Exact Recovery of Community Detection in k-Community Gaussian Mixture Model

TL;DR

This paper establishes the precise conditions under which exact community detection is achievable in a Gaussian mixture model with multiple communities, varying intensities, and unequal community sizes, extending to hypergraph applications.

Contribution

It introduces a novel analysis of community detection thresholds in a generalized Gaussian mixture model with non-uniform community sizes and entry-dependent intensities.

Findings

Derived explicit threshold for exact recovery

Applicable to hypergraph community detection

Generalizes previous models with unequal community sizes

Abstract

We study the community detection problem on a Gaussian mixture model, in which vertices are divided into $k\geq 2$ distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the threshold for the exact recovery of the maximum likelihood estimation. Applications include the community detection on hypergraphs.