Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-12T02:32:00.273Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes Graph theory Limit theorems

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: [email protected]
*Postal address: Department of Statistics, 24-29 St Giles’, Oxford OX1 3LB, United Kingdom.
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Aldous, D. J. (1981). Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11, 581–598.CrossRefGoogle Scholar
Ayed, F., Lee, J. and Caron, F. (2019). Beyond the Chinese restaurant and Pitman–Yor processes: statistical models with double power-law behavior. Proc. Machine Learning Res. 97, pp. 395–404.Google Scholar
Ayed, F., Lee, J. and Caron, F. (2020). The normal-generalised gamma-Pareto process: a novel pure-jump Lévy process with flexible tail and jump-activity properties. Preprint. Available at https://arxiv.org/abs/2006.10968.Google Scholar
Bickel, P. J. and Chen, A. (2009). A nonparametric view of network models and Newman–Girvan and other modularities. Proc. Nat. Acad. Sci. USA 106, 21068–21073.Google Scholar
Bickel, P. J., Chen, A. and Levina, E. (2011). The method of moments and degree distributions for network models. Ann. Statist. 39, 2280–2301.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3–122.Google Scholar
Bollobás, B. and Riordan, O. (2009). Metrics for sparse graphs. In Surveys in Combinatorics 2009, eds S. Huczynska, J. Mitchell and C. Roney-Dougal, Cambridge University Press, pp. 211–288.Google Scholar
Borgs, C., Chayes, J. T., Cohn, H. and Holden, N. (2018). Sparse exchangeable graphs and their limits via graphon processes. J. Machine Learning Res. 18, 1–71.Google Scholar
Borgs, C., Chayes, J. T., Cohn, H. and Veitch, V. (2019). Sampling perspectives on sparse exchangeable graphs. Ann. Prob. 47, 2754–2800.CrossRefGoogle Scholar
Borgs, C., Chayes, J. T., Dhara, S. and Sen, S. (2019). Limits of sparse configuration models and beyond: graphexes and multi-graphexes. Preprint. Available at https://arxiv.org/abs/1907.01605.Google Scholar
Caron, F. and Fox, E. (2017). Sparse graphs using exchangeable random measures. J. R. Statist. Soc. B [Statist. Methodology] 79, 1295–1366.CrossRefGoogle ScholarPubMed
Caron, F., Panero, F. and Rousseau, J. (2022). On sparsity, power-law, and clustering properties of graphs based of graphex processes: supplementary material. Tech. Rep., University of Oxford.Google Scholar
Chatterjee, S. (2015). Matrix estimation by universal singular value thresholding. Ann. Statist. 43, 177–214.Google Scholar
Diaconis, P. and Janson, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. Appl. 28, 33–61.Google Scholar
Gao, C., Lu, Y. and Zhou, H. (2015). Rate-optimal graphon estimation. Ann. Statist. 43, 2624–2652.Google Scholar
Gnedin, A., Hansen, B. and Pitman, J. (2007). Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws. Prob. Surveys 4, 146–171.CrossRefGoogle Scholar
Herlau, T., Schmidt, M. N. and Mørup, M. (2016). Completely random measures for modelling block-structured sparse networks. In NIPS’16: Proceedings of the 30th International Conference on Neural Information Processing Systems, Association for Computing Machinery, New York, pp. 4267–4275.Google Scholar
Hoover, D. N. (1979). Relations on probability spaces and arrays of random variables. Preprint, Institute for Advanced Study, Princeton, NJ.Google Scholar
Janson, S. (2016). Graphons and cut metric on sigma-finite measure spaces. Preprint. Available at https://arxiv.org/abs/1608.01833.Google Scholar
Janson, S. (2017). On convergence for graphexes. Preprint. Available at https://arxiv.org/abs/1702.06389.Google Scholar
Kallenberg, O. (1990). Exchangeable random measures in the plane. J. Theoret. Prob. 3, 81–136.Google Scholar
Kolaczyk, E. D. (2009). Statistical Analysis of Network Data. Springer, New York.Google Scholar
Last, G., Peccati, G. and Schulte, M. (2016). Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Prob. Theory Relat. Fields 165, 667–723.Google Scholar
Latouche, P. and Robin, S. (2016). Variational Bayes model averaging for graphon functions and motif frequencies inference in W-graph models. Statist. Comput. 26, 1173–1185.Google Scholar
Lloyd, J., Orbanz, P., Ghahramani, Z. and Roy, D. (2012). Random function priors for exchangeable arrays with applications to graphs and relational data. In NIPS’12: Proceedings of the 25th International Conference on Neural Information Processing Systems, Association for Computing Machinery, New York, pp. 998–1006.Google Scholar
Loève, M. (1977). Probability Theory I, 4th edn. Springer, New York.Google Scholar
Lovász, L. and Szegedy, B. (2006). Limits of dense graph sequences. J. Combinatorial Theory B 96, 933–957.Google Scholar
Naulet, Z., Sharma, E., Veitch, V. and Roy, D. M. (2017). An estimator for the tail-index of graphex processes. Preprint. Available at https://arxiv.org/abs/1712.01745.Google Scholar
Newman, M. E. J. (2010). Networks: An Introduction. Oxford University Press.Google Scholar
Nowicki, K. and Snijders, T. (2001). Estimation and prediction for stochastic blockstructures. J. Amer. Statist. Assoc. 96, 1077–1087.Google Scholar
Orbanz, P. and Roy, D. M. (2015). Bayesian models of graphs, arrays and other exchangeable random structures. IEEE Trans. Pattern Anal. Machine Intellig. 37, 437–461.CrossRefGoogle ScholarPubMed
Palla, G., Lovász, L. and Vicsek, T. (2010). Multifractal network generator. Proc. Nat. Acad. Sci. USA 107, 7640–7645.Google Scholar
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
Reitzner, M. and Schulte, M. (2013). Central limit theorems for U-statistics of Poisson point processes. Ann. Prob. 41, 3879–3909.Google Scholar
Resnick, S. (1987). Extreme Values, Point Processes and Regular Variation. Springer, New York.Google Scholar
Todeschini, A., Miscouridou, X. and Caron, F. (2020). Exchangeable random measures for sparse and modular graphs with overlapping communities. J. R. Statist. Soc. B [Statist. Methodology] 82, 487–520.Google Scholar
Veitch, V. and Roy, D. M. (2015). The class of random graphs arising from exchangeable random measures. Preprint. Available at https://arxiv.org/abs/1512.03099.Google Scholar
Veitch, V. and Roy, D. M. (2019). Sampling and estimation for (sparse) exchangeable graphs. Ann. Statist. 47, 3274–3299.Google Scholar
Willmot, G. E. (1990). Asymptotic tail behaviour of Poisson mixtures by applications. Adv. Appl. Prob. 22, 147–159.Google Scholar
Wolfe, P. J. and Olhede, S. C. (2013). Nonparametric graphon estimation. Preprint. Available at https://arxiv.org/abs/1309.5936.Google Scholar
Supplementary material: File

Caron et al. supplementary material

Caron et al. supplementary material 1

Download Caron et al. supplementary material(File)
File 87.6 KB
Supplementary material: PDF

Caron et al. supplementary material

Caron et al. supplementary material 2

Download Caron et al. supplementary material(PDF)
PDF 451.3 KB