Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure

TL;DR

This paper develops an exact theoretical framework for analyzing eigenvalue outliers in the spectra of sparse non-Hermitian random matrices with a local tree structure, relevant for understanding complex network dynamics.

Contribution

It introduces a general and exact theory for eigenvalue outliers in non-Hermitian matrices with local tree structures, providing analytical formulas for spectral properties.

Findings

Derived analytical expressions for eigenvalue outliers.

Identified universal spectral observables across a broad class of matrices.

Provided insights into stability and dynamics of processes on graphs.

Abstract

Spectra of sparse non-Hermitian random matrices determine the dynamics of complex processes on graphs. Eigenvalue outliers in the spectrum are of particular interest, since they determine the stationary state and the stability of dynamical processes. We present a general and exact theory for the eigenvalue outliers of random matrices with a local tree structure. For adjacency and Laplacian matrices of oriented random graphs, we derive analytical expressions for the eigenvalue outliers, the first moments of the distribution of eigenvector elements associated with an outlier, the support of the spectral density, and the spectral gap. We show that these spectral observables obey universal expressions, which hold for a broad class of oriented random matrices.