  Synchronization of networked oscillators is known to depend fundamentally on
the interplay between the dynamics of the graph's units and the microscopic
arrangement of the network's structure. For non identical elements, the lack of
quantitative tools has hampered so far a systematic study of the mechanisms
behind such a collective behavior. We here propose an effective network whose
topological properties reflect the interplay between the topology and dynamics
of the original network. On that basis, we are able to introduce the
"synchronization centrality", a measure which quantifies the role and
importance of each network's node in the synchronization process. In
particular, we use such a measure to assess the propensity of a graph to
synchronize explosively, thus indicating a unified framework for most of the
different models proposed so far for such an irreversible transition. Taking
advantage of the predicting power of this measure, we furthermore discuss a
strategy to induce the explosive behavior in a generic network, by acting only
upon a small fraction of its nodes.
