Analysis Of Complex Network By Using Random Matrix Theory

For the large-scale, strong-nonlinear, high complexity and variety of complexnetworks, analytical and numerical methods are not well suited to analysis andresearch complex networks. Thus, the statistical methods go into people’s view. Asone of the basic tools of statistical analysis, random matrix theory sets up a bridgebetween microscopic and macroscopic properties of complex network.Eigenvalues of complex network (or after mapping) adjacency matrixcorresponding to the energy level of random matrix theory, so the characteristics ofcomplex networks is reflected in the volatility of the eigenvalues of a sequence. Weuse random matrix theory to analyze complex eigenvalues of the network, in orderto find the relationship between structure and properties.By mapping the adjacency matrix of a complex network to Hamiltonian of aquantum system, the statistical properties of the spectra and eigenstates are analyzed.The spectral statistics, i.e. the nearest-neighbor spacing distribution, the numbervariance and the spectral form factor, are analyzed numerically. The results showthat when the rewiring probability of small-world network model is lower, thespectral properties are consistent with those of quantum integrable systems. Whenthe rewiring probability is higher than a certain threshold, its energy spectrumproperties are similar to those of the Gaussian orthogonal ensembles in randommatrix theory. Then, using random matrix theory to analyze the Western UnitedStates power grid network, the result is consistent with the analysis of theconclusions of scale-free networks. These results hint that certain analogies mayexist between the spatial topology of complex networks and the temporal evolutionproperties of quantum dynamical systems.Similarly, by using non-Hermitian random matrix theory, the spectra of sometypical directed complex networks are analyzed. Both the short-range andlong-range correlations in the eigenvalues are numerically calculated and comparedwith the predictions of random matrix theory. The spectral density, the nearestneighbor spacing distribution and the number variance show good agreements withWigner-Dyson ensemble when the adjacency matrices of directed complex networksare in the weak non-Hermitian regime. When the adjacency matrices are stronglynon-Hermitian, the statistics follows the predictions of Ginibre’s ensemble. Meanwhile, using random matrix theory to analyze a P2P network, the result isconsistent with the analysis of the conclusions of scale-free networks.Therefore, non-Hermitian random matrix provides a new way to model andstudy the properties of directed complex networks.