFERENCE
Direct Inference. To predict the sign of an unseen edge, we may perform inference by directly sampling from the fitted conditional distribution, given the known labels in Eoha- Relying on fundamental property that conditional distribution of Gaussian given another Gaussian is also a Gaussian, we may perform naive sampling analytically from our Gaussian copula. Specifically, (u,)), the conditional Gaussian distribution from for 2miss == (Φ (um+1). 1), T (um42)," Tb +-'((,)) which we sample is given by:
Zmiss Robs (R Zobs. R11 R10 (Ro)).
(14)
following the CopulaGNN (Ma et al., 2021).
To be more specific, to perform inference on an unobserved edge ej for j E {m + 1.....n] we first draw element of sample, 2, E IR (corresponding edge ej), from the conditional distribution in
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Eq. (14). This element is then converted into probabilistic score within the domain of marginal distribution (; (:: aj, ty), which indicates the probability positive edge sign. The predicted label 0 and the probabilistic score 8j jointly expressed as:
+1
if 8 2 0.5
such that 8y:= = F3 (f());)) 1z).
(15)
if 83 < 0.5
However, this direct sampling from the conditional distribution is prohibitively inefficient, mainly due to the high computational and memory cost of inverting Roo E [-1, 1]mmxm In practice naively performing this inference step on large dataset often leads to out-of-memory.
Woodbury Reformulation for Efficient Inference. To address this inefficiency, we start with the following decomposition for the correlation matrix R. which is enabled by the property that the normalization function v(-) is congruence transformation (Eq. (8)):
R = D ED- PP + K, such that == Q and K = cD
(16)
Then, we apply the following Woodbury matrix identity to R-1 1 to derive the equivalent result, not by inverting huge m m matrix, but much smaller one.
Theorem 3.1 (Woodbury Matrix Identity (Woodbury, 1950)). Provided the matrices are con- formable, the following equation holds:
(A + AUCV = A-1 A----C-U VA -A-'U))-VA"
(17)
Applying the Woodbury matrix identity, the inverse of correlation is reformulated as:
R (K (pT))''=K- - 'P' K-1 such that S = I, + PP. (18)
Therefore, the computational bottleneck is simplified to finding the inverse of a significantly smaller matrix S. Note that the positive definiteness of guarantees the existence of S by the matrix determinant lemmu (Ding & Zhou, 2007). This makes our approach particularly desirable, since the computational overhead relies only on the edge embedding size d, which is controllable hyperparameter, instead of the graph size
With a slight abuse of notation, the correlation matrix R is constructed as:
Roo R10
Roi R11
POP T + Ko P.P.
PoP P.P. K
R
=
(19)
allowing the conditional distribution to be efficiently derived as a low-rank multivariate Gaussian. We term this method the Woodbury Reformulation.
Theorem 3.2 (Woodbury Reformulation). To improve computational efficiency, Eq. (14) can be reformulated to the low-rank multivariate Gaussian distribution as follows:
miss Robs NPP1SP K. Zobsi P + K1).
(20)
Proof. See Appendix D.
4 CONVERGENCE ANALYSIS
In this section, we theoretically analyze the convergence our proposed method. We show that approach achieves linear convergence, which means the error in successive steps of the it- erative optimization process decreases by constant factor. Starting with the formal definition of linear convergence, we prove that our method satisfies this property. Its practical impact the fast convergence is empirically demonstrated in Sec. 5.
Definition 4.1 (Linear Convergence). A function f Ω + R is said to be linearly convergent there exists convergence rate E (0, 1), for each optimization step k N. such that:
f(xx) 8r S ((f((x=1) -f"), where f min f(x). #e
(21)

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If consider our loss function L (Eq. (13)) f above, and iterate Eq. (21) with the gradient descent optimization steps k = 1,.... ; K, then we obtain the following inequality:
LK - L S S (Co C*),
(22)
where Cs indicates the loss value after the step and L is the lowest possible loss. is important to note that term Lo - L can be interpreted as the initial model error before training, while LK - represents the model error achieved after K epochs of training. Therefore, if the loss function achieves linear convergence, it implies that the error of model decreases at an exponentially fast rate (r*) throughout the entire training process. Here, a smaller value of r indicates faster convergence. Consequently, we can verify our hypothesis-thar directly modeling the inter-edge correlation leads to an accelerated convergence rate rate-by proving that our loss is linearly convergent.
To prove the linear convergence of our loss function L, we introduce the following theorem.
Theorem 4.2 (Linear Convergence under the Gradient Descent (Karimi et al., 2016)), A function satiisfyng both L-smoothness and the p-PL PPolyak-Lojasievv a convergence rate r = - |u/ I. under the gradient descent optimization.
condition converges linearly with
We theoretically justify the linear convergence of our model by demonstrating that its loss function satisfies both T--smothnness and p--PI. (Polyak, 1963; Lojasiewicz, 1963) condition, with respect to the correlation R. Here, L-smoothness signifies that the gradient is L-Lipschitz continuous, and μ- PL condition is known as a weaker version of strong convexity. It is worth to note that the Gramian structure (Eq. (8) and label smoothing (Eq. (12)) we introduce are essential Theorem 4.2, which in turn establishes the theoretical soundness of our proposed method. The formal definitions of L-smoothness, μo-P condition, and the proof of Theorem 4.2 are provided Appendix
EXPERIMENTS
To strictly validate the effectiveness of proposed method, conduct comprehensive evalu- ation covering performance and scalability analysis, ablation studies to verify the contribution of each model component, and hyperparameter sensitivity analysis.
5.1 EXPERIMENTAL SETUP
We compare our method with state-of ofate-of-the-an baselines, including GCN (Kipf & Welling, 2017), SGCN (Derr et al.. 2018), SNEA (Li et al., 2020), SDGNN (Huang al., 2021), TrustSGCN (Kim et al., 2023), SL.GNN (Li et al., 2023), SE-SGformer (L.i et al., 2025), and SGAAE (Nakis et al., 2025), on multiple real-world datasets including BitcoinAlpha (Kumar et al., 2016), BitcoinOTC (Kumar et al., 2018), WikiElec (Leskovec et al., 2010b), WikiRfa (West et al., 2014), SlashDot, and Epinions (Leskovee et al., 2010a). We report AUC and (Macro) Fl as our primary metrics, consis- tent with previous studies. All results averaged over 10 distinct data splits ensure statistical reliability. Further experimental details and supplementary results are provides in Appendix F.
