The Random Connection Model on the Torus

We study the diameter of a family of random graphs on the torus that can be used to model wireless networks. In the random connection model two points x and y are connected with probability g(y − x) where g is a given function. We prove that the diameter of the graph is bounded by a constant, that only depends on ‖g‖1, with high probability as the number of vertices in the graph tends to infinity.