  Respondent driven sampling (RDS) is a method often used to estimate
population properties (e.g. sexual risk behavior) in hard-to-reach populations.
It combines an effective modified snowball sampling methodology with an
estimation procedure that yields unbiased population estimates under the
assumption that the sampling process behaves like a random walk on the social
network of the population. Current RDS estimation methodology assumes that the
social network is undirected, i.e. that all edges are reciprocal. However,
empirical social networks in general also have non-reciprocated edges. To
account for this fact, we develop a new estimation method for RDS in the
presence of directed edges on the basis of random walks on directed networks.
We distinguish directed and undirected edges and consider the possibility that
the random walk returns to its current position in two steps through an
undirected edge. We derive estimators of the selection probabilities of
individuals as a function of the number of outgoing edges of sampled
individuals. We evaluate the performance of the proposed estimators on
artificial and empirical networks to show that they generally perform better
than existing methods. This is in particular the case when the fraction of
directed edges in the network is large.
