Phase transitions in normalized cut of social networks

Abstract

One of the main goals of network analysis and data science is to identify community structures in a network. Community structure is a typical feature of social networks and community detection is considered to be crucial to understand the structure and function of social networks in cyberspace. The community detection problem of social networks can be viewed as a graph partitioning problem whose objective is to identify the external edges as the edge cut that separates the network as two subnetworks by recursive method. The degree-normalized adjacency matrix is more convenient than traditional Laplacian matrix in calculating eigenvalues and eigenvectors. In this work, based on random model we first prove the existence of an abrupt phase transition for community detection in terms of the degree-normalized adjacency matrix with accurate critical value. In detail, the detectability of community detection has changed from high to low when the critical value of phase transition q approaches p1p2 almost surely, where pi(i=1,2) is the intra-community edge connection probability of subnetworks Ci, respectively. Then, the validity of the theoretical results is verified by numerical experiments.