Distances in power-law random graphs

In many real-world networks, such as the Internet and social networks, power-law degree sequences have been observed. This means that, when the graph is large, the proportion of vertices with degree k is asymptotically proportional to k−τ , for some τ ≥ 1. These networks are often small worlds, which means that distances in these networks are small. We will study two random graph models, the configuration model and the preferential attachment model, which will have power-law degree sequences when the number of vertices tends to infinity. An overview is given of known results about distances in these graph models. Also some new results will be presented, among which a log log lower bound on the diameter of preferential attachment graphs with τ > 2.