Complex networks of interacting stochastic tipping elements: cooperativity of phase separation in the large-system limit

TL;DR

This paper models the collective behavior of large networks of interacting stochastic tipping elements, deriving analytical predictions for their response to perturbations and validating these with numerical simulations across various network types.

Contribution

It introduces an analytical framework for understanding emergent phase separation in large networks of tipping elements, accounting for network size, connectivity, and coupling.

Findings

Analytical prediction matches numerical simulations.

Large networks exhibit emergent collective tipping behavior.

Network topology influences the response dynamics.

Abstract

Tipping elements in the Earth System receive increased scientific attention over the recent years due to their nonlinear behavior and the risks of abrupt state changes. While being stable over a large range of parameters, a tipping element undergoes a drastic shift in its state upon an additional small parameter change when close to its tipping point. Recently, the focus of research broadened towards emergent behavior in networks of tipping elements, like global tipping cascades triggered by local perturbations. Here, we analyze the response to the perturbation of a single node in a system that initially resides in an unstable equilibrium. The evolution is described in terms of coupled nonlinear equations for the cumulants of the distribution of the elements. We show that drift terms acting on individual elements and offsets in the coupling strength are sub-dominant in the limit of large networks, and we derive an analytical prediction for the evolution of the expectation (i.e., the first cumulant). It behaves like a single aggregated tipping element characterized by a dimensionless parameter that accounts for the network size, its overall connectivity, and the average coupling strength. The resulting predictions are in excellent agreement with numerical data for Erd\"os-R\'enyi, Barab\'asi-Albert and Watts-Strogatz networks of different size and with different coupling parameters.