Minimal fatal shocks in multistable complex networks

Abstract

Multistability is a common phenomenon which naturally occurs in complex networks. often one of the coexisting stable states can be identified as being the desired one for a particular application. We present here a global approach to identify the minimal perturbation which will instantaneously kick the system out of the basin of attraction of its desired state and hence induce a critical or fatal transition we call shock-tipping. the corresponding Minimal Fatal Shock is a vector whose length can be used as a global stability measure and whose direction in state space allows us to draw conclusions on weaknesses of the network corresponding to critical network motifs. We demonstrate this approach in plant-pollinator networks and the power grid of Great Britain. in both system classes, tree-like substructures appear to be the most vulnerable with respect to the minimal shock perturbation. Many processes in nature can be well described in terms of nonlinear dynamical systems possessing multiple stable states for constant external conditions 1,2. As long as perturbations are small, it is sufficient to consider the linearized problem around an attracting state to evaluate its 'local' stability. However, in real systems, the most threatening perturbations are usually not small but large, even constituting extreme events like storms, earthquakes or financial crises 3-5. Obviously, large perturbations in multistable systems call for a 'global' stability paradigm. Adequate frameworks have been introduced in the ecological literature-by Holling 6,7-as well as in the engineering literature-by Soliman and Thompson 8 (see 9 for a recent review)-who both argued that the local approach to stability needs to be accompanied by non-local measures capturing the characteristics of a state's basin of attraction. Recently, this global approach has been picked up in the field of complex networks in which multistability and large disturbances occur naturally as well 10-14. Suitable aspects of a basin of attraction are its size, shape and depth, which until now have all been used-separately 15-19 or in combination 20-22-to measure the stability of multistable systems. The suitability of a certain stability criterion for a specific problem also depends on the shape or distribution of the perturbations that occur. If perturbations are best described as noise, a suitable approach is to analyze the most likely escape path from the basin of attraction 23-25. This path is determined by the position of the saddle point on the basin boundary possessing the lowest barrier to escape 26. A very different situation occurs if perturbations are singular, large and abrupt. The response of nonlinear systems to such perturbations has been considered in terms of linear response theory as well as transient dynamics in climate science 27 , fluid dynamics 28,29 and energy networks 18,30,31. In networks, the most frequently applied characteristic to measure a system's stability against large abrupt perturbations is the relative basin size which is usually esti...

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