On the Dynamics of Bounded-Degree Automata Networks

2025-11-14

AI Summary

This paper investigates the dynamical realizability problem for finite automata networks over bounded-degree communication graphs, focusing on constructing and characterizing canonical dynamics such as single-cycle behavior and Gray-code mappings. Methodologically, it integrates graph theory, finite dynamical systems theory, and isomorphism analysis, leveraging explicit cycle-structure constructions and complexity-theoretic reductions. The work establishes fundamental impossibility results: several key dynamical parameters—including the number of fixed points and network rank—are provably unrealizable under bounded-degree constraints. Moreover, it constructs, for the first time, a minimal realization exhibiting exactly one fixed point and a single cycle covering all remaining configurations. Collectively, these results precisely characterize the dynamical capability frontier of bounded-degree networks, establish tight computational complexity bounds (both upper and lower) for associated realizability decision problems, and provide foundational theoretical support for modeling distributed systems under topological constraints.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

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.

Problem

• Investigates dynamics realizability under bounded-degree communication graphs
• Explores constraints on fixed points and cycle structures in networks
• Analyzes complexity of recognizing bounded-degree automata network dynamics

Innovation

• Studied bounded-degree automata network dynamics
• Realized single fixed point with degree two
• Proved impossibility of Gray code bounded realization