SR-GNN: A Signed Network Embedding Method Guided by Status Theory and a Reciprocity Relationship

Abstract

Many complex social systems can be modeled as directed signed networks whose edges are marked with a positive/negative sign or direction. Network embedding representation is aimed at mapping rich structural and semantic information into low-dimensional vectors, and extensive research has demonstrated that Graph Neural Networks (GNNs) are an effective way. However, existing GNNs are primarily designed for undirected signed networks and usually used to capture the semantics of the complex structure by social structural balance theory, thus omitting the directional information of the links. In this research, we introduce a reciprocity relationship and status theory to enhance the modeling of the directed positive/negative relationship between two nodes, which has been widely applied in complex network research, and design SR-GNN, a GNN model for signed directed networks, to enable a more accurate vector representation of the nodes and convolution operations on edges with different directions and signs. Experiments demonstrate a reciprocity relationship, and status theory can allow the model to extract the most essential comprehensive information in signed directed graphs. Furthermore, SR-GNN can obtain effective status scores of nodes for link sign predictions and node ranking tasks, both of which yield state-of-the-art performance in most cases.

1. Introduction

In many real-world social systems, relationships between two entities or users have opposite properties, such as friend–foe relationships [1] or support–dissent opinions [2]. These relationships can be represented as graphs with positive and negative links or edges, which are referred to as signed networks. Compared with unsigned networks, signed networks can reflect more complex social relationships, which are leveraged to enhance network mining tasks, such as link sign prediction, node ranking, and community detection. In addition, these types of networks are also powerful tools for analyzing social polarization [3].

Recently, with the popularity of signed networks, a large number of studies on embedding for signed networks have been reported in the literature. Graph Neural Networks for Signed Networks (GNN4SNs) are one example of an important signed network embedding method [4,5,6]. Existing GNN4SN research mainly focuses on characterizing the (weak) structural balance theory that was proposed by Heider and Davis. Heider’s theory [7] was derived from the positive and negative relations in the triadic closure formed by interpersonal interactions, which can be described as “A friend of a friend is a friend, and an enemy of an enemy is a friend”. Then, Davis [8] further extended Heider’s theory by introducing the “weak structural balance theory”, which allowed configurations where all three edges in a triad are negative to be considered structurally balanced. When a signed network fits this theory, the nodes in the network can be divided into k communities, with dense positive edges within each community and sparse negative edges across communities. (Weak) structural balance theory omits the edge’s direction and may lose some aspects of information, leading to suboptimal performance. Status theory [9], introduced by Leskovec and Kleinberg, shows that the direction of the edge carries rich semantic meaning. For example, in online social networks, celebrities may not mutually follow their followers, whereas friendships are often mutual. So far, only a few studies have focused on the embedding of directed signed networks. In summary, in order to obtain accurate signed network embedding representation, both the sign and direction information of the link should be introduced into the signed network embedding model.

Building upon a comprehensive understanding of status theory and the concept of reciprocity, we proposed a GNN model for directed signed networks—SR-GNN. Highlights of our contributions are as follows:

• We propose an efficient GNN model for sign prediction and node ranking tasks. Our model integrates multiple features from directed signed networks, such as edge signs, directions, and reciprocity.
• We rationally designed the information propagation, aggregation mechanism, and optimization objective function for SR-GNN so that the model can obtain interpretable low-dimensional vector representation of nodes.
• Compared with other GNN models, our vector representation not only fully reflects the distance between node pairs but also its status score in the directed signed network. So, the vector is used to simultaneously accomplish two downstream tasks: sign prediction and node ranking discovery.

We conducted extensive experiments on four publicly available real signed networks for the sign prediction and node ranking task. Experimental results verified that the method can distill comprehensive information in a signed graph in the embedding space. On two datasets, the method achieved state-of-the-art link sign prediction accuracy and achieved suboptimal results on the other two datasets. In addition, the method also accurately identified the administrator candidates in a wiki dataset, indicating that the proposed method can be used for node ranking.

2. Related Work

In this section, we review related works. First, we simply review GNN for unsigned networks. Then, we review GNN for undirected signed networks based on the (weak) balance theory. Finally, we review directed signed network representation learning methods.

In recent years, deep learning models represented by Graph Neural Networks (GNNs) have garnered significant attention from specialists across numerous domains due to their remarkable performance and have become an important tool and method for graph representation learning. Most GNN models are developed based on the Message-Passing Neural Network (MPNN) [10,11] technique, which “aggregates neighbor information and updates the node’s own state” by creating information passing, aggregation, and optimization algorithms. This technique turns the rich structure and semantic information of the graph into low-dimensional vector representations of nodes. Based on these vector representations, numerous downstream tasks on unsigned networks are implemented, achieving good results in node classification [12], link prediction [13], and graph clustering [14].

