Exact Recovery in the Geometric Hidden Community Model
Julia Gaudio, Andrew Jin
TL;DR
The paper studies exact recovery in the Geometric Hidden Community Model (GHCM), a distance-dependent spatial analogue of hidden-community problems like SBM and submatrix localization. It derives an information-theoretic threshold governed by the CH-divergence $D_+(\theta_i \Vert \theta_j; \pi, g)$ and shows that exact recovery is impossible below this threshold while being achievable above it. The achievability is established by a polynomial-time MAP-based algorithm with seed labeling, block-wise propagation, and a refinement step, adapted to handle distance-dependent observations. The work thus extends exact-recovery results to spatially embedded models and provides a principled threshold and algorithm for exact recovery in networks where information diminishes with distance, with potential applications to sensor networks and spatial data analysis.
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.