Eigenvalue spectra of complex networks (original) (raw)

2005, Journal of Physics A: Mathematical and General

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/µ 2λ−1 for large µ. In the limit λ → ∞ we recover known results for the Erdös-Rényi random graph.