### Abstract

Language | English |
---|---|

Pages | 1-27 |

Number of pages | 27 |

Journal | Chaos |

Volume | 023109 |

Publication status | Published - 15 Feb 2017 |

### Fingerprint

### Keywords

- Gaussian matrix function
- adjacency matrix
- eigenvalue
- biclique

### Cite this

*Chaos*,

*023109*, 1-27.

}

*Chaos*, vol. 023109, pp. 1-27.

**Exploring the “Middle Earth” of network spectra via a Gaussian matrix function.** / Estrada, Ernesto; Alhomaidhi, Alhanouf Ali; Al-Thukair, Fawzi.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Exploring the “Middle Earth” of network spectra via a Gaussian matrix function

AU - Estrada, Ernesto

AU - Alhomaidhi, Alhanouf Ali

AU - Al-Thukair, Fawzi

N1 - 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, 023109, 1-27.

PY - 2017/2/15

Y1 - 2017/2/15

N2 - 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.

AB - 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.

KW - Gaussian matrix function

KW - adjacency matrix

KW - eigenvalue

KW - biclique

UR - http://aip.scitation.org/journal/cha/

M3 - Article

VL - 023109

SP - 1

EP - 27

JO - Chaos

T2 - Chaos

JF - Chaos

SN - 1054-1500

ER -