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

TL;DR

This paper establishes an essentially optimal bound on the $s$-energy in averaging systems, introducing twist systems and applying the results to improve convergence rate analysis in opinion dynamics, flocking, and synchronization.

Contribution

It introduces the concept of twist systems and derives a sharp bound on $s$-energy, enhancing understanding of convergence in network-based dynamics.

Findings

Derived an optimal bound of $(3/ ho s)^{n-1}$ on $s$-energy.

Applied the bound to improve convergence rate estimates in opinion dynamics.

Introduced twist systems as a new dynamic framework.

Abstract

The {\em $s$-energy} is a generating function of wide applicability in network-based dynamics. We derive an (essentially) optimal bound of $(3/\rho s)^{n-1}$ on the $s$-energy of an $n$-agent symmetric averaging system, for any positive 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 show how to use the new bound on the $s$-energy to tighten the convergence rate of systems in opinion dynamics, flocking, and synchronization.