Small-world behaviour in a system of mobile elements

We analyze the propagation of activity in a system of mobile automata. A number ρLd of elements move as random walkers on a lattice of dimension d, while with a small probability p they can jump to any empty site in the system. We show that this system behaves as a Dynamic Small World (DSW) and present analytic and numerical results for several quantities. Our analysis shows that the persistence time T* (equivalent to the persistence size L* of small-world networks) scales as T* ~ (ρp)−τ, with τ = 1/(d + 1).