Graph fibrations and symmetries of network dynamics

Dynamical systems with a network structure can display collective behaviour such as synchronisation. Golubitsky and Stewart observed that all the robustly synchronous dynamics of a network is contained in the dynamics of its quotient networks. DeVille and Lerman have recently shown that the original network and its quotients are related by graph fibrations and hence their dynamics are conjugate. This paper demonstrates the importance of self-fibrations of network graphs. Self-fibrations give rise to symmetries in the dynamics of a network. We show that every homogeneous network admits a lift with self-fibrations and that every robust synchrony in this lift is determined by the symmetries of its dynamics. These symmetries moreover impact the global dynamics of network systems and can be used to explain and predict generic scenarios for synchrony breaking. We also discuss networks with interior symmetries and nonhomogeneous networks.