Connectivity in Random Annulus Graphs and the Geometric Block Model

TL;DR

This paper establishes new connectivity thresholds for vertex-random graphs and uses these results to develop an efficient community detection algorithm for the geometric block model, a spatial network model that better captures community structure.

Contribution

It provides the first connectivity results for random annulus graphs and applies these to derive conditions for exact community recovery in the geometric block model.

Findings

Connectivity thresholds for random annulus graphs established.

Necessary and sufficient conditions for community recovery in GBM derived.

An efficient algorithm for exact community detection in GBM proposed.

Abstract

We provide new connectivity results for {\em vertex-random graphs} or {\em random annulus graphs} which are significant generalizations of random geometric graphs. Random geometric graphs (RGG) are one of the most basic models of random graphs for spatial networks proposed by Gilbert in 1961, shortly after the introduction of the Erd\H{o}s-R\'{en}yi random graphs. They resemble social networks in many ways (e.g. by spontaneously creating cluster of nodes with high modularity). The connectivity properties of RGG have been studied since its introduction, and analyzing them has been significantly harder than their Erd\H{o}s-R\'{en}yi counterparts due to correlated edge formation. Our next contribution is in using the connectivity of random annulus graphs to provide necessary and sufficient conditions for efficient recovery of communities for {\em the geometric block model} (GBM). The GBM is a probabilistic model for community detection defined over an RGG in a similar spirit as the popular {\em stochastic block model}, which is defined over an Erd\H{o}s-R\'{en}yi random graph. The geometric block model inherits the transitivity properties of RGGs and thus models communities better than a stochastic block model. However, analyzing them requires fresh perspectives as all prior tools fail due to correlation in edge formation. We provide a simple and efficient algorithm that can recover communities in GBM exactly with high probability in the regime of connectivity.