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.

How motifs condition critical thresholds: Linking Micro-to Macro-scales In this study, we investigate how specific micro interaction structures (motifs) affect the occurrence of tipping cascades on networks of stylized tipping elements. We compare the properties of cascades in Erdős-Rényi networks and an exemplary moisture recycling network of the Amazon rainforest. Within these networks, decisive small-scale motifs are the feed forward loop, the secondary feed forward loop, the zero loop and the neighboring loop.Of all motifs, the feed forward loop motif stands out in tipping cascades since it decreases the critical coupling strength necessary to initiate a cascade more than the other motifs.We find that for this motif, the reduction of critical coupling strength is 11% less than the critical coupling of a pair of tipping elements. For highly connected networks, our analysis reveals that coupled feed forward loops coincide with a strong 90% decrease of the critical coupling strength.For the highly clustered moisture recycling network in the Amazon, we observe regions of very high motif occurrence for each of the four investigated motifs suggesting that these regions are more vulnerable. The occurrence of motifs is found to be one order of magnitude higher than in a random Erdős-Rényi network.This emphasizes the importance of local interaction structures for the emergence of global cascades and the stability of the network as a whole. a) These authors equally contributed to this study. Correspondences should be addressed to: nico.wunderling@pikpotsdam.de 2 How motifs condition critical thresholds: Linking Micro-to Macro-scales Tipping elements are nonlinear systems, where a small perturbation can be sufficient to induce a qualitative change of the whole system as soon as a critical threshold (tipping point) is crossed. Coupled tipping elements exist for instance in connected lake systems, in the Earth's climate system or in social systems. Here, we investigate networks of interacting tipping elements, where each node consists of a stylized tipping element and explore important interaction structures on the micro scale of the network, the so-called motifs. Such motifs in complex networks have been found in multiple systems such as cell metabolism, food webs or neural networks and are known to be significantly overexpressed in real-world compared to random networks. However, motifs have not yet been studied extensively in complex networks, where nodes have their own dynamics. In our study, we find that tipping cascades occur more often at locations with high motif frequency revealing locations (nodes) of decreased robustness.