Multi-node basin stability in complex dynamical networks

Chiranjit Mitra, Anshul Choudhary, Sudeshna Sinha, Jürgen Kurths, Reik V. Donner

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Abstract

Dynamical entities interacting with each other on complex networks often exhibit multistability. The stability of a desired steady regime (e.g., a synchronized state) to large perturbations is critical in the operation of many real-world networked dynamical systems such as ecosystems, power grids, the human brain, etc. This necessitates the development of appropriate quantifiers of stability of multiple stable states of such systems. Motivated by the concept of basin stability (BS) (Menck et al., Nature Physics 9, 89 (2013)), we propose here the general framework of \emph{multi-node basin stability} for gauging global stability and robustness of networked dynamical systems in response to non-local perturbations simultaneously affecting multiple nodes of a system. The framework of multi-node BS provides an estimate of the critical number of nodes which when simultaneously perturbed, significantly reduces the capacity of the system to return to the desired stable state. Further, this methodology can be applied to estimate the minimum number of nodes of the network to be controlled or safeguarded from external perturbations to ensure proper operation of the system. Multi-node BS can also be utilized for probing the influence of spatially localised perturbations or targeted attacks to specific parts of a network. We demonstrate the potential of multi-node BS in assessing the stability of the synchronized state in a deterministic scale-free network of R\"{o}ssler oscillators and a conceptual model of the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics.
Original languageEnglish
Article number032317
Pages (from-to)1-9
Number of pages9
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Volume95
Issue number3
DOIs
Publication statusPublished - 16 Mar 2017

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Complex Dynamical Networks
Vertex of a graph
Perturbation
perturbation
dynamical systems
Dynamical system
Grid
grids
Multistability
Scale-free Networks
United Kingdom
Conceptual Model
Quantifiers
Global Stability
Ecosystem
ecosystems
Complex Networks
Estimate
estimates
attack

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Multi-node basin stability in complex dynamical networks. / Mitra, Chiranjit; Choudhary, Anshul; Sinha, Sudeshna; Kurths, Jürgen; Donner, Reik V.

In: Physical Review. E, Statistical, Nonlinear and Soft Matter Physics, Vol. 95, No. 3, 032317, 16.03.2017, p. 1-9.

Research output: Contribution to journalArticle

Mitra, Chiranjit ; Choudhary, Anshul ; Sinha, Sudeshna ; Kurths, Jürgen ; Donner, Reik V. / Multi-node basin stability in complex dynamical networks. In: Physical Review. E, Statistical, Nonlinear and Soft Matter Physics. 2017 ; Vol. 95, No. 3. pp. 1-9.
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N1 - ACKNOWLEDGMENTS C.M. and R.V.D. have been supported by the German Federal Ministry of Education and Research (BMBF) via the Young Investigators Group CoSy-CC2 (Grant No. 01LN1306A). A.C. acknowledges the JC Bose Fellowship (SB/S2/JCB-013/2015) for financial support. J.K. and R.V.D. acknowledge support from the IRTG 1740/TRP 2011/50151-0, funded by the DFG/FAPESP. The authors gratefully acknowledge the European Regional Development Fund (ERDF), the German Federal Ministry of Education and Research (BMBF), and the Land Brandenburg for supporting this project by providing resources on the high performance computer system at the Potsdam Institute for Climate Impact Research. The authors thank Paul Schultz for providing the data on the topology of the United Kingdom power grid.

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N2 - Dynamical entities interacting with each other on complex networks often exhibit multistability. The stability of a desired steady regime (e.g., a synchronized state) to large perturbations is critical in the operation of many real-world networked dynamical systems such as ecosystems, power grids, the human brain, etc. This necessitates the development of appropriate quantifiers of stability of multiple stable states of such systems. Motivated by the concept of basin stability (BS) (Menck et al., Nature Physics 9, 89 (2013)), we propose here the general framework of \emph{multi-node basin stability} for gauging global stability and robustness of networked dynamical systems in response to non-local perturbations simultaneously affecting multiple nodes of a system. The framework of multi-node BS provides an estimate of the critical number of nodes which when simultaneously perturbed, significantly reduces the capacity of the system to return to the desired stable state. Further, this methodology can be applied to estimate the minimum number of nodes of the network to be controlled or safeguarded from external perturbations to ensure proper operation of the system. Multi-node BS can also be utilized for probing the influence of spatially localised perturbations or targeted attacks to specific parts of a network. We demonstrate the potential of multi-node BS in assessing the stability of the synchronized state in a deterministic scale-free network of R\"{o}ssler oscillators and a conceptual model of the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics.

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