How motifs condition critical thresholds for tipping cascades in complex networks: Linking micro- to macro-scales

Abstract

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.

Interacting tipping elements increase risk of climate domino effects under global warming

“…3). However, the size, the timing and the occurrence of cascades can also depend critically on the specific initial conditions (Wunderling et al, 2020b), which are not varied in the experiments presented here. In an exemplary simulation, we show how a global mean surface temperature increase from 1.5 to 1.6 • C triggers the Greenland Ice Sheet to transition to an ice-free state in one realisation of our Monte Carlo ensemble at low interaction strength (Fig.…”

Network dynamics of drought-induced tipping cascades in the Amazon rainforest

“…For instance such changes can be critical for the tipping of cells that are not coupled directly, but via an intermediary cell, where partial changes are decisive for the emergence of a tipping cascade. Indirect effects have been found to account for 10% and more, already in very simple interaction structures in so called motifs 62 .…”

“…The state x j must be divided by 2 since the distance from minimum to maximum state is 2. Similar forms of the network and the differential equation have already been used in earlier studies in the literature, but in a way more simplified form compared to this work 62,38 .…”

Modelling nonlinear dynamics of interacting tipping elements on complex networks: the PyCascades package

“…Since there is increasing interest in modelling interacting tipping elements within the context of complex systems [27][28][29], we bring these two strands of research together since tipping elements on networks can not only tip themselves but also imply tipping of neighbouring systems or even the network as a whole. Building upon recent developments in studying interacting nonlinear dynamics on complex networks [30][31][32][33][34], we here introduce the unified Python package PyCascades.…”