Year: 2025
Type: article
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 sizenis generated with two communities of equal size, where each nodeiis associated with a group elementg∗i∈GMfor some finite groupGMof orderM. The nodes are connected with probabilitypif they are in the same community, andqotherwise. In addition, a group transformationgijis observed at each edge, wheregij=g∗i(g∗j)−1if the nodesiandjare in the same community, andgij∼ Unif(GM) otherwise. The goal is to recover both the underlying communities and group elements. Whenp=alogn/nandq=blogn/nwitha,b> 0, we establish the sharp information-theoretic threshold for exact recovery: (i) :a+b/2 − √ab/M> 1 and (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.
Authors: Yifeng Fan, Zhizhen Zhao
Institution: University of Illinois Urbana-Champaign
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7087160937