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We study the synchronization transition of Kuramoto oscillators in scale-freenetworks that are characterized by tunable local properties. Specifically, weperform a detailed finite size scaling analysis and inspect how the criticalproperties of the dynamics change when the clustering coefficient and theaverage shortest path length are varied. The results show that the onset ofsynchronization does depend on these properties, though the dependence issmooth. On the contrary, the appearance of complete synchronization isradically affected by the structure of the networks. Our study highlights theneed of exploring the whole phase diagram and not only the stability of thefully synchronized state, where most studies have been done up to now.Comment: 5 pages and 3 figures. APS style. Paper to be published in IJBC  (special issue on Complex Networks' Structure and Dynamics

SYNCHRONIZATION OF NETWORKS WITH VARIABLE LOCAL PROPERTIES