Extension of Modularity Density for overlapping community structure

Modularity is widely used to effectively measure the strength of the disjoint community structure found by community detection algorithms. Although several overlapping extensions of modularity were proposed to measure the quality of overlapping community structure, there is lack of systematic comparison of different extensions. To fill this gap, we overview overlapping extensions of modularity to select the best. In addition, we extend the Modularity Density metric to enable its usage for overlapping communities. The experimental results on four real networks using overlapping extensions of modularity, overlapping modularity density, and six other community quality metrics show that the best results are obtained when the product of the belonging coefficients of two nodes is used as the belonging function. Moreover, our experiments indicate that overlapping modularity density is a better measure of the quality of overlapping community structure than other metrics considered.