Normalized modularity optimization method for community identification with degree adjustment

As a fundamental problem in network study, community identification has attracted much attention from different fields. Representing a seminal work in this area, the modularity optimization method has been widely applied and studied. However, this method has issues in resolution limit and extreme degeneracy and may not perform well for networks with unbalanced structures. Although several methods have been proposed to overcome these limitations, they are all based on the original idea of defining modularity through comparing the total number of edges within the putative communities in the observed network with that in an equivalent randomly generated network. In this paper, we show that this modularity definition is not suitable to analyze some networks such as those with unbalanced structures. Instead, we propose to define modularity through the average degree within the communities and formulate modularity as comparing the sum of average degree within communities of the observed network to that of an equivalent randomly generated network. In addition, we also propose a degree-adjusted approach for further improvement when there are unbalanced structures. We analyze the theoretical properties of our degree adjusted method. Numerical experiments for both artificial networks and real networks demonstrate that average degree plays an important role in network community identification, and our proposed methods have better performance than existing ones.