The underlying mathematical model of many simulation models is graph dynamical systems (GDS). This dynamical system, its implementation, and analyses on each will be the focus of this paper. When using a simulation model to answer a research question, it is important to describe this underlying mathematical model in which we are operating for verification and validation. In this paper we discuss analyses commonly used in simulation models. These include sensitivity analyses and uncertainty quantification, which provide motivation for stability and structure-to-function research in GDS. We review various results in these areas, which contribute toward validation and computationally tractable analyses of our simulation model. We then present two new areas of research - stability of transient structure with respect to update order permutations, and an application of GDS in which a time-varying generalized cellular automata is implemented as a simulation model.