Combinatorial approach to Modularity

TL;DR

This paper offers a combinatorial analysis of modularity in community detection, using the configurational model to understand its properties, limitations, and statistical significance in network analysis.

Contribution

It introduces a novel combinatorial approach to analyze modularity, including enumeration of null model partitions and probability distribution calculations.

Findings

Provides a method to compute the distribution of modularity scores.

Analyzes the resolution limit and statistical significance of network partitions.

Offers insights into the extremal behavior of modularity in random graphs.

Abstract

Communities are clusters of nodes with a higher than average density of internal connections. Their detection is of great relevance to better understand the structure and hierarchies present in a network. Modularity has become a standard tool in the area of community detection, providing at the same time a way to evaluate partitions and, by maximizing it, a method to find communities. In this work, we study the modularity from a combinatorial point of view. Our analysis (as the modularity definition) relies on the use of the configurational model, a technique that given a graph produces a series of randomized copies keeping the degree sequence invariant. We develop an approach that enumerates the null model partitions and can be used to calculate the probability distribution function of the modularity. Our theory allows for a deep inquiry of several interesting features characterizing modularity such as its resolution limit and the statistics of the partitions that maximize it. Additionally, the study of the probability of extremes of the modularity in the random graph partitions opens the way for a definition of the statistical significance of network partitions.