Preprint 2025-24
Julio Aracena, Florian Bridoux, Maximilien Gadouleau, Pierre Guillon, Kevin Perrot, Adrien Richard, Guillaume Theyssier:
On the Dynamics of Bounded-Degree Automata Networks
Abstract:
Automata networks can be seen as bare finite dynamical systems, but their growing theory has shown the importance of the underlying communication graph of such networks. This paper tackles the question of what dynamics can be realized up to isomorphism if we suppose that the communication graph has bounded degree. We prove several negative results about parameters like the number of fixed points or the rank. We also show that we can realize with degree 2 a dynamics made of a single fixed point and a cycle gathering all other configurations. However, we leave open the embarrassingly simple question of whether a dynamics consisting of a single cycle can be realized with bounded degree, although we prove that it is impossible when the network become acyclic by suppressing one node, and that realizing precisely a Gray code map is impossible with bounded degree. Finally we give bounds on the complexity of the problem of recognizing such dynamics.
