  We study XY spin systems on small world lattices for a variety of graph
structures, e.g. Poisson and scale-free, superimposed upon a one dimensional
chain. In order to solve this model we extend the cavity method in the one
pure-state approximation to deal with real-valued dynamical variables. We find
that small-world architectures significantly enlarge the region in parameter
space where synchronization occurs. We contrast the results of population
dynamics performed on a truncated set of cavity fields with Monte Carlo
simulations and find excellent agreement. Further, we investigate the
appearance of replica symmetry breaking in the spin-glass phase by numerically
analyzing the proliferation of pure states in the message passing equations.
