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

Jan Kohler, Nico Wunderling, Jonathan F. Donges, and Jürgen Vollmer

Phys. Rev. E 104, 044301 – Published 1 October 2021

Abstract

Tipping elements in the Earth system have received increased scientific attention over 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 subdominant 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 Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks of different size and with different coupling parameters.

Received 13 April 2021
Accepted 26 August 2021

DOI:https://doi.org/10.1103/PhysRevE.104.044301

Physics Subject Headings (PhySH)

Bifurcations Dynamics of networks

Collective dynamics Dynamical systems Stochastic dynamical systems

Nonlinear Dynamics

Authors & Affiliations

Jan Kohler^1,2, Nico Wunderling ^2,3,4,*, Jonathan F. Donges ^2,5, and Jürgen Vollmer^1,†

¹Institute for Theoretical Physics, University of Leipzig, 04103 Leipzig, Germany, EU
²Earth System Analysis, Potsdam-Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany, EU
³Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany, EU
⁴Department of Physics, Humboldt University of Berlin, 12489 Berlin, Germany, EU
⁵Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden, EU

^*Author to whom all correspondence should be addressed: wunderling@pik-potsdam.de
^†juergen.vollmer@uni-leipzig.de

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Vol. 104, Iss. 4 — October 2021

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Images

Figure 1
Dynamics of the nodes in a network of tipping elements. (a) Bifurcation diagram of Eq. (1) where solid and dashed red lines show the positions of the stable and unstable steady states, respectively. In the range between $r_{crit, low}$ and $r_{crit, high}$ there are two stable states and one unstable state. At the borders of the interval, a stable state merges with the unstable state, and they disappear in a saddle-node bifurcation, and systems at that position will decay to the remaining stable state. The trembling blue line illustrates the stochastic fluctuations around the stable state that we are using here; see Eq. (2a). (b) Nodes in an exemplary network of interacting tipping elements [see Eq. (2a)], where each node is of the form shown in panel (a).
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Figure 2
Asymptotic values of the evolution. (a) Phase-space flow of the dynamical system, Eq. (9), for $\sqrt{α} ≃ 1 / 3$ . The analytical solutions of the nullclines are marked by gray lines, and fixed points are marked by black circles. The red line shows the numerical solution for $N = 80$ tipping elements that are connected by an ER network with $d = 0.1$ and $p = 0.25$ . The numerical solutions for the blue vectors have been computed using the function matplotlib.pyplot.streamplot in python. (b) The red and the green lines show the time evolution of the expectation and the variance of the numerical solution shown in panel (a). The thinner blue line shows the prediction, Eq. (13), for $x (0) = 1 / N$ (for an explanation see Sec. 4b). The dotted line shows the same function shifted by $Δ τ = 1.24$ . The shifted curve provides an excellent description of the late-time evolution of the expectation. (c) The data points show the asymptotic values $x^{*}$ of numerical simulations of Eq. (2a) with $p = 0.25$ and couplings $d_{k l}$ drawn from a uniform distribution $d_{k l} \in (0, 0.2)$ . Error bars indicate the standard deviation derived from 50 realizations of the network configuration and the noise. The connectivity $λ_{d}$ is varied by increasing $N$ from 20 to 400 [cf. Eq. (7)]. The numerical values agree very well with the prediction Eq. (10) that is shown by a solid blue line. The parameter value of the trajectory shown in panels (a) and (b) is indicated by a vertical red line. For small networks, the system occasionally tips into the direction opposite to the perturbation. Hence, we plot the absolute value of $x^{*}$ .
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Figure 3
Collective tipping behavior of different network types. The left and right panels show the evolution of the order parameter and relative deviation of numerical data from Eq. (2a) and the prediction from Eq. (13). The rows show data for (a,b) Erdös-Rényi, (c,d) Barabási-Albert, and (e,f) Watts-Strogatz networks with reconnection probability $β = 0.8$ . The other parameters of the simulations are fixed to $N = 100, p = 0.1, d = 1.5$ , and $ξ = 0.1$ , such that $λ_{d} ≃ 16$ . The data points and the error bars represent the expectation and the standard deviation over 50 realizations of the network and the noise, respectively. The prediction, Eq. (13), is indicated by a solid line in the left panels. It is shifted by $Δ τ$ to match the data at $x = 0.5$ (see the main text for details).
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Figure 4
Trajectories for different initial displacements $x (0) = ν / N$ . Calculations using Eq. (2a) have been conducted for ER networks with $d = 0.1, p = 0.25, ξ = 0.1, N \in {100, 200, 300, 800}$ , and $ν \in {1, 2, 3}$ , with couplings $d_{k l}$ drawn from a uniform distribution $d_{k l} \in (0, 0.2)$ . This amounts to connectivities of $λ_{d} = 3.475$ , 5.975, 10.975, and 20.975, respectively. The data and their error bars in the upper panels represent the expectation and standard deviation of 100 simulations runs with independent realizations of the network and the noise. The corresponding lower panels show the relative deviation of the trajectory from the prediction Eq. (15) with $x_{0} = ν / N$ and a time shift $Δ τ$ provided in the respective legends.
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Figure 5
Asymptotic fixed point $z^{*}$ (a) and the critical points (b) for networks with an offset $n$ and $r_{k l} = 0$ in Eq. (24). All calculations are conducted for an ER network with $N = 100$ nodes, couplings $d_{k l}$ drawn from a uniform distribution $d_{k l} \in (0, 1.2), λ_{d} = 16$ , and noise strength $ξ = 0.1$ . (a) For fixed values $n = m = 1$ and different drift $r_{D}$ we follow the dynamics for initial conditions $x_{init} = - 1$ and one excited node at $z = 1$ (orange) and $x_{init} = 2$ and one node at $z = - 1$ (blue), respectively. The dots and the error bars indicate the average and the standard deviation of the late-time asymptotic positions of 50 simulations with different realizations of the network and noise. They lie nicely on the prediction for the positions of the fixed points, Eq. (25a), that is provided by the solid red line. The positions of the critical points, $r_{crit}$ , are identified as having the largest distance between subsequent data points. (b) The $n$ -dependence of $r_{crit}$ determined from data as shown in panel (a), $m = 1$ , and different values of $n$ . The red lines provide the predicted positions, Eq. (25b), of the saddle-node bifurcations.
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