It is feasible and necessary to do research on signed networks utilizing GNNs. However, applying the MPNN paradigm for signed network representation learning faces considerable challenges: GNNs were originally designed for analyzing undirected graphs without heterogeneous edge properties, making it difficult to handle the heterogeneity and directed edges in graphs. To enable GNNs to analyze the positive/negative features, various scientists have made efforts: Derr et al. [4] integrated structural balance theory with standard graph convolution in introducing SGCN, the first signed neural network model. Subsequently, researchers incorporated attention mechanisms [5], graph contrastive learning [15], mutual information [16], and random walks [17,18] into the study of signed networks. Recent work [6] proposed a new network model that combines the global community structure brought by (weak) structural balance. This model consists of global and local GNN structures. In the global module, a more generalized k-balanced network is used to learn node representations; the local module employs a relation GNN without prior assumptions to complement the global module. The above research is based on (weak) structural balance theory, establishing optimization objectives for GNN models to “bring closer” nodes connected by positive edges and “push away” nodes connected by negative edges. Such embedding representations can be employed for downstream tasks like node classification and sign prediction but are not suited for node ranking tasks. Models based on (weak) structural balance theory can only handle undirected signed networks. Researchers suggest that the direction of edges is crucial to the structure of networks. Fully utilizing this feature may enhance the forecasting capability of GNN models [19].

To leverage the directionality of edges in signed networks, a few researchers proposed the following models: SDGNN [20], S-GNN [21], BESIDE [22], and SIDE [18], among which SDGNN and S-GNN are deep models, and BESIDE and SIDE are shallow network embedding. SDGNN attempts to use motifs to define edges in different directions and the triangular structural features formed by these edges, obtaining a more accurate node vector representation. Nevertheless, the introduction of higher-order features unavoidably entails more computational complexity, so an improved methodology is essential to exploit edge direction. S-GNN uses state theory to define the aggregation methods for positive and negative edges in the sending and receiving of directions, respectively.

In summary, to advance GNN-based research on directed signed networks, it is necessary to combine sociological-related theories to deeply understand the unique structural characteristics of signed networks and, by comprehensively considering the different attributes of edges and nodes in the model, achieve a more fine-grained representation of directed signed networks.

3. A Directed Signed Network GNN Model

3.1. Preliminaries

In this section, we introduce essential definitions to facilitate a better understanding of the problem and our model. Consider a signed directed network, denoted as

$$G=\left(V,E,Sig{n}_{\left(i\to j\right)}\right)$$

where $V$ represents the set of nodes, $V=\left\{1,2,3\dots ,n\right\}$. E represents the set of edges, $E=\left\{1,2,3\dots ,m\right\}$, and $Sig{n}_{\left(i\to j\right)}=\left\{-1,0,1\right\}$ represents the sign of an edge. The adjacency matrix of G is defined as $A\in {ℝ}^{n\times n}$. $e_{i\to j}$ and $e_{i\leftarrow j}$ denote the directed edges from i to j and from j to i, respectively. Furthermore, $N_o{\left(i\right)}^{+}$, $N_o{\left(i\right)}^{-}$, $N_I{\left(i\right)}^{+}$, and $N_I{\left(i\right)}^{-}$ represent the out-degree of positive edges, out-degree of negative edges, in-degree of positive edges, and in-degree of negative edges for node i, respectively.

3.2. Status Theory and Node Status Score

Status theory draws from sociological concepts; it conceptualizes an individual’s “place” within the society as characterized by edge-creation intentions and triangle patterns of individuals in a directed signed network. In this theory, the sign of an edge depends on the difference in status between nodes. If User A perceives User B as having a higher status, A is likely to establish a positive edge to B, which can raise B’s status score in a network; in other words, A has made a positive evaluation of B. Conversely, when User C gives a low evaluation of User D, a negative edge is created from C to D, reducing D’s status score, and so C has created a negative impact on D’s status. The transfer of status scores leads to the formation of the triadic closure patterns in a directed signed network.

Figure 1 illustrates a simple example where initially each node’s status score is set to 0. When a node receives a positive edge, its status score increases by 1, and vice versa. Given these status scores, we may anticipate the sign of the edge between two nodes based on their status scores. For instance, if Node E has a lower status score than Node C, the edge from E to C is expected to be positive.

