Modularity and Heavy-Tailed Degree Distributions

Identifying clusters of vertices in graphs continues to be an important problem, and modularity continues to be used as a tool for solving the problem. Modularity, which measures the quality of a division of the vertices into clusters, explicitly treats vertices of different degrees differently, imposing a larger penalty when high-degree vertices are put in the same cluster. We claim that this unequal treatment negatively impacts the performance of clustering algorithms based on modularity for graphs with heavy-tailed degree distributions. We used the Greedy Modularity hill-climb to find clusters in graphs with power-law degree distributions and observed that it performed poorly clustering low-degree vertices. We propose a simple variant of modularity that we call flat modularity. We found that, using the same algorithm with the modified score instead, we improved the performance of the clustering algorithm on low-degree vertices and the overall performance as well. We believe that the small change -- from modularity to flat modularity -- could improve the output in the many real-world processes that rely on modularity to measure the quality of a division into clusters with only a small change, which is really a simplification, to the implementation.