Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties

Abstract

The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random graphs with an arbitrary degree distribution in the high-connectivity limit, from which we perform a thorough analysis of the impact of degree fluctuations on the spectral density, the inverse participation ratio, and the distribution of the local density of states. We show that all eigenvectors are extended and that the spectral density exhibits a logarithmic or a power-law divergence when the variance of the degree distribution is large enough. We elucidate this singular behaviour by showing that the distribution of the local density of states at the center of the spectrum displays a power-law tail determined by the variance of the degree distribution. In the regime of weak degree fluctuations the spectral density has a finite support, which promotes the stability of large complex systems on random graphs.

22 pages, 6 figures

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• Spectral properties, localization transition and multifractal eigenvectors of the Laplacian on heterogeneous networks
• Robust extended states in Anderson model on partially disordered random regular graphs
• Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise
• A path integral approach to sparse non-Hermitian random matrices
• Statistical field theory of random graphs with prescribed degrees
• Condensation of vanishing photon emission rates in random atomic clouds