Abstract. In our version of Watts and Strogatz's small world model, space is a d-dimensional torus in which each individual has in addition exactly one long-range neighbor chosen at random from the grid. This modification is natural if one thinks of a town where an individual's interactions at school, at work, or in social situations introduces long-range connections. However, this change dramatically alters the behavior of the contact process, producing two phase transitions. We establish this by relating the small world to an infinite ``big world" graph where the contact process behavior is similar to the contact process on a tree.
Preprint (pdf file) of paper to appear in Stochastic Processes and Their Applications