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 ${\Delta }_3$ statistic of RMT as well. It follows RMT prediction of linear behavior in semilogarithmic scale with the slope being $\sim 1∕{\pi }^2$. 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.

• Received 18 March 2007

DOI:https://doi.org/10.1103/PhysRevE.76.046107