Spectra of "real-world" graphs: Beyond the semicircle law

Illés J. Farkas*, Imre Derényi, Albert László Barabási, Tamás Vicsek

*Corresponding author for this work

Research output: Contribution to Book/Report typesChapterpeer-review

Abstract (may include machine translation)

Many natural and social systems develop complex networks that are usually modeled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semicircle law is known to describe the spectral densities of uncorrected random graphs, much less is known about the spectra of real-world graphs, describing such complex systems as the Internet, metabolic pathways, networks of power stations, scientific collaborations, or movie actors, which are inherently correlated and usually very sparse. An important limitation in addressing the spectra of these systems is that the numerical determination of the spectra for systems with more than a few thousand nodes is prohibitively time and memory consuming. Making use of recent advances in algorithms for spectral characterization, here we develop methods to determine the eigenvalues of networks comparable in size to real systems, obtaining several surprising results on the spectra of adjacency matrices corresponding to models of real-world graphs. We find that when the number of links grows as the number of nodes, the spectral density of uncorrected random matrices does not converge to the semicircle law. Furthermore, the spectra of real-world graphs have specific features, depending on the details of the corresponding models. In particular, scale-free graphs develop a trianglelike spectral density with a power-law tail, while small-world graphs have a complex spectral density consisting of several sharp peaks. These and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks.

Original languageEnglish
Title of host publicationThe Structure and Dynamics of Networks
PublisherPrinceton University Press
Pages372-384
Number of pages13
Volume9781400841356
ISBN (Electronic)9781400841356
ISBN (Print)0691113572, 9780691113579
DOIs
StatePublished - 23 Oct 2011
Externally publishedYes

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