  Previous literature on random matrix and network science has traditionally
employed measures derived from nearest-neighbor level spacing distributions to
characterize the eigenvalue statistics of random matrices. This approach,
however, depends crucially on eigenvalue unfolding procedures, which in many
situations represent a major hindrance due to constraints in the calculation,
specially in the case of complex spectra. Here we study the spectra of directed
networks using the recently introduced ratios between nearest- and
next-to-nearest eigenvalue spacing, thus circumventing the shortcomings imposed
by spectral unfolding. Specifically, we characterize the eigenvalue statistics
of directed Erd\H{o}s-R\'enyi (ER) random networks by means of two adjacency
matrix representations; namely (i) weighted non-Hermitian random matrices and
(ii) a transformation on non-Hermitian adjacency matrices which produces
weighted Hermitian matrices. For both representations, we find that the
distribution of spacing ratios becomes universal for a fixed average degree, in
accordance with undirected random networks. Furthermore, by calculating the
average spacing ratio as a function of the average degree, we show that the
spectral statistics of directed ER random networks undergoes a transition from
Poisson to Ginibre statistics for model (i) and from Poisson to Gaussian
Unitary Ensemble statistics for model (ii). Eigenvector delocalization effects
of directed networks are also discussed.
