Phase Space Structures of k-threshold Sequential Dynamical Systems

TL;DR

This paper investigates the phase space structure of k-threshold sequential dynamical systems on graphs, revealing how component connectivity relates to thresholds, update sequences, and fixed-point reachability, with bounds depending on graph size.

Contribution

It provides novel insights into the connected component structure of phase space in k-threshold SDS, linking it to graph properties and update sequences.

Findings

Relations between component structure and threshold values

Fixed-point reachability from garden of eden configurations

Upper bounds on path lengths depending on graph size

Abstract

Sequential dynamical systems (SDS) are used to model a wide range of processes occurring on graphs or networks. The dynamics of such discrete dynamical systems is completely encoded by their phase space, a directed graph whose vertices and edges represent all possible system configurations and transitions between configurations respectively. Direct calculation of the phase space is in most cases a computationally demanding task. However, for some classes of SDS one can extract information on the connected component structure of phase space from the constituent elements of the SDS, such as its base graph and vertex functions. We present a number of novel results about the connected component structure of the phase space for k-threshold dynamical system with binary state spaces. We establish relations between the structure of the components, the threshold value, and the update sequence. Also fixed-point reachability from garden of eden configurations is investigated and upper bounds for the length of paths in the phase space are shown to only depend on the size of the vertex set of the base graph.