Advances in Applied Probability

First passage percolation on random geometric graphs and an application to shortest-path trees

C. Hirsch, D. Neuhäuser, C. Gloaguen, and V. Schmidt

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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.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 2 (2015), 328-354.

Dates
First available in Project Euclid: 25 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1435236978

Digital Object Identifier
doi:10.1239/aap/1435236978

Mathematical Reviews number (MathSciNet)
MR3360380

Zentralblatt MATH identifier
1355.60018

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 05C80: Random graphs [See also 60B20] 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 82B43: Percolation [See also 60K35]

Keywords
First passage percolation shape theorem shortest-path tree longest shortest path random geometric graph

Citation

Hirsch, C.; Neuhäuser, D.; Gloaguen, C.; Schmidt, V. First passage percolation on random geometric graphs and an application to shortest-path trees. Adv. in Appl. Probab. 47 (2015), no. 2, 328--354. doi:10.1239/aap/1435236978. https://projecteuclid.org/euclid.aap/1435236978


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