  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.
