# Global Robustness vs. Local Vulnerabilities in Complex Synchronous Networks

## TL;DR

This paper analyzes how complex networks of coupled oscillators respond to local perturbations, identifying vulnerable nodes and proposing spectral topological indices to measure and enhance global robustness.

## Contribution

It introduces resistance centralities and Kirchhoff indices derived from spectral analysis as novel tools to assess and improve network robustness against perturbations.

## Key findings

- Resistance centralities identify most fragile oscillators.
- Kirchhoff indices quantify global robustness.
- Inertia has minimal impact on robustness in homogeneous systems.

## Abstract

In complex network-coupled dynamical systems, two questions of central importance are how to identify the most vulnerable components and how to devise a network making the overall system more robust to external perturbations. To address these two questions, we investigate the response of complex networks of coupled oscillators to local perturbations. We quantify the magnitude of the resulting excursion away from the unperturbed synchronous state through quadratic performance measures in the angle or frequency deviations. We find that the most fragile oscillators in a given network are identified by centralities constructed from network resistance distances. Further defining the global robustness of the system from the average response over ensembles of homogeneously distributed perturbations, we find that it is given by a family of topological indices known as generalized Kirchhoff indices. Both resistance centralities and Kirchhoff indices are obtained from a spectral decomposition of the stability matrix of the unperturbed dynamics and can be expressed in terms of resistance distances. We investigate the properties of these topological indices in small-world and regular networks. In the case of oscillators with homogeneous inertia and damping coefficients, we find that inertia only has small effects on robustness of coupled oscillators. Numerical results illustrate the validity of the theory.