Percolation and Connectivity in the Intrinsically Secure Communications Graph

TL;DR

This paper analyzes the global connectivity properties of the intrinsically secure communications graph (iS-graph), demonstrating phase transitions for percolation and deriving asymptotic and explicit formulas for full connectivity in large networks, considering eavesdropper effects.

Contribution

It provides the first rigorous analysis of percolation and full connectivity in the iS-graph, including phase transition proofs and explicit connectivity probability formulas.

Findings

Existence of a phase transition for percolation in the iS-graph.

Asymptotic behavior of full connectivity as legitimate node density grows.

Explicit expressions for the probability of full connectivity in finite regions.

Abstract

The ability to exchange secret information is critical to many commercial, governmental, and military networks. The intrinsically secure communications graph (iS-graph) is a random graph which describes the connections that can be securely established over a large-scale network, by exploiting the physical properties of the wireless medium. This paper aims to characterize the global properties of the iS-graph in terms of: (i) percolation on the infinite plane, and (ii) full connectivity on a finite region. First, for the Poisson iS-graph defined on the infinite plane, the existence of a phase transition is proven, whereby an unbounded component of connected nodes suddenly arises as the density of legitimate nodes is increased. This shows that long-range secure communication is still possible in the presence of eavesdroppers. Second, full connectivity on a finite region of the Poisson iS-graph is considered. The exact asymptotic behavior of full connectivity in the limit of a large density of legitimate nodes is characterized. Then, simple, explicit expressions are derived in order to closely approximate the probability of full connectivity for a finite density of legitimate nodes. The results help clarify how the presence of eavesdroppers can compromise long-range secure communication.