Percolation and Range Bounds in Log-Normal Shadowing Modeled Networks

We study wireless network graphs where nodes are spatially distributed according to a homogeneous Poisson point process and connected based on the log-normal shadowing model. Well-known theoretical upper and lower bounds on the critical node density for percolation are investigated and generalized. We show empirically that the known lower and especially upper bound are still very conservative in the log-normal shadowing setting. We derive an expression, which provably lies below the known upper bound. Moreover, we study log-normal shadowing modeled graphs with a cutoff distance, i.e., all neighbors beyond a certain distance are discarded. We show how to find a small distance such that the graph still percolates. We adapt the known upper bound and as well our derived expression to that cutoff distance. We compare our expression with that lower and upper bound empirically for both with and without cutoff distances, and observe that our expression comes much closer to the actual transition point.