SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Langzhang Liang  
Harbin Institute of Technology, Shenzhen  
lazylzliang@gmail.com
Xiangjing Hu  
Harbin Institute of Technology, Shenzhen  
starry.hxj@gmail.com
Zenglin Xu*  
Harbin Institute of Technology, Shenzhen & Pengcheng Lab  
zenglin@gmail.com
Zixing Song  
The Chinese University of Hong Kong  
zxsong@cse.cuhk.edu.hk
Irwin King  
The Chinese University of Hong Kong  
king@cse.cuhk.edu.hk
Graph Neural Networks (GNNs) have been shown to achieve remarkable performance on node classification tasks by exploiting both graph structures and node features. The majority of existing GNNs rely on the implicit homophily assumption. Recent studies have demonstrated that GNNs may struggle to model heterophilous graphs where nodes with different labels are more likely connected. To address this issue, we propose a generic GNN applicable to both homophilous and heterophilous graphs, namely Low-Rank Graph Neural Network (LRGNN). Our analysis demonstrates that a signed graph's global label relationship matrix has a low rank. This insight inspires us to predict the label relationship matrix by solving a robust low-rank matrix approximation problem, as prior research has proven that low-rank approximation could achieve perfect recovery under certain conditions. The experimental results reveal that the solution bears a strong resemblance to the label relationship matrix, presenting two advantages for graph modeling: a block diagonal structure and varying distributions of within-class and between-class entries.
Graphs (or networks) are ubiquitous in various fields, such as social networks [Tang et al., 2013, Xu et al., 2015, 2019b], biology [Guzzi and Zitnik, 2022], and chemistry [Gilmer et al., 2017]. Many real-world networks follow the Homophily assumption, i.e., linked nodes tend to share the same label or have similar features; while for graphs with heterophily, nodes with different labels are more likely to form a link. For example, many people tend to connect with people of the opposite sex in dating networks. Graph Neural Networks (GNNs) [Kipf and Welling, 2017, Velickovic et al., 2018] have shown significant success in tackling a diverse set of graph mining tasks, such as NLP [Song and King, 2022, Chen et al., 2023] and clustering [Huang et al., 2023, Kang et al., 2020]. There are also studies dedicated to enhancing the representational power of GNNs [Liang et al., 2023, Xu et al., 2019a]. Recently, researchers have observed that GNNs may face difficulties in learning on graphs
Figure 1: Undirected signed triads. Structural balance theory states that  $T_{1}$  and  $T_{2}$  are balanced, whereas  $T_{3}$  and  $T_{4}$  are unbalanced. Weak balance states that only  $T_{3}$  is unbalanced. Clearly, triads of signed graphs obey weak balance. Consequently, the label relationship matrix  $\hat{\mathbf{Z}}$  exhibits a low-rank structure. Then  $\hat{\mathbf{Z}}$  can be recovered by low-rank approximation.
with heterophily due to the smoothing operation, which tends to generate similar node representations for adjacent nodes, even though their labels differ [Abu-El-Haija et al., 2019, Zhu et al., 2020].
Various designs [Zhu et al., 2020, Chien et al., 2021, Lim et al., 2021] have been proposed to enhance the performance of GNNs under heterophilous scenarios (see Zheng et al. [2022] for a survey). Among them, high-pass filters are the most frequently used components as they can push away a node from its neighbors in the embedding space, aligning with the characteristic of heterophily. Spectral-based methods [Chien et al., 2021, Luan et al., 2021] combine high-pass filters with low-pass ones by fusing the outputs from intermediate layers. However, these methods cannot capture the node-level homophily ratio, as they use only one convolutional filter type in each layer. Alternatively, some methods [Bo et al., 2021, Yang et al., 2021] enable neural networks to learn the aggregation coefficients. Specifically, these methods update node representation by computing a learned signed weighted combination of neighbors' representations by adopting graph attention function [Velickovic et al., 2018]. However, the graph attention function has been proven to compute a form of static attention: the ranking of attention scores remains unconditioned on the query [Brody et al., 2022].
According to Zhu et al. [2020], under specific conditions, a node's 2-hop neighborhood is expected to be predominantly homophilic. To capitalize on this result, they suggest amplifying a node's influence using its 2-hop neighbors. Building on this, GloGNN [Li et al., 2022] introduces a subspace clustering approach to derive a global coefficient matrix. Nonetheless, Liu et al. [2010] proves that this matrix's between-class entries are almost zero. This indicates that while GloGNN taps into global homophily, it rarely employs remote heterophilous nodes for feature propagation. Moreover, the subspace clustering approach ensures dense within-class matrix entries but does not account for the sign of these entries.
