# Recovery time after localized perturbations in complex dynamical networks

## TL;DR

This paper introduces the single-node recovery time (SNRT) framework to estimate how long it takes for nodes in complex networks to return to synchrony after perturbations, helping identify critical nodes and assess network resilience.

## Contribution

The paper proposes SNRT as a novel measure for transient dynamics, relating local recovery times to global relaxation and network topology, with applications to oscillator and power grid networks.

## Key findings

- SNRT effectively identifies slow and critical nodes in networks.
- Explicit relationships between SNRT and global relaxation time are established.
- SNRT demonstrates practical utility in oscillator and power grid models.

## Abstract

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.