  We study the synchronization transition of Kuramoto oscillators in scale-free
networks that are characterized by tunable local properties. Specifically, we
perform a detailed finite size scaling analysis and inspect how the critical
properties of the dynamics change when the clustering coefficient and the
average shortest path length are varied. The results show that the onset of
synchronization does depend on these properties, though the dependence is
smooth. On the contrary, the appearance of complete synchronization is
radically affected by the structure of the networks. Our study highlights the
need of exploring the whole phase diagram and not only the stability of the
fully synchronized state, where most studies have been done up to now.
