Spectral Density of Complex Networks with a Finite Mean Degree

TL;DR

This paper analyzes the spectral density of adjacency matrices in scale-free complex networks with finite mean degree using replica and EMA methods, deriving new integral equations and corrections to known asymptotic formulas.

Contribution

It introduces a new integral equation for spectral density and analytically computes $1/p$ corrections for scale-free networks with finite mean degree.

Findings

Derived a new integral equation for spectral density.

Reproduced known asymptotic formulas as $p o obreak \infty$.

Calculated $1/p$ corrections analytically.

Abstract

In order to clarify the statistical features of complex networks, the spectral density of adjacency matrices has often been investigated. Adopting a static model introduced by Goh, Kahng and Kim, we analyse the spectral density of complex scale free networks. For that purpose, we utilize the replica method and effective medium approximation (EMA) in statistical mechanics. As a result, we identify a new integral equation which determines the asymptotic spectral density of scale free networks with a finite mean degree $p$. In the limit $p \to \infty$, known asymptotic formulae are rederived. Moreover, the $1/p$ corrections to known results are analytically calculated by a perturbative method.