Finding network communities using modularity density

Many real-world complex networks exhibit a community structure, in which the modules correspond to actual functional units. Identifying these communities is a key challenge for scientists. A common approach is to search for the network partition that maximizes a quality function. Here, we present a detailed analysis of a recently proposed function, namely modularity density. We show that it does not incur in the drawbacks su. ered by traditional modularity, and that it can identify networks without ground-truth community structure, deriving its analytical dependence on link density in generic random graphs. In addition, we show that modularity density allows an easy comparison between networks of di. erent sizes, and we also present some limitations that methods based on modularity density may su. er from. Finally, we introduce an e. cient, quadratic community detection algorithm based on modularity density maximization, validating its accuracy against theoretical predictions and on a set of benchmark networks.