Backbone Encoder. Our proposed method is agnostic to the choice of backbone SGNN encoder. To achieve superior scalability in terms of both time and memory compared to alternatives, we se- lect SNEA (Li et al., 2020) as our primary backbone encoder. SNEA is graph attention framework grounded balance theory that learns two distinct representations for each node: a balanced embed- ding and an unbalanced embedding. These embeddings are updated hierarchically using signed graph attentional layer. Specifically, a node's balanced embedding is updated by aggregating the balanced embeddings from its positive neighbors and the unbalanced embeddings from its negative neighbors. Conversely, the unbalanced embedding is updated by aggregating unbalanced embed- dings from positive neighbors and balanced embeddings from negative neighbors. Finally, SNEA concatenates these two learned representations to produce the final node embedding.
5.2 PERFORMANCE AND SCALABILITY
In this section, we evaluate the scalability and link sign prediction performance of our method. We first verify the computational efficiency and convergence speed specifically against the SNEA backbone. Then, we demonstrate the method's efficiency and competitive performance through a comprehensive comparison with ofate-of-the-art baselines.

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Table Performance and scalability comparison: SNEA (backbone) vs. CopulaLSP (ours).
Bitcoin
BiteoinOTC
WikiElee
WikiRfa
SlashDot
Epinions
SNEA
Ours
SNEA
Ours
SNEA
Ours
SNEA
Ours
SNEA
Ours
SNEA
Ours
Epoch converge
325.5
56.7
284.4
65.1
511.2
201.6
505.8
164.1
519.9
514.8
59.7
Traning
Time per opoch (sec)
0.40
0.05
0.61
0.05
2.57
0.08
4.29
0.11
8.99
0.29
12.74
0.41
time (sec)
131.2
174.9
1316.1
2174.1
4676.5
12.4
6574,4
Time speedup
44x
48x
81x
121x
379x
266x
GPU memory (GB)
0.60
0.74
0.68
0.82
1.34
1.47
1,90
2.08
4,96
5.08
7.05
7.20
AUC
0.866
0.864
0,886
0.885
0.872
0.877
0.859
0.862
0.885
0.884
0.898
0.899
Infevence
Macro Fl
0.669
0.716
0.755
0.771
0.754
0.768
0.738
0.751
0.770
0.771
0.809
0.810
Total time
1.12
0.07
1.54
0.07
2.22
0.09
2.87
0.10
5.35
0.25
6.98
0.31
Time speedup
16x
22x
25x
29
23x
23x
GPU memory (GB)
0.55
0.69
0.62
0.76
1,20
1.33
1.68
1.85
4.28
4.40
6.03
6.26
Table 2: Overall link sign prediction performance. OOM indicates out-of-memory.
BitcoinAAlha
BitcoinOTC
WikiElee
WikiRfa
SlashDot
Epinions
AUC
Fl
AUC
F1
AUC
F1
AUC
F1
AUC
F1
AUC
Fl
GCN
0.755
0.562
0.789
0.636
0.746
0.597
0.720
0.569
0.647
0.505
0.760
0.614
SGCN
0.846
0.673
0.880
0.737
0.876
0.749
0.831
0.712
0.737
0.624
0.880
0.752
SNEA
0,866
0.669
0.886
0.755
0.872
0.754
0.859
0.738
0.885
0.770
0.898
0.809
SDGNN
0.831
0.674
0.869
0.750
0.874
0.751
0.856
0.731
0OM
00M
0OM
OOM
TrustGGCN
0.871
0.667
0.887
0,750
0.858
0.724
0.844
0.709
OOM
0OM
00M
OOM
SLGNN
0,864
0,716
0.903
0.818
0.887
0.770
0.874
0.748
0.890
0.771
0OM
OOM
SGAAE
0.839
0.683
0.875
0.759
0.873
0.754
0,851
0.724
OOM
0OM
00M
OOM
SE-SGformer
0.848
0.699
0.866
0.756
0.832
0.714
0.794
0.678
OOM
0OM
OOM
OOM
CopulaI (ours)
0.864
0.716
0.885
0.771
0.877
0.768
0.862
0.751
0.884
0.771
0.899
0.810
Table 3: Overall time and memory scalability comparison on WikiElec and WikiRfa.
WikiFlee
WikiRfa
Training (sec)
Inference (sec)
GPU (GB)
Training (sec)
Inference (sec)
GPU (GB)
GCN
22.4
2,80
27.7
3.62
2.23
SGCN
2261.8
0.08
1.46
4718.8
0.11
2.29
SNEA
1316.1
2.22
1.34
2174.1
2.87
1.90
SDGNN
66.2
5.73
2.20
92.5
7.17
3.79
TrisSGCNN
640.6
29.34
1.71
1561.0
46.60
1.90
SLGNN
59.0
0.06
1.83
87.0
0.09
3.00
SGAAE
40.1
0.57
2.09
37.6
0.97
3.27
SE-SGformer
1136.6
0.62
6.32
1610.0
0.78
15.42
CopulaLSP (ours)
16.2
0.09
1.47
18.0
0.10
2,08
Comparative Analysis with SNEA. As reported in Tab. 1, our method converges significantly faster than SNEA training. This highlights the exceptional time scalability of our method, which is a critical advantage for real-world applications that involve large-scale graphs. Our method is also significantly faster than baselines at inference. SNEA relies on a two-stage procedure, learning node embeddings and then training a separate classifier them, while our method simplifies this process into a single step of computing and sampling from a conditional probability distribution. Regarding memory consumption, our method requires slightly (about 200MB) more GPU memory than SNEA. However, this additional cost is primarily attributable to the linear projection (Eq. (10)) with a fixed number parameters, not scaling with the graph size. To sum up, our method offers a decent trade- off, achieving significantly faster training and inference, paying this modest additional memory cost. Moreover, the performance gain relative the SNEA backbone provides compelling evidence that our correlation modeling yields substantial practical benefits beyond mere computational efficiency.
Overall Performance and Scalability. We evaluate the overall link sign prediction performance and scalability of our method against a broader range of baselines, including the state-of-the-an model SLGNN. As shown in Tab. 2, our model achieves AUC and F1 scores that are competitive with most baselines. This is particularly significant when interpreted in the context of scalability on large-scale graphs. Most recent models encounter out-of-memory (OOM) issues and fail to operate on large datasets like SlashDot and Epinions. In contrast, our method demonstrates remarkable efficiency, as evidenced by the results on the Wiki dataset in Tab. 3. This highlights that our method not only maintains competitive performance but also possesses the robust scalability required for real-world applications, an advantage that remains consistent across other datasets (see Appendix F.1 for additional experimental results).
