  We study a general framework for broadcast gossip algorithms which use
companion variables to solve the average consensus problem. Each node maintains
an initial state and a companion variable. Iterative updates are performed
asynchronously whereby one random node broadcasts its current state and
companion variable and all other nodes receiving the broadcast update their
state and companion variable. We provide conditions under which this scheme is
guaranteed to converge to a consensus solution, where all nodes have the same
limiting values, on any strongly connected directed graph. Under stronger
conditions, which are reasonable when the underlying communication graph is
undirected, we guarantee that the consensus value is equal to the average, both
in expectation and in the mean-squared sense. Our analysis uses tools from
non-negative matrix theory and perturbation theory. The perturbation results
rely on a parameter being sufficiently small. We characterize the allowable
upper bound as well as the optimal setting for the perturbation parameter as a
function of the network topology, and this allows us to characterize the
worst-case rate of convergence. Simulations illustrate that, in comparison to
existing broadcast gossip algorithms, the approaches proposed in this paper
have the advantage that they simultaneously can be guaranteed to converge to
the average consensus and they converge in a small number of broadcasts.
