Extending the definition of modularity to directed graphs with overlapping communities

TL;DR

This paper extends Newman's modularity measure to directed graphs with overlapping communities, proposes a method for detecting such communities, and introduces a new benchmark dataset for evaluation.

Contribution

It introduces a novel extension of modularity for directed, overlapping communities and provides a community detection method along with a new benchmark dataset.

Findings

Extended modularity captures directed, overlapping community structures

Proposed method effectively identifies overlapping communities

Benchmark dataset facilitates evaluation of community detection algorithms

Abstract

Complex networks topologies present interesting and surprising properties, such as community structures, which can be exploited to optimize communication, to find new efficient and context-aware routing algorithms or simply to understand the dynamics and meaning of relationships among nodes. Complex networks are gaining more and more importance as a reference model and are a powerful interpretation tool for many different kinds of natural, biological and social networks, where directed relationships and contextual belonging of nodes to many different communities is a matter of fact. This paper starts from the definition of modularity function, given by M. Newman to evaluate the goodness of network community decompositions, and extends it to the more general case of directed graphs with overlapping community structures. Interesting properties of the proposed extension are discussed, a method for finding overlapping communities is proposed and results of its application to benchmark case-studies are reported. We also propose a new dataset which could be used as a reference benchmark for overlapping community structures identification.