On Spectral Analysis of Signed and Dispute Graphs: Application to Community Structure

Abstract

This paper presents a spectral analysis of signed networks from both theoretical and practical aspects. On the theoretical aspect, we conduct theoretical studies based on results from matrix perturbation for analyzing community structures of complex signed networks and show how the negative edges affect distributions and patterns of node spectral coordinates in the spectral space. We prove and demonstrate that node spectral coordinates form orthogonal clusters for two types of signed networks: graphs with dense inter-community mixed sign edges and $k$ -dispute graphs where inner-community connections are absent or very sparse but inter-community connections are dense with negative edges. The cluster orthogonality pattern is different from the line orthogonality pattern (i.e., node spectral coordinates form orthogonal lines) observed in the networks with $k$ -block structure. We show why the line orthogonality pattern does not hold in the spectral space for these two types of networks. On the practical aspect, we have developed a clustering method to study signed networks and $k$ -dispute networks. Empirical evaluations on both synthetic networks (with up to one million nodes) and real networks show our algorithm outperforms existing clustering methods on signed networks in terms of accuracy and efficiency.