Heuristic Modularity Maximization Algorithms for Community Detection Rarely Return an Optimal Partition or Anything Similar

TL;DR

This study evaluates heuristic algorithms for community detection based on modularity maximization, revealing they rarely find optimal solutions and often produce dissimilar partitions, highlighting the need for exact or approximate optimization methods.

Contribution

The paper provides a comprehensive empirical comparison of heuristic modularity maximization algorithms against exact methods, exposing their limitations in achieving optimal community partitions.

Findings

Heuristic algorithms find optimal partitions in only 19.4% of cases.

Sub-optimal partitions often significantly differ from optimal ones.

Near-optimal solutions can be substantially dissimilar to true optimal partitions.

Abstract

Community detection is a fundamental problem in computational sciences with extensive applications in various fields. The most commonly used methods are the algorithms designed to maximize modularity over different partitions of the network nodes. Using 80 real and random networks from a wide range of contexts, we investigate the extent to which current heuristic modularity maximization algorithms succeed in returning maximum-modularity (optimal) partitions. We evaluate (1) the ratio of the algorithms' output modularity to the maximum modularity for each input graph, and (2) the maximum similarity between their output partition and any optimal partition of that graph. We compare eight existing heuristic algorithms against an exact integer programming method that globally maximizes modularity. The average modularity-based heuristic algorithm returns optimal partitions for only 19.4% of the 80 graphs considered. Additionally, results on adjusted mutual information reveal substantial dissimilarity between the sub-optimal partitions and any optimal partition of the networks in our experiments. More importantly, our results show that near-optimal partitions are often disproportionately dissimilar to any optimal partition. Taken together, our analysis points to a crucial limitation of commonly used modularity-based heuristics for discovering communities: they rarely produce an optimal partition or a partition resembling an optimal partition. If modularity is to be used for detecting communities, exact or approximate optimization algorithms are recommendable for a more methodologically sound usage of modularity within its applicability limits.