Navigation on a Poisson point process

Details Navigation on a Poisson point process Navigation on a Poisson point process

2006-01-06

arxiv.org/abs/math/0601122v3

Annals of Applied Probability 2008, Vol. 18, No. 2, 708-746

Mathematics

Probability

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

Scientific paper

10.1214/07-AAP472

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.

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