Gossip along the way: order-optimal consensus through randomized path averaging

Gossip algorithms have recently received significant attention, mainly because they constitute simple and robust algorithms for distributed information processing over networks. However for many topologies that are realistic for wireless ad-hoc and sensor networks (like grids and random geometric graphs), the standard nearest-neighbor gossip converges very slowly. A recently proposed algorithm called geographic gossip improves gossip efficiency by a $\sqrt{n / \log n}$ factor for random geometric graphs, by exploiting geographic information of node locations. In this paper we prove that a variation of geographic gossip that averages along routed paths, improves efficiency by an additional $\sqrt{n / \log n}$ factor and is order optimal for grids and random geometric graphs. Our analysis provides some general techniques and can be used to provide bounds on the performance of randomized message passing algorithms operating over various graph topologies.