Many multi-agent systems evolve by repeatedly updating each state to a weighted average of its neighbors, a process known as averaging dynamics, whose behavior becomes difficult to analyze when the interaction network varies over time. In recent years, the $s$-energy has emerged as a useful tool for bounding the convergence rates of such systems, complementing the classical techniques that rely on fixed graphs. We derive new bounds on the $s$-energy under minimal connectivity assumptions. As a consequence, we obtain convergence guarantees for several models of collective dynamics and resolve a number of open questions in the areas. Our results highlight the dependence of the $s$-energy on the connectivity of the underlying networks and use it to explain the exponential gap in the convergence rates of stationary and time-varying consensus systems.

翻译：许多多智能体系统通过反复将每个状态更新为其邻居状态的加权平均值而演化，这一过程被称为平均化动力学。当交互网络随时间变化时，其行为分析变得困难。近年来，s-能量已成为约束此类系统收敛速率的有用工具，它补充了依赖于固定图的经典技术。我们在最小连通性假设下推导了s-能量的新界。作为结果，我们获得了多个集体动力学模型的收敛性保证，并解决了该领域中的若干开放性问题。我们的结果突出了s-能量对底层网络连通性的依赖性，并利用它解释了静态与时变共识系统收敛速率之间的指数级差距。