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Respondent-Driven Sampling on Sparse Erdös-Rényi Graphs

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Abstract

We study the exploration of an Erdös-Rényi random graph by a respondent-driven sampling method, where discovered vertices reveal their neighbors. Some of them receive coupons to reveal in their turn their own neighborhood. This leads to the study of a Markov chain on the random graph that we study. For sparse Erdös-Rényi graphs of large sizes, this process correctly renormalized converges to the solution of a deterministic curve, solution of a system of ODEs absorbed on the abscissa axis. The associated fluctuation process is also studied, providing a functional central limit theorem, with a Gaussian limiting process. Simulations and numerical computation illustrate the study.

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References

  1. Abbe, E.: Community detection and stochastic block models. Foundations and Trends® in Communications and Information Theory 14(1–2), 1–162 (2018)

  2. Ball, F., Britton, T., Larédo, C., Pardoux, E., Sirl, D., Tran, V.: Stochastic Epidemic Models with Inference. Springer, MathBiosciences (2019)

    Google Scholar 

  3. Billingsley, P.: Convergence of Probability Measures. John Wiley and Sons, New York (1968)

    MATH  Google Scholar 

  4. Billingsley, P.: Probability and Measure, 3rd edn. John Wiley and Sons, New York (1995)

    MATH  Google Scholar 

  5. Bollobás, B.: Random Graphs. Cambridge University Press, 2 edition (2001)

  6. Bollobás, B., Riordan, O.: Asymptotic normality of the size of the giant component via a random walk. J. Combinatorial Theory Serie B 102(1), 53–61 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clémençon, S., Arazoza, H.D., Rossi, F., Tran, V.: A statistical network analysis of the HIV/AIDS epidemics in Cuba. Socl. Net. Anal. Mining 5, Art. 58 (2015)

  8. Condamine-Ducreux, I., Tran, V.C., Dumont, C., Cousien, A., Barin, F., Stefic, K., Deuffic-Burban, S., Chollet, A., Feuillet, P., Jangal, C., Le Breton, M., Brodsky, G., Dhersin, J.S., Yazdanpanah, Y., Jauffret-Roustide, M.: Social supports and networks linked to HCV and HIV exposure among people who inject drugs in Paris and the Paris suburbs. (2023) Submitted

  9. Cousien, A., Tran, V., Deuffic-Burban, S., Jauffret-Roustide, M., Dhersin, J., Yazdanpanah, Y.: Hepatitis C treatment as prevention of viral transmission and level-related morbidity in persons who inject drugs. Hepatology 63(4), 1090–1101 (2016)

    Article  Google Scholar 

  10. Cousien, A., Tran, V., Jauffret-Roustide, M., Dhersin, J., Deuffic-Burban, S., Yazdanpanah, Y.: Dynamic modelling of HCV transmission among people who inject drugs: a methodological review. J. Viral Hepatitis 22(3), 213–229 (2015)

    Article  Google Scholar 

  11. Crawford, F.W., Wu, J., Heimer, R.: Hidden population size estimation from respondent-driven sampling: a network approach. Journal of the American Statistical Association 113(522), 755–766 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley Series in Probability and Statistics John Wiley & Sons Inc., New York (1986)

    Book  MATH  Google Scholar 

  13. der Hofstad, R.V.: Random graphs and complex networks. Volume 1 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2017)

  14. Enriquez, N., Faraud, G., Ménard, L.: Limiting shape of the depth first search tree in an Erdős-Rényi graph. Random Structures & Algorithms 56(2), 501–516 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin (1987)

    Book  MATH  Google Scholar 

  16. Frost, D.M., Parsons, J.T., Nanin, J.E.: Stigma, concealment and symptoms of depression as explanations for sexually transmitted infections among gay men. J. Health Psychol. 12(4), 636–640 (2007)

    Article  Google Scholar 

  17. Gile, K.J.: Improved inference for respondent-driven sampling data with application to HIV prevalence estimation. J. Amer. Statist. Assoc. 106(493), 135–146 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Goodman, L.A.: Snowball sampling. Ann. Math. Statist. 32(1), 148–170 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  19. Handcock, M., Gile, K., Mar, C.: Estimating hidden population size using respondent-driven sampling data. Electron. J. Stat. 8(1), 1491–1521 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Heckathorn, D.: Respondent-driven sampling: a new approach to the study of hidden populations. Soc. Prob. 44(2), 174–199 (1997)

    Article  Google Scholar 

  21. Hellard, M., Rolls, D.A., Sacks-Davis, R., Robins, G., Pattison, P., Higgs, P., Aitken, C., McBryde, E.: The impact of injecting networks on hepatitis C transmission and treatment in people who inject drugs. Hepatology 60(6), 1861–1870 (2014)

    Article  Google Scholar 

  22. Jakubowski, A.: On the Skorokhod topology. Annales de l’Institut Henri Poincaré 22(3), 263–285 (1986)

    MathSciNet  MATH  Google Scholar 

  23. Le Gall, J.F.: Mouvement Brownien, Martingales et Calcul Stochastique. Mathématiques & Applications (Berlin), 71. Springer, Heidelberg (2013)

