Mesoscale symmetries explain dynamical equivalence of food webs

A goal of complex system research is to identify the dynamical implications of network structure. While early results focused mainly on local or global structural properties, there is now growing interest in mesoscale structures that comprise more than one node but not the whole network. A central challenge is to discover under what conditions the occurrence of a specific mesoscale motif already allows conclusions on the dynamics of a network as a whole. In this paper, we investigate the dynamics of ecological food webs, complex heterogeneous networks of interacting populations. Generalizing the results of MacArthur and Sánchez-García (2009 Phys. Rev. E 80 26117), we show that certain mesoscale symmetries imply the existence of localized dynamical modes. If these modes are unstable the occurrence of the corresponding mesoscale motif implies dynamical instability regardless of the structure of the embedding network. In contrast, if the mode is stable it means that the symmetry can be exploited to reduce the number of nodes in the model, without changing the dynamics of the system. This result explains a previously observed dynamical equivalence between food webs containing a different number of species. (paper)