Spectra of random graphs with community structure and arbitrary degrees

Abstract: Using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure. Both reveal interesting spectral features, including deviations from the Wigner semicircle distribution and phase transitions in the spectra of community structured networks. In this paper we generalize both calculations, giving a prescription for calculating the spectrum of a network with both community structure and an arbitrary degree distribution. In general the spectrum has two parts, a continuous spectral band, which can depart strongly from the classic semicircle form, and a set of outlying eigenvalues that indicate the presence of communities.