Exact Recovery in the Geometric Hidden Community Model

TL;DR

This paper establishes the precise information-theoretic threshold for exact recovery in the Geometric Hidden Community Model, a spatially-embedded extension of the stochastic block model that accounts for distance-dependent observations.

Contribution

It generalizes the Geometric SBM to include arbitrary pairwise distributions and determines the exact recovery threshold under mild assumptions.

Findings

Identifies the information-theoretic threshold for exact recovery.

Extends the model to include distance-dependent pairwise distributions.

Completes the understanding of exact recovery in spatially-embedded hidden community models.

Abstract

Hidden community problems, such as community detection in the Stochastic Block Model (SBM), submatrix localization, and $\mathbb{Z}_2$ synchronization, have received considerable attention in the probability, statistics, and information-theory literature. Motivated by transitive behavior in social networks, which tend to exhibit high triangle density, recent works have considered hidden community models in spatially-embedded networks. In particular, Baccelli and Sankararaman proposed the Geometric SBM, a spatially-embedded analogue of the standard SBM with dramatically more triangles. In this paper, we consider the problem of exact recovery for the Geometric Hidden Community Model (GHCM) of Gaudio, Guan, Niu, and Wei, which generalizes the Geometric SBM to allow for arbitrary pairwise observation distributions. Under mild technical assumptions, we find the information-theoretic threshold for exact recovery in the ``distance-dependent'' GHCM, which allows the pairwise distributions to depend on distance as well as community labels, thus completing the picture of exact recovery in spatially-embedded hidden community models.