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

#### 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

https://projecteuclid.org/euclid.aap/1435236978

Digital Object Identifier
doi:10.1239/aap/1435236978

Mathematical Reviews number (MathSciNet)
MR3360380

Zentralblatt MATH identifier
1355.60018

#### 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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