Advances in Applied Probability

On the relation between graph distance and Euclidean distance in random geometric graphs

J. Díaz, D. Mitsche, G. Perarnau, and X. Pérez-Giménez

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Abstract

Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).

Article information

Source
Adv. in Appl. Probab., Volume 48, Number 3 (2016), 848-864.

Dates
First available in Project Euclid: 19 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1474296318

Mathematical Reviews number (MathSciNet)
MR3568895

Zentralblatt MATH identifier
1348.05188

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35]

Keywords
Random geometric graph graph distance Euclidean distance diameter

Citation

Díaz, J.; Mitsche, D.; Perarnau, G.; Pérez-Giménez, X. On the relation between graph distance and Euclidean distance in random geometric graphs. Adv. in Appl. Probab. 48 (2016), no. 3, 848--864. https://projecteuclid.org/euclid.aap/1474296318


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