Minimal coupled cell networks

It is known that non-isomorphic coupled cell networks can have equivalent dynamics. Such networks are said to be ODE-equivalent and are related by a linear algebra condition involving their graph adjacency matrices. A network in an ODE-equivalence class is said to be minimal if it has a minimum number of edges. When studying a given network in an ODE-class it can be of great value to study instead a minimal network in that class. Here we characterize the minimal networks of an ODE-equivalence class-the canonical normal forms of the ODE-class. Moreover, we present an algorithm that computes the canonical normal forms for a given ODE-class. This goes through the calculation of vectors with minimum length contained in a cone of a lattice described in terms of the adjacency matrices of any network in the ODE-class.