Spectral Density of Complex Networks with a Finite Mean Degree

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.

arXiv comment18 pages, 1 figure, minor corrections made

References in corpus (8)

• Emergence of scaling in random networks
• Universal Behavior of Load Distribution in Scale-free Networks
• Spectra of complex networks
• Sparse random matrices: the eigenvalue spectrum revisited
• A single defect approximation for localized states on random lattices
• Universality in Complex Networks: Random Matrix Analysis
• Spin Glass Phase Transition on Scale-Free Networks
• Spectral Density of Sparse Sample Covariance Matrices

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