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2021 Estimation of dense stochastic block models visited by random walks
Viet Chi Tran, Thi Phuong Thuy Vo
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Electron. J. Statist. 15(2): 5855-5887 (2021). DOI: 10.1214/21-EJS1899

Abstract

We are interested in recovering information on a stochastic block model from the subgraph discovered by an exploring random walk. Stochastic block models correspond to populations structured into a finite number of types, where two individuals are connected by an edge independently from the other pairs and with a probability depending on their types. We consider here the dense case where the random network can be approximated by a graphon. This problem is motivated from the study of chain-referral surveys where each interviewee provides information on her/his contacts in the social network. First, we write the likelihood of the subgraph discovered by the random walk: biases are appearing since hubs and majority types are more likely to be sampled. Even for the case where the types are observed, the maximum likelihood estimator is not explicit any more. When the types of the vertices is unobserved, we use an SAEM algorithm to maximize the likelihood. Second, we propose a different estimation strategy using new results by Athreya and Röllin. It consists in de-biasing the maximum likelihood estimator proposed in Daudin et al. and that ignores the biases.

Funding Statement

This work was supported by the GdR GeoSto 3477, by the 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.

Acknowledgments

The authors thank the two anonymous Referees whose comments contributed much to improve the paper. They also thank Mahendra Mariadassou who gave the solution of Section 4.2.2Case where both Xi and Zi are unobservedsubsubsection.4.2.2. They are also indebted to Jean-Stéphane Dhersin, Sophie Donnet, Stéphane Robin, Adrian Röllin and Timothée Tabouy for discussions.

Citation

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Viet Chi Tran. Thi Phuong Thuy Vo. "Estimation of dense stochastic block models visited by random walks." Electron. J. Statist. 15 (2) 5855 - 5887, 2021. https://doi.org/10.1214/21-EJS1899

Information

Received: 1 July 2020; Published: 2021
First available in Project Euclid: 27 December 2021

Digital Object Identifier: 10.1214/21-EJS1899

Subjects:
Primary: 05C80 , 05C81 , 60J20 , 62D05
Secondary: 82C20

Keywords: chain-referral survey , EM estimation , graphon , incomplete likelihood , random graph , random walk exploration , respondent driven sampling , sampling bias , stochastic approximation expectation-maximization

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Vol.15 • No. 2 • 2021
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