On the influence of the interaction graph on a finite dynamical system

TL;DR

This paper explores how the structure of the interaction graph influences key properties of finite dynamical systems, providing bounds and classifications for their dynamics based on graph features and alphabet size.

Contribution

It introduces the concept of absolute minimum rank and offers the first meaningful bounds, along with a comprehensive survey of related results and open questions.

Findings

Minimum rank decreases with alphabet size.

Bounds on absolute minimum rank are established.

Classification of graphs with extreme rank values.

Abstract

A finite dynamical system (FDS) is a system of multivariate functions over a finite alphabet, that is typically 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. As such, a major problem is to determine the effect of the interaction graph on the dynamics of the FDS. In this paper, we are interested in three main properties of an FDS: the number of images (the so-called rank), the number of periodic points (the so-called periodic rank) and the number of fixed points. In particular, we investigate the minimum, average, and maximum number of images (or periodic points, or fixed points) of FDSs with a prescribed interaction graph and a given alphabet size; thus yielding nine quantities to study. The paper is split into two parts. The first part considers the minimum rank, for which we derive the first meaningful results known so far. In particular, we show that the minimum rank decreases with the alphabet size, thus yielding the definition of an absolute minimum rank. We obtain lower and upper bounds on this absolute minimum rank, and we give classification results for graphs with very low (or highest) rank. The second part is a comprehensive survey of the results obtained on the nine quantities described above. We not only give a review of known results, but we also give a list of relevant open questions.