Spectral Theory of Sparse Non-Hermitian Random Matrices

TL;DR

It is shown how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations, and the use of these methods to obtain both analytic and numerical results for the spectrum.

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.

Figures (10)

• Figure 1: Left: eigenvalues of a random matrix of dimension $n=10^3$, with three non-zero entries per row and column, placed uniformly at random. Right: spectrum of the non-Hermitian operator $\mathcal{A}$ acting on the space of sequences indexed by elements of the free group $\mathcal{F}_3$ on generators $\{\alpha,\beta,\gamma\}$, defined by $\mathcal{A}|x\rangle_{w}=|x\rangle_{w\alpha}+|x\rangle_{w\beta}+|x\rangle_{w\gamma}$. The group $\mathcal{F}_3$ consists of all finite length strings of symbols $\{\alpha,\beta,\gamma\}$, or their inverses $\{\alpha^{-1},\beta^{-1},\gamma^{-1}\}$, after cancellation of adjacent reciprocal pairs. To see that $\mathcal{A}$ is non-Hermitian, note that $\mathcal{A}^{\dag}|x\rangle_{w}=|x\rangle_{w\alpha^{-1}}+|x\rangle_{w\beta^{-1}}+|x\rangle_{w\gamma^{-1}}$. The spectrum of $\mathcal{A}$ comprises a circle of radius $\sqrt{3}$ around the origin, and the eigenvalue $\lambda=3$.
• Figure 2: Left: white circles show the eigenvalues of the "bull's head" matrix $\mathbf{A}$ of dimension $n=50$, colour map shows the regularised spectral distribution of $\mathbf{A}$ with $\eta=0.01$. Right: eigenvalues of 50 realisations of the perturbed bull's head matrix $\mathbf{A}+\eta \mathbf{P}\mathbf{Q}^{-1}$, where $\mathbf{P}$ and $\mathbf{Q}$ have independent complex Gaussian entries. The bull's head matrix is a banded Toeplitz matrix (i.e. constant along its diagonals) with rows $(\cdots 0,\, 2i,\, 0,\, 0,\, 1,\, 7,\, 0 \cdots)$. As illustrated here, its spectrum is highly sensitive to perturbations, even at large values of $n$. This implies that the limits $\eta\to0$ and $n\to\infty$ do not commute for this example.
• Figure 3: Illustration of the local tree-like structure of sparse graphs (adapted from rogers2008cavity). In large tree-like networks, removal of a site de-correlates the neighbours, leading to an almost exact statistical recursion relation. Here, nodes $k_1$ and $k_2$ become disconnected from each other after removal of node $j$, moreover, they are also insensitive to the additional removal of node $\ell$.
• Figure 4: (a) A simple four-node graph $G$. (b) A piece of the infinite directed tree $\mathcal{T}$ of non-backtracking walks on $G$. The five edges of $G$ give rise to ten equivalence classes of nodes in $\mathcal{T}$ indexed by the two different directions each edge may be traversed, illustrated here by the five distinct node designs.
• Figure 5: Spectral distribution of the one-dimensional lattice with IID random weights $J_{kj},J_{jk}\in\{-1,1\}$, computed using population dynamics simulation of the cavity equations.