# SR-GNN: A Signed Network Embedding Method Guided by Status Theory and a Reciprocity Relationship
## 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 (GNN4 SNs) are one example of an important signed network embedding method [4, 5, 6]. Existing GNN4 SN research mainly focuses on characterizing the (weak) structural balance theory that was proposed by [PERSON] and [PERSON]. [PERSON]'s theory [7] wasderived 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, [PERSON] [8] further extended [PERSON]'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 [PERSON] and [PERSON], 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.