Phase ordering on small-world networks with nearest-neighbor edges

We investigate global phase coherence in a system of coupled oscillators on small-world networks constructed from a ring with nearest-neighbor edges. The effects of both thermal noise and quenched randomness on phase ordering are examined and compared with the global coherence in the corresponding [formula presented] model without quenched randomness. It is found that in the appropriate regime phase ordering emerges at finite temperatures, even for a tiny fraction of shortcuts. The nature of the phase transition is also discussed.