Graph-Theoretic Concepts in Computer Science

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. 1 Complex Networks and Random Graphs: A Motivation In this survey, we discuss random graph models for complex networks, which are large and highly heterogeneous real-world graphs such as the Internet, the WorldWide Web, social networks, collaboration networks, citation networks, the neural network of the brain, etc. Such networks have received enormous attention in the past decades, partly because they appear in virtually all domains in science. This is also due to the fact that such networks, even though they arise in highly different fields in science and society, share some fundamental properties. Let us describe the two most important ones now. c © Springer International Publishing AG 2017 H.L. Bodlaender and G.J. Woeginger (Eds.): WG 2017, LNCS 10520, pp. 1–17, 2017. https://doi.org/10.1007/978-3-319-68705-6_1 2 R. van der Hofstad Scale-Free Phenomenon. The first, maybe quite surprising, fundamental property of many real-world networks is that the number of vertices with degree at least k decays slowly for large k. This implies that degrees are highly variable, and that, even though the average degree is not so large, there exist vertices with extremely high degree. Often, the tail of the empirical degree distribution seems to fall off as an inverse power of k. This is called a ‘power-law degree sequence’, and resulting graphs often go under the name ‘scale-free graphs’. It is visualized for the AS graph in Fig. 1, where the degree distribution of the Autonomous System (AS) graph is plotted on a log-log scale. The vertices of the AS graph correspond to groups of routers controlled by the same operator. Thus, we see a plot of log k → log nk, where nk is the number of vertices with degree k. When nk is proportional to an inverse power of k, i.e., when, for some normalizing constant cn and some exponent τ ,