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June 2015 First passage percolation on random geometric graphs and an application to shortest-path trees
C. Hirsch, D. Neuhäuser, C. Gloaguen, V. Schmidt
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Adv. in Appl. Probab. 47(2): 328-354 (June 2015). DOI: 10.1239/aap/1435236978

Abstract

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.

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C. Hirsch. D. Neuhäuser. C. Gloaguen. V. Schmidt. "First passage percolation on random geometric graphs and an application to shortest-path trees." Adv. in Appl. Probab. 47 (2) 328 - 354, June 2015. https://doi.org/10.1239/aap/1435236978

Information

Published: June 2015
First available in Project Euclid: 25 June 2015

zbMATH: 1355.60018
MathSciNet: MR3360380
Digital Object Identifier: 10.1239/aap/1435236978

Subjects:
Primary: 60D05
Secondary: 05C10 , 05C80 , 82B43

Keywords: first passage percolation , longest shortest path , Random geometric graph , shape theorem , shortest-path tree

Rights: Copyright © 2015 Applied Probability Trust

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Vol.47 • No. 2 • June 2015
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