Spectra of modular and small-world matrices

Reimer Kühn, Jort van Mourik

    Research output: Contribution to journalArticlepeer-review


    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
    Issue number16
    Publication statusPublished - 11 Apr 2011

    Bibliographical note

    ©2011 IOP Publishing Ltd.


    • spectra
    • symmetric random matrices
    • modular structure
    • arbitrary inter-module
    • arbitrary intra-module
    • degree distributions
    • finite mean connectivities
    • small-world matrices
    • Lifshitz-type singular behaviour


    Dive into the research topics of 'Spectra of modular and small-world matrices'. Together they form a unique fingerprint.

    Cite this