Abstract
Although the spectra of random networks have been studied for a long time, the influence of network topology on the dense limit of network spectra remains poorly understood. By considering the configuration model of networks with four distinct degree distributions, we show that the spectral density of the adjacency matrices of dense random networks is determined by the strength of the degree fluctuations. In particular, the eigenvalue distribution of dense networks with an exponential degree distribution is governed by a simple equation, from which we uncover a logarithmic singularity in the spectral density. We also derive a relation between the fourth moment of the eigenvalue distribution and the variance of the degree distribution, which leads to a sufficient condition for the breakdown of the Wigner semicircle law for dense random networks. Based on the same relation, we propose a classification scheme of the distinct universal behaviours of the spectral density in the dense limit. Our theoretical findings should lead to important insights on the mean-field behaviour of models defined on graphs.
Highlights
A random network or graph is a collection of nodes joined by edges following a probabilistic rule
The study of a broad range of problems amounts to linearizing a large set of differential equations [4,5,6], coupled through a random network, around a stationary state, whose stability and transient dynamics are determined by the eigenvalue distribution of the adjacency matrix
We analyze four examples of degree distributions in which all moments are finite and the variances scale differently with the average degree, and we show that the dense limit of the eigenvalue distribution is determined by the degree fluctuations
Summary
A random network or graph is a collection of nodes joined by edges following a probabilistic rule. With the exception of graphs with a power-law degree distribution [19,21,35] whose moments are divergent, one expects that the eigenvalue distribution of undirected random networks converges, in the dense limit, to the Wigner semicircle distribution of random matrix theory, reflecting a high level of universality. [44] provides an approach to compute the spectral density for specific degree distributions In spite of these rigorous results, a more detailed understanding of how the structure of a dense random network influences its spectrum, and the universal status of the Wigner semicircle law, is still lacking.
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