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

Published online by Cambridge University Press:  30 January 2018

Rahul Vaze*
Tata Institute of Fundamental Research
Srikanth Iyer*
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:
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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.

Research Article
© Applied Probability Trust 


Dousse, O., Baccelli, F. and Thiran, P. (2005). Impact of interferences on connectivity in ad hoc networks. IEEE/ACM Trans. Networking 13, 425436.Google Scholar
Dousse, O. et al. (2006). Percolation in the signal to interference ratio graph. J. Appl. Prob. 43, 552562.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, 388404.Google Scholar
Haenggi, M. (2008). The secrecy graph and some of its properties. In Proc. IEEE Internat. Symp. Inf. Theory ISIT 2008, IEEE, pp. 539543.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, 17161730.Google Scholar
Sarkar, A. and Haenggi, M. (2013). Percolation in the secrecy graph. Discrete Appl. Math. 161, 21202132.Google Scholar
Vaze, R. (2012). Percolation and connectivity on the signal to interference ratio graph. In Proc. IEEE INFOCOM 2012, IEEE, pp. 513521.Google Scholar
Wyner, A. D. (1975). The wire-tap channel. Bell Syst. Tech. J. 54, 13551387.Google Scholar