Controlled Hopwise Averaging and Its Convergence Rate

This technical note develops Ideal Controlled Hopwise Averaging (ICHA) and Controlled Hopwise Averaging (CHA), two asynchronous distributed averaging algorithms for wireless networks, which attempt to “make the most” out of each iteration by fully exploiting the broadcast nature of wireless medium and incorporating feedback control of when to initiate an iteration. The latter feature, enabled by a common quadratic Lyapunov function, is novel among existing schemes and represents a new way to apply Lyapunov stability theory. We also derive deterministic upper bounds on the exponential convergence rate of ICHA on general and specific graphs, express the bounds explicitly in terms of the graph invariants, and show that they outperform the stochastic convergence rate of Pairwise Averaging on some common graphs of opposing densities. Finally, we obtain upper bounds on the convergence rate of CHA and show that CHA is capable of closely mimicking the behavior of ICHA, while being practical.