  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.
