Characterization of Fundamental Networks

In the framework of coupled cell systems, a coupled cell network describes graphically the dynamical dependencies between individual dynamical systems, the cells. The fundamental network of a network reveals the hidden symmetries of that network. Subspaces defined by equalities of coordinates which are flow-invariant for any coupled cell system consistent with a network structure are called the network synchrony subspaces. Moreover, for every synchrony subspaces, each network admissible system restricted to that subspace is a dynamical systems consistent with a smaller network. The original network is then said to be a lift of the smaller network. We characterize networks such that: its fundamental network is a lift of the network; the network is a subnetwork of its fundamental network, and the network is a fundamental network. The size of cycles in a network and the distance of a cell to a cycle are two important properties concerning the description of the network architecture. In this paper, we relate these two architectural properties in a network and its fundamental network.