The asymptotical error of broadcast gossip averaging algorithms

In problems of estimation and control which involve a network, efficient distributed computation of averages is a key issue. This paper presents theoretical and simulation results about the accumulation of errors during the computation of averages by means of iterative “broadcast gossip” algorithms. Using martingale theory, we prove that the expectation of the accumulated error can be bounded from above by a quantity which only depends on the mixing parameter of the algorithm and on few properties of the network: its size, its maximum degree and its spectral gap. Both analytical results and computer simulations show that in several network topologies of applicative interest the accumulated error goes to zero as the size of the network grows large.

💡 Research Summary

The paper addresses a fundamental problem in distributed estimation and control over networks: how to compute the global average of node states efficiently and accurately using a broadcast gossip protocol. Unlike traditional pairwise gossip, where nodes exchange information only with a single neighbor per iteration, broadcast gossip allows a node to transmit its current value to all of its neighbors simultaneously with probability p. Each receiving node then updates its state by taking a convex combination of its own value and the average of the received values, controlled by a mixing parameter α. This mechanism reduces the number of communication rounds needed for convergence, but the stochastic nature of the broadcasts introduces a bias that can accumulate over time.

The authors model the error e(t) = (1/N)∑ₙ xₙ(t) − x̄, where x̄ is the true network average, as a martingale. They prove that the one‑step increment Δe(t) = e(t+1) − e(t) has zero conditional expectation given the past, establishing a martingale difference sequence. By bounding the magnitude of Δe(t) in terms of the network’s maximum degree Δ_max, the broadcast probability p, and the mixing parameter α, they are able to apply Doob’s inequality and the Azuma–Hoeffding concentration bound. The resulting probabilistic bound

\