Distributed Averaging in the presence of a Sparse Cut

We consider the question of averaging on a graph that has one sparse cut separating two subgraphs that are internally well connected. While there has been a large body of work devoted to algorithms for distributed averaging, nearly all algorithms involve only {\it convex} updates. In this paper, we suggest that {\it non-convex} updates can lead to significant improvements. We do so by exhibiting a decentralized algorithm for graphs with one sparse cut that uses non-convex averages and has an averaging time that can be significantly smaller than the averaging time of known distributed algorithms, such as those of \cite{tsitsiklis, Boyd}. We use stochastic dominance to prove this result in a way that may be of independent interest.