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April 2021 Optimal subgraph structures in scale-free configuration models
Remco van der Hofstad, Johan S. H. van Leeuwaarden, Clara Stegehuis
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Ann. Appl. Probab. 31(2): 501-537 (April 2021). DOI: 10.1214/20-AAP1580

Abstract

Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced subgraph. We obtain these results by analyzing the configuration model with degree exponent τ(2,3) and introducing a novel class of optimization problems. For any given subgraph, the unique optimizer describes the degrees of the vertices that together span the subgraph. We find that subgraphs typically occur between vertices with specific degree ranges. In this way, we can count and characterize all subgraphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the erased configuration model.

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Remco van der Hofstad. Johan S. H. van Leeuwaarden. Clara Stegehuis. "Optimal subgraph structures in scale-free configuration models." Ann. Appl. Probab. 31 (2) 501 - 537, April 2021. https://doi.org/10.1214/20-AAP1580

Information

Received: 1 November 2017; Revised: 1 August 2019; Published: April 2021
First available in Project Euclid: 1 April 2021

Digital Object Identifier: 10.1214/20-AAP1580

Subjects:
Primary: 05C80, 05C82

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2021
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