Eigenvalue spectra of complex networks

Abstract

We examine the eigenvalue spectrum, of the adjacency matrix of a random scale-free network with an average of p edges per vertex using the replica method. We show how in the dense limit, when p , one can obtain two relatively simple coupled equations whose solution yields for an arbitrary complex network. For scale-free graphs, with degree distribution exponent , we obtain an exact expression for the eigenvalue spectrum when = 3 and show that ~ 1/2-1 for large . In the limit we recover known results for the ErdösRényi random graph.

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