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Table 4: Ablation on the correlation matrix: Identity (no correlation) vs. Gramian (ours)
BitcoinAlpha
BitcoinOTC
WikiFlee
WikiRfa
SlashDot
Epinions
Conelation
Identity
Gram
Ideatity
Gram
Ideatity
Gram
Identity
Gram
Idontity
Gram
Identity
Gram
0.877
0.861
0.862
0.885
0.884
0,900
FI
0.665
0.713
0,747
0.770
0.737
0.766
0.720
0.749
0.735
0.768
0.782
0.810
Epoch to converge
158.7
55.8
202.2
61.2
136.5
204.6
119.7
135.6
105.3
70.8
127.2
58.2
Training time (sec)
8.21
2.97
10.98
3.41
11.75
16.51
15.72
15.25
35.35
21.87
58.67
24.91
Table 5: Ablation on the Woodbury reformulation. OOM indicates out-of-memory.
BitcoinAlpha
BitcoinOTC
WikiElee
WikRIaa
SlashDot
Epinions
Woodbury refomulation
w/o
w/
w/o
v/o
w/o
w/
w/o
AUC
0.864
0.865
0.884
0.885
00M
0.877
OOM
0.862
00M
0.884
00M
0.899
F1
0.712
0.713
0.772
0.770
0OM
0.766
00M
0,749
00M
0.768
0OM
0.810
Inference (sec)
0.26
0.11
Inference speetlup
4x
9x
Inference GPU usage (GB)
3.12
0.62
6.47
70.18
597.19
4.33
1182.54
6.18
Inference GPU gain
5x
19x 40x
138
191x
5.3 ABLATION STUDY
We conduct an ablation study to validate the effectiveness of model's key components: the Gramian-based correlation matrix (Eq. (8)) and the Woodbury reformulation (Fq. (20)).
Gramian-based Correlation. We compare the performance of using our Gramian- based correlation against fixing it as an identity matrix (i.e., no inter-edge correlation). As shown in Tab. 4, learning the correlation matrix indeed improves the classification performance (FI). Furthermore, it also accelerates the convergence speed (Epoch to converge) in most cases, with the exception of the Wiki dataset. These empirical findings align well with our theoretical convergence analysis Sec. 4.
Woodbury Reformulation. We evaluate the effect of the Woodbury reformulation, which was intro- duced reduce the computational cost of matrix inversion. Tab. 5 confirms that this reformulation leads to a significant reduction in time and memory costs at inference, with no degradation in model performance.
5.4 HYPERPARAMETER STUDY
We primarily analyze the impact of the label smoothing on convergence speed and performance. See Appendix F.1 for more results on other hyperparameters (e and embedding size d).
Convergence Study. We examine how affects model convergence speed. Fig. 3 demonstrates
that increasing y) accelerates the convergence speed. This empirical observation supported by our theoretical analysis; as derived in Appendix E, larger 17 constrains the norm of the latent variables, which consequently lowers the upper bound of the matrix norm | This reduction leads to smaller convergence rate thereby resulting in faster convergence.
Performance Study. We further evaluate the sensitivity of the model performance with respect to Fig. demonstrates that the model maintains robust performance across wide range of 1) values.
Training time (ec)
BitcoinAlpha
Eaech to aataangg
Trainin time (tec)
BItcoinOTC
Trainin time (te))
WiktElec
Epoch to cotverge
15.
50
550
10
200
Epech
10
20
80
100
0.5
20
60
60
100
100
7
1x1001)
(x10-1
n(x10
Traning time bed
Spech to canrerge seed
Doa
Epech carverge Training time secd
Epinions
100
Spech to conrerge
20
200
30
200
100
20
100
6.5
40
60
60
100
7 (x10
(x1001)
(x110-)
Training time (sec)
Epoch converge
Figure 3: Convergence of CopulaLSP varying hyperparameter 17.

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Published as a conference paper at ICLR 2026
BitcoinAlpha
BitcoinOTC
WikiElec
0.95
0.95
0.90
0.8
Macco
0.90
0.8
Macro
0.50
Mocro F1
0.85
0.85
0.65
0.80
0.80
0.80
80
200
80
100
(x100
/1 1x100-)
(x10-
WikiRfa
SlashDot
Epinions
0.95
0.90
Macio F
0.90
Macroo F
0.90
Macro M
0.85
0.85
0.85
0.90
10
100
100
0.80
60
80
100
60
80
1000
7
[x100-
(x
n (K10
AUC
Mecro
Figure 4: Performance of CopulaLSP varying hyperparameter
6 RELATED WORK
Signed Graph Neural Networks. A key challenge in signed graph learning is that negative edges violate the graph homophily assumption. Early work, such as SGCN (Derr al., 2018) and SDGNN (Huang et al., 2021), addressed this by incorporating balance theory (Heider, 1946) or status theory (Easley & Kleinberg, 2010). Subsequently, SiGAT (Huang et al., 2019) and SNEA (Li et al., 2020) further enhanced representation quality by applying the graph attention (Velickovic et al., 2018). MSGNN (He et al., 2022) and SL.NNN (Li et al., 2023) refines representation by emphasizing low- frequency signals through the utilization of graph spectral filtering (Defferrard et al., 2016; Kim et al., 2025a). More recent efforts, such as TrustSGCN (Kim et al., 2023), SE-SGformer (Li et al., 2025), and SGAAE (Nakis et al., 2025), have shifted focus toward reliability and explainabilityy using trust-aware propagation and explainable architectures. Despite these advances, a common limitation across these methods is their separate handling of positive and negative edges during aggregation, which overlooks the statistical dependencies that may exist between them.
Copula in Deep Learning. A key advantage of copula functions (Sklar, 1959; Tagasovska al., 2019) is their ability to separate a joint probability distribution into two distinct components: the univariate marginal distributions and the dependency structure that couples them. This decomposi- tion is particularly powerful for modeling complex data distributions, applied to diverse tasks such as density estimation (Ling al., 2020; Huk al., 2025), representation learning (Wieczorek al., 2018; Lu & Peltonen, 2022), generative modeling (Ng et al.. 2021; Liu et al., 2025), and forecasting on time-series (Drouin et al., 2022: Ashok et al., 2024). For graph representation learning, Copu- laGNN (Ma et al., 2021) ) leveraged a Gaussian copula to explicitly model the correlation structure among nodes. Their demonstrated performance gain provides a strong motivation for our work.
CONCLUSION
In this work, we present CopulaLSP,, a novel and scalable framework that models the inter-edge correlation for link sign prediction. Our approach takes the Gaussian copula to couple marginal relaxed Bernoulli distributions with the edge correlation derived from the Gramian of edge embed- dings. Furthermore, we introduce the Woodbury reformulation to maximize the time and memory efficiency of the sampling-baseed based inference. We not just theoretically verify fast convergence of the proposed method, but also empirically demonstrate its competitive performance and memory ef- ficiency on real-world data. We helieve this work opens promising avenues for applying efficient inter-edge correlation modeling to a wider range of graph-based tasks.
Limitations. We have assumed a static graph structure throughout this work, as it is inherent to semi-supervised learning on graphs. Applying our approach to dynamic and bipartite graphs would be a promising future direction, e.g., for social recommendation systems.
