Robustness of Synchrony in Complex Networks and Generalized Kirchhoff Indices

TL;DR

This paper introduces generalized Kirchhoff indices derived from network eigenmodes to quickly assess the vulnerability of complex networks to specific or averaged perturbations, enhancing network robustness analysis.

Contribution

It develops a novel approach linking eigenmode overlaps to network vulnerability, introducing generalized Kirchhoff indices for rapid assessment.

Findings

Response depends on eigenmode overlap with perturbation vector.

Averaged response characterized by new topological indices.

Method enables fast evaluation of network vulnerability.

Abstract

In network theory, a question of prime importance is how to assess network vulnerability in a fast and reliable manner. With this issue in mind, we investigate the response to parameter changes of coupled dynamical systems on complex networks. We find that for specific, non-averaged perturbations, the response of synchronous states critically depends on the overlap between the perturbation vector and the eigenmodes of the stability matrix of the unperturbed dynamics. Once averaged over properly defined ensembles of such perturbations, the response is given by new graph topological indices, which we introduce as generalized Kirchhoff indices. These findings allow for a fast and reliable method for assessing the specific or average vulnerability of a network against changing operational conditions, faults or external attacks.