Spectral transitions in networks

Details Spectral transitions in networks Spectral transitions in networks

Dec 2006

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006njph....8..307p&link_type=abstract

New Journal of Physics, Volume 8, Issue 12, pp. 307 (2006).

Physics

6

Scientific paper

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 Erds Rényi (E R) random graph is determined by the average degree langkrang and p(s) undergoes a dramatic change when langkrang is varied around the critical point of the percolation transition, langkrang = 1. When langkrang Gt 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 langkrang = 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 regularized 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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