Power Optimization in Random Wireless Networks

In this paper, we analyze the problem of power control in large, random wireless networks that are obtained by "erasing" a finite fraction of nodes from a regular d-dimensional lattice of N transmit-receive pairs. In this model, which has the important feature of a minimum distance between transmitter nodes, we find that when the network is infinite, power control is always feasible below a positive critical value of the users' signal-to-interference-plus-noise ratio (SINR) target. Drawing on tools and ideas from statistical physics, we show how this problem can be mapped to the Anderson impurity model for diffusion in random media. In this way, by employing the so-called coherent potential approximation method, we calculate the average power in the system (and its variance) for 1-D and 2-D networks. This approach is equivalent to traditional techniques from random matrix theory and is in excellent agreement with the numerical simulations; however, it fails to predict when power control becomes infeasible. In this regard, even though infinitely large systems are always unstable beyond a critical value of the users' SINR target, finite systems remain stable with high probability even beyond this critical SINR threshold. We calculate this probability by analyzing the density of low lying eigenvalues of an associated random Schrodinger operator, and we show that the network can exceed this critical SINR threshold by at least O((log N)(-2/d)) before undergoing a phase transition to the unstable regime. Finally, using the same techniques, we also calculate the tails of the distribution of transmit power in the system and the rate of convergence of the Foschini-Miljanic power control algorithm in the presence of random erasures.