  We study the popular distributed consensus method over networks composed of a
number of densely connected clusters with a sparse connection between them. In
these cluster networks, the method often constitutes two-time-scale dynamics,
where the internal nodes within each cluster reach consensus quickly relative
to the aggregate nodes across clusters. Our main contribution is to provide the
rate of the distributed consensus method, which characterize explicitly the
impacts of the internal and external graphs on the performance of this method.
Our main result shows that this rate converges exponentially and only scales
with a few number of nodes, which is relatively small to the size of the
network.
  The key technique in our analysis is to consider a Lyapunov function which
captures the impacts of different time-scale dynamics on the convergence of the
method. Our approach avoids using model reduction, which is the typical way
according to singular perturbation theory and relies on relatively simple
definitions of the slow and fast variables. In addition, Lyapunov analysis
allows us to derive the rate of distributed consensus methods over cluster
networks, which is missing from the existing works using singular perturbation
theory. We illustrate our theoretical results by a number of numerical
simulations over different cluster networks.
