Random matrix analysis of spectral properties in directed complex networks

By using non-Hermitian random matrix theory, the spectra of adjacency matrices of directed complex networks are analyzed. Both the short-range and long-range correlations in the eigenvalues are numerically calculated for typical directed complex networks and compared with the predictions of Ginibre´s Ensemble. The spectral density ρ(λ), the nearest neighbor spacing distribution p(s) and the number variance Σ²(L) show good agreements with Ginibre´s ensemble when the adjacency matrices of directed complex networks are in the strongly non-Hermitian regime. Therefore, non-Hermitian random matrix theory provides a new way to model and study the spectral properties of directed complex networks.