Distributed Asynchronous Algorithms for Solving Positive Definite Linear Equations over Networks—Part II: Wireless Networks

—In this Part II of the two-part paper, we study the interplay among wireless communications, distributed algorithms, and control in solving symmetric positive definite systems of linear equations over multi-hop wireless networks with fixed topologies. Building on the results from Part I, we develop and analyze Pairwise, Groupwise, Random Hopwise, and Controlled Hopwise Equalizing (PE, GE, RHE, and CHE), showing along the way how the broadcast nature of wireless transmissions may be fully utilized, how undesirable overlapping iterations may be avoided, and how iterations may be feedback controlled in a greedy, decentralized, Lyapunov-based fashion, leading to CHE. We show that CHE yields a networked dynamical system with state-dependent switching, provable exponential convergence, and quantifiable worst-case convergence rate. Finally, through extensive simulation on random geometric graphs, we show that GE, RHE, and CHE are dramatically more efficient and scalable than two existing, average-consensus-based schemes, with CHE having the best performance.