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Cluster Assignment in Multi-Agent Systems

Miel Sharf, Daniel Zelazo

TL;DR

The paper addresses designing network Topologies to enforce prescribed cluster partitions in homogeneous diffusively coupled MAS by leveraging graph automorphisms. It introduces OS-type graphs, where the automorphism group action has orbit sizes equal to the desired cluster sizes, and provides tight lower and upper bounds on the number of edges, along with a constructive method to realize such graphs. The main contributions include formalizing OS$(r_1,\ldots,r_k)$ graphs, deriving edge-bounds $m$ and $M$, and presenting an algorithmic synthesis procedure with numerical examples to demonstrate cluster enforcement. This symmetry-based synthesis enables principled design of distributed clustering in MAS while minimizing inter-agent connections, with broad implications for scalable coordination in engineering and networked systems.

Abstract

We study cluster assignment in multi-agent networks. We consider homogeneous diffusive networks, and focus on design of the graph that ensures the system will converge to a prescribed cluster configuration, i.e., specifying the number of clusters and agents within each cluster. Leveraging recent results from cluster synthesis, we show that it is possible to design an oriented graph such that the action of the automorphism group of the graph has orbits of predetermined sizes, guaranteeing that the network will converge to the prescribed cluster configuration. We provide upper and lower bounds on the number of edges that are needed to construct these graphs along with a constructive approach for generating these graphs. We support our analysis with some numerical examples.

Cluster Assignment in Multi-Agent Systems

TL;DR

The paper addresses designing network Topologies to enforce prescribed cluster partitions in homogeneous diffusively coupled MAS by leveraging graph automorphisms. It introduces OS-type graphs, where the automorphism group action has orbit sizes equal to the desired cluster sizes, and provides tight lower and upper bounds on the number of edges, along with a constructive method to realize such graphs. The main contributions include formalizing OS graphs, deriving edge-bounds and , and presenting an algorithmic synthesis procedure with numerical examples to demonstrate cluster enforcement. This symmetry-based synthesis enables principled design of distributed clustering in MAS while minimizing inter-agent connections, with broad implications for scalable coordination in engineering and networked systems.

Abstract

We study cluster assignment in multi-agent networks. We consider homogeneous diffusive networks, and focus on design of the graph that ensures the system will converge to a prescribed cluster configuration, i.e., specifying the number of clusters and agents within each cluster. Leveraging recent results from cluster synthesis, we show that it is possible to design an oriented graph such that the action of the automorphism group of the graph has orbits of predetermined sizes, guaranteeing that the network will converge to the prescribed cluster configuration. We provide upper and lower bounds on the number of edges that are needed to construct these graphs along with a constructive approach for generating these graphs. We support our analysis with some numerical examples.
Paper Structure (6 sections, 6 theorems, 5 equations, 3 figures, 1 algorithm)

This paper contains 6 sections, 6 theorems, 5 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Symmetries in Networked Systems
  3. Diffusively Coupled Networks
  4. Group Theory, Graph Automorphisms, and Symmetric MAS
  5. Cluster Assignment in MAS
  6. Conclusions

Key Result

Proposition 1

Let $\mathbb{G}$ be a group of functions $X\to X$. Then the set $X$ can be written as the union of disjoint orbits.

Figures (3)

  • Figure 1: A diffusively coupled network.
  • Figure 2: First example of graphs solving the cluster synthesis problem, achieved by running Algorithm \ref{['alg.BuildingGraphs']}.
  • Figure 3: First example of graphs solving the cluster synthesis problem, achieved by running Algorithm \ref{['alg.BuildingGraphs']}.

Theorems & Definitions (18)

  • Definition 1: Sharf2019b
  • Definition 2
  • Definition 3
  • Proposition 1
  • Definition 4
  • Theorem 1: Sharf2019b
  • Definition 5
  • Theorem 2
  • proof
  • Remark 1
  • ...and 8 more