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 adjacency matrix of various model networks, namely, random, scale-free and small-world networks. These distributions follow Gaussian orthogonal ensemble statistic of RMT. To probe long-range correlations in the eigenvalues we study spectral rigidity via ${\Delta }_3$ statistic of RMT as well. It follows RMT prediction of linear behavior in semi-logarithmic scale with slope being $\sim 1/{\pi }^2$. Random and scale-free networks follow RMT prediction for very large scale. Small-world network follows it for sufficiently large scale, but much less than the random and scale-free networks.

accepted in Phys. Rev. E (replaced with the final version)

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