Quotients and lifts of symmetric directed graphs

TL;DR

This paper characterizes quotient and lift graphs of symmetric directed graphs related to balanced equivalence relations, with applications to coupled cell systems and network synchrony.

Contribution

It provides a detailed description of how quotient and lift graphs preserve symmetry and balance, extending the understanding of graph symmetries in dynamical systems.

Findings

Characterization of symmetric quotient graphs

Conditions for lifts to be symmetric

Application to coupled cell network synchrony

Abstract

Given a directed graph, an equivalence relation on the graph vertex set is said to be balanced if, for every two vertices in the same equivalence class, the number of directed edges from vertices of each equivalence class directed to each of the two vertices is the same. In this paper we describe the quotient and lift graphs of symmetric directed graphs associated with balanced equivalence relations on the associated vertex sets. In particular, we characterize the quotients and lifts which are also symmetric. We end with an application of these results to gradient and Hamiltonian coupled cell systems, in the context of the coupled cell network formalism of Golubitsky, Stewart and Torok(Patterns of synchrony in coupled cell networks with multiple arrows. {SIAM Journal of Applied Dynamical Systems, 4 (1) (2005) 78-100).