  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.
