Limit Set Reachability in Asynchronous Graph Dynamical Systems

Abstract

Using the framework of sequential dynamical systems (SDSs), a class of asynchronous graph dynamical systems, we show how the notions of reachability and stability can be negatively correlated. Specifically, we show that certain threshold SDSs exhibit update sequence instability: Over certain graph classes, there exist initial configurations from where exponentially many fixed points are reachable under different update sequences, i.e., the ω-limit set has size Θ(2^|V|). We establish this first for treewidth bounded graphs and then for random graphs in the G(n,p) model of Erdős-Rényi for a large range of p. We also show that this update sequence instability is not present in dense graphs, suggesting that sparsity and tree-like structure plays an important role in the stability of the system. These dynamical systems arise in applications such as functional gene annotation, where threshold SDSs are employed to predict gene functions through a fixed point computation, based on an initial state (prediction) and a nongeneric choice of update sequence. The results in this paper should be viewed as cautionary advice in the construction and application of such algorithms. This paper also provides a starting point for a study of update sequence stochastic SDSs.