Deformed Gaussian-orthogonal-ensemble description of small-world networks

Transport in deformed centrosymmetric networks

Centrosymmetry often mediates perfect state transfer (PST) in various complex systems ranging from quantum wires to photosynthetic networks. We introduce the deformed centrosymmetric ensemble (DCE) of random matrices H(λ)≡H_{+}+λH_{-}, where H_{+} is centrosymmetric while H_{-} is skew-centrosymmetric. The relative strength of the H_{±} prompts the system size scaling of the control parameter as λ=N^{-γ/2}. We propose two quantities, P and C, quantifying centro and skewcentrosymmetry, respectively, exhibiting second-order phase transitions at γ_{P}≡1 and γ_{C}≡-1. In addition, DCE posses an ergodic transition at γ_{E}≡0. Thus equipped with a precise control of the extent of centrosymmetry in DCE, we study the manifestation of γ on the transport properties of complex networks. We propose that such random networks can be constructed using the eigenvectors of H(λ) and establish that the maximum transfer fidelity F_{T} is equivalent to the degree of centrosymmetry P.

Universality in the spectral and eigenfunction properties of random networks

By the use of extensive numerical simulations, we show that the nearest-neighbor energy-level spacing distribution P(s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ≡αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that the Brody distribution characterizes well P(s) in the transition from α=0, when the vertices in the network are isolated, to α=1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

Scattering and transport properties of tight-binding random networks

We study numerically scattering and transport statistical properties of tight-binding random networks characterized by the number of nodes N and the average connectivity α. We use a scattering approach to electronic transport and concentrate on the case of a small number of single-channel attached leads. We observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements <|S(mn)|(2)>, the conductance probability distribution w(T), the average conductance <T>, the shot noise power P, and the elastic enhancement factor F by varying α from small (α→0) to large (α→1) values. We also show that all these quantities are invariant for fixed ξ=αN. Moreover, we proposes a heuristic and universal relation between <|S(mn)|(2)>, <T>, and P and the disorder parameter ξ.

Random matrix analysis of localization properties of gene coexpression network

We analyze gene coexpression network under the random matrix theory framework. The nearest-neighbor spacing distribution of the adjacency matrix of this network follows Gaussian orthogonal statistics of random matrix theory (RMT). Spectral rigidity test follows random matrix prediction for a certain range and deviates afterwards. Eigenvector analysis of the network using inverse participation ratio suggests that the statistics of bulk of the eigenvalues of network is consistent with those of the real symmetric random matrix, whereas few eigenvalues are localized. Based on these IPR calculations, we can divide eigenvalues in three sets: (a) The nondegenerate part that follows RMT. (b) The nondegenerate part, at both ends and at intermediate eigenvalues, which deviates from RMT and expected to contain information about important nodes in the network. (c) The degenerate part with zero eigenvalue, which fluctuates around RMT-predicted value. We identify nodes corresponding to the dominant modes of the corresponding eigenvectors and analyze their structural properties.

Spectral analysis of deformed random networks

We study spectral behavior of sparsely connected random networks under the random matrix framework. Subnetworks without any connection among them form a network having perfect community structure. As connections among the subnetworks 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 Delta3 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 Delta3 statistics remain identical to the undeformed network. On the other hand the Delta3 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.