Dynamical Scaling Behavior of Percolation Clusters in Scale-free Networks

TL;DR

This paper analyzes the eigenvalue spectra of Laplacian matrices in scale-free networks near the percolation threshold, revealing characteristic power-law behaviors and providing an analytical framework for ensemble averaging.

Contribution

It introduces a replica formalism-based analytical method to determine eigenvalue spectra in tree-like scale-free networks near criticality.

Findings

Spectral density rho(lambda) follows power laws with different exponents below and at the percolation threshold.

Analytical results agree well with numerical diagonalization.

Characteristic scaling functions describe the spectral behavior near criticality.

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 tree-like 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) ~ lambda^alpha_1 or rho(lambda) ~ lambda^alpha_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.