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

François Caron, Francesca Panero and Judith Rousseau

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

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: community structure; generalised graphon; Networks; Poisson processes; power law; sparsity; subgraph counts; transitivity; EPSRC and MRC Centre for Doctoral Training in Statistical Science (grant code EP/L016710/1; European Union’s Horizon 2020 research and innovation programme (grant agreement no. 834175 (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Pages: 43 pages
Date: 2023-12-01
New Economics Papers: this item is included in nep-ecm and nep-net
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Published in Advances in Applied Probability, 1, December, 2023, 55(4), pp. 1211 - 1253. ISSN: 0001-8678

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