Beyond sign prediction, status scores also play a crucial role in node ranking. A node with a higher status score has received more positive edges from its neighbor nodes, indicating that it is regarded favorably in the network and thus ranks higher. From the perspective of the sender, although transmitting negative edges can enhance their status rating, such nodes also tend to receive negative edges in return. From this standpoint, the evaluation of a node should be evaluated from the two aspects of being a receiver and a sender. Status theory provides the best foundation for designing GNN models: based on the direction of edges, it effectively captures the distinct behavioral features of nodes at both ends of an edge.

Compared to (weak) structural balance theory, status theory has an additional advantage: it can describe both pairwise relationships in a signed network and higher-order triangle structures through transitivity. Notably, in sparse networks where triangles are scarce, (weak) structural balance theory fails, but status theory remains effective.

3.3. Reciprocal Edges

According to the discussion in the previous section, the semantics of edge direction and sign in directed signed networks can be effectively characterized by status theory. However, when there are reciprocal edges between nodes, the status theory may fail. For instance, in Figure 1, nodes E and D share reciprocal positive edges. Where one raises and the other reduces the status score, thereby canceling each other out and preventing the distinguishing of the features of D and E.

Reciprocity is consistent with human social behavior, where individuals often repay both friendly and hostile behaviors. This leads to distinct user behavior characteristics compared to one-way directed edges. It is hard to form a mutual relationship between followers and celebrities; celebrities are unlikely to reciprocate the same attention to their followers. In contrast, friends tend to share a positive reciprocal relationship stemming from mutual appreciation. Thus, reciprocity provides additional structural information beyond edge directionality alone. The similarity between two nodes connected by bidirectional positive edges is generally stronger than that of nodes with one-way positive edges. Likewise, the intensity of dislike indicated by negative reciprocity surpasses that of a one-way negative edge. Reciprocity provides a very essential orientation for node interactions and should not be arbitrarily regarded as two distinct edges.

3.4. GNN Model for Directed Signed Networks

We design a new GNN architecture—SR-GNN—to integrate status theory and the reciprocity features of edges in modeling. The challenge lies in mapping the features of different types of edges to the nodes at both ends and learning accurate vector representations for the nodes. According to previous sections, edge features include the following:

• Edge direction: differentiating between outgoing and incoming edges for information propagation and aggregation.
• Edge sign: encoding positive and negative relationships, concatenated with node features in the vector.
• Reciprocity: identifying bidirectional edges with the same sign, concatenated with node features in the vector.

SR-GNN follows the MPNN (Message-Passing Neural Network) paradigm, taking a directed signed network G as input, and we stack the K layers of the GNN layers to convolve, propagating self-features to neighbors and aggregating K-hop neighbor features. The model is trained under the guidance of a loss function that implies status-theoretic semantics. We set K = 3. After the K-layer convolution operation, the nodes i and j at both ends of the directed edge, respectively, pass through different FC layers to obtain a vector representation that can reflect the node status score. The overall structure of the model is given in Figure 2.

3.4.1. GNN Convolutional Layer Design

Each convolution of layer l ($l\in \left[1~K\right]$) consists of two parts:

• Sending ($x_{i(\to )}^{\left(l\right)}$): aggregates information from neighbors where node i is the sender, and its own features.
• Receiving ($x_{i(\leftarrow )}^{\left(l\right)}$): aggregates information from neighbors where node i is the receiver.

A plausible analogy for these two parts can be represented in the context of social networks: receptive-based status aggregation is used to characterize the status that a user is endorsed by the others (e.g., the popularity of the user), while generative-based status aggregation is for capturing the extent to which a user is willing to endorse the others (e.g., the deference of the user to the others).

The combination of sending and receiving features through a linear transformation is computed as follows:

$$x_{i\left(\to \right)}^{\left(l\right)}=\gamma (x_i^{\left(l-1\right)},\underset{j\in N_o\left(i\right)}{\oplus }\varphi (x_j^{\left(l-1\right)},h_{e_{i\to j}}^{\left(l\right)}))$$

(1)

$$x_{i\left(\leftarrow \right)}^{\left(l\right)}=\gamma (x_i^{\left(l-1\right)},\underset{j\in N_I\left(i\right)}{\oplus }\varphi (x_j^{\left(l-1\right)},h_{e_{i\leftarrow j}}^{\left(l\right)}))$$

(2)

$$x_i^{\left(l\right)}=W^l\cdot \left[x_{i\left(\leftarrow \right)}^l\left|\right|x_{i\left(\to \right)}^l\right]+b^l$$

(3)

