Distances in the configuration model

An interesting feature of many real world networks is that they have short paths, meaning that the average distance scales as the logarithm of the size of the graph. This phenomenon is usually referred to as the small world property. In this talk I will discuss how distances behave in the well-studied configuration model. I will start with result for the undirected case, mostly due to work by van den Esker, van der Hofstad, Hooghiemstra and Van Mieghem. I will show how the behavior of the shortest path between two randomly sampled nodes changes with the existence of moments of the degree distribution. Then I will move to more recent work on distances in the directed configuration model, which is based on joint work with Mariana Olvera-Cravioto. I will show that here distances behave similarly to the undirected case but that it is the joint distribution of the in- and out-degree of a vertex that plays an important role. I will end with some result on universality of distances in graphs and some interesting open questions.