  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 Roellin. It consists in de-biasing
the maximum likelihood estimator proposed in Daudin et al. and that ignores the
biases.