In this paper, we aim at recovering a global label relationship matrix  $\hat{\mathbf{Z}}$  such that  $\hat{\mathbf{Z}}_{i,j} = 1$  if node  $v_i$  and  $v_j$  have the same label (homophilous) and  $-1$  otherwise (heterophilous). In this case, we exploit global homophily and heterophily by assigning proper signs. Determining  $\hat{\mathbf{Z}}_{i,j}$ , which requires knowledge of node labels, seems impractical due to the unavailability of many labels during training. Here a question arises: can we infer the unknown  $\hat{\mathbf{Z}}_{i,j}$  from a small set of observations?
Unlike previous neural network-based methods, we employ sign inference to solve the problem. Signed social networks (SSNs) utilize positive links to denote friendship between two users and negative links to represent enmity. Sign inference is the task of predicting the signs of unknown links using the knowledge of available ones. The structural balance theory [Cartwright and Harary, 1956], a fundamental principle of SSN, postulates that individuals in signed networks are disposed to following specific patterns of "an enemy of my friend is my enemy" (e-f-e), "a friend of my friend is my friend" (f-f-f), "a friend of my enemy is my enemy" (f-e-e), and "an enemy of my enemy is my friend" (e-e-f). However, the last assumption could contradict actual scenarios. Therefore, a more general notion, called weak balance, was introduced to address this issue [Davis, 1967].
We study signed graphs, wherein a negative edge signifies that the two nodes possess distinct labels. Figure 1 illustrates that triads in signed graphs follow the weak balance theory, which suggests that the signed graph has a global low-rank structure [Hsieh et al., 2012]. An intriguing finding [Candes and Tao, 2010] demonstrates that if a matrix has a low rank, it can be accurately recovered from a
limited number of observations. Motivated by this, we approximate  $\hat{\mathbf{Z}}$  with the solution to a low-rank matrix approximation problem, which is then used for feature propagation.
Since the solution is a low-rank matrix, our new model is termed Low-Rank Graph Neural Network (LRGNN). We summarize the contributions of this paper as:
- We show that triads in signed graphs obey weak balance, based on which the unknown label relationship between two nodes can be inferred by looking up the known ones.
- We prove that a variant of signed GNNs shows a tendency to follow the structural balance theory, which can be enhanced by eliminating the faulty assumption from the model design.
- We propose an effective model LRGNN. Using both real-world and synthetic datasets, we provide comprehensive experimental results, thereby highlighting the superior performance of LRGNN over other state-of-the-art methods.
We first introduce the notations used in this paper.
Notations. Denote by  $\mathcal{G} = (\mathcal{V},\mathcal{E})$  an undirected graph, where  $\mathcal{V}$  and  $\mathcal{E}$  denote the node set and edge set, respectively. The nodes are described by a node feature matrix  $\mathbf{X}\in \mathbb{R}^{n\times f}$ , where  $n$  and  $f$  are the number of nodes and number of features per node, respectively.  $\mathbf{Y}\in \mathbb{R}^{n\times c}$  is the node label matrix. The neighbor set of node  $v_{i}$  is denoted by  $\mathcal{N}_i$ . Let  $\tilde{\mathbf{A}}\in \mathbb{R}^{n\times n}$  denote the adjacency matrix where  $\mathbf{A}_{i,j} = 1$  if  $(i,j)\in \mathcal{E}$  and 0 otherwise. We define a label relationship matrix  $\hat{\mathbf{Z}}\in \mathbb{R}^{n\times n}$  for a signed graph, where  $\hat{\mathbf{Z}}_{i,j} = 1$  if  $(i,j)$  share the same label and  $-1$  otherwise. Let  $\tilde{\mathbf{A}}$  be a signed adjacency matrix indicated by an observation set  $\Omega$ , with  $\tilde{\mathbf{A}}_{i,j} = \hat{\mathbf{Z}}_{i,j}$  if  $(i,j)\in \Omega$  and 0 otherwise. The Frobenius norm of a matrix is given by  $\| \cdot \| _F^2$ .  $\mathbf{A}_{i,:}$  represents the  $i$ -th row of matrix  $\mathbf{A}$ .
