Spectral statistics in directed complex networks and universality of the Ginibre ensemble

• Spectral distributions for directed complex networks follow generalized circle law.
• For strongly non-Hermitian adjacency matrix, its spectra follow Ginibre statistics.
• For weakly non-Hermitian adjacency matrix, its spectra follow Poisson statistics.

Spectra of the adjacency matrices of directed complex networks are analyzed by using non-Hermitian random matrix theory. Both the short-range and long-range correlations in the eigenvalues are calculated numerically for directed model complex networks and real-world networks. The results are compared with predictions of Ginibre’s ensemble. The spectral density ρ(λ)ρ(λ), the nearest neighbor spacing distribution p(s)p(s) and the level-number variance Σ2(L)Σ2(L) show good agreement with Ginibre’s ensemble when the adjacency matrices of directed complex networks are in the strongly non-Hermitian regime.