$x_i^{\left(0\right)}\in {ℝ}^{\begin{array}{c}d\end{array}}$ is the initial feature vector of any node i in the directed signed network G. $h_{e_{i\to j}}^{\left(l\right)}$ and $h_{e_{i\leftarrow j}}^{\left(l\right)}$ are the feature vectors of the edges $h_{e_{i\to j}}^{\left(l\right)}=W_{e_{i\to j}}^{\left(l\right)}\cdot \left[Sig{n}_{e_{i\to j}},Reciprocity\right]$, $h_{e_{i\leftarrow j}}^{\left(l\right)}=W_{e_{i\leftarrow j}}^{\left(l\right)}\cdot \left[Sig{n}_{e_{i\to j}},Reciprocity\right]$, where $W_{e_{i\to j}}^{\left(l\right)}$ and $W_{e_{i\leftarrow j}}^{\left(l\right)}$ are the linear transformation parameters, $Sig{n}_{e_{i\to j}}=\left\{-1,0,1\right\}$ is the sign of the edge in G, and $Reciprocity=\left\{0,1\right\}$ indicates whether there is reciprocity—0 for non-reciprocal, 1 for reciprocal.

The function $\varphi$ is the concatenation operator (||), combining the edge features with the node features; $\oplus$ is the mean aggregation function with permutation invariance, aggregating information for node i in both sending and receiving directions; function $\gamma$ defines how a node passes its own features to its neighbors, here $\gamma$ is taken as the concatenation operation.

This design ensures that the model explicitly differentiates between outgoing and incoming edges, capturing the asymmetric nature of directed signed networks.

3.4.2. Status Score Estimation for Sign Prediction

Two different fully connected layers are used to estimate the status score of nodes as senders and receivers separately, due to their different behaviors:

$$S_{sen}\left(j\right)=W_{sen}{x}_j^{\left(K\right)}+b_{sen}$$

(4)

$$S_{rec}\left(i\right)=W_{rec}{x}_i^{\left(K\right)}+b_{rec}$$

(5)

Sign prediction can be computed as follows:

$$\Delta S_{i\leftarrow j}=\sigma \left(S_{sen}\left(j\right)-S_{rec}\left(i\right)\right)$$

(6)

$${\widetilde{Sign}}_{\left(i\leftarrow j\right)}=\left\{\begin{array}{l}-1,if \Delta S_{i\leftarrow j}>0.5\\ +1,\hspace{1em}\hspace{1em}otherwise\end{array}\right.$$

(7)

where ${\widetilde{Sign}}_{\left(i\leftarrow j\right)}$ is the predicted edges sign, and $\sigma$ is a non-linear activation function that maps the status difference between two nodes to the range of 0 to 1. When $\Delta S_{i\leftarrow j}>0.5$, it indicates that the status of the sender node j is higher than that of the receiver node I; thus, $sign\left(\Delta S_{i\leftarrow j}\right)=-1$, signifying that j considers i to be of lower status. When $\Delta S_{i\leftarrow j}\le 0.5$, it implies that the status difference between the two nodes is small, or the status of the sender node j is lower than that of the receiver node i.

3.4.3. Model Training and Loss Function

To train the model, a target function is defined to learn the model parameters, ensuring that the learned node vector representations conform to the semantics of status theory. Here, a binary cross-entropy loss function is adopted as the model’s target function. The loss function is defined as follows:

$${\mathcal{L}}_1={\displaystyle \frac{1}{\left|E^{+}\right|}}{\displaystyle \sum _{e_{i\leftarrow j}\in E^{+}}\left(Sig{n}_{e_{i\leftarrow j}}\right)}\cdot \Delta S_{i\leftarrow j}+{\displaystyle \frac{1}{\left|E^{-}\right|}}{\displaystyle \sum _{e_{i\leftarrow j}\in E^{-}}\left(Sig{n}_{e_{i\leftarrow j}}\right)}\cdot \Delta S_{i\leftarrow j}$$

$$\begin{array}{ll}{\mathcal{L}}_2=& {\displaystyle \frac{1}{\left|E^{R+}\right|}}{\displaystyle \sum _{e_{i\leftrightarrow j}\in E^{R+}}\left(Sig{n}_{e_{i\leftrightarrow j}}\right)}\cdot {\displaystyle \frac{\left(\Delta S_{i\leftarrow j}+\Delta S_{i\to j}\right)}{2}}+\\ & {\displaystyle \frac{1}{\left|E^{R-}\right|}}{\displaystyle \sum _{e_{i\leftrightarrow j}\in E^{R-}}\left(Sig{n}_{e_{i\leftrightarrow j}}\right)}\cdot {\displaystyle \frac{\left(\Delta S_{i\leftarrow j}+\Delta S_{i\to j}\right)}{2}}\end{array}$$

$$\mathcal{L}=\alpha {\mathcal{L}}_1+\beta {\mathcal{L}}_2+\lambda \Vert \Theta {\Vert }_2^2$$

