  We consider the averaging process on a graph, that is the evolution of a mass
distribution undergoing repeated averages along the edges of the graph at the
arrival times of independent Poisson processes. We establish cutoff phenomena
for both the $L^1$ and $L^2$ distance from stationarity when the graph is a
discrete hypercube and when the graph is complete bipartite. Some general facts
about the averaging process on arbitrary graphs are also discussed.
