The $s$-Energy and Its Applications

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.