On scale-free and poly-scale behaviors of random hierarchical network

In this paper the question about statistical properties of block--hierarchical random matrices is raised for the first time in connection with structural characteristics of random hierarchical networks obtained by mipmapping procedure. In particular, we compute numerically the spectral density of large random adjacency matrices defined by a hierarchy of the Bernoulli distributions $\{q_1,q_2,...\}$ on matrix elements, where $q_{\gamma}$ depends on hierarchy level $\gamma$ as $q_{\gamma}=p^{-\mu \gamma}$ ($\mu>0$). For the spectral density we clearly see the free--scale behavior. We show also that for the Gaussian distributions on matrix elements with zero mean and variances $\sigma_{\gamma}=p^{-\nu \gamma}$, the tail of the spectral density, $\rho_G(\lambda)$, behaves as $\rho_G(\lambda) \sim |\lambda|^{-(2-\nu)/(1-\nu)}$ for $|\lambda|\to\infty$ and $0<\nu<1$, while for $\nu\ge 1$ the power--law behavior is terminated. We also find that the vertex degree distribution of such hierarchical networks has a poly--scale fractal behavior extended to a very broad range of scales.Comment: 11 pages, 6 figures (paper is substantially revised

Spectral analysis of deformed random networks

We study spectral behavior of sparsely connected random networks under the random matrix framework. Sub-networks without any connection among them form a network having perfect community structure. As connections among the sub-networks are introduced, the spacing distribution shows a transition from the Poisson statistics to the Gaussian orthogonal ensemble statistics of random matrix theory. The eigenvalue density distribution shows a transition to the Wigner's semicircular behavior for a completely deformed network. The range for which spectral rigidity, measured by the Dyson-Mehta $\Delta_3$ statistics, follows the Gaussian orthogonal ensemble statistics depends upon the deformation of the network from the perfect community structure. The spacing distribution is particularly useful to track very slight deformations of the network from a perfect community structure, whereas the density distribution and the $\Delta_3$ statistics remain identical to the undeformed network. On the other hand the $\Delta_3$ statistics is useful for the larger deformation strengths. Finally, we analyze the spectrum of a protein-protein interaction network for Helicobacter, and compare the spectral behavior with those of the model networks.Comment: accepted for publication in Phys. Rev. E (replaced with the final version

Random Matrices and Chaos in Nuclear Spectra

We speak of chaos in quantum systems if the statistical properties of the eigenvalue spectrum coincide with predictions of random-matrix theory. Chaos is a typical feature of atomic nuclei and other self-bound Fermi systems. How can the existence of chaos be reconciled with the known dynamical features of spherical nuclei? Such nuclei are described by the shell model (a mean-field theory) plus a residual interaction. We approach the question by using a statistical approach (the two-body random ensemble): The matrix elements of the residual interaction are taken to be random variables. We show that chaos is a generic feature of the ensemble and display some of its properties, emphasizing those which differ from standard random-matrix theory. In particular, we display the existence of correlations among spectra carrying different quantum numbers. These are subject to experimental verification.Comment: 17 pages, 20 figures, colloquium article, submitted to Reviews of Modern Physic

Random matrix analysis of complex networks

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