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Maintaining the synchronous motion of dynamical systems interacting oncomplex networks is often critical to their functionality. However, real-worldnetworked dynamical systems operating synchronously are prone to randomperturbations driving the system to arbitrary states within the correspondingbasin of attraction, thereby leading to epochs of desynchronized dynamics witha priori unknown durations. Thus, it is highly relevant to have an estimate ofthe duration of such transient phases before the system returns to synchrony,following a random perturbation to the dynamical state of any particular nodeof the network. We address this issue here by proposing the framework of\emph{single-node recovery time} (SNRT) which provides an estimate of therelative time scales underlying the transient dynamics of the nodes of anetwork during its restoration to synchrony. We utilize this in differentiatingthe particularly \emph{slow} nodes of the network from the relatively\emph{fast} nodes, thus identifying the critical nodes which when perturbedlead to significantly enlarged recovery time of the system before resumingsynchronized operation. Further, we reveal explicit relationships between theSNRT values of a network, and its \emph{global relaxation time} when startingall the nodes from random initial conditions. We employ the proposed conceptfor deducing microscopic relationships between topological features of nodesand their respective SNRT values. The framework of SNRT is further extended toa measure of resilience of the different nodes of a networked dynamical system.We demonstrate the potential of SNRT in networks of R\"{o}ssler oscillators onparadigmatic topologies and a model of the power grid of the United Kingdomwith second-order Kuramoto-type nodal dynamics illustrating the conceivablepractical applicability of the proposed concept.

Recovery time after localized perturbations in complex dynamical networks