Topological approach and analysis of clustering in consensus networks

We study clustering properties of networks of single-integrator nodes over a directed graph, in which the nodes converge to steady-state values. These values define clustering groups of nodes, which are considered dependent on interaction topology and edge weights. Focusing on the interaction topology of the network, in this paper, we introduce the notion of topological clusters, which are sets of nodes that converge to an identical value due to the topological characteristics of the network, independent of the value of the edge weights. We then investigate properties of topological clusters and present a necessary and sufficient condition for a set of nodes to form a topological cluster. We also provide an algorithm for finding topological clusters, which is validated by an example.