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Limit theorems for assortativity and clustering in null models for scale-free networks

Part of: Limit theorems Graph theory

Published online by Cambridge University Press:  03 December 2020

Remco van der Hofstad ,
Pim van der Hoorn ,
Nelly Litvak  and
Clara Stegehuis
Remco van der Hofstad*
Affiliation:
Eindhoven University of Technology
Pim van der Hoorn*
Affiliation:
Eindhoven University of Technology and Northeastern University
Nelly Litvak*
Affiliation:
Eindhoven University of Technology and University of Twente
Clara Stegehuis*
Affiliation:
University of Twente
*
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology.
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology.
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology.
***Postal address: Department of Electrical Engineering, Mathematics and Computer Science, University of Twente.
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Abstract

An important problem in modeling networks is how to generate a randomly sampled graph with given degrees. A popular model is the configuration model, a network with assigned degrees and random connections. The erased configuration model is obtained when self-loops and multiple edges in the configuration model are removed. We prove an upper bound for the number of such erased edges for regularly-varying degree distributions with infinite variance, and use this result to prove central limit theorems for Pearson’s correlation coefficient and the clustering coefficient in the erased configuration model. Our results explain the structural correlations in the erased configuration model and show that removing edges leads to different scaling of the clustering coefficient. We prove that for the rank-1 inhomogeneous random graph, another null model that creates scale-free simple networks, the results for Pearson’s correlation coefficient as well as for the clustering coefficient are similar to the results for the erased configuration model.

Keywords

Degree–degree correlationsclusteringconfiguration modellimit theorems

MSC classification

Primary: 05C80: Random graphs
Secondary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 4 , December 2020 , pp. 1035 - 1084
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Copyright
© Applied Probability Trust 2020

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