Spectra of modular and small-world matrices

Kühn, Reimer and van Mourik, Jort (2011). Spectra of modular and small-world matrices. Journal of Physics A: Mathematical and Theoretical, 44 (16), p. 165205.

Abstract

We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter- and intra-module degree distributions, subject only to the constraint of finite mean connectivities. We also evaluate spectra of a certain class of small-world matrices generated from random graphs by introducing shortcuts via additional random connectivity components. Both adjacency matrices and the associated graph Laplacians are investigated. For the Laplacians, we find Lifshitz-type singular behaviour of the spectral density in a localized region of small |?| values. In the case of modular networks, we can identify contributions of local densities of state from individual modules. For small-world networks, we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states, exhibiting only localized states in the band gaps. Results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances, and with direct diagonalization results.

Publication DOI: https://doi.org/10.1088/1751-8113/44/16/165205
Divisions: Engineering & Applied Sciences > Mathematics
Engineering & Applied Sciences > Systems analytics research institute (SARI)
Additional Information: ©2011 IOP Publishing Ltd.
Uncontrolled Keywords: spectra,symmetric random matrices,modular structure,arbitrary inter-module,arbitrary intra-module,degree distributions,finite mean connectivities,small-world matrices,Lifshitz-type singular behaviour,Mathematical Physics,Physics and Astronomy(all),Statistical and Nonlinear Physics,Modelling and Simulation,Statistics and Probability
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Related URLs: http://www.scop ... tnerID=8YFLogxK (Scopus URL)
http://iopscien ... 1/44/16/165205/ (Publisher URL)
Published Date: 2011-04-11
Authors: Kühn, Reimer
van Mourik, Jort

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