A lower bound for distributed averaging algorithms

We derive lower bounds on the convergence speed of a widely used class of distributed averaging algorithms. In particular, we prove that any distributed averaging algorithm whose state consists of a single real number and whose (possibly nonlinear) update function satisfies a natural smoothness condition has a worst case running time of at least on the order of $n^2$ on a network of $n$ nodes. Our results suggest that increased memory or expansion of the state space is crucial for improving the running times of distributed averaging algorithms.