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.Comment: Submitted for journal publicatio
