8,951 research outputs found
Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
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.Comment: 25 pages, 4 figure
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
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