Spectra of modular and small-world matrices

Reimer Kühn, Jort van Mourik

Research output: Contribution to journalArticle

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.
Original languageEnglish
Article number165205
Pages (from-to)165205
Number of pages1
JournalJournal of Physics A: Mathematical and Theoretical
Volume44
Issue number16
DOIs
Publication statusPublished - 11 Apr 2011

Fingerprint

Small-world networks
Spectral density
Small World
Energy gap
modules
Satellites
Thermodynamics
Module
Connectivity
matrices
Graph Laplacian
Small-world Network
Diagonalization
Spectral Density
Degree Distribution
Thermodynamic Limit
Adjacency Matrix
Band Gap
Density of States
Random Matrices

Bibliographical note

©2011 IOP Publishing Ltd.

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

Cite this

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Spectra of modular and small-world matrices. / Kühn, Reimer; van Mourik, Jort.

In: Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 16, 165205, 11.04.2011, p. 165205.

Research output: Contribution to journalArticle

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