Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-19T01:46:52.572Z Has data issue: false hasContentIssue false
  • English
  • Français

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

Part of: Graph theory Equilibrium statistical mechanics Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

C. Hirsch ,
D. Neuhäuser ,
C. Gloaguen  and
V. Schmidt
C. Hirsch*
Affiliation:
Ulm University
D. Neuhäuser*
Affiliation:
Ulm University
C. Gloaguen*
Affiliation:
Orange Labs
V. Schmidt*
Affiliation:
Ulm University
*
∗ Postal address: Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
∗ Postal address: Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
∗∗∗ Postal address: Orange Labs, 38-40 rue du Général Leclerc, 92794 Issy-les-Moulineaux, France.
∗ Postal address: Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

Keywords

First passage percolationshape theoremshortest-path treelongest shortest pathrandom geometric graph

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 05C80: Random graphs 05C10: Planar graphs; geometric and topological aspects of graph theory 82B43: Percolation
Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 328 - 354
Copyright
© Applied Probability Trust 

References

Aldous, D. J. (2009). “Which connected spatial networks on random points have linear route-lengths?” Preprint. Available at http://arxiv.org/abs/0911.5296.Google Scholar
Aldous, D. J. and Kendall, W. S. (2008). “Short-length routes in low-cost networks via Poisson line patterns.“Adv. Appl. Prob. 40, 1–21.Google Scholar
Aldous, D. J. and Shun, J. (2010). “Connected spatial networks over random points and a route-length statistic.” Statist. Sci. 25, 275–288.CrossRefGoogle Scholar
Antal, P. and Pisztora, A. (1996). “On the chemical distance for supercritical Bernoulli percolation.“Ann. Prob. 24, 1036–1048.CrossRefGoogle Scholar
Baccelli, F., Tchoumatchenko, K. and Zuyev, S. (2000). “Markov paths on the Poisson–Delaunay graph with applications to routing in mobile networks.“Adv. Appl. Prob. 32, 1–18.CrossRefGoogle Scholar
Bertin, E., Billiot, J.-M. and Drouilhet, R. (2002). “Continuum percolation in the Gabriel graph.“Adv. Appl. Prob. 34, 689–701.CrossRefGoogle Scholar
Billiot, J.-M., Corset, F. and Fontenas, E. (2010). “Continuum percolation in the relative neighborhood graph.” Preprint. Available at http://arxiv.org/abs/1004.5292.Google Scholar
Calka, P. (2002). “The distributions of the smallest disks containing the Poisson–Voronoi typical cell and the Crofton cell in the plane.“Adv. Appl. Prob. 34, 702–717.Google Scholar
Daley, D. J. and Last, G. (2005). “Descending chains, the lilypond model, and mutual-nearest-neighbour matching.“Adv. Appl. Prob. 37, 604–628.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). “An Introduction to the Theory of Point Processes, Vol. I, Elementary Theory and Methods, 2nd edn.” Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). “An Introduction to the Theory of Point Processes, Vol. II, General Theory and Structure, 2nd edn.” Springer, New York.Google Scholar
Deuschel, J.-D. and Pisztora, A. (1996). “Surface order large deviations for high-density percolation.“Prob. Theory Relat. Fields 104, 467–482.CrossRefGoogle Scholar
Garet, O. and Marchand, R. (2007). “Large deviations for the chemical distance in supercritical Bernoulli percolation.“Ann. Prob. 35, 833–866.CrossRefGoogle Scholar
Gloaguen, C., Voss, F. and Schmidt, V. (2011). “Parametric distributions of connection lengths for the efficient analysis of fixed access networks.” Ann. Telecommun. 66, 103–118.Google Scholar
Hammersley, J. M. and Welsh, D. J. A. (1965). “First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory.“In Proceedings of the International Research Seminar, Statistics Lab, University of California, Berkeley, Springer, New York, pp. 61–110.Google Scholar
Hirsch, C., Neuhäuser, D. and Schmidt, V. (2013). “Connectivity of random geometric graphs related to minimal spanning forests.“Adv. Appl. Prob. 45, 20–36.CrossRefGoogle Scholar
Hirsch, C., Neuhäuser, D. and Schmidt, V. (2015). “Moderate deviations for shortest-path lengths on random geometric graphs.” Submitted.Google Scholar
Jeulin, D. (1997). “Dead leaves models: from space tessellation to random functions.“In Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets, World Scientific, River Edge, NJ, pp. 137–156.CrossRefGoogle Scholar
Kesten, H. (1986). “Aspects of first passage percolation.“In École d'Été de Probabilités de Saint-Flour, XIV, Springer, Berlin, pp. 125–264.CrossRefGoogle Scholar
Kingman, J. F. C. (1973). “Subadditive ergodic theory.“Ann. Prob. 1, 883–909.Google Scholar
Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). “Domination by product measures.“Ann. Prob. 25, 71–95.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978). “Infinitely Divisible Point Processes. “John Wiley, Chichester.Google Scholar
Molchanov, I. (2005). “Theory of Random Sets. “Springer, London.Google Scholar
Møller, J. (1992). “Random Johnson–Mehl tessellations.“Adv. Appl. Prob. 24, 814–844.CrossRefGoogle Scholar
Neuhäuser, D., Hirsch, C., Gloaguen, C. and Schmidt, V. (2012). “On the distribution of typical shortest-path lengths in connected random geometric graphs.“Queueing Systems 71, 199–220.Google Scholar
Neuhäuser, D., Hirsch, C., Gloaguen, C. and Schmidt, V. (2013). “A parametric copula approach for modelling shortest-path trees in telecommunication networks.“In Analytical and Stochastic Modeling Techniques and Applications, eds Dudin, A. and De Turck, K., Springer, Berlin, pp. 324–336.Google Scholar
Penrose, M. (2003). “Random Geometric Graphs.” Oxford University Press.Google Scholar
Pimentel, L. P. R. (2011). “Asymptotics for first-passage times on Delaunay triangulations.” Combin. Prob. Comput. 20, 435–453.Google Scholar
Pisztora, A. (1996). “Surface order large deviations for Ising, Potts and percolation models.“Prob. Theory Relat. Fields 104, 427–466.CrossRefGoogle Scholar
Rousselle, A. (2015). “Recurrence or transience of random walks on random graphs generated by point processes in R d .“To appear in Stoch. Process. Appl. Google Scholar
Schneider, R. and Weil, W. (2008). “Stochastic and Integral Geometry. “Springer, Berlin.Google Scholar
Vahidi-Asl, M. Q. and Wierman, J. C. (1990). “First-passage percolation on the Voronoı˘ tessellation and Delaunay triangulation.“In Random Graphs '87, John Wiley, Chichester, pp. 341–359.Google Scholar
Vahidi-Asl, M. Q. and Wierman, J. C. (1992). “A shape result for first-passage percolation on the Voronoı˘ tessellation and Delaunay triangulation.“In Random Graphs '89, Vol. 2, John Wiley, New York, pp. 247–262.Google Scholar
Voss, F., Gloaguen, C. and Schmidt, V. (2009). “Capacity distributions in spatial stochastic models for telecommunication networks.” Image Anal. Stereol. 28, 155–163.Google Scholar
Yao, C.-L., Chen, G. and Guo, T.-D. (2011). “Large deviations for the graph distance in supercritical continuum percolation.“J. Appl. Prob. 48, 154–172.Google Scholar