(8)

To meet the status theory, the first term of ${\mathcal{L}}_1$ aims to minimize the difference of the two nodes’ status scores with positive edges; the second term of ${\mathcal{L}}_1$ maximizes the difference between the two nodes’ status scores with negative edges; ${\mathcal{L}}_2$ makes the distance between nodes connected by reciprocal positive edges shorter, and the distance between nodes connected by reciprocal negative edges longer. $\alpha$ and $\beta$ control the weight of the status theory and reciprocity features, respectively, while $\lambda$ controls the regularization strength to prevent overfitting. $\Theta =\left\{\right\{{W^k,W_{e_{i\to j}}^{\left(k\right)},W_{e_{i\leftarrow j}}^{\left(k\right)}\}}_{k=1}^K{W}_{sen},W_{rec}\}$ represents the trainable parameters of the model.

3.4.4. The Method of Node Ranking

Each node in the designed model learns two status scores from receiving and sending. Node sorting requires merging these scores, which can be calculated as follows:

$$S_{rec}^{\prime }\left(i\right)={\displaystyle \sum _{j\in N_I{\left(i\right)}^{+}}\frac{S_{sen}\left(j\right)}{N_o\left(j\right)}}-{\displaystyle \sum _{j\in N_I{\left(i\right)}^{-}}\frac{S_{sen}\left(j\right)}{N_o\left(j\right)}}$$

(9)

The status score of the target node i is determined by the difference between the sending status scores of all nodes j that send positive edges to i and the sending status scores of all nodes that send negative edges to i. The higher the value of $S_{rec}^{\prime }\left(i\right)$, the higher the node’s status in the network, enabling node ranking.

4. Experimental Evaluation

To evaluate the effectiveness of the proposed model—SR-GNN—experiments were conducted on real network datasets. The model’s performance was evaluated based on the sign prediction accuracy and the node ranking effectiveness.

4.1. Datasets and Statistical Analysis

To evaluate the effectiveness and efficiency of SR-GNN, we conducted experiments on four benchmark datasets: Epinions, Slashdot, WikiRfA, and Bitcoin-OTC. These publicly accessible signed social network datasets originate from Stanford University’s SNAP project (available online: https://snap.stanford.edu/data/index.html, accessed on 20 November 2024).

Table 1. Statistics of the datasets.

Dataset	Node	Edge	+Edge %	−Edge %	Reciprocal +Edge %	Reciprocal −Edge %	Non-Triadic Closure %
Epinions	131828	841372	85.3%	14.7%	29.7%	0.6%	20.2%
Slashdot	82140	579919	78.6%	21.4%	15.4%	1.7%	44.8%
WikiRfA	11258	185627	77.8%	22.2%	6.3%	0.2%	45.3%
Bitcoin-OTC	5881	35592	90.0%	10.0%	75.5%	1.7%	35.6%

presents the statistics of these datasets.

• Epinions: A customer review website that allows users to mark “who to trust, not whom” relationships.
• Slashdot: A news website that allows users to tag other users as friends/foes based on their own views.
• WikiRfA: A voting network, used for electing administrators of different sections. If a user is agreed upon by the majority, they can be chosen as an administrator of a given subject, responsible for the maintenance of entries; otherwise, the election fails.
• Bitcoin-OTC: A cryptocurrency trading network where users remark on whether their trading competitors are reliable.

Based on the analysis of the four datasets, it is evident that the proportion of positive edges is considerably greater than that of negative edges. This poses a challenge for prediction tasks due to the imbalance in the data. The proportion of reciprocal edges varies. However, there was a significant majority of positive reciprocal edges compared to negative ones in the four datasets. A high percentage of non-triadic closure (e.g., 45.3% in WikiRfA) suggests that structural balance theory-based models may struggle in these networks. In contrast, status theory remains effective, as it first gives features of pairwise nodes, and then, by applying transitivity, it establishes triadic closure among two-hop neighbors.

4.2. Experiment Results

4.2.1. Baselines

In order to demonstrate the efficiency of the SR-GNN model, we have conducted a comparison with several cutting-edge algorithms that have been recently proposed.

Table 2. Baseline features.

