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

Izaak Neri, Fernando Lucas Metz

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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.
Original languageEnglish
Article number224101
Number of pages6
JournalPhysical Review Letters
Issue number22
Early online date23 Nov 2016
Publication statusPublished - 25 Nov 2016


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