Ergodic Randomized Algorithms and Dynamics Over Networks

Algorithms and dynamics over networks often involve randomization and randomization can induce oscillating dynamics that fail to converge in a deterministic sense. Under assumptions of independence across time and linearity of the updates, we show that the oscillations are ergodic if the expected dynamics is stable. We apply this result to three problems of network systems, namely, the estimation from relative measurements, the PageRank computation, and the dynamics of opinions in social networks. In these applications, the randomized dynamics is the asynchronous counterpart of a deterministic (stable) synchronous one. By ergodicity, the deterministic limit can be recovered via a time-averaging operation, which can be performed locally by each node of the network.