Hierarchical mutual information for the comparison of hierarchical community structures in complex networks

TL;DR

This paper introduces hierarchical mutual information, a new measure for comparing hierarchical community structures in complex networks, validated through extensive experiments on artificial and real networks.

Contribution

It presents the hierarchical mutual information as a novel similarity measure for hierarchical structures, extending traditional mutual information to better compare complex network hierarchies.

Findings

Hierarchical mutual information behaves consistently across various experiments.

It effectively compares different community detection methods.

It helps analyze the robustness and evolution of network hierarchies.

Abstract

The quest for a quantitative characterization of community and modular structure of complex networks produced a variety of methods and algorithms to classify different networks. However, it is not clear if such methods provide consistent, robust and meaningful results when considering hierarchies as a whole. Part of the problem is the lack of a similarity measure for the comparison of hierarchical community structures. In this work we give a contribution by introducing the {\it hierarchical mutual information}, which is a generalization of the traditional mutual information, and allows to compare hierarchical partitions and hierarchical community structures. The {\it normalized} version of the hierarchical mutual information should behave analogously to the traditional normalized mutual information. Here, the correct behavior of the hierarchical mutual information is corroborated on an extensive battery of numerical experiments. The experiments are performed on artificial hierarchies, and on the hierarchical community structure of artificial and empirical networks. Furthermore, the experiments illustrate some of the practical applications of the hierarchical mutual information. Namely, the comparison of different community detection methods, and the study of the consistency, robustness and temporal evolution of the hierarchical modular structure of networks.