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The paper considers the consensus problem in large networks represented bytime-varying directed graphs. A practical way of dealing with large-scalenetworks is to reduce their dimension by collapsing the states of nodesbelonging to densely and intensively connected clusters into aggregatevariables. It will be shown that under suitable conditions, the states of theagents in each cluster converge fast toward a local agreement. Local agreementscorrespond to aggregate variables which slowly converge to consensus. Existingresults concerning the time-scale separation in large networks focus on fixedand undirected graphs. The aim of this work is to extend these results to themore general case of time-varying directed topologies. It is noteworthy that inthe fixed and undirected graph case the average of the states in each clusteris time-invariant when neglecting the interactions between clusters. Therefore,they are good candidates for the aggregate variables. This is no longerpossible here. Instead, we find suitable time-varying weights to compute theaggregate variables as time-invariant weighted averages of the states in eachcluster. This allows to deal with the more challenging time-varying directedgraph case. We end up with a singularly perturbed system which is analyzed byusing the tools of two time-scales averaging which seem appropriate to thissystem.

Time scale modeling for consensus in sparse directed networks with time-varying topologies