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

Author

Listed:
  • Caron, François
  • Panero, Francesca
  • Rousseau, Judith

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.

Suggested Citation

  • Caron, François & Panero, Francesca & Rousseau, Judith, 2023. "On sparsity, power-law, and clustering properties of graphex processes," LSE Research Online Documents on Economics 119794, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:119794
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    File URL: http://eprints.lse.ac.uk/119794/
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    References listed on IDEAS

    as
    1. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
    2. François Caron & Emily B. Fox, 2017. "Sparse graphs using exchangeable random measures," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1295-1366, November.
    3. Adrien Todeschini & Xenia Miscouridou & François Caron, 2020. "Exchangeable random measures for sparse and modular graphs with overlapping communities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(2), pages 487-520, April.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    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;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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