  Maintaining the synchronous motion of dynamical systems interacting on
complex networks is often critical to their functionality. However, real-world
networked dynamical systems operating synchronously are prone to random
perturbations driving the system to arbitrary states within the corresponding
basin of attraction, thereby leading to epochs of desynchronized dynamics with
a priori unknown durations. Thus, it is highly relevant to have an estimate of
the duration of such transient phases before the system returns to synchrony,
following a random perturbation to the dynamical state of any particular node
of the network. We address this issue here by proposing the framework of
\emph{single-node recovery time} (SNRT) which provides an estimate of the
relative time scales underlying the transient dynamics of the nodes of a
network during its restoration to synchrony. We utilize this in differentiating
the particularly \emph{slow} nodes of the network from the relatively
\emph{fast} nodes, thus identifying the critical nodes which when perturbed
lead to significantly enlarged recovery time of the system before resuming
synchronized operation. Further, we reveal explicit relationships between the
SNRT values of a network, and its \emph{global relaxation time} when starting
all the nodes from random initial conditions. We employ the proposed concept
for deducing microscopic relationships between topological features of nodes
and their respective SNRT values. The framework of SNRT is further extended to
a 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 on
paradigmatic topologies and a model of the power grid of the United Kingdom
with second-order Kuramoto-type nodal dynamics illustrating the conceivable
practical applicability of the proposed concept.
