Spectra of complex networks

Details Spectra of complex networks Spectra of complex networks

2003-06-12

arxiv.org/abs/cond-mat/0306340v2

Phys. Rev. E 68, 046109 (2003)

Physics

Condensed Matter

Statistical Mechanics

10 pages, 4 figures

Scientific paper

10.1103/PhysRevE.68.046109

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.

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