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Journal of Probability and Statistics
Volume 2011 (2011), Article ID 350382, 21 pages
http://dx.doi.org/10.1155/2011/350382
Research Article

On the One Dimensional Poisson Random Geometric Graph

Institut Télécom, Télécom ParisTech, CNRS LTCI, 75634 Paris, France

Received 24 March 2011; Revised 20 July 2011; Accepted 22 July 2011

Academic Editor: Kelvin K. W. Yau

Copyright © 2011 L. Decreusefond and E. Ferraz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process, and edges exist between two points if and only if their distance is less than a fixed given threshold. We compute explicitly the distribution of the number of connected components of this graph. The proof relies on inverting some Laplace transforms.