Dynamical scaling behavior of percolation clusters in scale-free networks.

Abstract

In this work we investigate the spectra of Laplacian matrices that determine many dynamic properties of scale-free networks below and at the percolation threshold. We use a replica formalism to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of treelike networks. Close to the percolation threshold we find characteristic scaling functions for the density of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic power laws rho (lambda) approximately lambda (alpha(1) ) or rho (lambda) approximately lambda (d(2) ) for small lambda, where alpha(1) holds below and alpha(2) at the percolation threshold. In the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results.