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

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