Enumerating ODE Equivalent Homogeneous Networks

TL;DR

This paper introduces a graph-theoretic criterion for ODE equivalence in homogeneous coupled networks, simplifying proofs and enabling enumeration of such networks up to equivalence without linear algebra.

Contribution

It provides a new, purely graph-theoretic criterion for ODE equivalence and a formula for counting minimal homogeneous coupled networks up to this equivalence.

Findings

A simple criterion for ODE equivalence in homogeneous networks.

A proof of a key theorem from Aquiar and Dias using this criterion.

A formula for enumerating networks up to ODE equivalence.

Abstract

We give an simple criterion for ODE equivalence in identical edge homogeneous coupled cell networks. This allows us to give a simple proof of Theorem 10.3 of Aquiar and Dias "Minimal Coupled Cell Networks", which characterizes minimal identical edge homogeneous coupled cell networks. Using our criterion we give a formula for counting homogeneous coupled cell networks up to ODE equivalence. Our criterion is purely graph theoretic and makes no explicit use of linear algebra.