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 connectivity are considered, one where the number of intermodule connections per-node diverges, and one where this number remains finite in the infinite module-size limit. In the first case, 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 of consisting of just two coupled modules. Analytic results are presented for the infinite module-size limit; they are well corroborated by numerical simulations.

16 pages, 4 figures. to appear in J. Phys. A

References in corpus (7)

• Emergence of scaling in random networks
• The Bethe lattice spin glass revisited
• Spectra and eigenvectors of scale-free networks
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Optimisation problems and replica symmetry breaking in finite connectivity spin-glasses
• Spectra of Sparse Random Matrices
• Cavity approach to the spectral density of non-Hermitian sparse matrices