Evolution of synchrony under combination of coupled cell networks

A natural way of modelling large coupled cell networks is to combine smaller networks through binary network operations. In this paper, we consider several non-product binary operations on networks such as join and coalescence, and examine the evolution of the lattice of synchrony subspaces under these operations. Classification results are obtained for synchrony subspaces of the combined network, which clarify the relation between the lattice of synchrony subspaces of the combined network and its components. Yet, in the case when the initial networks have the same edge type, this classification only applies to those synchrony subspaces that are compatible with respect to the considered operation. Based on the classification results, we give examples to show how the lattice of synchrony subspaces of the combined network can be reconstructed using the initial ones. Also, we show how the classification results can be applied to analyse the evolutionary fitness of synchrony patterns.