The Geometric Block Model

TL;DR

This paper introduces the Geometric Block Model, a new community detection framework that better captures geometric features in networks and demonstrates the near-optimality of a simple triangle-counting algorithm for community detection.

Contribution

The paper proposes the Geometric Block Model as a natural extension of existing models and shows that a triangle-counting algorithm is near-optimal for community detection within this model.

Findings

The geometric block model better explains practical community structures.

A simple triangle-counting algorithm performs near-optimally in sparse regimes.

The algorithm outperforms methods based on the stochastic block model.

Abstract

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model generalizes the random geometric graphs in the same way that the well-studied stochastic block model generalizes the Erdos-Renyi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancement in community detection. While being a topic of fundamental theoretical interest, our main contribution is to show that many practical community structures are better explained by the geometric block model. We also show that a simple triangle-counting algorithm to detect communities in the geometric block model is near-optimal. Indeed, even in the regime where the average degree of the graph grows only logarithmically with the number of vertices (sparse-graph), we show that this algorithm performs extremely well, both theoretically and practically. In contrast, the triangle-counting algorithm is far from being optimum for the stochastic block model. We simulate our results on both real and synthetic datasets to show superior performance of both the new model as well as our algorithm.