Network Critical Slowing Down: Data-Driven Detection of Critical Transitions in Nonlinear Networks

Scheffer et al. (2009) presented a novel data-driven framework to predict critical transitions in complex systems. These transitions, which may stem from failures, degradation, or adversarial actions, have been attributed to bifurcations in the nonlinear dynamics. Their approach was built upon the phenomenon of
critical slowing down
, i.e., slow recovery in response to small perturbations near bifurcations. We extend their approach to detect and localize critical transitions in nonlinear networks. By introducing the notion of
network critical slowing down
, the objective of this article is to detect that the network is undergoing a bifurcation only by analyzing its signatures from measurement data. We focus on two classes of widely used nonlinear networks: Kuramoto model for the synchronization of coupled oscillators, and attraction–repulsion dynamics in swarms, each of which presents a specific type of bifurcation. Based on the phenomenon of
critical slowing down
, we study the asymptotic behavior of the perturbed system away and close to the bifurcation and leverage this fact to develop a deterministic method to detect and identify critical transitions in nonlinear networks. Furthermore, we study the state covariance matrix subject to a stochastic noise process away and close to the bifurcation and use it to develop a stochastic framework for detecting critical transitions. Our simulation results show the strengths and limitations of the methods.