Engineering mesoscale structures with distinct dynamical implications

The dynamics of networks of interacting systems depends intricately on the interaction topology. When the dynamics is explored, generally the whole topology has to be considered. However, here we show that there are certain mesoscale subgraphs that have precise and distinct consequences for the system-level dynamics. In particular, if mesoscale symmetries are present then eigenvectors of the Jacobian localize on the symmetric subgraph and the corresponding eigenvalues become insensitive to the topology outside the subgraph. Hence, dynamical instabilities associated with these eigenvalues can be analysed without considering the topology of the embedding network. While such instabilities are thus generated entirely in small subgraphs, they generally do not remain confined to the subgraph once the instability sets in and thus have system-level consequences. Here we illustrate the analytical investigation of such instabilities in an ecological metapopulation model consisting of a network of delay-coupled delay oscillators.