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.