Information Limits of Joint Community Detection and Finite Group Synchronization

Yifeng Fan, Zhizhen Zhao

Abstract: The emerging problem of joint community detection and group synchronization, with applications in signal processing and machine learning, has been extensively studied in recent years. Previous research on this topic has focused predominantly on a statistical model that extends the stochastic block model (SBM) by incorporating additional group transformations. In its simplest form, a random network of size n is generated with two communities of equal size, where each node i is associated with a group element g∗i∈GM for some finite group GM of order M. The nodes are connected with probability p if they are in the same community, and q otherwise. In addition, a group transformation gij is observed at each edge, where gij=g∗i(g∗j)−1 if the nodes i and j are in the same community, and gij∼Unif(GM) otherwise. The goal is to recover both the underlying communities and group elements. When p=alognn and q=blognn with a,b>0 , we establish the sharp information-theoretic threshold for exact recovery: mathfonts (i):a+b2−abM−−√>1and(ii):a>2 where exact recovery of communities is possible only if (i) is satisfied, and recovery of group elements is achieved only if both (i) and (ii) hold. Our theory indicates the recovery of communities greatly benefits from the group elements, and demonstrates a significant performance gap between the information limit and existing approaches.