Spectral properties of complex networks

TL;DR

This paper develops a theoretical framework using the replica method to analyze the spectral properties of adjacency and Laplacian matrices in complex networks, including weighted and correlated networks, advancing beyond previous approximations.

Contribution

It introduces a novel approach combining the replica method and a population algorithm to solve non-linear integral equations for spectra density in sparse random matrices.

Findings

Provides a solution for spectra density in sparse random matrices

Extends spectral analysis to weighted networks with correlations

Advances beyond the effective medium approximation (EMA)

Abstract

We derive the spectral properties of adjacency matrix of complex networks and of their Laplacian by the replica method combined with a dynamical population algorithm. By assuming the order parameter to be a product of Gaussian distributions, the present theory provides a solution for the non linear integral equations for the spectra density in random matrix theory of the spectra of sparse random matrices making a step forward with respect to the effective medium approximation (EMA) . We extend these results also to weighted networks with weight-degree correlations