Origins of Instability in Dynamical Systems on Undirected Networks

TL;DR

This paper investigates how the stability of networked dynamical systems is influenced by the spectral properties of their Jacobian matrices, revealing that instability can originate from nodes with extreme degrees and varies with network size and structure.

Contribution

It provides a spectral analysis linking eigenvalue outliers to instability origins in undirected networks, with implications for predicting and preventing failures in complex systems.

Findings

Stability determined by spectral outliers with regime-dependent eigenvectors

Instability can originate from nodes with very high or low degrees

Finite-size effects depend on the degree distribution ensemble

Abstract

Robustness to perturbation is a key topic in the study of complex systems occurring across a wide variety of applications from epidemiology to biochemistry. Here we analyze the eigenspectrum of the Jacobian matrices associated to a general class of networked dynamical systems, which contains information on how perturbations to a stationary state develop over time. We find that stability is always determined by a spectral outlier, but with pronounced differences to the corresponding eigenvector in different regimes. We show that, depending on model details, instability may originate in nodes of anomalously low or high degree, or may occur everywhere in the network at once. Importantly, the dependence on extremal degrees results in considerable finite-size effects with different scaling depending on the ensemble degree distribution. Our results have potentially useful applications in network monitoring to predict or prevent catastrophic failures, and we validate our analytical findings through applications to epidemic dynamics and gene regulatory systems.