Dynamic transitions in small world networks: approach to equilibrium limit.

We study the transition to phase synchronization in a model for the spread of infection defined in a small world network. It was shown [Phys. Rev. Lett. 86, 2909 (2001)] that the transition occurs at a finite degree of disorder p, unlike equilibrium models where systems behave as random networks even at infinitesimal p in the infinite-size limit. We examine this system under variation of a parameter determining the driving rate and show that the transition point decreases as we drive the system more slowly. Thus it appears that the transition moves to p=0 in the very slow driving limit, just as in the equilibrium case.