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On sparsity, power-law, and clustering properties of graphex processes

Part of: Limit theorems Graph theory Stochastic processes

Published online by Cambridge University Press:  16 June 2023

François Caron ,
Francesca Panero  and
Judith Rousseau
François Caron*
Affiliation:
University of Oxford
Francesca Panero*
Affiliation:
London School of Economics and Political Science
Judith Rousseau*
Affiliation:
University of Oxford
*
*Postal address: Department of Statistics, 24-29 St Giles’, Oxford OX1 3LB, United Kingdom.
***Postal address: Department of Statistics, 69 Aldwych, London WC2B 4RR, United Kingdom. Email address: f.panero@lse.ac.uk
*Postal address: Department of Statistics, 24-29 St Giles’, Oxford OX1 3LB, United Kingdom.
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Abstract

This paper investigates properties of the class of graphs based on exchangeable point processes. We provide asymptotic expressions for the number of edges, number of nodes, and degree distributions, identifying four regimes: (i) a dense regime, (ii) a sparse, almost dense regime, (iii) a sparse regime with power-law behaviour, and (iv) an almost extremely sparse regime. We show that, under mild assumptions, both the global and local clustering coefficients converge to constants which may or may not be the same. We also derive a central limit theorem for subgraph counts and for the number of nodes. Finally, we propose a class of models within this framework where one can separately control the latent structure and the global sparsity/power-law properties of the graph.

Keywords

NetworkssparsityPoisson processescommunity structurepower lawgeneralised graphontransitivitysubgraph counts

MSC classification

Primary: 05C80: Random graphs
Secondary: 60F15: Strong theorems 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 4 , December 2023 , pp. 1211 - 1253
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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Footnotes

‡

This article was originally published without a data access statement. The data access information has been added and a correction notice prepared. All versions of the article have been updated.

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