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The $s$-Energy and Its Applications

Bernard Chazelle, Kritkorn Karntikoon

TL;DR

This work introduces the $s$-energy as a scale-aware invariant to analyze time-varying averaging dynamics, establishing bounds that scale polynomially in the number of agents $n$ and exponentially in the maximum number of connected components $m$. By proving tight upper and lower bounds for general, reversible, expanding, and random averaging systems, the authors derive convergence guarantees for diverse models, including multi-flock bird dynamics, opinion formation with stubborn agents, and distributed coordination under failures. A key insight is that network connectivity governs the convergence rate, explaining the observed exponential gap between fixed and time-varying networks and enabling uniform convergence analyses across heterogeneous, adversarial, or non-reversible updates. The results offer a general-purpose framework for bounding convergence across dynamic networks and provide concrete algorithmic implications for distributed control and collective behavior.

Abstract

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.

The $s$-Energy and Its Applications

TL;DR

This work introduces the -energy as a scale-aware invariant to analyze time-varying averaging dynamics, establishing bounds that scale polynomially in the number of agents and exponentially in the maximum number of connected components . By proving tight upper and lower bounds for general, reversible, expanding, and random averaging systems, the authors derive convergence guarantees for diverse models, including multi-flock bird dynamics, opinion formation with stubborn agents, and distributed coordination under failures. A key insight is that network connectivity governs the convergence rate, explaining the observed exponential gap between fixed and time-varying networks and enabling uniform convergence analyses across heterogeneous, adversarial, or non-reversible updates. The results offer a general-purpose framework for bounding convergence across dynamic networks and provide concrete algorithmic implications for distributed control and collective behavior.

Abstract

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 -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 -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 -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.

Paper Structure

This paper contains 19 sections, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Models
  3. Averaging dynamics.
  4. The $\bm{s}$-energy.
  5. Convergence rates.
  6. Energy Bounds for Averaging Systems
  7. General averaging systems: Theorem \ref{['s-Energ-Bounds']} (a)
  8. Lower bounds for twist systems.
  9. Reversible averaging systems: Theorem \ref{['s-Energ-Bounds']} (b)
  10. A lower bound for reversible systems.
  11. Expanding averaging systems: Theorem \ref{['s-Energ-Bounds']} (c)
  12. Random averaging systems: Theorem \ref{['s-Energ-Bounds']} (d)
  13. Applications
  14. Bird Flocking
  15. Single-flock dynamics
  16. ...and 4 more sections

Figures (3)

  • Figure 1: The graph $G_t$ is embedded on the real line (self-loops not shown) and $E_{s,t}= 3^s+4^s$.
  • Figure 2: A bird is influenced by its neighbors within distance $r$.
  • Figure 3: Simulation of a network of 900 robots in 3D with 60 of them (orange dots) pinned to a fixed plane $X=0$ and the rest (blue dots) are moving. The underlying network $G$ is a 30-by-30 grid graph, with edge failure probability equal to 0.3. The 60 nodes on two opposite sides of the grid are pinned to $X=0$.