Spectra of complex networks

TL;DR

This paper develops a general method to analyze the spectral properties of complex networks, deriving exact equations for uncorrelated networks and extending to correlated ones, with applications to real-world networks like the Internet.

Contribution

It introduces a unified approach to describe spectra of complex networks, including correlated cases, and provides analytical tools for spectral analysis of real-world networks.

Findings

Spectral density tail relates to degree distribution at large degrees.

Analytical approximation for spectra of various graphs.

Spectra of locally tree-like graphs can inform real-world network analysis.

Abstract

We propose a general approach to the description of spectra of complex networks. For the spectra of networks with uncorrelated vertices (and a local tree-like structure), exact equations are derived. These equations are generalized to the case of networks with correlations between neighboring vertices. The tail of the density of eigenvalues $\rho(\lambda)$ at large $|\lambda|$ is related to the behavior of the vertex degree distribution $P(k)$ at large $k$. In particular, as $P(k) \sim k^{-\gamma}$, $\rho(\lambda) \sim |\lambda|^{1-2\gamma}$. We propose a simple approximation, which enables us to calculate spectra of various graphs analytically. We analyse spectra of various complex networks and discuss the role of vertices of low degree. We show that spectra of locally tree-like random graphs may serve as a starting point in the analysis of spectral properties of real-world networks, e.g., of the Internet.