title: Eigenvalue Outliers of Non-Hermitian Random Matrices with a Local Tree Structure.
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.
authors.affiliation: Max Planck Institute for the Physics of Complex Systems, Nöthnitzerstraße 38, 01187 Dresden, Germany.|Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstraße 108, 01307 Dresden, Germany.
authors.forename: Izaak
authors.initials: I
authors.lastname: Neri
authors.affiliation: Departamento de Física, Universidade Federal de Santa Maria, 97105-900 Santa Maria, Brazil.
authors.forename: Fernando Lucas
authors.initials: FL
authors.lastname: Metz