Resolution limit revisited: community detection using generalized modularity density

Abstract

Various attempts have been made in recent years to solve the resolution limit (RL) problem in community detection by considering variants of modularity in the detection algorithms. These objective functions purportedly largely mitigate the RL problem and are preferable to modularity in many realistic scenarios. However, they are not generally suitable for analyzing weighted networks or for detecting hierarchical community structure. RL problems can be complicated, though, and in particular it can be unclear when it should be considered as problem. In this paper, we introduce an objective function that we call generalized modularity density Q g . Q g has a tunable parameter χ that enables structure to be resolved at any desired scale. Rather than being a problem, the scale associated with the RL can be used as a tool for finding hierarchical structure by varying χ. The definition of Q g is easily extended to study weighted networks. We also propose a benchmark test to quantify the RL problem, examine various modularity-like objective functions to show that Q g performs best, and demonstrate that it can be used to identify modular structure in real-world and artificial networks that is otherwise hidden.