Multiple-node basin stability in complex dynamical networks

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)], 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.

Figure 1

Schematic illustrating the concept of single-node BS. The region inside the box comprises the entire state space of the two-oscillator network.

Figure 2

(a) (Mean) single-node BS $\langle S_B^1\rangle$ of all the $N=81$ nodes of the three generations of the undirected deterministic scale-free network of $N$ identical Rössler oscillators. The first nine nodes comprise the first generation, the next 18 nodes the second generation, and the final 54 nodes the third generation. (b) Histogram of $\langle S_B^1\rangle$ of all the $N=81$ nodes. (c,d) Relationship of $\langle S_B^1\rangle$ with (c) degree and (d) betweenness centrality of the nodes, respectively.

Figure 3

Network topology of the undirected deterministic scale-free network of $N=81$ identical Rössler oscillators. The size of each node is proportional to the degree, and the color indicates the $\langle S_B^1\rangle$ value of the respective node. Note that in agreement with Fig. 2, the single-node BS values are very similar for all nodes.

Figure 4

(Mean) $m$-node BS $\langle S_B^m\rangle$ (blue dots) for $m$ varying from 1 to $N=81$ in the undirected deterministic scale-free network of $N$ identical Rössler oscillators. The shaded areas are representative of the standard deviations of the $m$-node BS values for the ensembles of $m$-node sets chosen for computing $\langle S_B^m\rangle$ for a particular value of $m$. The red line is an exponential fit of $\langle S_B^m\rangle \left[\approx 1.018exp\left(-0.037m\right)\right]$.

Figure 5

Network topology of the power transmission grid of the United Kingdom (comprising $N=120$ nodes) with second-order Kuramoto-type nodal dynamics. Circular nodes are net generators while square nodes are net consumers. The size of each node is proportional to the degree, and its color corresponds to the $\langle S_B^1\rangle$ value of the respective node.

Figure 6

As in Fig. 2 for the $N=120$ nodes of the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics.

Figure 7

(Mean) $m$-node BS $\langle S_B^m\rangle$ (blue dots) for $m$ varying from 1 to $N=120$ for the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics. The shaded background is representative of the standard deviation of the $m$-node BS values for the ensemble of $m$-node sets chosen for computing $\langle S_B^m\rangle$ for a particular value of $m$. The red line shows an exponential fit of $\langle S_B^m\rangle$.