Exact Recovery in the General Hypergraph Stochastic Block Model

摘要

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 $(p_{1},ldots , p_{k})$ . 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.