Random Overlapping Communities: Approximating Motif Densities of Large Graphs

TL;DR

This paper introduces the Random Overlapping Communities model, a generalized random graph model that effectively captures diverse clustering behaviors in large complex networks by generating graphs through dense, overlapping subgraphs.

Contribution

It proposes a novel graph model that extends Erdős-Rényi graphs to achieve a wide range of clustering coefficients and motif ratios, better reflecting real-world network properties.

Findings

Achieves desired clustering coefficients and motif ratios

Models local clustering through overlapping dense subgraphs

Provides explanations for clustering phenomena in complex networks

Abstract

A wide variety of complex networks (social, biological, information etc.) exhibit local clustering with substantial variation in the clustering coefficient (the probability of neighbors being connected). Existing models of large graphs capture power law degree distributions (Barab\'asi-Albert) and small-world properties (Watts-Strogatz), but only limited clustering behavior. We introduce a generalization of the classical Erd\H{o}s-R\'enyi model of random graphs which provably achieves a wide range of desired clustering coefficient, triangle-to-edge and four-cycle-to-edge ratios for any given graph size and edge density. Rather than choosing edges independently at random, in the Random Overlapping Communities model, a graph is generated by choosing a set of random, relatively dense subgraphs ("communities"). We discuss the explanatory power of the model and some of its consequences.