Community detection and percolation of information in a geometric setting

TL;DR

This paper extends stochastic block models to geometric random graphs, providing conditions for community detection and information percolation in a geometric setting, bridging graph theory and spatial models.

Contribution

It introduces a geometric model replacing community structures with spatial geometry and establishes conditions for recovering node locations and information percolation.

Findings

Conditions for location recovery in sparse geometric graphs

Criteria for percolation and non-percolation of information

Extension of stochastic block models to geometric spaces

Abstract

We make the first steps towards generalizing the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.