Characterization of Fundamental Networks

TL;DR

This paper explores the structural properties of coupled cell networks, focusing on their fundamental networks, symmetries, and synchrony subspaces, and how these relate to network architecture and cycle properties.

Contribution

It characterizes when a network is a lift or a subnetwork of its fundamental network, linking network architecture with cycle size and distance to cycles.

Findings

Fundamental networks can be characterized as lifts or subnetworks.

Network properties like cycle size relate to fundamental network structure.

Synchronization subspaces correspond to smaller networks within the original network.

Abstract

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.