Duality and Stability in Complex Multiagent State-Dependent Network Dynamics

TL;DR

This paper introduces a duality-based framework for analyzing the stability of complex multiagent systems with highly coupled state-network dynamics, connecting stability analysis with nonlinear optimization techniques.

Contribution

It extends previous work by applying a duality perspective to view network dynamics as dual variables, linking Lyapunov stability with primal-dual optimization algorithms.

Findings

Network evolution can be modeled as primal-dual algorithm iterates.

Stability analysis is connected to optimization methods like mirror descent and Newton's method.

Numerical simulations validate the theoretical framework.

Abstract

Despite significant progress on stability analysis of conventional multiagent networked systems with weakly coupled state-network dynamics, most of the existing results have shortcomings in addressing 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 that uses a sequential optimization framework. Building upon that, in this paper, we extend our results by providing another angle on multiagent network dynamics from a duality perspective, which allows us to view the network structure as dual variables of a constrained nonlinear program. Leveraging that idea, we show that the evolution of the coupled state-network multiagent dynamics can be viewed as iterates of a primal-dual algorithm for 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, the Newton method, and the subgradient method. As a result, we develop a systematic framework for analyzing the Lyapunov stability of state-dependent network dynamics using techniques from nonlinear optimization. Finally, we support our theoretical results through numerical simulations from social science.