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Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

Part of: Stochastic processes Graph theory Geometric probability and stochastic geometry

Published online by Cambridge University Press:  30 January 2018

Rahul Vaze  and
Srikanth Iyer
Rahul Vaze*
Affiliation:
Tata Institute of Fundamental Research
Srikanth Iyer*
Affiliation:
Indian Institute of Science
*
∗ Postal address: School of Technology and Computer Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India.
∗∗ Postal address: Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India. Email address: [email protected]
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Abstract

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We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R2 of intensities λ and λE, respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λE of eavesdropper nodes, there exists a critical intensity λc < ∞ such that for all λ > λc the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λc a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

Keywords

Percolationinformation-theoretic securitywireless communicationSINR graph

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G70: Extreme value theory; extremal processes
Secondary: 05C05: Trees 90C27: Combinatorial optimization
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 910 - 920
Copyright
© Applied Probability Trust 

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