On the Dynamics of Bounded-Degree Automata Networks

TL;DR

This paper investigates the possible dynamics of automata networks with communication graphs of bounded degree, revealing limitations and possibilities for fixed points and cycles, and establishing complexity bounds for recognizing such dynamics.

Contribution

It provides new theoretical results on the realizability of certain dynamics in bounded-degree automata networks and explores the complexity of recognizing these dynamics.

Findings

Impossible to realize Gray code maps with bounded degree.

Degree 2 networks can realize a fixed point and a cycle involving all configurations.

Certain simple cycles remain impossible to realize with bounded degree.

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.