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

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. *

Dynamical properties of disjunctive Boolean networks

“…It would be interesting to compare the disjunctive network on D with the other Boolean networks on D. Firstly, we want to investigate when the disjunctive network has as many image/periodic/fixed points as possible. There are three main upper bounds on the image/periodic/fixed rank of a Boolean network with interaction graph D, that depend on three graph parameters reviewed in [15]. The image rank of a Boolean network f on D is at most 2 α 1 (D) [14], while its periodic rank is at most 2 αn(D) [14], and its fixed rank is at most 2 τ (D) (the famous feedback bound [33,4]).…”

“…Since disjunctive networks are the closest to being constant by Theorem 4, one might expect that they should minimise the image rank over all networks with a given interaction graph. This is true in the following extreme cases, where the minimum rank is equal to 1, 2, or 2 n [15]. However, this turns out not to be the case in general: [15, Theorem 5] gives a counter-example, where the minimum image rank is not achieved by the disjunctive network, but can be actually achieved by another monotone network.…”

Threshold-based Network Structural Dynamics

“…These examples include network generation models [7,30], community detection [33], "life-like" cellular automata [28], robot motion [26] and go all the way up to fundamental physics as a candidate model for space [31,32]. In view of this recent trend, a stream of work is devoted to the study of such dynamics per se, without a particular application in mind (e.g., [14]). Motivated by such a plethora of examples, we study the stabilization properties of protocols that affect solely the structure of networks.…”

Expansive automata networks

“…In other words, the interaction graph represents the underlying network of entities and their influences on one another. A major topic of interest is to determine how the interaction graph affects different properties of the network, such as the number of fixed points or images (see [4] for a review of known results on the influence of the interaction graph). In particular, a stream of work aims to design networks with a prescribed interaction graph and with a specific dynamical property, such as a being bijective [5], or having many fixed points [6], or converging towards a fixed point [1].…”