Rates of Convergence for Greedy Gossip with Eavesdropping

Greedy gossip with eavesdropping (GGE) is a randomized gossip algorithm that exploits the broadcast nature of wireless communications to converge rapidly on grid-like network topologies without requiring that nodes know their geographic locations. When a node decides to gossip, rather than choosing one of its neighbors randomly, it greedily chooses to gossip with the neighbor whose values are most different from its own. We assume that all transmissions are wireless broadcasts so that nodes can keep track of their neighbors' values by eavesdropping on their communications. We have previously proved that GGE converges to the average consensus on connected network topologies. In this paper we study the rate of convergence of GGE, a non-trivial task due to the greedy, data-driven nature of the algorithm. We demonstrate that GGE outperforms standard randomized gossip, and we characterize the rate of convergence in terms of a topology- dependent constant analogous to the second-largest eigenvalue characterization for previous randomized gossip algorithms. Simulations demonstrate that the convergence rate of GGE is superior to existing average consensus algorithms such as geographic gossip.