Percolation on the information theoretic secure SINR graph: Upper and lower bounds

Connectivity in an information-theoretically secure graph is considered where both the legitimate and the eavesdropper nodes are distributed as Poisson point processes. To allow concurrent transmissions from multiple legitimate nodes, a signal-to-interference plus noise ratio secure graph is introduced, and its percolation (having an unbounded connected component) properties are studied. It is shown that for a fixed eavesdropper node density, percolation happens for large enough (but finite) legitimate node density and small enough interference suppression parameter of the legitimate nodes. Conversely, a concrete bound is obtained that shows that if the legitimate node density is below a fixed threshold, then the probability of percolation is zero.