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ACKNOWLEDGMENTS
This work was supported by SOFT Foundry Institute at SNU, Youlchon Foundation, National Research Foundation of Korea (NRF) grants (RS-2021-NRO5515, RS-2024-00336577, RS-2023- 0022663, RS-2025 25--2955223399 and Institute for Information & communication Technology Planning & evaluation (IITP) grants (RS-202--1220264, RS-2024-00353131) by the government of Korea.
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APPENDIX
A RELAXED BERNOULLI DISTRIBUTION
Note that simplified PDF of the relaxed Bernoulli distribution can be written
Ssimple (2; 0, = atxx' (1-x) (a(1 - x) +
and remark that is well-defined (0, 1).
Now, we can show that CDF of relaxed Bernoulli distribution F(-;a, a, () is well-defined by proving dF/dr fsimple (z; a, t). For the sake of simplicity, we introduce Q1 == x', Q2 = a(1 x),
Q3 = Q1 + Q2, and Q == dQ./dz.
From the quotient rule, we obtain:
dF' d Q1 1:03-Q194 (Qa)²
-
dr dx Q3
A straightforward calculation shows that:
QiQ== - Qid's = txx [(xt + a(1 x)²) - (x* ax(1 - x²)-
tx 1- [a( - x²² + ax - x)'-
= txx=-1 a(1 x)) [[1 - x) + +
= atx''1 (1 - x)--1
Therefore,
dF
atx' 1 (1- x)(=1
= Ssimple (2; 0 t).
dx (a(1 - x)x + x²)²
The derivation of the ICDF from the CDF is considerably more intuitive. Consider the following:
yd
F(y,a, 1) = x- a(1
if y + 0
1)
+ vt
a(1/y 1y
1
a
1
1
1
= a -1
(1 (--1)
11 1/1
1/t
= 1 a-1 1/t ( - 1) 1/
r²/t a 1/0(1 x) - (I;
F-
B REPRESENTATIONAL POWER OF GRAMIAN-BASER CORRELATION
We experimentally demonstrate that our Gramian-basee correlation design (Eq. (8) can behave as an ideal correlation matrix. To validate our approach, we define an ideal correlation R. E 1-1,0, 0, 10 1 derived from the entire graph dataset follows:
0 if e; ej have no shared node,
R
+1 if e,, ey have shared node and y Vj*
(23)
1 if dee, ey have shared node and y: # yj
where Vc. Mj ff1,,+1 +1} are the true labels of the edges C,, ej E E, respectively.
However, this matrix is an pos-dd correlation matrix because it is not guaranteed to be positive definite. To address this, we leveraged the theory of the nearest correlation matrix problem (Higham,

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Published as a conference paper at ICLR 2026
2002: Borsdorf & Higham, 2010). We solve the following optimization problem using gradient descent to find an approximated ideal factor Q E R=xd:
Q argmin Il (XXT + +el) RR XeRoxd
(24)
Now we obtain the approximated ideal correlation, R = v(QQQ(QQ@00 +eI), based on our Gramian- -based correlation design (Eq. (8)). Formally, R is the nearest (valid) correlation matrix of R
We then plug and fix this approximated ideal correlation into the Gaussian copula in our model to evaluate its performance. Notably, we observe near- pear-perfect performances, achieving an AUC of 0.99 and F1 of 0.95 on the BitcoinAlpha dataset.
This results show that the ideal form of correlation matrix is R", and our Stamian-bassee correlation design has sufficient representational power to mimic the ideal correlation matrix R
C LOW-RANK GAUSSIAN DISTRIBUTION
A low-rank Gaussian distribution refers to multivariate Gaussian distribution whose covariance matrix is expressed in the form A + B, where is a covariance factor (matrix) and B is covariance diagonal (matrix). This low-rank covariance structure allows for the computationally efficient implementation of algorithms due to the Woodbury matrix identity, For details, refer to the wowRankbultivariiteeeeo..
class in the PyTorch library'.
It is worthwhile to note that the Gramian-based correlation R can be derived as: R D- PP K such that P : D- and K := :==
(25)
Therefore, the Gaussian copula function of our method can be implemented in a computationally efficient manner.
also holds true for inference. When considering the Woodbury reformulation in Eq. (20), the correlation term of the re-derived conditional probability distribution is equivalent to P,S. P + K₁ Since So1 is positive definite and symmetric (thus allowing for Cholesky decomposition). P,S, PT K₁ can be re-derived into the following low-rank form:
P.Sp'P, 'P, + K PLLL [ + K₁ (P,L)(P,L) +K1
(26)
Again, the conditional distribution can be implemented efficiently.
D WOODBURY REFORMULATION
Theorem D.1 (Woodbury Reformulation). To improve computational efficiency, Eq. (14) can be reformulated to low-rank multivariate Gaussian distribution as follows:
Zuiss Zobs (PPPS Kn Zobs, P 'I. + Ki).
(27)
Proof. We aim reformulate the mean R10 Zobs and covariance Ru R1o Roi of the conditional distribution Zmiss Zobs-
First, we apply the Woodbury matrix identity to R Given the definition Roo = Kg + P.P. its inverse is:
R K2 K PL + -P.
(28)
Here, by defining So = I + Po K Po. the expression simplifies to:
R K
(29)
The conditional mean Rio Uniss. We substitute Rio0 = P.P.T and the derived expression for Rnd into this equation.
R1oR Zobs = (P.P.) (K,' K, PoS, ; P K, ') Zobs
-P,P K - P.P. K PP.S PP_ K Zobs
-P -PP,P SP) K Zoba-
'https://dccs. http://///oo..ppoorrroooo///eee stt.c/cccssssalleeeeeeeeeeecccea staale/diiitt e/diiee/iiiss www.diitrriioo sttitttrriuuiiii ons.i html
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From the definition of Su, the identity Po P Po = So La and substitute it into the expression.
R Zobs
(PiP) Pi - P0 K2
Zobs
= (PiPo K2 Pi - / Pp K Zobs
= (PIP. K. P.(L. S K Zobs
-PiPo K. (PIP. PiS ko)))
Zobs
= P,S Ko
Zoba
The conditional covariance is R11 R10 Rno Ro1- We substitute R11 = P.P K1, R10 = P P,Po and Roi = P
R11 RIo Roi (PiP) K2) (P.P) R (PoPT).
(30)
From the derivation of the mean, showed that R1o R P.P P.S.'PP K substitute this result.
We
R1 Rin R.
