Advances in Applied Probability

Percolation and connectivity in AB random geometric graphs

Srikanth K. Iyer and D. Yogeshwaran

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Abstract

Given two independent Poisson point processes Φ(1), Φ(2) in Rd, the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

Article information

Source
Adv. in Appl. Probab., Volume 44, Number 1 (2012), 21-41.

Dates
First available in Project Euclid: 8 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aap/1331216643

Digital Object Identifier
doi:10.1239/aap/1331216643

Mathematical Reviews number (MathSciNet)
MR2951545

Zentralblatt MATH identifier
1248.60016

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C80: Random graphs [See also 60B20]
Secondary: 82B43: Percolation [See also 60K35] 05C40: Connectivity

Keywords
Random geometric graph percolation connectivity wireless network secure communication

Citation

Iyer, Srikanth K.; Yogeshwaran, D. Percolation and connectivity in AB random geometric graphs. Adv. in Appl. Probab. 44 (2012), no. 1, 21--41. doi:10.1239/aap/1331216643. https://projecteuclid.org/euclid.aap/1331216643


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