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We are interested in recovering information on a stochastic block model fromthe subgraph discovered by an exploring random walk. Stochastic block modelscorrespond to populations structured into a finite number of types, where twoindividuals are connected by an edge independently from the other pairs andwith a probability depending on their types. We consider here the dense casewhere the random network can be approximated by a graphon. This problem ismotivated from the study of chain-referral surveys where each intervieweeprovides information on her/his contacts in the social network. First, we writethe likelihood of the subgraph discovered by the random walk: biases areappearing since hubs and majority types are more likely to be sampled. Even forthe case where the types are observed, the maximum likelihood estimator is notexplicit any more. When the types of the vertices is unobserved, we use an SAEMalgorithm to maximize the likelihood. Second, we propose a different estimationstrategy using new results by Athreya and Roellin. It consists in de-biasingthe maximum likelihood estimator proposed in Daudin et al. and that ignores thebiases.

Estimation of dense stochastic block models visited by random walks