  24. Li, X., Rohe, K.: Central limit theorems for network driven sampling. Electron. J. Stat. 11(2), 4871–4895 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Malmros, J., Liljeros, F., Britton, T.: Respondent-driven sampling and an unusual epidemic. J. Appl. Probab. 53, 518–530 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Massart, P.: Concentration inequalities and model selection. Ecole d’Eté de Probabilités de Saint-Flour XXXIII - 2003, Jean Picard eds. Springer, Lecture Notes in Mathematics, 1896 (2007)

  27. Métivier, M.: Semimartingales: A Course on Stochastic Processes. de Gruyter, Berlin, New-York (1982)

    Book  MATH  Google Scholar 

  28. Mouw, T., Verdery, A.: Network sampling with memory: a proposal for more efficient sampling from social networks. Sociol Methodol. 42, 206–256 (2012)

    Article  Google Scholar 

  29. Rebolledo, R.: La méthode des martingales appliquée à l’étude de la convergence en loi de processus. Number 62 in Mémoires de la Société Mathématique de France. Société Mathématique de France (1979)

  30. Riordan, O.: The phase transition in the configuration model. Combinatorics, Probability and Computing 21(1–2), 265–299 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Robineau, O., Gomes, M., Kendall, C., Kerr, L., Périssé, A., Boëlle, P.Y.: Model-based respondent driven sampling analysis for HIV prevalence in Brazilian MSM. Scientific Reports 10, 2646 (2020)

    Article  Google Scholar 

  32. Robineau, O., Velter, A., Barin, F., Boelle, P.Y.: HIV transmission and pre-exposure prophylaxis in a high risk MSM population: a simulation study of location-based selection of sexual partners. PLoS ONE 12(11), e0189002 (2017)

    Article  Google Scholar 

  33. Rolls, D.A., Sacks-Davis, R., Jenkinson, R., McBryde, E., Pattison, P., Robins, G., Hellard, M.: Hepatitis C transmission and treatment in contact networks of people who inject drugs. PLOS ONE 8(11), 1–15 (2013)

    Article  Google Scholar 

  34. Rolls, D.A., Wang, P., Jenkinson, R., Pattison, P., Robins, G., Sacks-Davis, R., Daraganova, G., Hellard, M., McBryde, E.: Modelling a disease-relevant contact network of people who inject drugs. Socl. Net. 35(4), 699–710 (2013)

    Article  Google Scholar 

  35. Vo, T.: Chain-referral sampling on stochastic block models. ESAIM: PS 24, 718–738 (2020)

  36. Vo, T.: Exploration of Random Graphs by the Respondent Driven Sampling Method. PhD thesis, Université Sorbonne Paris Nord, Paris, France (2020)

  37. Volz, E., Heckathorn, D.: Probability-based estimation theory for respondent-driven sampling. J. Offic. Stats. 24, 79–97 (2008)

    Google Scholar 

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Acknowledgements

The authors would like to thank the working group previously involved in the development of the model for HCV transmission among PWID: Sylvie Deuffic-Burban, Marie Jauffret-Roustide, and Yazdan Yazdanpanah. Last but not least, the authors thank Hélène Guérin and an anonymous referee for valuable corrections and relevant comments.

Funding

This work was partially funded by the French Agence Nationale de Recherche sur le Sida et les Hépatites virales (ANRS, http://www.anrs.fr), grant number 95146. V.C.T. and T.P.T.V. have been supported by the GdR GeoSto 3477, ANR Econet (ANR-18-CE02-0010) and by the Chair “Modélisation Mathématique et Biodiversité” of Veolia Environnement-Ecole Polytechnique-Museum National d’Histoire Naturelle-Fondation X. V.C.T. and T.P.T.V. acknowledge support from Labex Bézout (ANR-10-LABX-58).

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Authors and Affiliations

  1. Université Paris Cité, IAME, INSERM, Paris, F-75018, France

    Anthony Cousien

  2. UMR 7539 - LAGA, Univ. Paris 13, CNRS, 99 avenue J.-B. Clément, Villetaneuse, F-93430, France

    Jean-Stéphane Dhersin

  3. LAMA, Univ. Gustave Eiffel, Univ. Paris Est Creteil, CNRS, Marne-la-Vallée, F-91037, France

    Viet Chi Tran

  4. ENSIIE, LaMME, INRAE, CNRS UMR 8071, Université d‘Evry Val d‘Essonne, Université Paris-Saclay, Evry Courcouronnes, F-91037, France

    Thi Phuong Thuy Vo

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  1. Anthony Cousien
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  2. Jean-Stéphane Dhersin
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Correspondence to Viet Chi Tran.

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Cousien, A., Dhersin, JS., Tran, V.C. et al. Respondent-Driven Sampling on Sparse Erdös-Rényi Graphs. Acta Math Vietnam 48, 479–513 (2023). https://doi.org/10.1007/s40306-023-00510-8

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