Laplacian spectra of, and random walks on, complex networks: Are scale-free architectures really important?

We study the Laplacian operator of an uncorrelated random network and, as an application, consider hopping processes (diffusion, random walks, signal propagation, etc.) on networks. We develop a strict approach to these problems. We derive an exact closed set of integral equations, which provide the averages of the Laplacian operator’s resolvent. This enables us to describe the propagation of a signal and random walks on the network. We show that the determining parameter in this problem is the minimum degree q_m of vertices in the network and that the high-degree part of the degree distribution is not that essential. The position of the lower edge of the Laplacian spectrum λ_c appears to be the same as in the regular Bethe lattice with the coordination number q_m. Namely, λ_c>0 if q_m>2, and λ_c=0 if q_m≤2. In both of these cases the density of eigenvalues ρ(λ)→0 as λ→λ_c+0, but the limiting behaviors near λ_c are very different. In terms of a distance from a starting vertex, the hopping propagator is a steady moving Gaussian, broadening with time. This picture qualitatively coincides with that for a regular Bethe lattice. Our analytical results include the spectral density ρ(λ) near λ_c and the long-time asymptotics of the autocorrelator and the propagator.