## Journal of Applied Probability

- J. Appl. Probab.
- Volume 51, Number 4 (2014), 910-920.

### Percolation on the information-theoretically secure signal to interference ratio graph

#### 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 λ_{0} such that
for all
λ < λ_{0} ≤ λ_{c} a
suitable graph defined over eavesdropper node connections percolates that
precludes percolation in the graphs formed by the legitimate nodes.

#### Article information

**Source**

J. Appl. Probab., Volume 51, Number 4 (2014), 910-920.

**Dates**

First available in Project Euclid: 20 January 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1421763317

**Mathematical Reviews number (MathSciNet)**

MR3301278

**Zentralblatt MATH identifier**

1349.94134

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G70: Extreme value theory; extremal processes

Secondary: 05C05: Trees 90C27: Combinatorial optimization

**Keywords**

Percolation information-theoretic security wireless communication SINR graph

#### Citation

Vaze, Rahul; Iyer, Srikanth. Percolation on the information-theoretically secure signal to interference ratio graph. J. Appl. Probab. 51 (2014), no. 4, 910--920. https://projecteuclid.org/euclid.jap/1421763317