Resolution limit revisited: community detection using generalized modularity density

Various attempts have been made in recent years to solve the Resolution Limit (RL) problem in community detection by considering variants of the modularity metric in the detection algorithms. These metrics 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. Resolution limit problems can be complicated, though, and in particular it can be unclear when it should be considered as problem. In this paper, we introduce a metric that we call generalized modularity density Q_g that eliminates the RL problem at any desired resolution scale and is easily extendable to study weighted, directed, and hierarchical networks. We also propose a benchmark test to quantify the resolution limit problem, examine various modularity-like metrics to show that the new metric Q_g performs best, and show that Q_g can identify modular structures in real-world and artificial networks that are otherwise hidden.