00 Roi
PP.P K1) (PiS K2
(P)pP) (PoP))
= K₁ P1S (Po K, PP)
Again, we use the identity Po Kr Po So Idd
R11 R Roı P,P. + K1 P1S, (S, PTP
P + K1 PI So S PT
P.P. + K₁ - P S P
P.P. + +K₁ (PPP - P,S PPT)
-P,PI +K, P.P PPS. P
PiS, PT K
By combining the derived mean and covariance, the conditional distribution is reformulated as:
Zuniss Zobs ~ NPP.P Ko Zobs, P1S PP + K2).
E CONVERGENCE ANALYSIS
We introduce the formal definitions of L-smoothness and μ-PL (Polyak-Lojasiene)
condition.
Definition E.1 (L-smoothnesss) differentiable function f + RR is said to be L-smooth its gradient V f is L-Lipschitz continuous. In other words, there exists constant L > 0 such that:
[V) fyyil < L||x yl
Vz, y E Ω.
(31)
Definition E.2 (p-PL Condition (Polyak, 1963; Lojasiewiez, 1963)). Suppose a differentiable func- tion f E has minimum function value f*. Then, we say f satisfies the p-PL condition the following holds for some > 0:
[ff()) #(f(x) - f*)
Va Ω.
(32)
To examine the effect correlation on model convergence, we begin our proof with the gradient of the loss L (Eq, (13)) with respect to R:
VR R- R--C-RR---R---- RR1
(33)
should be noted that for the sake of simplicity, we have omitted the notation specifically introduced for the observed data.
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Lemma E.3. VRC is Lipschitz continuous.
Proof. Suppose that arbitrary two correlations R E R are given, where R. is set of our learnable correlations. Note that || |l the spectral norm operator and || || is the Frobenious norm operator. From the triangular inequality, we obtain:
2 V(VC)()) Vn & ||R - Rl' + |R_ 22 R R 22 R 1r. (34)
As our design covariance matrix Σ (Eq. (8)) constrains the minimum eigenvalue to c > 0. and so any correlation matrix R R also has a positive minimum eigenvalue by Sylvester's law of inertia (Sylvester, 1852), we can assume |R- S a for some constant a > 0. Since RT R
R R,-1(R, (R2 - R1 R we can utilize a sub-multiplicatiivty of matrix norm as follows:
R R lle & R
122 - R1Ila R lle S a² R1 - -R2l
(35)
Similarly, we can also derive:
J = R zz R 'z
zz T R = R2 zz (R²² -RT (R²² R
zz
R
(36)
Again, by sub- sub-multiplicatiiity:
|||| <
T || RR, R.
|| + || R| [R
(37)
Moreover, we can assume ||zz || S for some > 0. Consider the definition of 2, i.e. 2, t (w;). Since is derived from a univariate Gaussian CDF ((-) by the smooth label (Eq. (12)) which restricts the support of (-) to [7, 17], we can say u, closed and bounded. In hence, x- (u,) is also bounded and this implies ||zz || is bounded. For instance,
check
( 1 (le - 5) < 4.3, and exemplifies assumption [zz
|| S 3 is not strong. Hence, we have:
|| v 2 R - R S 20 3 R R2Ilr-
(38)
Therefore, VR is Lipschitz continuous with respect to R:
2[,,R((R,) - VRC(R2)] S || RI2²1 || ||]|
(39)
I < R2 - RRllr + 2a 2a B| R2ll
(40)
(a 2 2a 22 B)|R RellE.
(41)
Lemma E.4. L satisfies the Polyak-Lojasiewicz condition with respect to R.
Proof. Redefine with A = R-1 as follows:
((A) = L(R) = = log det A + = T (A - I)z.
(42)
Consider the gradient of å with respect to A:
1 VAC - _22 = RR(RRC)
(43)
We can utilize sub-multiplicativity of matrix norm as follows:
| F ||((RR()R|| < | Amax (R) V S IlI
(44)
where Amax RR) is the maximum eigenvalue of correlation matrix R. and remark that the maximum eigenvalues of correlation matrix has upperbound, in fact, which is m. It is worth noting that z Az is trivially convex, and - log det A is strongly convex with respect to A since R has a positive minimum eigenvalue. So, 2 is also strongly convex. Karimi et al. (2016) shows strong convexity implies a satisfaction of Lojsaiawoojsiiiiiz condition. Hence, E satisfies the Polyak-L.ojasiewicz condition for some constant ğ
|AACC ((((A) - -Z*),
(45)
2
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where L is the minimum value of É. Note that L is exactly same to the minimum value of L, denoted by L And so, we obtain the inequality from the Eqs. (44) and (45):
1 m VR
|V M(C(A) E*) (
(46)
2
Therefore, L satisfies the Polyak- Soayak-Lojasiwwizz condition with respect R:
|R
An me (C(R) -C)
(47)
Combining Lemmas E.3 and E.4, we conclude our loss function & converges linearly with a conver- gence rate E (0, 1):
2/r
1 m²(a2 + 2)
(48)
Notably, the practical design choices for model's implementation--peiiiiee correlation matrix and the label smoothing technique- play a pivotal role in these proofs. This demonstrates strong coherence between the practical design of our methodology and its theoretical underpinnings, suggesting that they are mutually reinforcing.
the structured
F MORE DETAILS ON EXPERIMENTS
Configurations. To ensure the reproducibility of scalability and performance, specify our key software and hardware configurations. We implement our model using PyTorch 2.7 with CUDA 12.4. The majority of experiments are conducted NVIDIA A6000 GPU (48GB VRAM), with the exception of the ablation studies, which are performed on an NVIDIA A5000 GPU (24GB VRAM).
Dataset and Preprocessing. To make sure our experiments are fair and can be replicated, we include our entire preprocessing pipeline in our code, which works directly with the data files from their original sources. Our preprocessing process has two main steps. First, we treat the data an undirected graph. If we find confficting signs for reciprocal edges between two nodes in the raw data, we clear up this confusion by giving the edge negative sign. This is based on the fact that negative edges are much less common across all datasets. Second, we pull out the largest connected component from each graph. The original graphs usually have one main component along with several smaller, disconnected subgraphs. We consider these smaller components noise and leave them out-this is a common approach in graph-based research that helps us focus on structurally coherent subgraph. You can find the statistics the preprocessed datasets we used for our experiments in Tab.
Table I: Statistics of the preprocessed datasets.
Dataset
Nodes
Positive Edges
Negative Edges
Positive Ratio
BitcoinAlpha
3,775
12,721
1,399
90.09 (%)
BitcoinOTC
5,875
18,230
3,259
84.83 (%)
WikiElee
7,066
78,441
22,253
77.90
WikiRfa
11,259
132,187
39,415
77.03 (%)
SlashDot
82,140
380,933
119,548
76.11 (%)
Epinions
119,130
583,957
120,462
82.90 (%)
Baselines. Graph Convolutional Networks (Kipf & Welling, 2017; Gilmer et al., 2017) laid the foundation for message passing on graphs by aggregating neighborhood information under the ho- mophily assumption. However, this assumption is violated in signed graphs due to the presence of negative edges. To address this, SGCN (Derr al., 2018) generalized GCN based on balance the- ory, and SDGNN (Huang et al., 2021) extended this idea to directed signed graphs by incorporating social theories into aggregation and loss functions.

