Update Sequence Stability in Graph Dynamical Systems

TL;DR

This paper investigates how the order of updates affects the stability of finite graph-based dynamical systems, introducing three notions of stability and analyzing their relation to graph structure.

Contribution

It introduces and compares three new notions of update sequence stability in graph dynamical systems, linking stability properties to graph structure.

Findings

Different stability notions lead to contrasting insights about the impact of update sequences.

The analysis reveals diverse relationships between stability and graph structural properties.

The work provides a comprehensive framework for understanding update sequence effects in graph dynamics.

Abstract

In this article, we study finite dynamical systems defined over graphs, where the functions are applied asynchronously. Our goal is to quantify and understand stability of the dynamics with respect to the update sequence, and to relate this to structural properties of the graph. We introduce and analyze three different notions of update sequence stability, each capturing different aspects of the dynamics. When compared to each other, these stability concepts yield vastly different conclusions regarding the relationship between stability and graph structure, painting a more complete picture of update sequence stability.