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Percolation and Connectivity in AB Random Geometric Graphs

Part of: Geometric probability and stochastic geometry Equilibrium statistical mechanics Graph theory

Published online by Cambridge University Press:  04 January 2016

Srikanth K. Iyer  and
D. Yogeshwaran
Srikanth K. Iyer*
Affiliation:
Indian Institute of Science
D. Yogeshwaran*
Affiliation:
INRIA/ENS
*
∗ Postal address: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India. Email address: [email protected]
∗∗ Current address: Faculty of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, 32000, Israel.
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Abstract

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Given two independent Poisson point processes Φ(1), Φ(2) in , 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.

Keywords

Random geometric graphpercolationconnectivitywireless networksecure communication

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 05C80: Random graphs
Secondary: 82B43: Percolation 05C40: Connectivity
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 21 - 41
Copyright
© Applied Probability Trust 

Footnotes

Research supported in part by UGC SAP-IV and DRDO, grant no. DRDO/PAM/SKI/593.

Supported in part by a grant from EADS, France, Israel Science foundation (grant no. 853/10), and AFOSR (grant no. FA8655-11-1-3039).

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