  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.
