Hierarchical clustering of dynamical networks using a saddle-point analysis

This paper studies cluster synchronization in dynamical networks. A class of cooperative dynamical networks that exhibit clustering in their asymptotic behavior is analyzed. The network nodes are equipped with heterogeneous dynamics and interact with a nonlinear and saturated interaction rule. It is proven that cluster synchronization appears asymptotically independent of the initial conditions. The clustering behavior of the dynamic network is shown to correspond to the solution of a static saddle-point problem, enabling a precise characterization of the clustering structure. We show how the clustering structure depends on the relation between the underlying graph, the node dynamics, and the saturation level of the interactions. This interpretation leads to deeper combinatorial insights related to clustering, including a generalization of classical network partitioning problems such as the inhibiting bisection problem, the min $s-t$-cut problem, and hierarchical clustering analysis. The theoretical results are applied for the analysis of a test-case network, inspired by the IEEE 30-bus system.