  We investigate the collective dynamics of bi-stable elements connected in
different network topologies, ranging from rings and small-world networks, to
scale-free networks and stars. We estimate the dynamical robustness of such
networks by introducing a variant of the concept of multi-node basin stability,
which allows us to gauge the global stability of the dynamics of the network in
response to local perturbations affecting a certain class of nodes of a system.
We show that perturbing nodes with high closeness and betweeness-centrality
significantly reduces the capacity of the system to return to the desired
state. This effect is very pronounced for a star network which has one hub node
with significantly different closeness/betweeness-centrality than all the
peripheral nodes. In such a network, perturbation of the single hub node has
the capacity to destroy the collective state. On the other hand, even when a
majority of the peripheral nodes are strongly perturbed, the hub manages to
restore the system to its original state, demonstrating the drastic effect of
the centrality of the perturbed node on the dynamics of the network. Further,
we explore explore Random Scale-Free Networks of bi-stable dynamical elements.
We exploit the difference in the distribution of betweeness centralities,
closeness centralities and degrees of the nodes in Random Scale-Free Networks
with m=1 and m=2, to probe which centrality property most influences the
robustness of the collective dynamics in these heterogeneous networks.
Significantly, we find clear evidence that the betweeness centrality of the
perturbed node is more crucial for dynamical robustness, than closeness
centrality or degree of the node. This result is important in deciding which
nodes to safeguard in order to maintain the collective state of this network
against targeted localized attacks.