2.1 Link Sign Prediction via Low-Rank Matrix Completion
Next, we discuss the problem of link sign prediction and the theoretical support for using low-rank matrix completion.
Link Sign Prediction. The link sign prediction task involves determining the signs of unobserved links in a network, where some links are already labeled as positive or negative. This work specifically considers scenarios where only one link exists between each node pair  $(i,j) \in \mathcal{V} \times \mathcal{V}$ . To be precise, given  $\tilde{\mathbf{A}}$  partly observed from  $\hat{\mathbf{Z}}$ , the objective is to predict the signs of the unknown entries of  $\hat{\mathbf{Z}}$ . Note that the observed entries can be constructed using available labels from the training set. This task is difficult in general because of the low supervised ratio. For instance, if the supervised ratio is  $30\%$ , we need to guess  $91\%$  of entries of  $\hat{\mathbf{Z}}$  using the knowledge of the rest  $9\%$  of entries.
Here is a fact that significantly changes the task's premise, rendering it practical. The rank of  $\hat{\mathbf{Z}}$  is equivalent to the number of classes if  $c\geq 2$ . This low-rank structure is described by the weak balance theory: for homophilous node pair  $(i,j)$  and any  $v_{k}\in \mathcal{V}$ , if  $\hat{\mathbf{Z}}_{i,k} = 1$  then  $\hat{\mathbf{Z}}_{j,k} = 1$  (f-f-f), if  $\hat{\mathbf{Z}}_{i,k} = -1$  then  $\hat{\mathbf{Z}}_{j,k} = -1$  (f-e-e); for heterophilous  $(i,j)$ , if  $\hat{\mathbf{Z}}_{i,k} = 1$  then  $\hat{\mathbf{Z}}_{j,k} = -1$  (e-f-e). Therefore,  $\hat{\mathbf{Z}}$  has  $c$  linearly independent distinct rows if  $c\geq 2$  ( $\mathrm{rank}(\hat{\mathbf{Z}}) = c$ ). Inspired by this finding, we employ Matrix Completion to address this challenge.
Matrix Completion. The task of matrix completion involves predicting the missing values in an unknown matrix based on a limited number of observed entries in the observation set  $\Omega$ . Exact recovery of a matrix from a small  $\Omega$  is impossible without making any assumption about the matrix. But the search for solutions becomes meaningful as soon as the unknown matrix has a low rank; numerous algorithms can accomplish a near-perfect recovery with high probability under mild assumptions. This is known as the Low-Rank Matrix Completion (LRMC) problem.
Low-Rank Matrix Completion. Given a signed adjacency matrix  $\tilde{\mathbf{A}}$  and the observation set  $\Omega$ , the task of LRMC is to find the lowest-rank solution among all the feasible solutions as follows,
$$
\min  \quad \operatorname {r a n k} (\mathbf {Z}), \quad s. t. \quad \mathrm {P} _ {\Omega} (\mathbf {Z}) = \mathrm {P} _ {\Omega} (\tilde {\mathbf {A}}), \tag {1}
$$
where  $\mathbf{Z}$  is the decision variable and  $\mathrm{P}_{\Omega}(\cdot)$  is the projection onto the observation set. Unfortunately, (1) is an NP-hard problem without known algorithms capable of solving problems in practical time [Candès and Tao, 2010]. A matrix of rank  $r$  has exactly  $r$  nonzero singular values. Hence, the rank function is simply the number of nonvanishing singular values. As the  $l_{1}$ -minimization is the tightest convex relaxation of the combinatorial  $l_{0}$ -minimization problem [Candès and Plan, 2010]. An alternative is the nuclear minimization problem,
$$
\min  \quad \| \mathbf {Z} \| _ {*}, \quad s. t. \quad \mathrm {P} _ {\Omega} (\mathbf {Z}) = \mathrm {P} _ {\Omega} (\tilde {\mathbf {A}}), \tag {2}
$$
where  $\| \mathbf{Z} \|_*$  denotes the nuclear norm of  $\mathbf{Z}$ , which is the sum of the singular values. Equivalently we can reformulate Eq.(2) in Lagrange form
$$
\min  \frac {1}{2} \| P _ {\Omega} (\mathbf {Z} - \tilde {\mathbf {A}}) \| _ {F} ^ {2} + \lambda \| \mathbf {Z} \| _ {*}, \tag {3}
$$
where  $\lambda \geq 0$  is a regularization parameter. A surprising result involving LRMC is that under certain assumptions, the missing entries of  $\tilde{\mathbf{A}}$  can be accurately predicted.
