A Symmetry-Based Classification of Synchrony in Tree Networks

TL;DR

This paper proves that exotic synchrony patterns do not occur in tree-structured networks and explores how the network's combinatorial structure influences the stability of synchronized states, especially in cherry configurations.

Contribution

It establishes that all synchrony patterns in tree networks are symmetry-induced and analyzes their stability, highlighting the role of leaves and local symmetries.

Findings

Exotic synchronies do not occur in tree networks.

Stability of synchrony is influenced by leaf configurations.

Local symmetries affect the stability of synchronized states.

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.