Navigation on a Poisson point process

Abstract

On a locally finite point set, a navigation defines a path through the point set from one point to another. The set of paths leading to a given point defines a tree known as the navigation tree. In this article, we analyze the properties of the navigation tree when the point set is a Poisson point process on ${\mathbb{R}}^d$. We examine 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 where new results are established.

Published in at http://dx.doi.org/10.1214/07-AAP472 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• The radial spanning tree of a Poisson point process
• Navigation on a Poisson point process
• Random oriented Trees: a Model of drainage networks
• Routing in Poisson small-world networks

Cited by in corpus (6)

• The radial spanning tree of a Poisson point process
• Navigation on a Poisson point process
• Optimal Paths on the Space-Time SINR Random Graph
• Transmission and navigation on disordered lattice networks, directed spanning forests and Brownian web
• Large and moderate deviations in Poisson navigations
• Efficiently navigating a random Delaunay triangulation