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

Part of: Equilibrium statistical mechanics Graph theory 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.
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Abstract

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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 

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