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.
| Original language | English |
|---|---|
| Article number | 434003 |
| Journal | Journal of Physics A: Mathematical and Theoretical |
| Volume | 52 |
| Issue number | 43 |
| Early online date | 26 Apr 2019 |
| DOIs | |
| Publication status | Published - 1 Oct 2019 |
Keywords
- cond-mat.stat-mech
- cond-mat.dis-nn
- math-ph
- math.MP
- non-Hermitian matrices
- random matrix theory
- complex networks
- sparse matrices
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Statistics and Probability
- Mathematical Physics
- Modelling and Simulation
Cite this
Spectral Theory of Sparse Non-Hermitian Random Matrices. / Metz, Fernando Lucas; Neri, Izaak; Rogers, Tim.
In: Journal of Physics A: Mathematical and Theoretical, Vol. 52, No. 43, 434003, 01.10.2019.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Spectral Theory of Sparse Non-Hermitian Random Matrices
AU - Metz, Fernando Lucas
AU - Neri, Izaak
AU - Rogers, Tim
N1 - Invited Topical Review
PY - 2019/10/1
Y1 - 2019/10/1
N2 - 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.
AB - 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.
KW - cond-mat.stat-mech
KW - cond-mat.dis-nn
KW - math-ph
KW - math.MP
KW - non-Hermitian matrices
KW - random matrix theory
KW - complex networks
KW - sparse matrices
UR - http://www.scopus.com/inward/record.url?scp=85069812488&partnerID=8YFLogxK
U2 - 10.1088/1751-8121/ab1ce0
DO - 10.1088/1751-8121/ab1ce0
M3 - Article
VL - 52
JO - Journal of Physics A: Mathematical and Theoretical
JF - Journal of Physics A: Mathematical and Theoretical
SN - 1751-8113
IS - 43
M1 - 434003
ER -