Network clustering: A dynamical systems and saddle-point perspective

This paper studies a class of cooperative networks that exhibit clustering in their steady-state behavior. We consider a collection of agents with heterogeneous dynamics and a bounded interaction rule between neighboring systems. We relate the steady state-behavior of the dynamical network to a static saddle-point problem. The saddle-point description of the system allows for a precise characterization of clustering. We show that the graph forms clusters along edges that are saturated and the corresponding cluster values depend only on these edges and the objective functions of each agent. We then provide a Lyapunov stability proof connecting the steady-state behavior of the dynamic system to the solution of the static saddle-point problem.