Spectra of modular random graphs

Abstract We compute spectra of symmetric random matrices defined on graphs exhibiting a modular structure. Modules are initially introduced as fully connected sub-units of a graph. By contrast, inter-module connectivity is taken to be incomplete. Two different types of inter-module connectivities are considered, one where the number of intermodule connections per-node diverges, and one where this number remains finite in the infinite modulesize limit. In the first case, the results can be understood as a perturbation of a superposition of semicircular spectral densities one would obtain for uncoupled modules. In the second case, matters can be more involved, and depend in detail on inter-module connectivities. For suitable parameters we even find near-triangular-shaped spectral densities, similar to those observed in certain scale-free networks, in a system consisting of just two coupled modules. Analytic results are presented for the infinite module-size limit; they are well corroborated by numerical simulations.