Exploring the “Middle Earth” of network spectra via a Gaussian matrix function

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Estrada, Ernesto and Alhomaidhi, Alhanouf Ali and Al-Thukair, Fawzi (2017) Exploring the “Middle Earth” of network spectra via a Gaussian matrix function. Chaos, 27 (2). 023109. ISSN 1054-1500 (https://doi.org/10.1063/1.4976015)

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Abstract

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrodinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index - an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Such method is a generalization of the so-called "Folded Spectrum Method" used in quantum molecular sciences. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patters, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs.

ORCID iDs

Estrada, Ernesto ORCID logoORCID: https://orcid.org/0000-0002-3066-7418, Alhomaidhi, Alhanouf Ali ORCID logoORCID: https://orcid.org/0000-0003-0762-5807 and Al-Thukair, Fawzi;
  • Item type:Article
    ID code:59970
    Dates:
    Date
    Event
    15 February 2017
    Published
    27 January 2017
    Accepted
    Notes:This is an author accepted manuscript that has been accepted for publication by AIP. Estrada, E., Alhomaidhi, A. A., & Al-Thukair, F. (2017). Exploring the “Middle Earth” of network spectra via a Gaussian matrix function. Chaos 27, 023109 (2017); https://doi.org/10.1063/1.4976015
    Subjects:Science > Mathematics > Probabilities. Mathematical statistics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:27 Feb 2017 10:07
    Last modified:11 Nov 2024 11:38
    Related URLs:
    URI:https://strathprints.strath.ac.uk/id/eprint/59970
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