Topological Uncertainty in Wireless Networks

This work provides an analysis of topological uncertainty in wireless networks. Here, we model a wireless network as a random geometric graph (RGG) and quantify topological uncertainty in terms of the Shannon entropy of the underlying graph model. Direct pairwise connections between nodes are probabilistic in general, and thus our analysis covers cases where channel randomness contributes to uncertainty in the network topology. We derive a simple bound on RGG entropy. We then consider a pairwise connection model based on small-scale Rayleigh fading and study the behavior of the entropy bound as the number of nodes n in the network grows large while the typical connection range r_0 increases or decreases. We present three key results that quantify the rate of growth or decay of r_0, as a function of the number of nodes n, that must be obeyed in order for the entropy bound to converge to a positive limit. Although the contributions of this paper are theoretical, they have applications in systems such as ad hoc networks employing opportunistic routing and device-to-device (D2D) networks for future cellular communication (5G and beyond).