Multiple-node basin stability in complex dynamical networks

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) [P. J. Menck et al., Nat. Phys. 9, 89 (2013)1745-247310.1038/nphys2516], we propose here the general framework of multiple-node basin stability for gauging the global stability and robustness of networked dynamical systems in response to nonlocal perturbations simultaneously affecting multiple nodes of a system. The framework of multiple-node BS provides an estimate of the critical number of nodes that, when simultaneously perturbed, significantly reduce 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. Multiple-node BS can also be utilized for probing the influence of spatially localized perturbations or targeted attacks to specific parts of a network. We demonstrate the potential of multiple-node BS in assessing the stability of the synchronized state in a deterministic scale-free network of Rössler oscillators and a conceptual model of the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics.

Timing of transients: quantifying reaching times and transient behavior in complex systems

“…In contrast, CSD is usually done with (local) noise only. The usage of shocks has been developed in the context of Basin stability [30,31] and its extensions [23,[39][40][41][42].…”

Multistability and variations in basin of attraction in power-grid systems

“…The overall transition shapes show a similar pattern for all nodes. The global basin stability, defined as the value of B averaged over all of the nodes [28] and denoted by á ñ B in figures 5(e) and (f), shows that such drastically different patterns for individual nodes cannot always be observed by just looking at the global basin stability.…”

“…The main question is whether the power-grid system retains synchronization after being perturbed or falls into desynchronization. The volume of the basin of attraction to synchronization in the phase space of perturbation is interpreted as the probability of synchronization recovery, so-called 'basin stability', which has been widely used to quantify the synchronization stability of power grids [16,17,20,21,[24][25][26][27][28][29][30].…”

Interval stability for complex systems

“…Recently, the concept of basin stability (BS) was suggested to describe the stability against large perturbations [15,[18][19][20]. The BS method was later used in various applications [21][22][23][24] and also confirmed experimentally [25].…”