Random matrix analysis of complex networks

Abstract

We study complex networks under random matrix theory (RMT) framework. Using nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the eigenvalues of the adjacency matrix of various model networks, namely, random, scale-free, and small-world networks. These distributions follow the Gaussian orthogonal ensemble statistic of RMT. To probe long-range correlations in the eigenvalues we study spectral rigidity via the Δ3 statistic of RMT as well. It follows RMT prediction of linear behavior in semilogarithmic scale with the slope being ∼1/π2. Random and scale-free networks follow RMT prediction for very large scale. A small-world network follows it for sufficiently large scale, but much less than the random and scale-free networks.