Spacing ratio characterization of the spectra of directed random networks

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 Erdős-Rényi (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.

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• From Ergodicity to Many-Body Localization in a One-Dimensional Interacting Non-Hermitian Stark System
• Spacing ratio statistics of multiplex directed networks
• Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry
• Boundary-driven many-body phase transitions in a non-Hermitian disordered fermionic chain
• Non-uniform random graphs on the plane: A scaling study
• Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos
• On the extremal spectral properties of random graphs
• Limitations of SVD-Based Diagnostics for Non-Hermitian Many-Body Localization with Time-Reversal Symmetry
• Statistical properties of mutualistic-competitive random networks