Theorem 1. [Candès and Tao, 2010] Let  $\hat{\mathbf{Z}}\in \mathbb{R}^{n\times n}$  be a fixed matrix of rank  $r = O(1)$  obeying the strong incoherence property with parameter  $\varepsilon$ . Suppose we observe  $m$  entries of  $\hat{\mathbf{Z}}$  with locations sampled uniformly at random. Then there is a positive numerical constant  $C$  such that if
$$
m \geq C \varepsilon^ {4} n (\log n) ^ {2} \tag {4}
$$
then  $\hat{\mathbf{Z}}$  is the unique solution to Eq.(2) with probability at least  $1 - n^{-3}$ . In other words: with high probability, nuclear-norm minimization recovers all the entries of  $\hat{\mathbf{Z}}$  with no error.
Solving Eq.(2) requires the computation of a SVD of a potentially large matrix, which can be computationally expensive. To reduce this cost, we can use an SVD-free low-rank matrix factorization (LRMF) problem formulated as follows:
$$
\min  _ {\mathbf {U}, \mathbf {V} \in \mathbb {R} ^ {n \times q}} \| \mathrm {P} _ {\Omega} \left(\mathbf {U} \mathbf {V} ^ {T} - \tilde {\mathbf {A}}\right) \| _ {F} ^ {2} + \lambda \left(\| \mathbf {U} \| _ {F} ^ {2} + \| \mathbf {V} \| _ {F} ^ {2}\right), \tag {5}
$$
where  $c \leq q \ll n$  is the operating rank. There is a remarkable fact that ties the LRMF problem and the nuclear-norm regularized problem (3) [Mazumder et al., 2010]: by selecting a  $q \geq c$ , the solution to Eq.(5) also provides a solution to Eq.(3). The equivalence between the solutions allows us to approximate  $\hat{\mathbf{Z}}$  by the solution to Eq.(5).
We present some fundamental facts that support the recovery of  $\hat{\mathbf{Z}}$ . The first fact is that the rank of  $\hat{\mathbf{Z}}$  is deterministic. Ideally,  $q = \text{rank}(\hat{\mathbf{Z}})$  should be the optimal choice [Hastie et al., 2015]. The second fact is that  $\hat{\mathbf{Z}}_{i,:} = \hat{\mathbf{Z}}_{j,:}$  holds for any homophilous  $(i,j)$ . This implies that class imbalance is the upper bound of the coherence parameter  $\varepsilon$ . In most cases,  $\varepsilon = O(1)$ . Furthermore, if  $\tilde{\mathbf{A}}_{i,j} = 1$  is observed,  $v_i$  and  $v_j$  can combine their observations. For example, if  $\tilde{\mathbf{A}}_{i,k}$  is observed, then  $\tilde{\mathbf{A}}_{j,k}$  is also ascertainable, regardless of whether it is directly observed or not ( $\tilde{\mathbf{A}}_{i,k} = \tilde{\mathbf{A}}_{j,k}$ ). As a result, there is a significant increase in the observation rate.
In real-life data, we typically witness a deviation from the assumption that observations exhibit uniform distribution. Nonetheless, this deviation only moderately influences the performance. Suppose a node lacks any observed samples. In that case, an exact recovery is theoretically impossible, but it would not hinder the recovery of other rows. Low-rank approximation methods have been applied to practical applications with a fair amount of success, such as the Netflix problem [Koren et al., 2009].
3 Structural Balance Theory and Signed GNNs
We consider a variant of signed GNNs [Bo et al., 2021] that updates node representations as
$$
h _ {i} ^ {(l + 1)} = \sum_ {j \in \mathcal {N} (i)} \alpha_ {i, j} ^ {(l)} h _ {j} ^ {(l)}, \quad \alpha_ {i, j} = f _ {\theta} \left(h _ {i} ^ {(l)}, h _ {j} ^ {(l)}\right), \tag {6}
$$
where  $f_{\theta}(\cdot, \cdot)$  refers to a neural network that measures the similarity between the inputs and returns a scalar ranging in  $[-1, 1]$ . Let  $y_{i,j} = 1$  if two neighboring nodes  $v_{i}$  and  $v_{j}$  belong to the same class and  $y_{i,j} = -1$  otherwise. Let  $< i, j, k >$  be a triad such that  $v_{j}$  is a neighbor of  $v_{i}$  and  $v_{k}$  is a neighbor of  $v_{j}$ , while  $v_{i}$  and  $v_{k}$  are not directly linked. We have the following result.
