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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.