Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?

TL;DR

This paper analyzes the Laplacian spectra of uncorrelated random networks to understand signal propagation and random walks, revealing that the minimum degree is crucial and that scale-free degree distributions are less important.

Contribution

The authors develop an exact integral equation approach to describe Laplacian spectra and dynamics on uncorrelated networks, emphasizing the role of minimum degree over degree distribution.

Findings

The spectral edge depends on the minimum degree q_m.

Signal propagation resembles a Gaussian broadening over time.

Spectral density near the edge is characterized analytically.

Abstract

We study the Laplacian operator of an uncorrelated random network and, as an application, consider hopping processes (diffusion, random walks, signal propagation, etc.) on networks. We develop a strict approach to these problems. We derive an exact closed set of integral equations, which provide the averages of the Laplacian operator's resolvent. This enables us to describe the propagation of a signal and random walks on the network. We show that the determining parameter in this problem is the minimum degree $q_m$ of vertices in the network and that the high-degree part of the degree distribution is not that essential. The position of the lower edge of the Laplacian spectrum $\lambda_c$ appears to be the same as in the regular Bethe lattice with the coordination number $q_m$. Namely, $\lambda_c>0$ if $q_m>2$, and $\lambda_c=0$ if $q_m\leq2$. In both these cases the density of eigenvalues $\rho(\lambda)\to0$ as $\lambda\to\lambda_c+0$, but the limiting behaviors near $\lambda_c$ are very different. In terms of a distance from a starting vertex, the hopping propagator is a steady moving Gaussian, broadening with time. This picture qualitatively coincides with that for a regular Bethe lattice. Our analytical results include the spectral density $\rho(\lambda)$ near $\lambda_c$ and the long-time asymptotics of the autocorrelator and the propagator.