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 Erdös–Rényi random graph.