Information Theory Meets Percolation Theory : Capacity Scaling in Wireless Relay Network

In this expository article we revisit the topic of capacity scaling in wireless relay networks from a percolation theory perspective. In particular, following the works of Franceschetti et al [2]., we give an outline of a construction to show that in a large wireless network with interference, it is possible, w.h.p., to sustain a non-zero throughput between any two source-destination pair chosen among any α fraction of nodes, 0 < α < 1. Then using Information Theoretic arguments and in particular, utilizing the Broadcast-Cut bound from multi terminal network information theory, we prove a converse result and show that w.h.p., there exists a set of strictly positive fraction of nodes such that it is impossible to sustain any given non-zero throughput between any source-destination pair in it. In this article, our contribution is to develop a new, short and simple yet rigorous proof for the converse utilizing Monotone Convergence Theorem and Campbell’s Theorem.