  Respondent-driven sampling (RDS) is a chain-referral method for sampling
members of a hidden or hard-to-reach population such as sex workers, homeless
people, or drug users via their social network. Most methodological work on RDS
has focused on inference of population means under the assumption that
subjects' network degree determines their probability of being sampled.
Criticism of existing estimators is usually focused on missing data: the
underlying network is only partially observed, so it is difficult to determine
correct sampling probabilities. In this paper, we show that data collected in
ordinary RDS studies contain information about the structure of the
respondents' social network. We construct a continuous-time model of RDS
recruitment that incorporates the time series of recruitment events, the
pattern of coupon use, and the network degrees of sampled subjects. Together,
the observed data and the recruitment model place a well-defined probability
distribution on the recruitment-induced subgraph of respondents. We show that
this distribution can be interpreted as an exponential random graph model and
develop a computationally efficient method for estimating the hidden graph. We
validate the method using simulated data and apply the technique to an RDS
study of injection drug users in St. Petersburg, Russia.
