Diffusion Processes on Power-Law Small-World Networks*

We consider diffusion driven processes on power-law small-world networks: a random walk process related to folded polymers and surface growth related to synchronization problems. The random links introduced in small-world networks often lead to mean-field coupling (as if the random links were annealed) but in some systems mean-field predictions break down, like diffusion in one dimension. This break-down can be understood treating the random links perturbatively where the mean field prediction appears as the lowest order term of a naive perturbation expansion. Our results were obtained using self-consistent perturbation theory \footnote{B. Kozma, M. B. Hastings, and G. Korniss, Phys. Rev. Lett. {\bf 92}, 108701 (2004).} and can also be understood in terms of a scaling theory. We find a rich phase diagram, with different transient and recurrent phases, including a critical line with continuously varying exponents.