Generalized modularity measure for evaluating community structure in complex networks

Discovering community structure is fundamental for uncovering the links between structure and function in complex networks and modularity optimization is the widely accepted method for this issue. However, there is no consensus criteria for measuring the community structure. In this paper, we propose a new quantitative function for community partition–i.e., generalized modularity or M value. We demonstrate that this quantitative function is superior to the widely used modularity and prove its equivalence with the objective functions of the weighted kernel k-means, symmetric matrix factorization and spectral clustering. Both theoretical and numerical results show that optimizing the new criterion not only can tolerate the resolution limit that modularity optimization approaches cannot achieve, but also can extract the number of communities and discovery reasonable communities with different sizes. keyword: complex networks; community structure; modularity; resolution limit