We modify Watts and Strogatz's small world model so that 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. This work is
joint with R. Durrett.
