Stochastic Modeling and Simulation for Redundancy and Coexistence in Graphs Resulting from Log-Normal Shadowing

Recent studies have identified redundancy and coexistence as a supporting graph structure for building connected intersection free planar drawings in wireless network graphs. Empirical evidence suggests that under certain conditions these properties can be assumed to hold with high probability. In this paper we advance insight on these probabilities with a rigorous stochastic treatment studying randomly generated network graphs pertaining to the log-normal shadowing model. We derive nested integral expressions to compute probabilities of redundancy and coexistence numerically under the standard uncorrelated log-normal shadowing model. For a recent model extension including correlation among network links our findings provide a means for efficient stochastic simulation of these probabilities. We illustrate numerical and simulation application of the derived formulae with a comprehensive parameter study for redundancy and coexistence under correlated and uncorrelated log-normal shadowing modeled randomly generated graphs. We also demonstrate how far support for these properties can be improved by artificially cutting communication distance while keeping the network well connected in terms of percolation bounds.