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

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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.

Original languageEnglish
Title of host publicationGraph-Theoretic Concepts in Computer Science
Subtitle of host publication43rd International Workshop, WG 2017, Eindhoven, The Netherlands, June 21-23, 2017, Revised Selected Papers
EditorsH.L. Bodleander, G.J. Woeginger
Place of PublicationDordrecht
Number of pages17
ISBN (Electronic)978-3-319-68704-9
ISBN (Print)978-3-319-68704-9
Publication statusPublished - 2017
Event43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017) - Kapellerput, Eindhoven, Netherlands
Duration: 21 Jul 201723 Jul 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10520 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Workshop43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017)
Abbreviated titleWG 2017
Internet address

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