Geographic Gossip on Geometric Random Graphs via Affine Combinations

In recent times, a considerable amount of work has been devoted to the development and analysis of gossip algorithms in Geometric Random Graphs. In a recently introduced model termed "Geographic Gossip," each node is aware of its position but possesses no further information. Traditionally, gossip protocols have always used convex linear combinations to achieve averaging. We develop a new protocol for Geographic Gossip, in which counter-intuitively, we use {\it non-convex affine combinations} as updates in addition to convex combinations to accelerate the averaging process. The dependence of the number of transmissions used by our algorithm on the number of sensors $n$ is $n \exp(O(\log \log n)^2) = n^{1 + o(1)}$. For the previous algorithm, this dependence was $\tilde{O}(n^{1.5})$. The exponent 1+ o(1) of our algorithm is asymptotically optimal. Our algorithm involves a hierarchical structure of $\log \log n$ depth and is not completely decentralized. However, the extent of control exercised by a sensor on another is restricted to switching the other on or off.