Super Critical and Sub Critical Regimes of Percolation with Secure   Communication
  Percolation in an information-theoretically secure graph is considered where both the legitimate and the eavesdropper nodes are distributed as Poisson point processes. For both the path-loss and the path-loss plus fading model, upper and lower bounds on the minimum density of the legitimate nodes (as a function of the density of the eavesdropper nodes) required for non-zero probability of having an unbounded cluster are derived. The lower bound is universal in nature, i.e. the constant does not depend on the density of the eavesdropper nodes. 
Rahul Vaze
cs.IT math.IT