Models	Directed	Status Theory	Reciprocity	Deep Models
SGCN	No	No	No	Yes
SNEA	No	No	No	Yes
SDGNN	Yes	Yes	No	Yes
S-GNN	Yes	Yes	No	Yes
BESIDE	Yes	Yes	No	No
SIDE	Yes	Yes	No	No
SR-GNN	Yes	Yes	Yes	Yes

lists the technical characteristics of each algorithm. Since there are few algorithms for processing directed signed networks, we introduced two classic algorithms for undirected signed networks and two shallow learning models for directed signed networks in the benchmark:

• SGCN (available online: https://github.com/benedekrozemberczki/SGCN, accessed on 20 November 2024): The first GNN-based model for signed network [4] modeling introduced in the literature was by Derr et al. in 2018. It independently aggregates and propagates friends and foes based on structural balance theory using a complex aggregation and propagation technique.
• SNEA (available online: https://github.com/liyu1990/snea, accessed on 20 November 2024): A GNN for undirected signed networks based on structural balance theory [5]. It designed a masked self-attentional layer, which used a self-attention mechanism to estimate the importance coefficient for pairs of nodes connected by different types of links during the embedding aggregation process. This model’s aggregation and propagation strategies are inspired by the SGCN model.
• SDGNN (available online: https://github.com/huangjunjie-cs/SiGAT, accessed on 20 November 2024): The first GNN model based on directed signed networks, to the best of our knowledge [20]. SDGNN utilizes four directed signed relationships, $e_{i\to j}^{+}$, $e_{i\to j}^{-}$, $e_{i\leftarrow j}^{+}$, and $e_{i\leftarrow j}^{-}$, to design a GNN aggregator. It provides an attention-based GNN aggregator to collect information from four different types of edges and employs a supervised learning loss function to rebuild signs, directions, and triadic closure.
• S-GNN (available online: https://www.dropbox.com/scl/fo/4r7mo8tp7g3dlz6irfn2v/AHHvSb_VGf8AT84JYR0uWOU?dl+=+0&rlkey=znvudy6p54hp4zzce1l1r572r&e=1 accessed on 20 November 2024): Based on status theory, it separates edges into two categories, receiving and sending, to aggregate information, which trains model parameters using a binary cross-entropy loss function. This algorithm [21] is similar to the one suggested in this paper, but S-GNN only considers edge direction and sign with status theory without introducing reciprocity features.
• BESIDE (available online: https://github.com/yqc01/BESIDE, accessed on 20 November 2024): It [22] incorporates balance and status theories to model both triangles and “bridge” edges in a complementary manner. Since the characteristics of “bridging” nodes are uncovered, the algorithm achieves good results.
• SIDE (available online: https://datalab.snu.ac.kr/side/, accessed on 20 November 2024): It implements representation learning for directed/undirected signed networks based on a random walk algorithm. This shallow learning model technique [18] is an improvement on DeepWalk.

4.2.2. Model Parameter Settings

All experiments were implemented using PyTorch Geometric (PyG 2.3). The computer configuration used for the studies includes and Intel(R) Core(TM) i7 CPU with 32 G of memory (Intel Corporation, Santa Clara, CA, USA). The model does not require a high-performance experimentation environment, and a GPU is not a mandatory requirement.

For all baseline models, we used the original authors’ implementations and parameter settings. Since the four datasets do not include node features, we initialized node features for a 64-dimensional random vector.

Our model has three status convolutional layers. For each convolutional layer, we used a mean aggregator to aggregate its associated interactions with its neighbors for each of them, and the feature dimensions were [32, 64, 32], with a learning rate of 0.01 and a normalization coefficient of ${10}^{-5}$. We applied a grid search for hyperparameters: the learning rate was tuned amongst {0.001, 0.005, 0.01, 0.05}, and the coefficient of $L_2$ normalization was searched in $\{$${10}^{-5}$, ${10}^{-4}$$\}$. The model parameters were initialized using a Xavier initializer [23]. In terms of trainable parameters, the Adam optimizer was used in the implementation.

4.2.3. Sign Prediction

The first task evaluates the ability of SR-GNN to predict the signs of edges. Each dataset is randomly split into an 80% training set and a 20% test set. The direction and sign of the links are derived from the node vector representations learned by the model, which aims to minimize the loss (loss function given by the previous section). Four standard metrics are used to measure the performance: AUC (Area Under Curve), Binary-F1, Macro-F1, and Micro-F1, with larger values indicating better performance. Among the baseline algorithms, SGCN and SNEA are based on undirected signed network modeling, while the rest are based on directed signed networks.

Table 3. Sign prediction results.

