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

  • Rahul Vaze (a1) and Srikanth Iyer (a2)

Abstract

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 R 2 of intensities λ and λ E , 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 λ E of eavesdropper nodes, there exists a critical intensity λ c < ∞ such that for all λ > λ c 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 ≤ λ c a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

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Copyright

Corresponding author

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

References

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[1] Dousse, O., Baccelli, F. and Thiran, P. (2005). Impact of interferences on connectivity in ad hoc networks. IEEE/ACM Trans. Networking 13, 425436.
[2] Dousse, O. et al. (2006). Percolation in the signal to interference ratio graph. J. Appl. Prob. 43, 552562.
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[5] Haenggi, M. (2008). The secrecy graph and some of its properties. In Proc. IEEE Internat. Symp. Inf. Theory ISIT 2008, IEEE, pp. 539543.
[6] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.
[7] Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.
[8] Pinto, P. and Win, M. Z. (2012). Percolation and connectivity in the intrinsically secure communications graph. IEEE Trans. Inf. Theory 58, 17161730.
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[10] Vaze, R. (2012). Percolation and connectivity on the signal to interference ratio graph. In Proc. IEEE INFOCOM 2012, IEEE, pp. 513521.
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Keywords

MSC classification

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

  • Rahul Vaze (a1) and Srikanth Iyer (a2)

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