Stochastic Block Model for Hypergraphs: Statistical limits and a semidefinite programming approach

TL;DR

This paper investigates community detection in hypergraphs using a stochastic block model, identifying a phase transition threshold for exact recovery and proposing a semidefinite programming algorithm for efficient community detection.

Contribution

It introduces the stochastic block model for hypergraphs, analyzes the phase transition for exact recovery, and proposes a semidefinite relaxation method for community detection.

Findings

Sharp phase transition threshold identified for community recovery

Efficient semidefinite programming algorithm achieves near-perfect recovery above threshold

Theoretical analysis confirms the limits of community detection in hypergraphs

Abstract

We study the problem of community detection in a random hypergraph model which we call the stochastic block model for $k$-uniform hypergraphs ($k$-SBM). We investigate the exact recovery problem in $k$-SBM and show that a sharp phase transition occurs around a threshold: below the threshold it is impossible to recover the communities with non-vanishing probability, yet above the threshold there is an estimator which recovers the communities almost asymptotically surely. We also consider a simple, efficient algorithm for the exact recovery problem which is based on a semidefinite relaxation technique.