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Consensus on Open Multi-Agent Systems Over Graphs Sampled from Graphons

Renato Vizuete, Julien M. Hendrickx

TL;DR

This work introduces a graphon-based framework for open multi-agent systems under linear consensus, where topology evolves due to arrivals, departures, and replacements. The authors analyze the disagreement descriptor $V(t)=\frac{1}{n}||x(t)||^2-\bar{x}^2(t)$ and derive upper bounds on $\limsup_{t\to\infty} \mathbb{E}[V(t)]$ that depend on spectral properties $\mathbb{E}[e^{−2\gamma\mu_2}]$ of graphs sampled from a fixed graphon. They show how to bound these terms for replacements and arrivals/departures, and introduce tractable approximations for SBM graphons by reducing the spectrum computation to a small matrix of size $m$ that depends only on the graphon. A piecewise Lipschitz graphon bound further connects $\mathbb{E}[e^{−2\gamma\mu_2}]$ to the limiting eigenvalue $\bar{μ}_2$, enabling scalable analysis for large $n$ and complex networks.

Abstract

We show how graphons can be used to model and analyze open multi-agent systems, which are multi-agent systems subject to arrivals and departures, in the specific case of linear consensus. First, we analyze the case of replacements, where under the assumption of a deterministic interval between two replacements, we derive an upper bound for the disagreement in expectation. Then, we study the case of arrivals and departures, where we define a process for the evolution of the number of agents that guarantees a minimum and a maximum number of agents. Next, we derive an upper bound for the disagreement in expectation, and we establish a link with the spectrum of the expected graph used to generate the graph topologies. Finally, for stochastic block model (SBM) graphons, we prove that the computation of the spectrum of the expected graph can be performed based on a matrix whose dimension depends only on the graphon and it is independent of the number of agents.

Consensus on Open Multi-Agent Systems Over Graphs Sampled from Graphons

TL;DR

This work introduces a graphon-based framework for open multi-agent systems under linear consensus, where topology evolves due to arrivals, departures, and replacements. The authors analyze the disagreement descriptor and derive upper bounds on that depend on spectral properties of graphs sampled from a fixed graphon. They show how to bound these terms for replacements and arrivals/departures, and introduce tractable approximations for SBM graphons by reducing the spectrum computation to a small matrix of size that depends only on the graphon. A piecewise Lipschitz graphon bound further connects to the limiting eigenvalue , enabling scalable analysis for large and complex networks.

Abstract

We show how graphons can be used to model and analyze open multi-agent systems, which are multi-agent systems subject to arrivals and departures, in the specific case of linear consensus. First, we analyze the case of replacements, where under the assumption of a deterministic interval between two replacements, we derive an upper bound for the disagreement in expectation. Then, we study the case of arrivals and departures, where we define a process for the evolution of the number of agents that guarantees a minimum and a maximum number of agents. Next, we derive an upper bound for the disagreement in expectation, and we establish a link with the spectrum of the expected graph used to generate the graph topologies. Finally, for stochastic block model (SBM) graphons, we prove that the computation of the spectrum of the expected graph can be performed based on a matrix whose dimension depends only on the graphon and it is independent of the number of agents.

Paper Structure

This paper contains 12 sections, 8 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Model and descriptor
  4. Sampling from graphons
  5. Replacements
  6. Arrivals and departures
  7. Process for arrivals and departures
  8. Disagreement subject to arrivals and departures
  9. Impact of the graphon structure
  10. Estimation of $\mathbb{E}\left[e^{-2\gamma\mu_{2}}\right]$
  11. Computation of $\bar{\mu}_2$ for stochastic block model (SBM) graphons
  12. Conclusions and future work

Key Result

Proposition 1

If $x$ follows the linear consensus dynamics eq:linear_consensus, the disagreement $V$ satisfies: where $V(t^k)$ and $V^+(t^k)$ denote the values of $V$ before and after the occurrence of event $k$ respectively, $\lambda_2(t^k)$ is the second smallest Laplacian eigenvalue at the time instant $t^k$, and $\Delta t^{k,k+1}$ is the interval of time between $t^k$ and $t^{k+1}$.

Figures (1)

  • Figure 1: Stochastic process used to model the evolution of the number of agents $n(t^k)$. The process resembles a birth-death process with a minimum number of agents $n_m$ and a maximum number of agents $n_M$ such that $n(t^k)\in[n_m,n_M]$ for all $k\in\mathbb N$.

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Definition 1: Sampled Graph avella2018centrality
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Theorem 2
  • proof
  • Definition 2: Piecewise Lipschitz graphon
  • ...and 6 more