Linear stability analysis for large dynamical systems on directed random graphs

TL;DR

This work derives an exact, universal theory for the linear stability of stationary states of large dynamical systems defined on random directed graphs with prescribed indegree–outdegree distributions. By analyzing the leading eigenvalue of the adjacency-derived Jacobian in the limit $n\to\infty$, the authors show stability can persist for heavy-tailed directed graphs and establish a universal phase diagram controlled by the effective connectivity $c(1+\rho)$, the coupling-variance $v_J$, and the ratio $\alpha=\langle J\rangle/d$. The spectrum splits into a deterministic part governed by the giant strongly connected component and a potential random part from oriented rings, with explicit formulas for the boundary of the continuous spectrum, outliers, and the leading eigenvector statistics; these results are validated by finite-size numerical experiments and extended to diagonal disorder and nondirected graphs. The findings have broad implications for stability analyses in networks, including ecological, financial, and information-spreading systems, and provide a framework for studying cycles and feedback in large directed networks.

Abstract

We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally tree-like and oriented, as well as, for the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops.

Figures (8)

• Figure 1: Topology of directed graphs. Graphical illustration of the connected components of directed graphs (bow-tie diagram, see also Refs. broder2000graphdorogovtsev2001gianttimar2017mapping): largest strongly connected component (SCC), largest incomponent (IN), largest outcomponent (OUT), largest weakly connected component (WC), and isolated components (IC), which consist of isolated trees and cycles.
• Figure 2: Distribution of the leading eigenvalue. Sketch of the distribution $p_{\lambda_1}$ of the leading eigenvalue $\lambda_1$ of random matrices $\mathbf{A}$, as defined in Sec. \ref{['eq:modelDef']}, in the regime $c(\rho+1)>1$. The distribution consists of a delta distribution at the typical value $\lambda^\ast$ given by Eq. (\ref{['eq:lamddaAst']}) and a continuous distribution $p_{\rm cycle}$ with a total weight $\nu\approx 0$.
• Figure 3: Universal phase diagram for the stability of dynamical systems on random directed graph with positive $\langle J\rangle$. Black solid line and black dashed line separate the unstable phase at large effective connectivity $c(\rho+1)$ from the stable phase at small connectivity $c(\rho+1)$ for two given values of $\alpha = \langle J\rangle/d$. The red dotted line separates the gapped phase at small $v_J$ from a gapless phase at high $v_J$, which can also be considered a transition line from a ferromagnetic phase (gapped) to a spin-glass phase (gapless).
• Figure 4: Universal phase diagram diagram for the stability of dynamical systems on random directed graph with negative $\langle J\rangle$. Similar as in figure \ref{['fig5']}, but now for negative $\alpha$. In this case there is no gapped (or ferromagnetic) phase.
• Figure 5: Effect of negative $\rho$ on the spectral properties of the adjacency matrices of random directed graphs. Spectral properties for the adjacency matrices of Poissonian [see Eq. (\ref{['eq:neg1']})] or geometric [see Eq. (\ref{['eq:neg2']})] random directed graphs with a mean degree $c=2$ and negative $\rho$ are presented. Direct diagonalization results for matrices of size $n=4000$ (markers) are compared with the theoretical results for infinitely large matrices (lines) derived in Sec. \ref{['eq:theory']}. Panels (a) and (b): eigenvalues $\lambda_j(\mathbf{A})$ of the adjacency matrices of two Poissonian random graphs with $\rho=0$ [Panel (a)] and $\rho=-0.3$ [Panel (b)], respectively, are plotted and compared with the theoretical boundary $\lambda_b$ for the spectrum given by Eq. (\ref{['eq:boundary']}). Panel (c): Mean values of the leading eigenvalue $\overline{\lambda}_1$ and real part of the subleading eigenvalue $\overline{{\rm Re}[\lambda_2]}$ are plotted as a function of $\rho$ and compared with theoretical results $\lambda_{\rm isol} = 2(\rho+1)$ and $|\lambda_{\rm b}| = \sqrt{2 (\rho+1)}$ if $\rho>-0.5$ and $\langle \lambda_1 \rangle = 1- [1-c (\rho +1)] e^{c (\rho +1)}$ if $\rho<-0.5$. Panel (d): Mean value $\overline{\mathcal{R}_1}$ for the entries of the right eigenvector associated with the leading eigenvalue are plotted as a function of $\rho$ and compared with the theoretical results $\frac{\langle R_1\rangle}{\sqrt{\langle |R_1|^2\rangle}} = \sqrt{ \frac{1 + 2\rho }{ 2+\rho -2\rho^2 } }$ and $\frac{\langle R_1 \rangle }{\sqrt{\langle R^2_1 \rangle}} = \sqrt{\frac{1+2\rho }{ 2 (2 +\rho - 2\rho^2)} }$ for the Poissonian and geometric ensemble, respectively, when $\rho\geq-0.5$, and with $\frac{\langle R_1\rangle}{\sqrt{\langle |R_1|^2\rangle}} = 0$ when $\rho<-0.5$. In Panels (c) and (d), direct diagonalization results are the sample means over $1000$ matrix realizations and error bars represent the sample standard deviations.