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

Ernesto Estrada, Alhanouf Ali Alhomaidhi, Fawzi Al-Thukair

Research output: Contribution to journalArticle

2 Citations (Scopus)

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.
LanguageEnglish
Pages1-27
Number of pages27
JournalChaos
Volume023109
Publication statusPublished - 15 Feb 2017

Fingerprint

Gaussian Function
Matrix Function
Earth (planet)
matrices
Bicliques
Adjacency Matrix
Schrodinger equation
eigenvalues
Hamiltonians
Eigenvalue
Star Graph
Schrodinger Equation
Complete Bipartite Graph
Stars
Zero
Graph in graph theory
Erdös
Simple Graph
Random Graphs
Subgraph

Keywords

  • Gaussian matrix function
  • adjacency matrix
  • eigenvalue
  • biclique

Cite this

Estrada, Ernesto ; Alhomaidhi, Alhanouf Ali ; Al-Thukair, Fawzi. / Exploring the “Middle Earth” of network spectra via a Gaussian matrix function. In: Chaos. 2017 ; Vol. 023109. pp. 1-27.
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Exploring the “Middle Earth” of network spectra via a Gaussian matrix function. / Estrada, Ernesto; Alhomaidhi, Alhanouf Ali; Al-Thukair, Fawzi.

In: Chaos, Vol. 023109, 15.02.2017, p. 1-27.

Research output: Contribution to journalArticle

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