Will a time-varying complex system be stable?

TL;DR

This paper extends classical complexity-stability theory to time-varying systems, showing that temporal variability can enhance stability and delay instability in complex dynamical systems.

Contribution

It derives exact stability bounds for non-autonomous systems with stochastic parameters, revealing how temporal variability can improve stability beyond static interaction predictions.

Findings

Temporal variability allows systems to stay stable despite instantaneous instability predictions.

The theory applies exactly to neural network models and numerically to Lotka-Volterra equations.

Time-varying interactions postpone the onset of replica-symmetry breaking.

Abstract

Randomly-assembled dynamical systems are theoretically predicted to be unstable upon crossing a critical threshold of complexity, as first shown by May. Yet, empirical complex systems exhibit remarkable stability, indicating the presence of additional mechanisms playing a stabilizing role. The relation between complexity and stability is typically assessed by assuming fixed interactions, whereas real systems often evolve in intrinsically time-dependent states. To understand how this affects stability, we linearize a general non-autonomous dynamics around a reference operating state and model the resulting parameters as stochastic processes, which represent the minimal extension of static random interactions to time-varying ones. We derive exact stability bounds that generalize complexity-stability theory to dynamically varying systems. Notably, we find that temporal variability allows systems to remain stable even when their instantaneous Jacobian would predict instability. We compare our results against a non-linear neural network model, where our theory applies exactly, and the generalized Lotka-Volterra equations, where we numerically find that time-varying interactions systematically postpone the onset of replica-symmetry breaking. Overall, our results indicate that temporal variability systematically improves stability, demonstrating a general mechanism by which complex systems can violate classical complexity-stability bounds.