Theorem 2. Consider that we apply a  $L$ -layer signed GNN  $(L \geq 2)$  on a triad. The output of the signed GNN is equivalent to  $\mathbf{H}^{(L)} = \mathbf{Z}\mathbf{H}^{(0)}$ , where  $\mathbf{Z}$  is a matrix. Assume that each coefficient  $\alpha_{i,j}$  is independent of other coefficients and the probability that the model can precisely predict the sign of each coefficient is  $p$ , namely  $Pr(\mathrm{sign}(\alpha_{i,j}^{(l)}) = y_{i,j}) = p$ . Also, assume that all the self-coefficients are positive. The probability that at least one of  $\mathbf{Z}_{i,j}$  and  $\mathbf{Z}_{j,k}$  are correct in sign and the product  $\mathbf{Z}_{i,j}\mathbf{Z}_{j,k}\mathbf{Z}_{i,k} > 0$  is given by  $p_b$ . Then  $p_b$  is monotonically increasing concerning  $p$ . Especially, if  $p = 1$ ,  $\mathbf{Z}_{i,j}\mathbf{Z}_{j,k}\mathbf{Z}_{i,k} > 0$  always holds.
Remark 1. A triad is said to be balanced iff its product is positive as the four types of triads given by balance theory have two or zero negative signs.
The above result suggests that multi-layer signed GNNs may implicitly implement the structural balance theory as they frequently create balanced triads. When  $p$  equals 1, all triads are balanced, indicating that the model is following the idea that "a heterophilous neighbor of my heterophilous neighbor is a homophilous node." Nonetheless, this notion does not necessarily hold under multi-class classification tasks. Hence, even if the model can predict the label relationship between direct neighbors accurately, it will still likely predict the relationship between 2-hop neighbors inaccurately. This suggests that multiple-layer signed GNNs are sub-optimal. Signed Graph Convolutional Networks (SignedGNNs) [Derr et al., 2018] utilizes the balance theory to model signed graphs. By eliminating the assumption "e-e-f" (see Supplement for implementation
Figure 2: Over-smoothing analysis.
tion details), we obtain a weakly balanced version of SignedGNNs termed WB-SignedGNNs. To evaluate their vulnerability to over-smoothing, we compared the performance of WB-SignedGCN and SignedGCN at different depths, ranging from 4 to 24. As Figure 2 illustrates, SignedGCN suffer from over-smoothing, and this problem can be largely resolved by dropping the incorrect assumption in the model's design. Such a minute adjustment in the design can result in substantial performance enhancement for 24-layer SignedGCN (80% vs. 53% on Wisconsin dataset).
In this section, we present the overall framework of the Low-Rank Graph Neural Networks.
4.1 Predicting the label relationship matrix via robust low-rank approximation
Note that in  $\hat{\mathbf{Z}}$ , the information on the original graph topology is lost: we cannot tell whether two given nodes are linked or not since all entries are nonzero. To preserve the graph topology information, following Lim et al. [2021], we first apply MLPs to fuse feature matrix and adjacency matrix into a lower-dimensional matrix  $\mathbf{H}^{(0)} \in \mathbb{R}^{n \times c}$ ,
$$
\mathbf {H} ^ {(0)} = (1 - \mu) M L P _ {X} (\mathbf {X}) + \mu M L P _ {A} (\mathbf {A}), \tag {7}
$$
where  $0 < \mu < 1$  is the balance term. Next, we utilize LRMF to recover  $\hat{\mathbf{Z}}$  with the edge set being the observation set  $(\Omega = \mathcal{E})$ , matching the time complexity of vanilla GCNs [Kipf and Welling, 2017]. Furthermore, an attention term [Bahdanau et al., 2015] is incorporated into Eq.(5) as a supplement to the matrix completion. We approximate  $\hat{\mathbf{Z}}$  with  $\mathbf{Z} = \mathbf{U}_{*}\mathbf{V}_{*}^{T}$  by solving the following robust low-rank matrix factorization problem,
