Dynamical Equivalence of Network Architecture for Coupled Dynamical Systems I: Asymmetric Inputs

We give a simple necessary and sufficient condition for the dynamical equivalence of two coupled cell networks with different network architectures. The results are applicable to both continuous and discrete dynamical systems and are framed in terms of what we term input and output equivalence. We also give an algorithm that allows explicit construction of the cells in a system with a given network architecture in terms of the cells from an equivalent system with different network architecture. Details of proofs are provided for the case of cells with asymmetric inputs — details for the case of symmetric inputs are provided in a companion paper.