The Annals of Applied Probability

Navigation on a Poisson point process

Charles Bordenave

Full-text: Open access


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 ℝd. 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.

Article information

Ann. Appl. Probab., Volume 18, Number 2 (2008), 708-746.

First available in Project Euclid: 20 March 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C05: Trees
Secondary: 90C27: Combinatorial optimization 60G55: Point processes

Random spanning trees Poisson point process local weak convergence small-world phenomenon stochastic geometry


Bordenave, Charles. Navigation on a Poisson point process. Ann. Appl. Probab. 18 (2008), no. 2, 708--746. doi:10.1214/07-AAP472.

Export citation


  • Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin.
  • Athreya, K. and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493–501.
  • Baccelli, F. and Bordenave, C. (2007). The radial spanning tree of a Poisson point process. Ann. Appl. Probab. 17 305–359.
  • Baccelli, F. and Brémaud, P. (2003). Elements of Queuing Theory, 2nd ed. Springer, Berlin.
  • Baccelli, F., Tchoumatchenko, K. and Zuyev, S. (2000). Markov paths on the Poisson–Delaunay graph with applications to routing in mobile networks. Adv. in Appl. Probab. 32 1–18.
  • Baum, L. and Katz, M. (1965). Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120 108–123.
  • Bordenave, C. Navigation on a Poisson point process. Available at
  • Draief, M. and Ganesh, A. (2006). Efficient routeing in Poisson small-world networks. J. Appl. Probab. 43 678–686.
  • Ferrari, P., Landim, C. and Thorisson, H. (2004). Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. Poincaré Probab. Statist. 40 141–152.
  • Franceschetti, M. and Meester, R. (2006). Navigation in small-world networks: A scale-free continuum model. J. Appl. Probab. 43 1173–1180.
  • Ganesh, A., Kermarrec, A. and Massoulie, L. (2003). Peer-to-peer membership management for gossip-based protocols. IEEE Trans. Comp. 52 139–149.
  • Ganesh, A. and Xue, F. (2005). On the connectivity and diameter of small-world networks. MSR Technical report.
  • Gangopadhyay, S., Roy, R. and Sarkar, A. (2004). Random oriented trees: A model of drainage networks. Ann. Appl. Probab. 14 1242–1266.
  • Gut, A. (1980). Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices. Ann. Probab. 8 298–313.
  • Howard, C. and Newman, C. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab. 29 577–623.
  • Kleinberg, J. (2000). The small-world phenomenon: An algorithmic perspective. In Proc. 32nd Annual ACM Symposium on the Theory of Computing 163–170. ACM, New York.
  • Ko, Y. and Vaidya, N. (2000). Location-aided routing (lar) in mobile ad hoc networks. Wireless Networks 6 307–321.
  • Kranakis, E., Singh, H. and Urrutia, J. (1999). Compass routing on geometric networks. In Proc. 11th Canadian Conference on Computational Geometry 51–54.
  • Lee, S. (1997). The central limit theorem for Euclidean minimal spanning trees. I. Ann. Appl. Probab. 7 996–1020.
  • Liebeherr, J., Nahas, M. and Si, W. (2002). Application-layer multicasting with Delaunay triangulation overlays. IEEE J. Selected Areas in Communications 20 1472–1488.
  • Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Morin, P. (2001). Online routing in geometric graphs. Ph.D. thesis, School of Computer Science, Carleton Univ.
  • Penrose, M. and Wade, A. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. in Appl. Probab. 36 691–714.
  • Penrose, M. and Yukich, J. (2001). Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 272–301.
  • Penrose, M. and Yukich, J. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277–303.
  • Pimentel, L. (2005). Asymptotics for first-passage times on Delaunay triangulations. Available at
  • Plaxton, G., Rajaraman, R. and Richa, A. (1997). Accessing nearby copies of replicated objects in a distributed environment. In ACM Symposium on Parallel Algorithms and Architectures 311–320. ACM, New York.
  • Vahidi-Asl, M. and Wierman, J. (1990). First-passage percolation on the Voronoĭ tessellation and Delaunay triangulation. In Random Graphs’87 (Poznań, 1987) 341–359. Wiley, Chichester.
  • Vahidi-Asl, M. and Wierman, J. (1992). A shape result for first-passage percolation on the Voronoĭ tessellation and Delaunay triangulation. In Random Graphs 2 (Poznań, 1989) 247–262. Wiley, New York.