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Published as a conference paper at ICLR 2026
With the growing interest in attention mechanisms, SNEA (Li al., 2020) adaptes GAT (Velickovic et al., 2018) to signed graphs, enabling the model to learn separate attention weights for positive and negative neighbors. To the best of our knowledge, SLGNN (Li et al., 2023) represents the current state-of-the-ar in signed graph neural networks, combining graph spectral filtering and SGNNs. Its self-gating mechanism adaptively captures information from both balanced and imbalanced struc- tures.
Other approach is introduced by TrustSGCN (Kim et al., 2023), which incorporated trustworthi- ness measures to weight message passing more reliably than using raw edge signs alone. Be- yond performance-orientt
approaches, recent works such as SE-SGformer (Li al., 2025) and SGAAE (Nakis et al., 2025) explore explainability by leveraging Transformer architectures and auto-encoding techniques better interpret signed network behaviors.
Nevertheless, common limitation across these methods is their separate handling of positive and negative edges during aggregation, which overlooks the statistical dependencies that may exist be- tween them. Motivated by this, we propose to directly model these inter-edge relationships by learning a correlation matrix. This approach not only provides a more holistic view of the graph structure but also, our results show, leads to dramatic acceleration in model convergence.
Details of Experimental Setting. One of the major hurdles in the field of link sign prediction is the absence of a standardized evaluation protocol. Previous studies often vary in crucial elements that affect performance, such as how data is split and how features are managed, which makes it tough to make fair comparisons. To tackle this problem, we unify experimental options to reassess baseline models using their official implementations. This protocol is guided by the following principles:
To simulate real-world node attributes, which are often absent in publicly avail- able signed graph datasets, we initialize node features from a standard normal distribution with- out scaling. Consistent with this simulation of pre-exte eristxxiting attributes, these features remain fixed (i.e., non-learnable) throughout the training process.
Node features:
Data split: We establish robust evaluation protocol to ensure the reliability and generalizabil- ity of our results. First, we partition the data into training, validation, and test sets using an 8:1:1 ratio. This deviates from the 8:2 split used in some prior work by incorporating a dedi- cated validation set, which is crucial for systematic hyperparameter tuning and early stopping to mitigate overfitting. Furthermore, to account for performance variance arising from data splits, we conduct each experiment on 10 distinct data splits. The final reported metrics are the aver- age of these runs. This multi-run evaluation provides more stable and accurate assessment of model generalization, a step that is particularly vital for smaller datasets like BitcoinAlpha and BitcoinOTC where results can be sensitive to the specific data split.
Linear probing: Current research in link sign prediction is broadly divided into two paradigms: end-to-end frameworks with integrated classifiers (Derr et al., 2018; Li et al., 2023; 2025), and two-stage approaches that first learn node embeddings for evaluation with a separate classifier (Li et al., 2020; Huang al., 2021). This methodological divergence presents challenge for direct and fair performance comparisons. To address this, we adopt a standardized evaluation protocol. For end-to-end models, we utilize their native classifiers as designed. For all two-stage, umbedding-basee methods, we standardize the downstream task by employing single, fixed logistic regression classifier from the scikit-learn library to evaluate the quality of the learned representations.
F.1 ADDITIONAL EXPERIMENTAL RESULTS
Embedding Size Study. We report the effect of different embeeding sizes d on model performance and scalability in Fig. I. As expected, a larger embedding size tends to demand more computational resources (e.g., training time per epoch, inference time, and GPU memory). However, a larger embedding size does not necessarily increase the total training time, as often leads to faster con- vergence in fewer epochs. Although the model remains robust to variations in d overall, instances of underfitting or overfitting are observed in specific datasets where d is extremely small or large, respectively.
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AUC
Macro
Epoch to converge
Time per epoch (sec)
0.95
400
1.0
0.5
64
256
128
256
16
12a
216
32
256
Embedding Size Training time (sec)
Embedding Size
Embedding
Embedding Size
Inference time (sec)
Training GPU (MB)
Inference GPU (MB)
20000
60
0.50
20000
30
0.25
10000
64
128
254
128
16
128
256
52
256
Embedding Size
Embedding Size
Embedding Size
Embedding Size
BitcoinAlpi
BitcoinOTC
WikeBee
Wikit/s
SlashDot
Epinioon
Figure Embedding size study of CopulaI.SP.
Performance and Convergence Study on 6. We investigate the sensitivity of the performance and convergence with respect to variations in As indicated in these tables and visualized in Fig. II, the model maintains consistent performance across a broad range of parameter values, confirming its robustness. As shown in Fig. III, discernible trend is observed in convergence speed concerning changes in e. The optimal hyperparameter configurations for each dataset are detailed Tab. II.
BitcoinÁlpha
BitcoinOTC
WikiElec
0.90
0.8
Merro
0.90
AVC
0.90
Nattro
0.85
0.85
0.85
150
550
200
200
(X110 ?)
[x110 ")
[x10 "1
WikiRfa
SlashDot
Epintons
0.95
0.95
0.95
0.8
AUC
0.90
0.8
AVC
MACIOO F1
9.85
0.80
0.50
0.80
8
80
100
150
90
100
150
200
100
150
200
(x100-
((x10->)
[
AUC
Macro F1
Figure Performance of CopulaL varying hyperparameter e.
BitcoinAlpha
Traning time (se))
10
BitcoinOTC
Epoch to converge
Taaingg time (eo))
WikiElec
Epoch converge
Traning time
100
100
Epoch to
1.5
100
10
20
50
19
80
100
50
100
150
200
10
60
100
150
C(x10
[x110
WikiRfa
training time (we)
SlashDot
Spoch to converge Training time (we)
150
Epinions
Spach comwagee
Trainin time
80
200
20
50
50
3
Spech
90
500
150
200-
50
100
150
200
70
00
100-
250
200
x(x10-))
(x100-)
(xx10-))
Teaining time (Sec)
Epoch to converge
Figure III: Convergence of CopulaL varying hyperparameter
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Table Best hyperparameters and r for CopulaLSP.
Dataset
Best
Best 1/
BitcoinAlpha
0.04
0.0008
BitcuinOTC
0.05
0.0001
WikiElec
0.04
0.005
WikiRfa
0.04
0.01
SlashDot
0.04
0.01
Epinion
0.04
0,001
Scalability Analysis Other Datasets. To demonstrate the robustness of our method beyond the Wiki dataset analyzed the main text, we present scalability comparisons for other datasets.
