  We investigate the effects of structural perturbations of both, undirected
and directed diffusive networks on their ability to synchronize. We establish a
classification of directed links according to their impact on
synchronizability. We focus on adding directed links in weakly connected
networks having a strongly connected component acting as driver. When the
connectivity of the driver is not stronger than the connectivity of the slave
component, we can always make the network strongly connected while hindering
synchronization. On the other hand, we prove the existence of a perturbation
which makes the network strongly connected while increasing the
synchronizability. Under additional conditions, there is a node in the driving
component such that adding a single link starting at an arbitrary node of the
driven component and ending at this node increases the synchronizability.
