  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.
