Eigenvalue spectra of complex networks

Abstract We examine the eigenvalue spectrum, ρ(μ), 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 ρ(μ) for an arbitrary complex network. For scale-free graphs, with degree distribution exponent λ, we obtain an exact expression for the eigenvalue spectrum when λ = 3 and show that ρ(μ) ∼ 1/μ2λ−1 for large μ. In the limit λ → ∞ we recover known results for the Erdös–Rényi random graph.