Maximum Rank and Asymptotic Rank of Finite Dynamical Systems

TL;DR

This paper characterizes the maximum and average ranks of finite dynamical systems based on their interaction graphs, extending results to non-Boolean alphabets and various update schedules, providing insights into their structural complexity.

Contribution

It determines the maximum and average ranks of finite dynamical systems for any alphabet size and interaction graph, including Boolean cases and different update schedules.

Findings

Maximum rank and periodic rank are characterized for systems over any alphabet.

Average rank approaches maximum as alphabet size grows.

Results extend to various update schedules beyond parallel updates.

Abstract

A finite dynamical system is a system of multivariate functions over a finite alphabet used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. The rank of a finite dynamical system is the cardinality of its image; the periodic rank is the number of its periodic points. In this paper, we determine the maximum rank and the maximum periodic rank of a finite dynamical system with a given interaction graph over any non-Boolean alphabet. We also obtain a similar result for Boolean finite dynamical systems (also known as Boolean networks) whose interaction graphs are contained in a given digraph. We then prove that the average rank is relatively close (as the size of the alphabet is large) to the maximum. The results mentioned above only deal with the parallel update schedule. We finally determine the maximum rank over all block-sequential update schedules and the supremum periodic rank over all complete update schedules.