Duality and Stability in Complex Multiagent State-Dependent Network Dynamics

Despite significant progress for stability analysis of conventional multiagent networked systems with weakly coupled state-network dynamics, most of the existing results have shortcomings to address multiagent systems with highly coupled state-network dynamics. Motivated by numerous applications of such dynamics, in our previous work [1], we initiated a new direction for stability analysis of such systems using a sequential optimization framework. Building upon that, in this paper, we complete our results by providing another angle to multiagent network dynamics from a duality perspective, which allows us to view the network structure as dual variables of a constrained nonlinear program. Leveraging this idea, we show that the evolution of the coupled state-network multiagent dynamics can be viewed as iterates of a primal-dual algorithm to a static constrained optimization/saddle-point problem. This view bridges the Lyapunov stability of state-dependent network dynamics and frequently used optimization techniques such as block coordinated descent, mirror descent, Newton method, and subgradient method. As a result, we develop a systematic framework to analyze the Lyapunov stability of state-dependent network dynamics using well-developed techniques from nonlinear optimization. Finally, we support our theoretical results through numerical simulations.