Synchronization on small-world networks

We investigate collective synchronization in a system of coupled oscillators on small-world networks. The order parameters that measure synchronization of phases and frequencies are introduced and analyzed by means of dynamic simulations and finite-size scaling. Phase synchronization is observed to emerge in the presence of even a tiny fraction P of shortcuts and to display saturated behavior for $P\gtrsim \mathrm{0.5.}$ This indicates that the same synchronizability as the random network $\mathrm{}(P=1\mathrm{})\mathrm{}$ can be achieved with relatively small number of shortcuts. The transient behavior of the synchronization, obtained from the measurement of the relaxation time, is also discussed.