Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-19T06:38:04.980Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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: skiyer@math.iisc.ernet.in
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Dousse, O., Baccelli, F. and Thiran, P. (2005). Impact of interferences on connectivity in ad hoc networks. IEEE/ACM Trans. Networking 13, 425–436.Google Scholar
Dousse, O. et al. (2006). Percolation in the signal to interference ratio graph. J. Appl. Prob. 43, 552–562.Google Scholar
Grimmett, G. (1980). Percolation. Springer.Google Scholar
Gupta, P. and Kumar, P. R. (2000). The capacity of wireless networks. IEEE Trans. Inf. Theory 46, 388–404.Google Scholar
Haenggi, M. (2008). The secrecy graph and some of its properties. In Proc. IEEE Internat. Symp. Inf. Theory ISIT 2008, IEEE, pp. 539–543.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
Pinto, P. and Win, M. Z. (2012). Percolation and connectivity in the intrinsically secure communications graph. IEEE Trans. Inf. Theory 58, 1716–1730.Google Scholar
Sarkar, A. and Haenggi, M. (2013). Percolation in the secrecy graph. Discrete Appl. Math. 161, 2120–2132.Google Scholar
Vaze, R. (2012). Percolation and connectivity on the signal to interference ratio graph. In Proc. IEEE INFOCOM 2012, IEEE, pp. 513–521.Google Scholar
Wyner, A. D. (1975). The wire-tap channel. Bell Syst. Tech. J. 54, 1355–1387.Google Scholar