Spectral properties of networks with community structure

In this paper, we discuss the eigenspectra of networks with community structure. It is shown that in many cases, the spectrum of eigenvalues of the adjacency matrix of a network with community structure gives a clear indication of the number of communities in the network. In particular, for a network with N nodes and N-c communities, there will typically be N-c eigenvalues that are significantly larger than the magnitudes of all the other (N-N-c) eigenvalues. We discuss this property as well as its use and limitations for determining N-c.