Information Theoretic Limits of Exact Recovery in Sub-hypergraph Models for Community Detection

TL;DR

This paper establishes the fundamental information-theoretic limits for exact community recovery in sub-hypergraph models, providing tight bounds for when recovery is possible or impossible.

Contribution

It introduces the $m$-ShSBM model and derives tight bounds for community detection success and failure regions using Fano's inequality and MLE analysis.

Findings

Identifies parameter regions where exact recovery is impossible.

Determines conditions under which MLE achieves exact recovery.

Provides tight bounds applicable to various hypergraph models.

Abstract

In this paper, we study the information theoretic bounds for exact recovery in sub-hypergraph models for community detection. We define a general model called the $m-$uniform sub-hypergraph stochastic block model ($m-$ShSBM). Under the $m-$ShSBM, we use Fano's inequality to identify the region of model parameters where any algorithm fails to exactly recover the planted communities with a large probability. We also identify the region where a Maximum Likelihood Estimation (MLE) algorithm succeeds to exactly recover the communities with high probability. Our bounds are tight and pertain to the community detection problems in various models such as the planted hypergraph stochastic block model, the planted densest sub-hypergraph model, and the planted multipartite hypergraph model.