Density Analysis of Network Community Divisions

We present a compact matrix formulation of the modularity, a commonly used quality measure for the community division in a network. Using this formulation we calculate the density of modularities, a statistical measure of the probability of finding a particular modularity for a random but valid community division into $C$ communities. We present our results for some well--known and some artificial networks, and we conclude that the general features of the modularity density are quite similar for the different networks. From a simple model of the modularity we conclude that all nnected networks must show similar shapes of their modularity densities. The general features of this density may give valuable information in the search for good optimization schemes of the modularity.