Counting graphs and null models of complex networks:Configuration model and extensions

Due to its ease of use, as well as its enormous flexibility in its degree structure, the configuration model has become the network model of choice in many disciplines. It has the wonderful property, that, conditioned on being simple, it is a uniform random graph with the prescribed degrees. This is a beautiful example of a general technique called the probabilistic method that was pioneered by Erdős. It allows us to count rather precisely how many graphs there are with various degree structures. As a result, the configuration model is often used as a null model in network theory, so as to compare real-world network data to. When the degrees are sufficiently light-tailed, the asymptotic probability of simplicity for the configuration model can be explicitly computed. Unfortunately, when the degrees vary rather extensively and vertices with very high degrees are present, this method fails. Since such degree sequences are frequently reported in empirical work, this is a major caveat in network theory. In this survey, we discuss recent results for the configuration model, including asymptotic results for typical distances in the graph, asymptotics for the number of self-loops and multiple edges in the finite-variance case. We also discuss a possible fix to the problem of non-simplicity, and what the effect of this fix is on several graph statistics. Further, we discuss a generalization of the configuration model that allows for the inclusion of community structures. This model removes the flaw of the locally tree-like nature of the configuration model, and gives a much improved fit to real-world networks.