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The {\em $s$-energy} is a generating function of wide applicability innetwork-based dynamics. We derive an (essentially) optimal bound of $(3/\rhos)^{n-1}$ on the $s$-energy of an $n$-agent symmetric averaging system, for anypositive real $s\leq 1$, where~$\rho$ is a lower bound on the nonzero weights.This is done by introducing the new dynamics of {\em twist systems}. We showhow to use the new bound on the $s$-energy to tighten the convergence rate ofsystems in opinion dynamics, flocking, and synchronization.

A Sharp Bound on the $s$-Energy and Its Applications to Averaging Systems