Abstract
Complex systems are challenging to control because the system responds to the controller in a nonlinear fashion, often incorporating feedback mechanisms. Interdependence of systems poses additional difficulties, as cross-system connections enable malicious activity to spread between layers, increasing systemic risk. In this paper we explore the conditions for an optimal control of cascading failures in a system of interdependent networks. Specifically, we study the Bak–Tang–Wiesenfeld sandpile model incorporating a control mechanism, which affects the frequency of cascades occurring in individual layers. This modification allows us to explore sandpile-like dynamics near the critical state, with supercritical region corresponding to infrequent large cascades and subcritical zone being characterized by frequent small avalanches. Topological coupling between networks introduces dependence of control settings adopted in respective layers, causing the control strategy of a given layer to be influenced by choices made in other connected networks. We find that the optimal control strategy for a layer operating in a supercritical regime is to be coupled to a layer operating in a subcritical zone, since such condition corresponds to reduced probability of inflicted avalanches. However this condition describes a parasitic relation, in which only one layer benefits. Second optimal configuration is a mutualistic one, where both layers adopt the same control strategy. Our results provide valuable insights into dynamics of cascading failures and and its control in interdependent complex systems.
Highlights
Complex systems are challenging to control because the system responds to the controller in a nonlinear fashion, often incorporating feedback mechanisms
Our goal is to identify conditions for optimal control exerted through modification of the dynamical rules defining cascading process
A wide range of complex systems, including biological, physiological, financial, ecological and social systems have been identified as operating at or near criticality[28]. This dynamic state poised between order and disorder is believed to arise through evolutionary-like mechanisms as it is characterized by optimal trade-off between robustness and flexibility[29], the highest level of computational capabilities[30] and optimal dynamic range and m emory[31]
Summary
Complex systems are challenging to control because the system responds to the controller in a nonlinear fashion, often incorporating feedback mechanisms. We study the Bak–Tang–Wiesenfeld sandpile model incorporating a control mechanism, which affects the frequency of cascades occurring in individual layers This modification allows us to explore sandpile-like dynamics near the critical state, with supercritical region corresponding to infrequent large cascades and subcritical zone being characterized by frequent small avalanches. Recent studies have determined that the density of interlayer connections affects occurrence of cascading failures in interconnected systems[13,14], while correlations present in the multilayer structure facilitate cooperation in evolutionary games[12] or decrease of epidemic thresholds[15] Those studies focus on the impact the structure of a system has on observed dynamics, while in many realistic scenarios large scale changes to the network structure are not a preferable or available method of exerting control. The latter rule leads to accumulation of load on the network, which in turn results in very infrequent, but highly destructive failures
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