Spectra of random graphs with given expected degrees

Abstract

In the study of the spectra of power-law graphs, there are basically two competing approaches. One is to prove analogues of Wigner's semicircle law, whereas the other predicts that the eigenvalues follow a power-law distribution. Although the semicircle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power-law graph follow the semicircle law, whereas the spectrum of the adjacency matrix of a power-law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. Of interest are a number of (new) values for the exponent ␤, where phase transitions for eigenvalue distributions occur. The spectrum distributions have direct implications to numerous graph algorithms such as, for example, randomized algorithms that involve rapidly mixing Markov chains. E igenvalues of graphs are useful for controlling many graph properties and consequently have numerous algorithmic applications including low rank approximations, ‡ information retrieval (1), and computer vision. § Of particular interest is the study of eigenvalues for graphs with power-law degree distributions (i.e., the number of vertices of degree j is proportional to j Ϫ␤ for some exponent ␤). It has been observed by many research groups (2-8, ¶) that many realistic massive graphs including Internet graphs, telephone-call graphs, and various social and biological networks have power-law degree distributions.For the classical random graphs based on the Erdös-Rényi model, it has been proved by Füredi and Komlós that the spectrum of the adjacency matrix follows the Wigner semicircle law (9). Wigner's theorem (10) and its extensions have long been used for the stochastic treatment of complex quantum systems that lie beyond the reach of exact methods. The semicircle law has extensive applications in statistical and solid-state physics (21,22). Here we intend to reconcile these two schools of thought on eigenvalue distributions. To begin with, there are in fact several ways to associate a matrix to a graph. The usual adjacency matrix A associated with a (simple) graph has eigenvalues quite sensitive to the maximum degree (which is a local property). The combinatorial Laplacian D Ϫ A, with D denoting the diagonal degree matrix, is a major tool for enumerating spanning trees and has numerous applications (13, 14). Another matrix associated with a graph is the (normalized) Laplacian L ϭ I Ϫ D Ϫ1/2 AD Ϫ1/2 , which controls the expansion͞isoperimetrical properties (which are global) and essentially determines the mixing rate of a random walk on the graph. The traditional random matrices and random graphs are regular or almost regular, thus the spectra of all the above three matrices are basically the same (with possibly a scaling factor or a linear shift). However, for graphs with...

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