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



Author: G. Ergun, R. Kuehn

Source: https://arxiv.org/







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