Reaching consensus about gossip: convergence times and costs

Gossip algorithms have recently received significant attention, mainly because they constitute simple and robust methods for distributed information processing over networks. However, for many topologies that are realistic for wireless adhoc and sensor networks (such as grids and random geometric graphs), the standard nearest-neighbor gossip converges however slowly. Moreover we show that convergence of gossip algorithms consists of a transient and a steady state phase, that have not been distinguished so far. In this paper, we first introduce a metric for convergence time and cost that allow us to clearly characterize the steady state regime of the convergence, not only for i.i.d. but for all stationary and ergodic time-varying networks. This metric is based on Oseledec’s theorem, which gives an almost-sure description of the algorithm’s convergence rate. We next describe a variation of geographic gossip that averages along routed paths, and which is proven to be order optimal (cost of O(n) messages for a network of n nodes) for grids and random geometric graphs, in sharp contrast with standard nearest-neighbor gossip (O(n) messages). This paper summarizes some of the results in [1] and [2].