dorsal/arxiv
View SchemaSpectral transitions in networks
| Authors | Gergely Palla, Gabor Vattay |
|---|---|
| Categories | |
| ArXiv ID | physics/0701054 |
| URL | https://arxiv.org/abs/physics/0701054 |
| DOI | 10.1088/1367-2630/8/12/307 |
| Journal | New J. Phys. 8 307 (2006) |
Abstract
We study the level spacing distribution p(s) in the spectrum of random networks. According to our numerical results, the shape of p(s) in the Erdos-Renyi (E-R) random graph is determined by the average degree <k>, and p(s) undergoes a dramatic change when <k> is varied around the critical point of the percolation transition, <k>=1. When <k> > 1, the p(s) is described by the statistics of the Gaussian Orthogonal Ensemble (GOE), one of the major statistical ensembles in Random Matrix Theory, whereas at <k>=1 it follows the Poisson level spacing distribution. Closely above the critical point, p(s) can be described in terms of an intermediate distribution between Poisson and the GOE, the Brody-distribution. Furthermore, below the critical point p(s) can be given with the help of the regularised Gamma-function. Motivated by these results, we analyse the behaviour of p(s) in real networks such as the Internet, a word association network and a protein protein interaction network as well. When the giant component of these networks is destroyed in a node deletion process simulating the networks subjected to intentional attack, their level spacing distribution undergoes a similar transition to that of the E-R graph.
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"abstract": "We study the level spacing distribution p(s) in the spectrum of random\nnetworks. According to our numerical results, the shape of p(s) in the\nErdos-Renyi (E-R) random graph is determined by the average degree \u003ck\u003e, and\np(s) undergoes a dramatic change when \u003ck\u003e is varied around the critical point\nof the percolation transition, \u003ck\u003e=1. When \u003ck\u003e \u003e 1, the p(s) is described by\nthe statistics of the Gaussian Orthogonal Ensemble (GOE), one of the major\nstatistical ensembles in Random Matrix Theory, whereas at \u003ck\u003e=1 it follows the\nPoisson level spacing distribution. Closely above the critical point, p(s) can\nbe described in terms of an intermediate distribution between Poisson and the\nGOE, the Brody-distribution. Furthermore, below the critical point p(s) can be\ngiven with the help of the regularised Gamma-function. Motivated by these\nresults, we analyse the behaviour of p(s) in real networks such as the\nInternet, a word association network and a protein protein interaction network\nas well. When the giant component of these networks is destroyed in a node\ndeletion process simulating the networks subjected to intentional attack, their\nlevel spacing distribution undergoes a similar transition to that of the E-R\ngraph.",
"arxiv_id": "physics/0701054",
"authors": [
"Gergely Palla",
"Gabor Vattay"
],
"categories": [
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"doi": "10.1088/1367-2630/8/12/307",
"journal_ref": "New J. Phys. 8 307 (2006)",
"title": "Spectral transitions in networks",
"url": "https://arxiv.org/abs/physics/0701054"
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