Random graphs: From static to dynamic

Many empirical studies on real-life networks show that many networks are small worlds, meaning that typical distances in these networks are small, and many of them have power-law degree sequences, meaning that the number of nodes with degree k falls off as kˆ (-τ) for some exponent τ>1. These networks are modeled by means of scale-free random graphs. One way to construct such a random graph is to start with a fixed number of nodes and randomly add edges between pairs of nodes. Using a growth model is a second way to construct a random graph. In such a model one starts with a given graph, and at each discrete time step a new node is added to the graph and the node is connected to some of the old nodes, where nodes with a high number of edges are preferred (preferential attachment). In this thesis two types of random graphs are considered: static random graphs and dynamic random graphs. A static random graph aims to describe a network and its topology at a given time instant, and a dynamical random graph aims to explain how the network came to be as it is. In this thesis two static random graphs are studied which produce power-law degree sequences: 'the configuration model' and 'the inhomogeneous random graph'. Two dynamic random graphs are introduced: 'the preferential attachment model with random initial degrees' and 'the geometric preferential attachment model with fitness'. In this thesis the degree sequence, the typical distance and the diameter for each of the models is considered, which are influenced by the power-law exponent τ. If τ>3, then each node in the graph has the same kind of neighborhood and the typical distance is proportional to log(n) if the graph consists of n nodes. If τ∈(2,3), then nodes with a high degree will appear and the typical distance is proportional to loglog(n). If τ∈(1,2), then the graph has a star-shaped structure and the distance is bounded by some constant