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.