Detectability of communities in heterogeneous networks

TL;DR

This paper develops a theory to understand when communities in heterogeneous networks can be detected using modularity maximization, revealing that degree heterogeneity, especially in scale-free networks with degree exponent less than 2.5, aids in community detection.

Contribution

It establishes necessary conditions for community detectability via modularity, highlighting the positive role of degree heterogeneity in real networks.

Findings

Heterogeneity in degree distribution enhances modularity's ability to detect communities.

Modularity can reliably detect communities in scale-free networks with degree exponent less than 2.5.

The developed theory applies to various network models to determine detectability conditions.

Abstract

Communities are fundamental entities for the characterization of the structure of real networks. The standard approach to the identification of communities in networks is based on the optimization of a quality function known as "modularity". Although modularity has been at the center of an intense research activity and many methods for its maximization have been proposed, not much it is yet known about the necessary conditions that communities need to satisfy in order to be detectable with modularity maximization methods. Here, we develop a simple theory to establish these conditions, and we successfully apply it to various classes of network models. Our main result is that heterogeneity in the degree distribution helps modularity to correctly recover the community structure of a network and that, in the realistic case of scale-free networks with degree exponent $\gamma < 2.5$, modularity is always able to detect the presence of communities.