Robustness to perturbation is a key topic in the study of complex systems occurring across a wide variety of applications from epidemiology to biochemistry. In this talk I will examine 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. I will show that stability is always determined by a spectral outlier, but with pronounced differences to the corresponding eigenvector in different regimes. Depending on model details, instability may originate in nodes of anomalously low or high degrees, or may occur everywhere in the network at once. Our results have potentially useful applications in network monitoring to predict or prevent catastrophic failures.
