Network community detection using modularity density measures

Modularity, since its introduction, has remained one of the most widely used metrics to assess the quality of community structure in a complex network. However the resolution limit problem associated with modularity limits its applicability to networks with community sizes smaller than a certain scale. In the past various attempts have been made to solve this problem. More recently a new metric, modularity density, was introduced for the quality of community structure in networks in order to solve some of the known problems with modularity, particularly the resolution limit problem. Modularity density resolves some communities which are otherwise undetectable using modularity. However, we find that it does not solve the resolution limit problem completely by investigating some cases where it fails to detect expected community structures. To address this problem, we introduce a variant of this metric and show that it further reduces the resolution limit problem, effectively eliminating the problem in a wide range of networks.