Power optimization on a network: The effects of randomness

Consider a wireless network of transmitter-receiver pairs. The transmitters adjust their powers to maintain a particular SINR target at the corresponding receiver in the presence of interference from neighboring transmitters. In this paper we analyze the optimal power vector that achieves this target in the presence of randomness in the network. Specifically, starting from a regular lattice of transmitter-receiver pairs we randomly turn off a finite fraction of them. We apply random matrix theory to evaluate the asymptotic optimal power per link, as well as the variance of powers in the optimal power vector in the limit of a large number of links. Our analytical results show remarkable agreement with numerically generated networks, both in one- and two-dimensional network geometries. Interestingly, we observe that unlike regular lattices, the optimal power in random networks has a discontinuity at a finite value, while the variance of its powers diverges at that value. Beyond that critical point, no feasible power solution exists. We discuss the relevance of these results in realistic networks.