Spacing ratio characterization of the spectra of directed random networks

TL;DR

This paper introduces a new method using eigenvalue spacing ratios to analyze the spectra of directed networks, avoiding spectral unfolding issues, and reveals universal behaviors and phase transitions in their eigenvalue statistics.

Contribution

It applies spacing ratio measures to directed networks, demonstrating universality and spectral transitions, and compares different matrix representations to characterize eigenvalue statistics.

Findings

Spacing ratio distribution becomes universal for fixed average degree.

Spectral statistics transition from Poisson to Ginibre or GUE depending on matrix representation.

Eigenvector delocalization effects are discussed.

Abstract

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.