Performance of modularity maximization in practical contexts

TLDR

Modularity maximization is widely used, yet its behavior and accuracy in practical settings remain poorly understood. The study aims to broadly characterize the performance of modularity maximization in real‑world contexts. The authors revisit the resolution limit phenomenon and propose mitigation strategies, such as aggregating degenerate solutions and employing generative models. They find that the modularity function exhibits extreme degeneracies, lacks a clear global optimum, and that its maximum depends on network size and module count, leading to divergent high‑scoring partitions in real networks and necessitating cautious interpretation.

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 Qmax 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.