$$
\begin{array}{l} F _ {c} (\mathbf {U}, \mathbf {V}) = \sum_ {(i, j) \in \mathcal {E}} \min  \left(\left(\left(\mathbf {U} \mathbf {V} ^ {T}\right) _ {i, j} - \tilde {\mathbf {A}} _ {i, j}\right) ^ {2}, \tau_ {i, j}\right) + \lambda \| \mathbf {U} \| _ {F} ^ {2} + \lambda \| \mathbf {V} \| _ {F} ^ {2} + \\ \gamma \sum_ {i, j} \left(h _ {i} ^ {(0) ^ {T}} \mathbf {W} h _ {j} ^ {(0)} - \left(\mathbf {U V} ^ {T}\right) _ {i, j}\right) ^ {2}, \tag {8} \\ \end{array}
$$
where  $\lambda > 0$  and  $\gamma > 0$  are hyper-parameters,  $\mathbf{W}$  a parameter matrix,  $h_i^{(0)^T} \mathbf{W} h_j^{(0)}$  an attention coefficient,  $\tau_{i,j}$  a parameter of the capped norm. Low-rank matrix approximation models are prone
to outliers due to the square loss function [Keshavan et al., 2010]. To mitigate this issue, this work applies capped norm to limit the contribution of the approximation error  $(\left(\mathbf{U}\mathbf{V}^T\right)_{i,j} - \tilde{\mathbf{A}}_{i,j})^2$ . This strategy has been extensively utilized by researchers [Nie et al., 2014, 2017, Jiang et al., 2015]. Note that here each sample  $(i,j)$  has an independent parameter  $\tau_{i,j}$  which will be explained in detail later. The min operator renders the function non-differentiable, and the projection function prevents us from obtaining a closed-form solution. It seems that solving Eq.(8) is infeasible. Using the surrogate function and Majorization-Minimization algorithm [Hastie et al., 2015], we can ensure that the objective function does not increase after each iteration. The optimization algorithm, initialization method, and proof of convergence can be found in the Supplement.
Once we have derived  $\mathbf{U}_*\mathbf{V}_*^T$  by minimizing Eq.(8), the output of the model is formulated as
$$
\mathbf {H} _ {\text {o u t}} = \left(\left(1 - \beta\right) \mathbf {Z} + \beta \mathbf {I} _ {n}\right) \mathbf {H} ^ {(0)}, \tag {9}
$$
where the identity matrix  $\mathbf{I}_n$  is used to augment the ego-information and  $0 < \beta < 1$  is a hyperparameter. A 1-layer network is sufficient to aggregate all nodes' representations for a node, it also averts the implicit usage of the balance theory.
Remark 2. The time complexity of LRGNN is proportional to the number of edges. A detailed discussion can be found in the Supplement.
4.2 Parameterized outlier detection
This section outlines a practical algorithm for detecting outliers based on the reliability of the estimated  $\tilde{\mathbf{A}}_{i,j}$ . To generate the signed adjacency matrix  $\tilde{\mathbf{A}}$ , we can use any off-the-shelf neural network classifier to generate pseudo labels. Additionally, the known node labels in training set  $\mathcal{T}_{\mathbf{V}}$  and label matrix  $\mathbf{Y}$  can also be utilized. Similar to Zhu et al. [2021], the pseudo labels are generated as follows,
$$
\bar {\mathbf {Y}} = \mathbf {O} \odot \mathbf {Y} + (1 - \mathbf {O}) \odot \hat {\mathbf {Y}}, \hat {\mathbf {Y}} = \operatorname {s o f t m a x} (\mathbf {P}), \mathbf {P} = f _ {N N} (\mathbf {X}), \tag {10}
$$
where  $f_{NN}(\cdot)$  denotes a trained neural network named estimator,  $\odot$  the Hadamard product, and  $\mathbf{O}_{i,:} = \mathbf{1}$  if  $i \in \mathcal{T}_{\mathcal{V}}$  and  $\mathbf{0}$  otherwise. The signed adjacency matrix is defined as
$$
\tilde {\mathbf {A}} _ {i, j} = \left\{ \begin{array}{c c} \langle \bar {\mathbf {Y}} _ {i,:}, \bar {\mathbf {Y}} _ {j,:} \rangle - \delta & (i, j) \in \mathcal {E} \\ 0 & \text {o t h e r w i s e} \end{array} \right. \tag {11}
$$