Respondent-driven sampling (RDS) is a chain-referral method for sampling members of hidden or hard-to-reach populations, such as sex workers, homeless people, or drug users, via their social networks. 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 article, the author shows that data collected in ordinary RDS studies contain information about the structure of the respondents’ social network. The author constructs 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. The author shows that this distribution can be interpreted as an exponential random graph model and develops a computationally efficient method for estimating the hidden graph. The author validates the method using simulated data and applies the technique to an RDS study of injection drug users in St. Petersburg, Russia.