Table III: Overall time and memory efficiency comparison on BitcoinAApha and BitcoinOTC.
Altc
BitcoinOTC
Training (sec)
Inference (sec)
GPU (GB)
Training (sec)
Inference (sec)
GPU (GB)
GCN
5.3
2.43
0.61
6.86
3.12
0.68
SGCN
136.3
0.07
0.62
341.46
0.07
0.72
SNEA
131.2
1.12
0,60
174.9
1.54
0,68
SDGNN
9.4
3.99
0.84
12.81
4.75
1.12
TrustSGCN
161.0
2.22
1.22
322.84
4.52
1.31
SLGNN
0.02
0.72
27.34
0.03
0,81
SGAAE
14.5
0.09
0.77
16.76
0.13
1.13
SR-SGformer
236.8
0.48
2.13
$23.19
0.40
4.51
CopulaLSP (ours)
3,0
0.07
0.74
3.6
0.07
0,82
Table IV: Overall time and memory efficiency comparison on SlashDot and Epinions. OOM indi- cates out-of aut-of-memory.
SlashDot Inference (soc)
Epinions
Training (soc)
GPU (GB)
Training (scc)
Inference (scc)
GPU (GB)
GCN
73.0
5.90
5.24
237.80
16.60
7.17
SGCN
5557.4
0.22
5.72
15008.90
0.28
8.48
SNEA
4676.5
5.35
4.96
6574.4
6.98
7.03
SDGNN
0OM
OOM
00M
OOM
OOM
OOM
TrustGGCN
0OM
OOM
OOM
OOM
OOM
OOM
SLGNN
281.8
0.23
33.38
OOM
OOM
OOM
SGAAE
OOM
OOM
OOM
OOM
OOM
OOM
SE-SGtormer
OOM
OOM
OOM
OOM
OOM
OOM
CopulaL (ours)
12.4
0.23
5.08
24.7
0.31
7.20
F.2 MODEL ANALYSIS ON SYNTHETIC DATASET
Model Performance on Synthetic Dataset. To demonstrate the superiority of CopulaLSP over node-centric models, utilize synthetic dataset comprising 40 nodes divided into distinct communities, as illustrated in Fig. IV. Nodes (-19 constitute Group 1, while nodes 20-39 consti- tute Group The graph is characterized by strong intra-group cohesion and inter-group hostility; specifically, it contains 183 and 178 intra-group positive edges for Group and Group 2, respec- tively, alongside 76 inter-group negative edges. The edge indices sorted such that intra-group positive edges appear first, followed by inter-group negative edges.
We train both our method and SNEA on this dataset for 10 epochs using an 8:2 train-test split. The experimental results reveal distinct contrast. SNEA predicts all edges as positive, failing to iden- tify inter-group hostility, whereas our method achieves perfect scores in both AUC and FI. This performance discrepancy stems from the node-centric mechanism of SNEA, which generates em- beddings by aggregating topological information from neighbors. Since both groups possess nearly identical intra-group structures, nodes in opposing groups generate virtually indistinguishable em- beddings despite being connected by negative edges. Consequently. SNEA fails to discern negative edges within such structurally symmetric configurations. In contrast, method effectively resolves this ambiguity by modeling the signs of individual edges via marginal probability distributions and explicitly capturing the correlations between edges.
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35
Figure IV: Topology of the synthetic signed graph comprising two symmetric communities.
Model Component Analysis Synthetic Dataset. To clucidate how CopulaLSP successfully distinguishes edge signs where baseline models fail, we visualize the learned components edge embeddings, marginal probability distributions, and the correlation matrix-in Fig. V.
First, the PCA visualization of edge embeddings demonstrates a clear separation between positive and negative edges. Notably, the positive embeddings form two distinct yet symmetrical clusters, accurately mirroring the underlying topology where two groups possess identical structures but dis- tinct community memberships.
Second, the analysis of the location parameter E (0, 0o) validates the efficacy of the relaxed Bernoulli distribution. We observe that is generally greater than for positive edges and less than for negative edges. Although a small subset of edges exhibits parameters misaligned with their actual signs, our model effectively mitigates these local discrepancies inference. This robustness achieved by incorporating the temperature parameter t (0. 1) and, more importantly, by leveraging explicit inter-edge correlations to compensate for individual uncertainties.
Finally, the estimated correlation matrix provides insight into how the model captures structural dependencies. Given the edge ordering uutra-Group 1, intra-Group 2, and inter-group edges), the matrix exhibits strong positive correlations within the diagonal blocks, confirming that edges within the same structural category share high mutual information. Interestingly, we observe weak posi- tive correlations between the blocks of Group and Group 2; this reflects that, despite belonging to conflicting communities, the two groups share topologically identical structures. Crucially, the correlations between intra-group (positive) and inter-group (negative) edges predominantly neg- ative. This indicates that CopulaLSP successfully learns to distinguish edges that share common nodes but possess opposing semantic meanings, thereby resolving the ambiguity that confounds node-centric models.
Edge Embeddings
Marginal Parameters
Inter-edge Correlation
1.00
Eage Sign Pookic Negative
Edge Sign
0.54
Pssitive
0.75
Negetive
0.4
0.50
0.2
0.25
PCA (dim
0.0
0.50
5009
0.00
-0.2
-4.25
-0.50
0.46
-.75
Edges
1.00
PCA (dim 1)
0
Figure V: Component analysis of CopulaLSP on synthetic dataset.
