  In this paper, we consider consensus problems over a network of nodes, where
the network is divided into a number of clusters. We are interested in the case
where the communication topology within each cluster is dense as compared to
the sparse communication across the clusters. Moreover, each cluster has one
leader which can communicate with other leaders in different clusters. The goal
of the nodes is to agree at some common value under the presence of
communication delays across the clusters.
  Our main contribution is to propose a novel distributed two-time-scale
consensus algorithm, which pertains to the separation in network topology of
clustered networks. In particular, one scale is to model the dynamic of the
agents in each cluster, which is much faster (due to the dense communication)
than the scale describing the slowly aggregated evolution between the clusters
(due to the sparse communication). We prove the convergence of the proposed
method in the presence of uniform, but possibly arbitrarily large,
communication delays between the leaders. In addition, we provided an explicit
formula for the convergence rate of such algorithm, which characterizes the
impact of delays and the network topology. Our results shows that after a
transient time characterized by the topology of each cluster, the convergence
of the two-time-scale consensus method only depends on the connectivity of the
leaders. Finally, we validate our theoretical results by a number of numerical
simulations on different clustered networks.
