Asymptotic distribution of modularity in networks

The structure of complex networks is an important aspect in the study of the real network data. Quite often, it is desirable to know the division of the network into communities. A large number of community detection algorithms have been proposed to probe the community structure of complex networks. For a specific partition of a given network, we show that the distribution of modularity under a null hypothesis of free labeling is asymptotically normal when the size of the network gets large. The significance of the partition is defined based on this asymptotic distribution, which can help assess its goodness. Two different partitions can also be compared statistically. Simulation studies and real data analyses are performed for illustration.