ix


Reviews:
[{"review_id": "spQV8j87Nk", "reviewer": "Reviewer_bz2b", "summary": "The authors study a problem of graph learning on signed graph, where existing GNNs struggle due to the homophily assumption. Thus, the authors borrow the idea from CopulaGNN. However, directly applying CopulaGNN is not applicable due to computational limitations. Two main ideas were introduced to tackle the limitations. First, (at the training phase) the correlation matrix is constructed as a Gramian of edge embeddings, which dramatically reduces the number of parameters. Second, the conditional probability distribution has been reformulated for reducing the inference cost. Experimental results show how CopulaLSP achieves significant reductions in computation time while maintaining the predictive performances from competitive models.", "questions": "Q1. The notion of missing edges can be further clarified. In conventional GNN or in graph learning models, missing edges refer to unobserved relationships. \n\nQ2. I wonder how their ‘label softening’ is different from ‘label smoothing’ which is well-established term. \n\nQ3. Can you further elaborate why the correlation matrix should satisfy the condition in line 180-181? It is unclear why this condition should be added supposedly matrix R is the same from line 127-128 which is directly borrowed from CopulaGNN.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "- The main problem/ task is one of the important topic in GNN. The problem formulation (in Section 2) is clear. The paper is well organized and relevant theorem has been provided to support the statements.  \n- The main idea: using the CopulaGNN for signed graph learning has been well justified theoretically. The idea of reducing the computational cost in two ways: training phase and inference phase, is straightforward and supported well. \n- For the experimental results, the computational reduction, both in time and memory, is significant. Ablation study was conducted thoroughly, which supports the efficacy of CopulaLSP.", "weaknesses": "- Some of the details are missing requiring further clarifications.(see questions)  \n- How does the low-rank multivariate Gaussian brings, in what extent, how significant ? The authors simply state that Woodbury reformulation is used to improve computational efficiency.  \n- The proposed model uses SNEA as their backbone and show the improvements in computational efficiency. However, from practical point, considering that the best prediction performances are achieved among TrustSGCN, SLGNN, further computational comparison in the Appendix can be added in Section 5. In some sense, the achievement (437 times faster convergence than baseline) can be viewed as an overstatement considering that the baseline is SNEA.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZKq0E7CIIn", "reviewer": "Reviewer_6GuR", "summary": "The paper proposes CopulaGNN for Link Sign Prediction (CopulaLSP). It builds on the prior work CopulaGNN for the task of node regression, the core idea of which is the model dependencies between nodes by learning a Gaussian copula. Here the model must instead learn correlations between edges, which is intractable with a naïve approach, so the authors propose two ideas to deal with this. First, the correlation matrix is parametrized as the Gramian of edge embeddings (which are learned with a prior GNN model for signed graphs, SNEA). Second, a Woodbury matrix identity is used to transform the inference-time inversion of a matrix of the size of the number of observed edges, to the inversion of the matrix of the size of the embedding dimension. Furthermore, the authors prove that their approach converges linearly. Finally, the method is evaluated against other recent methods on 4 real-world datasets, showing competitive prediction performance and faster convergence.", "questions": "- A \"relaxed Bernoulli distribution\" is defined in Eq. 6 as a continuous analog of the Bernoulli distribution. Has this distribution been studied before? Why are other, more commonly-used distributions on $(0,1)$, like the beta distribution, not suitable?\n- Have you evaluated using the graph Laplacian of the line graph for edge correlations, as an analog to use of the graph Laplacian itself in CopulaGNN?\n- The question about the modeling side of the approach above.\n- How does CopulaLSP work with other backbones than SNEA? What is preventing applying the CopulaLSP on top of the strongest prior method, SLGNN? Relatedly, what would be the performance if dropping the GNN encoder backbones and learning a simpler encoder (e.g., just a linear transformation or MLP of the node features, or directly learning node embeddings if there are no node features)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper has a clear and significant narrative: It considers the natural approach of edge-edge correlation modeling, which has some general relevance in graph modeling, identifies the scalability problem, and provides a solution.\n- The writing is clear and grammatical. The diagram of Figure 1 is helpful for understanding the core concepts.\n- The paper includes conceptual, theoretical, and experimental components.\n- Several ablations are provided to strengthen the paper's claims.", "weaknesses": "- The method is competitive with others in terms of prediction performance, but not clearly superior and arguably inferior to one other method on the chosen datasets.\n- The diversity of the datasets is limited, with two Bitcoin datasets and two Wikipedia datasets. \n- There is little discussion of how the prior methods work, including SNEA, which is used as the encoder backbone in this paper.\n- There is little discussion of the modeling side of the proposed approach, e.g., what do the learned embeddings, linear projection, and Gramian look like, on real or toy data?\n- The convergence result is welcome, but it seems to simply be an application of a prior result to the proposed loss function. I would favor more space in the main body of the paper on the prior two points.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NiAYoim634", "reviewer": "Reviewer_y8s8", "summary": "This paper introduces CopulaLSP, a scalable framework for link sign prediction in signed graphs. Based on the assumption of statistical dependencies between edges, this paper overcomes the inability of traditional GNNs to handle negative edges, extending the architecture from unsigned graphs to signed graphs. It uses a Gramian of lower-dimensional edge embeddings to model the correlation between edges, which significantly reduces the number of learnable parameters and memory consumption. Furthermore, it reformulates the conditional probability distribution using the Woodbury matrix identity, transforming the matrix inversion required during inference into the inversion of a smaller matrix, significantly reducing computational cost. The proposed method is proven to have linear convergence.", "questions": "1. Is the method insensitive to the backbone encoder?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper proposes an innovative hypothesis that there exists a statistical dependence between edges connected by common nodes, thus extending GNNs from unsigned to signed graphs.\n\n2. By introducing a Gramian-based correlation matrix for edge dependencies and a Woodbury matrix rewrite for computational efficiency, the paper significantly reduces memory usage and computational cost, greatly accelerates model convergence, and achieves good scalability while maintaining performance comparable to baseline models.\n\n3. The paper provides rigorous mathematical support for its core claims through numerous mathematical derivations.\n\n4. Ablation experiments on the two core components demonstrate the effectiveness of the proposed method.\n\n5. The paper has a clear and well-organized structure, with distinct separation of different components.", "weaknesses": "1. The paper contains a large number of mathematical formulas, which makes it somewhat difficult to read.\n\n2. The innovation of the framework mainly relies on the creative combination of existing tools (Gaussian Copula, Gramian construction, and the Woodbury identity), lacking some novelty.\n\n3. Model training depends on hyperparameters η and ε, making it difficult to directly obtain optimal model performance.\n\n4. Section 4 spends some length analyzing the model's convergence, which is not directly related to the main logic of the paper.\n\n5. The method only uses SNEA as the backbone encoder, which raises questions about its generalizability.\n\n6. Time and memory efficiency comparison is a key indicator for assessing the performance of the method, but the paper only provides comparisons with SNEA, making the experimental results less persuasive.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UZjwCxv1gh", "reviewer": "Reviewer_87t8", "summary": "This study proposes CopulaLSP, which employs a Gaussian copula to couple the marginal relaxed Bernoulli distributions with the correlation structure derived from the Gramian matrix of edge embeddings. It introduces the Woodbury reformulation to maximize the spatiotemporal efficiency of sampling-based inference and theoretically validates the rapid convergence of the method.", "questions": "1.When using the Woodbury reformulation for matrix inversion, how stable is the inversion process when the embedding dimension is large?\n2.To what extent does the setting of the label softening parameter ηimpact model performance, particularly convergence speed? While the appendix mentions a hyperparameter study, the main text lacks a sensitivity analysis of η.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The study features rigorous theoretical analysis, supplemented by visual illustrations to elucidate parameter functions. The integration of theoretical and visual elements ensures clarity in demonstrating methodological robustness and practical applicability.", "weaknesses": "The study exhibits an excessive emphasis on theoretical analysis, while the experimental section appears somewhat rudimentary, lacking in-depth investigation into the configuration of hyperparameters.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)