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We study a general framework for broadcast gossip algorithms which usecompanion variables to solve the average consensus problem. Each node maintainsan initial state and a companion variable. Iterative updates are performedasynchronously whereby one random node broadcasts its current state andcompanion variable and all other nodes receiving the broadcast update theirstate and companion variable. We provide conditions under which this scheme isguaranteed to converge to a consensus solution, where all nodes have the samelimiting values, on any strongly connected directed graph. Under strongerconditions, which are reasonable when the underlying communication graph isundirected, we guarantee that the consensus value is equal to the average, bothin expectation and in the mean-squared sense. Our analysis uses tools fromnon-negative matrix theory and perturbation theory. The perturbation resultsrely on a parameter being sufficiently small. We characterize the allowableupper bound as well as the optimal setting for the perturbation parameter as afunction of the network topology, and this allows us to characterize theworst-case rate of convergence. Simulations illustrate that, in comparison toexisting broadcast gossip algorithms, the approaches proposed in this paperhave the advantage that they simultaneously can be guaranteed to converge tothe average consensus and they converge in a small number of broadcasts.

Broadcast Gossip Algorithms for Consensus on Strongly Connected Digraphs