Exact Recovery in the General Hypergraph Stochastic Block Model

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.