Information Limits of Joint Community Detection and Finite Group Synchronization

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 has predominantly focused on a statistical model that extends the stochastic block model~(SBM) by incorporating additional group transformations. In its simplest form, the model randomly generates a network of size $n$ that consists of two equal-sized communities, where each node $i$ is associated with an unknown group element $g_i^{\ast }\in G_M$ for some finite group $G_M$ of order $M$. The connectivity between nodes follows a probability $p$ if they belong to the same community, and a probability $q$ otherwise. Moreover, a group transformation $g_{ij}\in G_M$ is observed on each edge $(i,j)$, where $g_{ij}=g_i^{\ast }-g_j^{\ast }$ if nodes $i$ and $j$ are within the same community, and $g_{ij}\sim \text{Uniform}(G_M)$ otherwise. The goal of the problem is to recover both the underlying communities and group elements. Under this setting, when $p=a\log n/n$ and $q=b\log n/n$ with $a,b>0$, we establish the following sharp information-theoretic threshold for exact recovery by maximum likelihood estimation~(MLE): $$(i):\frac{a+b}{2}-\sqrt{\frac{ab}{M}}>1\text{and}(ii):a>2$$ where the exact recovery of communities is possible only if $(i)$ is satisfied, and the recovery of group elements is achieved only if both $(i)$ and $(ii)$ are satisfied. Our theory indicates the recovery of communities greatly benefits from the extra group transformations. Also, it demonstrates a significant performance gap exists between the MLE and all the existing approaches, including algorithms based on semidefinite programming and spectral methods.