Network architecture and spatio-temporally symmetric dynamics

We examine the relation between the structure of a network and the spatio-temporally symmetric periodic dynamics it can support. For solutions in which no cell is stationary, we show that only networks in which all cells interact with each other, or which contain a single group of interacting cells which drive the remainder of the network can exhibit such dynamics robustly. These characteristics of network architecture are not captured by the typical statistical quantities used to describe network structure. We illustrate the existence of spatio-temporally periodic solutions through a direct construction using ideas from coupled cell theory and the theory of weakly coupled oscillators, and show that these solutions can be stable in a very large region of parameter space. While we consider only a special type of network behavior, these ideas extend to more general architectures and dynamics.