The performance of modularity maximization in practical contexts

TL;DR

This paper investigates the practical performance of modularity maximization in network analysis, revealing its degeneracies, limitations, and implications for interpreting community detection results.

Contribution

It provides a comprehensive analysis of modularity maximization's behavior in real-world networks, clarifies the resolution limit, and discusses strategies to address its degeneracies.

Findings

Modularity maximization often admits exponentially many high-scoring solutions.

The maximum modularity depends on network size and number of modules.

Degenerate solutions can differ significantly in partition properties.

Abstract

Although widely used in practice, the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts. Here, we present a broad characterization of its performance in such situations. First, we revisit and clarify the resolution limit phenomenon for modularity maximization. Second, we show that the modularity function Q exhibits extreme degeneracies: it typically admits an exponential number of distinct high-scoring solutions and typically lacks a clear global maximum. Third, we derive the limiting behavior of the maximum modularity Q_max for one model of infinitely modular networks, showing that it depends strongly both on the size of the network and on the number of modules it contains. Finally, using three real-world metabolic networks as examples, we show that the degenerate solutions can fundamentally disagree on many, but not all, partition properties such as the composition of the largest modules and the distribution of module sizes. These results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts. They also explain why many heuristics are often successful at finding high-scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network. We conclude by discussing avenues for mitigating some of these behaviors, such as combining information from many degenerate solutions or using generative models.