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

The s-energy is a generating function of wide applicability in network-based dynamics. We derive an (essentially) optimal bound of (3/ρs)
^n-1
on the s-energy of an n-agent symmetric averaging system, for any positive real s ≤ 1, where ρ is a lower bound on the nonzero weights. This is done by introducing the new dynamics of twist systems. We show how to use the new bound on the s-energy to tighten the convergence rates of systems in opinion dynamics, flocking, and synchronization.