Spectral theory of sparse non-Hermitian random matrices

Abstract

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 Erdős-Rényi graphs, and adjacency matrices of weighted oriented Erdős-Rényi 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.

Linear stability analysis of large dynamical systems on random directed graphs

“…(D1), we use the spectral theory for sparse, non-Hermitian, random matrices from Refs. [42,45]. As shown in those references, the spectral distribution μ(z) of matrices of the form given by Eq.…”

“…These recursion relations have first been derived in Ref. [44] using the cavity method [42,45,[50][51][52][53], a method borrowed from the statistical physics of spin glasses [54,55]. In the present paper, we present an alternative derivation for these recursion relations based on the Schur formula [56], which we believe is simpler to understand and thus more insightful.…”

“…[45] for an example ofμ(λ) in the case of directed Erdős-Rényi ensembles. Equation (45) implies that the algebraic multiplicity of the −d eigenvalue is equal to…”

“…Indeed, the leading right eigenvector R 1 is the stationary state of a spherical model defined on the graph represented by the adjacency matrix A, see equations (45) till (52) in Ref. [45]. The spherical model at zero temperature exhibits either a ferromagnetic phase or a spin-glass phase, see Ref.…”

“…[39][40][41], and recently also spectral properties of random, directed graphs have been studied, see Refs. [42][43][44][45][46], but the properties of the leading eigenvalue of the adjacency matrices of random, directed graphs have not been studied so far.…”