{
  "id": "2410.22341",
  "version": "v1",
  "published": "2024-10-12T06:48:47.000Z",
  "updated": "2024-10-12T06:48:47.000Z",
  "title": "The $s$-Energy and Its Applications",
  "authors": [
    "Bernard Chazelle",
    "Kritkorn Karntikoon"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:2207.00213",
  "categories": [
    "math.OC",
    "cs.MA"
  ],
  "abstract": "Averaging dynamics drives countless processes in physics, biology, engineering, and the social sciences. In recent years, the $s$-energy has emerged as a useful tool for bounding the convergence rates of time-varying averaging systems. We derive new bounds on the $s$-energy, which we use to resolve a number of open questions in the areas of bird flocking, opinion dynamics, and distributed motion coordination. We also use our results to provide a theoretical validation for the idea of the \"Overton Window\" as an attracting manifold of viable group opinions. Our new bounds on the $s$-energy highlight its dependency on the connectivity of the underlying networks. In this vein, we use the $s$-energy to explain the exponential gap in the convergence rates of stationary and time-varying consensus systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-12T06:48:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convergence rates",
      "applications",
      "averaging dynamics drives countless processes",
      "time-varying consensus systems",
      "energy highlight"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}