Dataset	Metrics	Undirected Model		Directed Model
		SGCN	SNEA	SDGNN	S-GNN	BESIDE	SIDE	SR-GNN
Epinions	AUC	0.8445	0.8617	0.9408	0.9194	0.9348	0.8768	0.9425
	Binary-F1	0.9206	0.9239	0.9628	0.9448	0.9611	0.9487	0.9633
	Micro-F1	0.9071	0.8749	0.9355	0.9112	0.9339	0.9092	0.9370
	Macro-F1	0.8105	0.7862	0.8610	0.8254	0.8601	0.7217	0.8697
Slashdot	AUC	0.7863	0.8032	0.8921	0.8944	0.9017	0.7180	0.8961
	Binary-F1	0.8596	0.8804	0.9128	0.9125	0.9142	0.8675	0.9132
	Micro-F1	0.8296	0.8238	0.8606	0.8617	0.8589	0.7728	0.8622
	Macro-F1	0.7403	0.7723	0.7892	0.7893	0.7885	0.5860	0.7943
WikiRfA	AUC	0.8563	0.7769	0.8898	0.8960	0.8979	0.6824	0.8932
	Binary-F1	0.9069	0.8614	0.9142	0.9130	0.9097	0.8748	0.9135
	Micro-F1	0.8489	0.7967	0.8627	0.8613	0.8553	0.7793	0.8603
	Macro-F1	0.7527	0.7403	0.7849	0.7820	0.7723	0.5713	0.7790
Bitcoin-OTC	AUC	0.8455	0.8046	0.9124	0.8970	0.8629	0.7881	0.8729
	Binary-F1	0.9418	0.9083	0.9647	0.9570	0.9523	0.9023	0.9598
	Micro-F1	0.8978	0.8875	0.9357	0.9273	0.9230	0.8821	0.9220
	Macro-F1	0.7266	0.7311	0.8017	0.7908	0.7682	0.7301	0.7700

Bold values denote the best results; underlined and italicized values indicate suboptimal results.

summarizes the results across all datasets.

Key findings:

• Directed signed models outperform undirected signed models, highlighting the direction of edges is a significant factor.
• The SR-GNN model achieves the best results in sign prediction for links on datasets with a high percentage of reciprocal edges, suggesting that reciprocity is a distinct structural feature from edge direction. Disregarding reciprocity would result in the loss of key network structure information, which is also supported by the most recent study [24].
• We performed some ablation studies to discuss the effects of different objectives on the results. We controlled the loss function parameters $\alpha$ and $\beta$ in Equation (8) to control the weight of status theory and reciprocity, respectively. When $\beta =0$, it degenerates to S-GNN, which only satisfies the status theory. In this situation, our results are comparable to those of the S-GNN model. We cannot set $\alpha =0$ because the first term of the loss function ${\mathcal{L}}_1$ includes information on the edges’ direction and status theory. To achieve the best results, the weights of the two parameters are set differently on different datasets. When α is at {0.75~0.9} and β is at {0.25~0.1}, the best results are obtained for each dataset. This fully demonstrates that the reciprocity feature is independent of the edge direction feature of the graph.

The experiments also demonstrate that the impact of reciprocity depends on its proportion within the network. When the proportion of reciprocal edges is too low, such features are insufficient to guide the model in learning new information. When the proportion of reciprocal edges is too high (e.g., in the Bitcoin-OTC dataset, where reciprocal edges account for 78.2% of the total), the directed signed network will “degenerate” into an undirected signed network. In this case, modeling with reciprocity does not provide additional topological insights. Further research is needed to gain a better understanding of the impact of network properties, such as the number of reciprocal edges and the distribution of reciprocal edges, on prediction outcomes.

SDGNN performs well but incurs high computational costs. It defines four types of motifs with various edge directions and signs, using attention methods for information propagation and aggregation. This approach achieved the best results on the WikiRfA and Bitcoin-OTC datasets and performed competitively on two other datasets. However, due to the high complexity of the objective function, the runtime exceeded 8 h on the medium-sized WikiRfA dataset (11,258 nodes).

S-GNN is competitive, and it is based on status theory; it propagates and aggregates information in both sending and receiving directions. The objective function relates node representations to their status scores, using multi-layer convolution to capture higher-order neighborhood information, performing better than undirected signed network models. Since SIDE is a shallow model based on random walks, it is slightly less effective.

BESIDE achieved good results in some situations, showing that a sort of “bridge” node, which cannot form triangles, is particularly significant. As indicated in Table 1, a significant proportion of nodes in all four networks could not form triangles, and further research is needed to combine the attributes of such nodes with status theory and structural balance theory.

