Navigation on Self-Organized Networks

On a locally finite point set, a navigation defines a path through the point set from a point to an other. The set of paths leading to a given point defines a tree, the navigation tree. In this article, we analyze the properties of the navigation tree when the point set is a Poisson point process on R^d. We examine the distribution of stable functionals, the local weak convergence of the navigation tree, the asymptotic average of a functional along a path, the shape of the navigation tree and its topological ends. We illustrate our work in the small world graphs, and new results are established. This work is motivated by applications in computational geometry and in self-organizing networks.