Spectra of Modular Random Graphs

G. Ergun; R. Kuehn

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arXiv · 0908.3155

Spectra of Modular Random Graphs

G. Ergun·R. Kuehn

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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.

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BibTeX RIS

G. Ergun, R. Kuehn. 2009-08-21. Spectra of Modular Random Graphs. https://arxiv.org/abs/0908.3155

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