Random graphs is a well-studied field of probability theory, and have proven very useful in a range of applications — modeling social networks, epidemics, and structures on the Internet to name a few. However, most random graphs are static in the sense that the network structure does not change over time. Furthermore, standard models also tend to consist of single-type objects. This puts restrictions on possible applications. The first part of this thesis concerns random graphs with a focus on dynamic and multi-type extensions of standard models. The second part of the thesis deals with random growth models. Random growth models are important objects in probability theory and, as the name suggests, models the random growth of some entity. Typical examples include infectious disease spread; how a liquid flows through a random medium; and tumor growth. The growth of these models, properly scaled by time, tends to be deterministic. The second theme of the thesis concerns the final shape of the growing entity for two standard random growth models.

In Paper I, we study a dynamic version of the famous Erdős-Rényi graph. The graph changes dynamically over time but still has the static Erdős-Rényi graph as its stationary distribution. In studying the dynamic graph we present two results. The first result concerns the time to stationarity, and the second concerns the time it takes for the graph to reach a certain number of edges. We also study the time until a large component emerges, as well as how it emerges.

In Paper II, we introduce and study an extension of the preferential attachment tree. The standard version is already dynamic, but its vertices are only allowed to be of one type. We introduce a multi-type analog of the preferential attachment tree and study its asymptotic degree distributions as well as its asymptotic composition.

Paper III concerns the configuration model — a random graph neither dynamic nor multi-type — and we break with the first theme of the thesis since no extensions are made to the model. Instead, we argue that the size of the largest component in the model does not depend on the tail of the degree distribution, but rather on the distribution over small degrees. This is quantified in some detail.

In Paper IV, we consider the frog model on Z^d and a two-type extension of it. For the one-type model, we show that the asymptotic shape does not depend on the initial set and the particle configuration there. For the two-type model, we show that the possibility of both types to coexist also does not depend on the initial sets and the particle configurations there.

Paper V is concerned with the predictability of the set of discovered sites generated by the first passage percolation model. First passage percolation has the property that the set of discovered sites, scaled properly by time, converges to some deterministic set as time grows. Typically, not much is known about this set, and to get an impression of it simulations are needed. Using simulated data we show that it is possible to use a neural network to adequately predict the shape, on this dataset, from some easily calculable properties of the passage times. The purpose of the paper is to give researchers a proof of concept of this method as wells as a new tool for quickly getting an impression of the shape.

In this article we study a dynamic Erdos-Renyi graph model, in which, independently for each vertex pair, edges appear and disappear according to a Markov on-off process.In studying the dynamic graph we present the following results. The first being on how long it takes for the graph to reach stationarity. We give an explicit expression for this time, as well as proving that this is the fastest time to reach stationarity among all strong stationary times.The main result concerns the time it takes for the dynamic graph to reach a certain number of edges. We give an explicit expression for the expected value of such a time, as well as study its asymptotic behavior. This can be used to determine how a large component emerges, is it through many edges being present or by an unlikely configuration of fewer edges? In the critical case, a large component emerges through an unlikely configuration of relatively few edges.

The size of the giant component in the configuration model, measured by the asymptotic fraction of vertices in the component, is given by a well-known expression involving the generating function of the degree distribution. In this note, we argue that the distribution over small degrees is more important for the size of the giant component than the precise distribution over very large degrees. In particular, the tail behavior of the degree distribution does not play the same crucial role for the size of the giant as it does for many other properties of the graph. Upper and lower bounds for the component size are derived for an arbitrary given distribution over small degrees d <= L and given expected degree, and numerical implementations show that these bounds are close already for small values of L. On the other hand, examples illustrate that, for a fixed degree tail, the component size can vary substantially depending on the distribution over small degrees.

Random graphs is a well-studied field of probability theory, and have proven very useful in a range of applications. However, most random graphs are \textit{static} in the sense that the network structure does not change over time; they also tend to consist of \textit{single-type} objects. This puts restrictions on possible applications. In this thesis we extend two standard models to a \textit{dynamic} and \textit{multi-type} setting, respectively.

In the first paper we study a dynamic version of the famous Erd\H{o}s-Rényi graph. The graph changes dynamically over time, but still has the static Erd\H{o}s-Rényi graph as its stationary distribution. In studying the dynamic graph we present two results. The first one concerns the time to stationarity, and the second one the time to reach a certain number of edges.

In the second paper we introduce and study an extension of the preferential attachment model. The standard preferential attachment model is already dynamic, but its vertices are only allowed to be of one type. We introduce a multi-type analogue of the preferential attachment model and study its asymptotic degree distributions as well as its asymptotic composition.