Exact Recovery in the General Hypergraph Stochastic Block Model

TL;DR

This paper establishes the fundamental limits for exact community recovery in the general hypergraph stochastic block model and introduces a polynomial-time algorithm that achieves these limits.

Contribution

It derives a sharp threshold for exact recovery in the d-uniform hypergraph SBM and proposes a two-stage spectral and local refinement algorithm.

Findings

Identifies a sharp threshold based on generalized Chernoff-Hellinger divergence.

Develops a polynomial-time algorithm that achieves exact recovery above the threshold.

Recovers prior results for special cases of the model.

Abstract

This paper investigates fundamental limits of exact recovery in the general d-uniform hypergraph stochastic block model (d-HSBM), wherein n nodes are partitioned into k disjoint communities with relative sizes (p1,..., pk). Each subset of nodes with cardinality d is generated independently as an order-d hyperedge with a certain probability that depends on the ground-truth communities that the d nodes belong to. The goal is to exactly recover the k hidden communities based on the observed hypergraph. We show that there exists a sharp threshold such that exact recovery is achievable above the threshold and impossible below the threshold (apart from a small regime of parameters that will be specified precisely). This threshold is represented in terms of a quantity which we term as the generalized Chernoff-Hellinger divergence between communities. Our result for this general model recovers prior results for the standard SBM and d-HSBM with two symmetric communities as special cases. En route to proving our achievability results, we develop a polynomial-time two-stage algorithm that meets the threshold. The first stage adopts a certain hypergraph spectral clustering method to obtain a coarse estimate of communities, and the second stage refines each node individually via local refinement steps to ensure exact recovery.