# Characterization of Fundamental Networks

#### Preprint

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

Manuela Aguiar

Ana Dias

Pedro Soares

### Publication

Year of publication: 2017