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. %of dynamics that can be realized with bounded degree of communication. We also give bounds on the complexity of the problem of recognizing such dynamics. However, we leave open the embarrassingly simple question of whether a dynamics consisting of a single cycle can be realized with bounded degree.

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