Sharp exact recovery threshold for two-community Euclidean random graphs

TL;DR

This paper establishes the exact recovery threshold for two-community Euclidean random graphs, specifically the Geometric Hidden Community Model, and introduces a linear-time algorithm that achieves this threshold without the previously assumed distribution distinctness condition.

Contribution

It proves the tightness of the recovery threshold for the two-community GHCM and presents a new two-phase, linear-time algorithm that works without the distinctness-of-distributions assumption.

Findings

The recovery threshold for the two-community GHCM is proven to be tight.

A two-phase, linear-time algorithm achieves exact recovery at the threshold.

The results extend to geometric formulations of inference problems like planted dense subgraph and submatrix localization.

Abstract

This paper considers the problem of label recovery in random graphs and matrices. Motivated by transitive behavior in real-world networks (i.e., ``the friend of my friend is my friend''), a recent line of work considers spatially-embedded networks, which exhibit transitive behavior. In particular, the Geometric Hidden Community Model (GHCM), introduced by Gaudio, Guan, Niu, and Wei, models a network as a labeled Poisson point process where every pair of vertices is associated with a pairwise observation whose distribution depends on the labels and positions of the vertices. The GHCM is in turn a generalization of the Geometric SBM (proposed by Baccelli and Sankararaman). Gaudio et al. provided a threshold below which exact recovery is information-theoretically impossible. Above the threshold, they provided a linear-time algorithm that succeeds in exact recovery under a certain ``distinctness-of-distributions'' assumption, which they conjectured to be unnecessary. In this paper, we partially resolve the conjecture by showing that the threshold is indeed tight for the two-community GHCM. We provide a two-phase, linear-time algorithm that explores the spatial graph in a data-driven manner in Phase I to yield an almost exact labeling, which is refined to achieve exact recovery in Phase II. Our results extend achievability to geometric formulations of well-known inference problems, such as the planted dense subgraph problem and submatrix localization, in which the distinctness-of-distributions assumption does not hold.