This paper presents a general class of gossip-based averaging algorithms, which are inspired from Uniform Gossip. While Uniform Gossip works synchronously on complete graphs, weighted gossip algorithms allow asynchronous rounds and converge on any connected, directed or undirected graph. Unlike most previous gossip algorithms, Weighted Gossip admits stochastic update matrices which need not be doubly stochastic. Double-stochasticity being very restrictive in a distributed setting, this novel degree of freedom is essential and it opens the perspective of designing a large number of new gossip-based algorithms. To give an example, we present one of these algorithms, which we call One-Way Averaging. It is based on random geographic routing, just like Path Averaging, except that routes are one way instead of round trip. Hence in this example, getting rid of double stochasticity allows us to add robustness to Path Averaging.
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