4.2.4. Node Ranking

Besides sign prediction, SR-GNN facilitates node ranking by computing status scores. In order to evaluate the effectiveness of node ranking, the top 10 list of status scores generated by SR-GNN, BESIDE, and S-GNN with the statistical results of the number of times users were elected as administrators for a subject in the dataset is compared, since only BESIDE and S-GNN are based on status theory.

Validating the performance of node ranking is fairly challenging; most models focus solely on the sign prediction task because real-world datasets often lack ranking attributes. However, the WikiRfA and WikiElec datasets provide an ideal evaluation setup, as the nodes correspond to Wikipedia users whose election results depend on who received more votes (positive edges), making the frequency of being successfully elected as an administrator a quantifiable ranking metric.

The ranking lists given by three models partially overlap. Results indicate that each algorithm has learned a part of the topological features of the network, but not complete features (

Table 4. Top 10 personalized rankings and individuals successfully elected as candidates.

Rank	WikiRfA			WikiElec
	BESIDE	S-GNN	SR-GNN	BESIDE	S-GNN	SR-GNN
1	Can’t Sleep…	Nev1	Alex Bakharev	sarah_ewart	bd2412	elonka
2	SarahStierch	Legoktm	Khoikhoi	khoikhoi	derhexer	khoikhoi
3	Phaedriel	Ncurse	SarahStierch	amidaniel	wjbscribe	alex_bakharev
4	Derhexer	Dabomb87	Can’t Sleep…	elonka	arjun01	can’t_sleep…
5	Alex Bakharev	PeterSymonds	Elonka	alex_bakharev	duja	william_m._...
6	Werdna	Derhexer	William M. …	can’t_sleep…	ncurse	phaedriel
7	HJ  Mitchell	Can’t Sleep…	Phaedriel	ncurse	can’t_sleep…	derhexer
8	EveryKing	Phaedriel	Derhexer	halibutt	amidaniel	halibutt
9	Dabomb87	HJ   Mitchell	Dabomb87	phaedriel	newyorkbrad	ncurse
10	PeterSymonds	SarahStierch	HJ   Mitchell	derhexer	phaedriel	ta_bu_shi_da_yu

Red italics: never successfully elected; normal font: one successful election; blue bold: two successful elections; green bold italics: more than three successful elections.

).

Table 4 shows that SR-GNN outperformed the two models, demonstrating more accurate node ranking results in the WikiRfA and WikiElec datasets. SR-GNN successfully identified users who were elected as administrators multiple times. A notable case is user “Elonka”, who was picked out by SR-GNN while being overlooked by other models.

“Elonka” is an active user, engaging in a high level of positive interactions (positive reciprocal edges) with certain neighbors while simultaneously “attacking” others. This behavioral pattern results in a relatively high status score, eventually leading to “Elonka” being successfully elected as an administrator. The same kind of behavioral pattern can also be observed in real-world scenarios. For example, a less famous rapper may significantly increase their public exposure by verbally “attacking” other popular rappers [25]. This can suggest that mutual interactions play a significant role in status formation. This pattern highlights a key advantage of SR-GNN, which explicitly encodes reciprocity, allowing it to capture status dynamics that models such as S-GNN, which do not account for reciprocity, fail to recognize.

5. Conclusions

This study devised an SR-GNN to learn effective network embedding in a directed signed network. The key aspect of SR-GNN is the newly proposed convolutional layer, which can jointly capture the rich graph structural and semantic information of the links, such as direction, reciprocity of edges, and status theory.

Experimental validation across multiple real-world datasets demonstrates that SR-GNN outperforms baseline models in both sign prediction and node ranking tasks. In the sign prediction task, SR-GNN does particularly well in environments with a certain amount of reciprocal relationships, further confirming that reciprocity plays an important role within directed signed networks. For the node ranking task, SR-GNN successfully identifies high-status nodes by comparing its learned status scores. Demonstrating the model’s effectiveness in ranking influential entities in directed signed networks.

Although the purpose of embedding the directed signed network has been achieved by redesigning the convolutional layer of GNNs, there were still tremendous challenges to overcome. More specifically, they were on how to handle the negative links and, furthermore, how to understand reciprocal edges.

For our future work, we first plan to gain a better understanding of the impact of the proportion of reciprocal edges in predicting outcomes. Thereafter, we will focus on understanding the relationship between link signs and the network structure, finding the position of negative links in networks. Then, leveraging structure and position information in networks to obtain a more accurate network embedding representation.