  In a Nature article, Scheffer et al. 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 paper 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: (1) Kuramoto model for the synchronization of coupled
oscillators and (2) 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.
