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A Symmetry-Based Classification of Synchrony in Tree Networks

Nicolas Brito, Miriam Manoel

TL;DR

It is proved that exotic synchronizations do not arise in tree-type networks, showing that every balanced coloring is a fixed-point coloration determined by graph automorphisms, and the importance of the role played by the leaves of a graph is identified.

Abstract

Coupled cell systems model interacting dynamical units and provide a natural framework for studying synchrony phenomena arising from collective behavior. Graph symmetries often induce such patterns, but certain networks exhibit additional synchronies not associated with automorphisms, commonly referred to as exotic synchronies. In undirected asymmetric graphs, any synchrony, if present, must be non-symmetry-induced, and determining when such exotic patterns occur remains a challenging structural problem. In this work, we address this question for networks whose underlying coupling graph is a tree, a class of graphs that naturally models hierarchical interactions among elements. We prove that exotic synchronizations do not arise in tree-type networks, showing that every balanced coloring is a fixed-point coloration determined by graph automorphisms. Furthermore, we identify the importance of the role played by the leaves of a graph in this context. Beyond existence results, we investigate the dynamical consequences of these structures by analyzing the linear stability of equilibria and the Lyapunov stability of synchrony subspaces for admissible vector fields defined on tree networks. Particular attention is devoted to cherry- type configurations, where local symmetries generated by leaves attached to a common vertex influence the stability properties of the associated synchronous states, thereby clarifying how the combinatorial architecture of trees constrains both the emergence and the stability of synchrony.

A Symmetry-Based Classification of Synchrony in Tree Networks

TL;DR

It is proved that exotic synchronizations do not arise in tree-type networks, showing that every balanced coloring is a fixed-point coloration determined by graph automorphisms, and the importance of the role played by the leaves of a graph is identified.

Abstract

Coupled cell systems model interacting dynamical units and provide a natural framework for studying synchrony phenomena arising from collective behavior. Graph symmetries often induce such patterns, but certain networks exhibit additional synchronies not associated with automorphisms, commonly referred to as exotic synchronies. In undirected asymmetric graphs, any synchrony, if present, must be non-symmetry-induced, and determining when such exotic patterns occur remains a challenging structural problem. In this work, we address this question for networks whose underlying coupling graph is a tree, a class of graphs that naturally models hierarchical interactions among elements. We prove that exotic synchronizations do not arise in tree-type networks, showing that every balanced coloring is a fixed-point coloration determined by graph automorphisms. Furthermore, we identify the importance of the role played by the leaves of a graph in this context. Beyond existence results, we investigate the dynamical consequences of these structures by analyzing the linear stability of equilibria and the Lyapunov stability of synchrony subspaces for admissible vector fields defined on tree networks. Particular attention is devoted to cherry- type configurations, where local symmetries generated by leaves attached to a common vertex influence the stability properties of the associated synchronous states, thereby clarifying how the combinatorial architecture of trees constrains both the emergence and the stability of synchrony.
Paper Structure (10 sections, 3 theorems, 37 equations, 7 figures)

This paper contains 10 sections, 3 theorems, 37 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Coupled Cell Systems
  3. Synchrony Patterns in Tree-type Networks
  4. Properties of Balanced Relations
  5. Fixed-point coloration on trees
  6. Quotient Networks of Trees
  7. Admissible Vector Fields on Trees
  8. Linear Stability
  9. Effects of Cherry Structures
  10. Lyapunov Stability of Cherry Synchrony

Key Result

theorem 1

Every balanced coloring on a tree $G$ is a fixed-point coloration.

Figures (7)

  • Figure 1: Example of an asymmetric tree of order $7$.
  • Figure 2: Example of a balanced coloring on a 10-cell undirected tree (left) and its quotient network (right).
  • Figure 3: Exotic patterns of synchrony on a 2-vertex network (left) and on a 3-vertex network (right).
  • Figure 4: 2-balanced coloring (left) and 3-balanced coloring (right) in the Frucht graph.
  • Figure 5: A schematic representation of the coupling process in the proof of Proposition \ref{['prop:reflected']}.
  • ...and 2 more figures

Theorems & Definitions (20)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • theorem 1
  • proof
  • proof
  • ...and 10 more