Exact Community Recovery in the Geometric SBM

TL;DR

This paper introduces a linear-time algorithm for exact community detection in the Geometric Stochastic Block Model, leveraging local density properties to achieve optimal recovery down to the theoretical limit.

Contribution

It presents the first linear-time algorithm for exact community recovery in GSBM that matches the information-theoretic threshold, confirming a prior conjecture.

Findings

Algorithm succeeds down to the information-theoretic threshold.

The method combines local density exploitation with a Poisson testing refinement.

The approach is computationally efficient, running in linear time.

Abstract

We study the problem of exact community recovery in the Geometric Stochastic Block Model (GSBM), where each vertex has an unknown community label as well as a known position, generated according to a Poisson point process in $\mathbb{R}^d$. Edges are formed independently conditioned on the community labels and positions, where vertices may only be connected by an edge if they are within a prescribed distance of each other. The GSBM thus favors the formation of dense local subgraphs, which commonly occur in real-world networks, a property that makes the GSBM qualitatively very different from the standard Stochastic Block Model (SBM). We propose a linear-time algorithm for exact community recovery, which succeeds down to the information-theoretic threshold, confirming a conjecture of Abbe, Baccelli, and Sankararaman. The algorithm involves two phases. The first phase exploits the density of local subgraphs to propagate estimated community labels among sufficiently occupied subregions, and produces an almost-exact vertex labeling. The second phase then refines the initial labels using a Poisson testing procedure. Thus, the GSBM enjoys local to global amplification just as the SBM, with the advantage of admitting an information-theoretically optimal, linear-time algorithm.