Dynamical Stability of Threshold Networks over Undirected Signed Graphs

In this paper we study the dynamic behavior of threshold networks on undirected signed graphs. While much attention has been given to the convergence and long-term behavior of this model, an open question remains: How does the underlying graph structure influence network dynamics? While similar papers have been carried out for threshold networks (as well as for other networks) these have largely focused on unsigned networks. However, the signed graph model finds applications in various real-world domains like gene regulation and social networks. By studying a graph parameter that we call "stability index," we search to establish a connection between the structure and the dynamics of threshold network. Interestingly, this parameter is related to the concepts of frustration and balance in signed graphs. We show that graphs that present negative stability index exhibit stable dynamics, meaning that the dynamics converges to fixed points regardless of threshold parameters. Conversely, if at least one subgraph has positive stability index, oscillations in long term behavior may appear. Finally, we generalize the analysis to network dynamics under periodic update schemes and we explore the case in which the stability index is positive for some subgraph finding that attractors with superpolynomial period on the size of the network may appear

Balancing Augmentation with Edge-Utility Filter for Signed GNNs

Signed graph neural networks (SGNNs) has recently drawn more attention as many real-world networks are signed networks containing two types of edges: positive and negative. The existence of negative edges affects the SGNN robustness on two aspects. One is the semantic imbalance as the negative edges are usually hard to obtain though they can provide potentially useful information. The other is the structural unbalance, e.g. unbalanced triangles, an indication of incompatible relationship among nodes. In this paper, we propose a balancing augmentation method to address the above two aspects for SGNNs. Firstly, the utility of each negative edge is measured by calculating its occurrence in unbalanced structures. Secondly, the original signed graph is selectively augmented with the use of (1) an edge perturbation regulator to balance the number of positive and negative edges and to determine the ratio of perturbed edges to original edges and (2) an edge utility filter to remove the negative edges with low utility to make the graph structure more balanced. Finally, a SGNN is trained on the augmented graph which effectively explores the credible relationships. A detailed theoretical analysis is also conducted to prove the effectiveness of each module. Experiments on five real-world datasets in link prediction demonstrate that our method has the advantages of effectiveness and generalization and can significantly improve the performance of SGNN backbones.Comment: 16 page

Trustworthiness-Driven Graph Convolutional Networks for Signed Network Embedding

The problem of representing nodes in a signed network as low-dimensional vectors, known as signed network embedding (SNE), has garnered considerable attention in recent years. While several SNE methods based on graph convolutional networks (GCN) have been proposed for this problem, we point out that they significantly rely on the assumption that the decades-old balance theory always holds in the real-world. To address this limitation, we propose a novel GCN-based SNE approach, named as TrustSGCN, which corrects for incorrect embedding propagation in GCN by utilizing the trustworthiness on edge signs for high-order relationships inferred by the balance theory. The proposed approach consists of three modules: (M1) generation of each node's extended ego-network; (M2) measurement of trustworthiness on edge signs; and (M3) trustworthiness-aware propagation of embeddings. Furthermore, TrustSGCN learns the node embeddings by leveraging two well-known societal theories, i.e., balance and status. The experiments on four real-world signed network datasets demonstrate that TrustSGCN consistently outperforms five state-of-the-art GCN-based SNE methods. The code is available at https://github.com/kmj0792/TrustSGCN.Comment: 12 pages, 8 figures, 9 table

Emergent Behaviors over Signed Random Networks in Dynamical Environments

We study asymptotic dynamical patterns that emerge among a set of nodes that interact in a dynamically evolving signed random network. Node interactions take place at random on a sequence of deterministic signed graphs. Each node receives positive or negative recommendations from its neighbors depending on the sign of the interaction arcs, and updates its state accordingly. Positive recommendations follow the standard consensus update while two types of negative recommendations, each modeling a different type of antagonistic or malicious interaction, are considered. Nodes may weigh positive and negative recommendations differently, and random processes are introduced to model the time-varying attention that nodes pay to the positive and negative recommendations. Various conditions for almost sure convergence, divergence, and clustering of the node states are established. Some fundamental similarities and differences are established for the two notions of negative recommendations