Dynamics of tipping cascades on complex networks

Tipping points occur in diverse systems in various disciplines such as ecology, climate science, economy, and engineering. Tipping points are critical thresholds in system parameters or state variables at which a tiny perturbation can lead to a qualitative change of the system. Many systems with tipping points can be modeled as networks of coupled multistable subsystems, e.g., coupled patches of vegetation, connected lakes, interacting climate tipping elements, and multiscale infrastructure systems. In such networks, tipping events in one subsystem are able to induce tipping cascades via domino effects. Here, we investigate the effects of network topology on the occurrence of such cascades. Numerical cascade simulations with a conceptual dynamical model for tipping points are conducted on Erdős-Rényi, Watts-Strogatz, and Barabási-Albert networks. Additionally, we generate more realistic networks using data from moisture-recycling simulations of the Amazon rainforest and compare the results to those obtained for the model networks. We furthermore use a directed configuration model and a stochastic block model which preserve certain topological properties of the Amazon network to understand which of these properties are responsible for its increased vulnerability. We find that clustering and spatial organization increase the vulnerability of networks and can lead to tipping of the whole network. These results could be useful to evaluate which systems are vulnerable or robust due to their network topology and might help us to design or manage systems accordingly.

Figure 1

Illustration of a tipping network. Each node represents a tipping element with a corresponding state variable $x_i$. A directed link corresponds to a positive linear coupling with strength $d$. The effective control parameter ${\tilde{r}}_i$ of a node depends on the state of the nodes it is coupled to. The equilibria with respect to the effective control parameter are qualitatively illustrated in the box.

Figure 2

Cascade simulations on ER networks of different sizes, an average degree of $\langle k\rangle \approx 5$, and a coupling strength of $d=0.2$. The time evolution of the fraction of tipped elements is shown.

Figure 3

Network size dependency of critical coupling strength in ER networks with $\langle k\rangle \approx 5$. Upper panel: Average cascade size with respect to the coupling strength in the transition region. Each average is calculated from 100 cascade simulations on different randomly generated networks with $N=100$. Error bars indicate the standard error. Lower panel: Approximate critical coupling strength (coupling strength where $\langle L\rangle \approx 0.5$) with respect to the network size $N$. The dashed line indicates the critical coupling strength $d_c\approx r_{\mathrm{crit}}=0.183$ for a simple unidirectional coupling of two elements.

Figure 4

Dependence of the transition region on the reciprocity $R$ (left panel) and on the clustering coefficient $\mathcal{C}$ (right panel). Each average is calculated from 100 cascade simulations on different randomly generated networks with $N=100$.

Figure 5

Network size dependency of the critical average degree $k_{\mathrm{c}}$ in ER networks with $d=0.15$. Upper panel: Average cascade size with respect to the average degree in the transition region. Each average is calculated from 100 cascade simulations on different randomly generated networks with $N=100$. Error bars indicate the standard error in both panels. Lower panel: Approximate critical average degree (average degree where $\langle L\rangle \approx 0.5$) with respect to the network size $N$.

Figure 6

Distributions of cascade sizes $L$ for different network topologies. A random topology generated with the ER model (first row), a small-world topology generated with the WS model and $\beta =0.1$ (second row), and a scale-free topology generated with the BA model (third row). Each distribution is an average of 10 distributions with 100 cascade simulations on different networks with $N=100$ and $\langle k\rangle \approx 5$. The (almost-invisible) error bars indicate the standard error across the 10 distributions. Three coupling strengths for each network topology are shown: one where almost no cascades occur; one where in about half of the simulations cascades are triggered; and one where in almost all simulations cascades are triggered.

Figure 7

Average cascade size $\langle L\rangle$ with respect to average degree $\langle k\rangle$ and coupling strength $d$ for three network topologies. Random networks generated with the ER model (left), small-world topology networks generated with the WS model and $\beta =0.1$ (center), and scale-free networks generated with the BA model (right). Each average is calculated from 100 cascade simulations on different randomly generated networks with $N=100$.

Figure 8

Shift of the transition (upper panel) and average clustering coefficient $\mathcal{C}$ (lower panel) with increasing rewiring probability $\beta$ for WS networks with $N=100$ and $\langle k\rangle \approx 5$. The shift of the transition towards higher coupling strengths for high rewiring probabilities corresponds to the decrease in the average clustering coefficient. The extent of the small black circles in the lower panel exceeds the standard error, which is therefore not visible.

Figure 9

Spatially organized network generated from atmospheric moisture-flow data ($2\times 2^{\circ }$-grid resolution) of the Amazon rainforest. The threshold is chosen such that $\langle k\rangle =5$. Total rainfall values for each node in 2014 are shown in the background.

Figure 10

Average cascade size $\langle L\rangle$ with respect to coupling strength for different networks with an average degree of $\langle k\rangle =5$. Upper panel: Results for the networks generated from the moisture-flow data with $1\times 1^{\circ }$-grid resolution (567 nodes) and $2\times 2^{\circ }$-grid resolution (160 nodes). For comparison, simulation results for ER networks with the same network sizes are shown. Lower panel: Simulation results for a directed configuration model and a stochastic block model are compared with the results of the Amazon network and the ER networks with $N=160$ for all networks. Error bars indicate the standard error. Note that the standard errors for the original moisture-flow networks are smaller than for the other network types. The reason is that all moisture-flow simulation results are based on the same network, whereas the other results are based on different randomly generated networks.

Figure 11

Distribution of cascade sizes analogous to the above distributions for different networks generated from moisture-flow simulations of the Amazon rainforest ($N=160$). Note that there is no standard error indicated (error bars) for the original moisture-flow networks, as there is only one distribution due to the deterministic network generation procedure.

Figure 12

Average cascade size $\langle L\rangle$ with respect to average degree and coupling strength for different networks generated with moisture-flow simulations of the Amazon rainforest ($N=160$).