  Modularity is widely used to effectively measure the strength of the disjoint
community structure found by community detection algorithms. Several
overlapping extensions of modularity were proposed to measure the quality of
overlapping community structure. However, all these extensions differ just in
the way they define the belonging coefficient and belonging function. Yet,
there is lack of systematic comparison of different extensions. To fill this
gap, we overview overlapping extensions of modularity and generalize them with
a uniform definition enabling application of different belonging coefficients
and belonging functions to select the best. In addition, we extend localized
modularity, modularity density, and eight local community quality metrics to
enable their usages for overlapping communities. The experimental results on a
large number of real networks and synthetic networks using overlapping
extensions of modularity, overlapping modularity density, and local 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, the
results may be used to guide researchers on which metrics to adopt when
measuring the quality of overlapping community structure.
