
Physics 2001
Spectra of "RealWorld" Graphs: Beyond the SemiCircle LawDOI: 10.1103/PhysRevE.64.026704 Abstract: Many natural and social systems develop complex networks, that are usually modelled 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 density of uncorrelated random graphs, much less is known about the eigenvalues of realworld 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 new 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 realworld graphs. We find that when the number of links grows as the number of nodes, the spectral density of uncorrelated random graphs does not converge to the semicircle law. Furthermore, the spectral densities of realworld graphs have specific features depending on the details of the corresponding models. In particular, scalefree graphs develop a trianglelike spectral density with a power law tail, while smallworld graphs have a complex spectral density function 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.
