Random Walks on Directed Networks: Inference and Respondent-driven Sampling

Respondent driven sampling (RDS) is often used to estimate population properties (e.g., sexual risk behaviour) in hard-to-reach populations. In RDS, already sampled individuals recruit population members to the sample from their social contacts in an efficient snowball-like sampling procedure. By assuming a Markov model for the recruitment of individuals, asymptotically unbiased estimates of population characteristics can be obtained. Current RDS estimation methodology assumes that the social network is undirected, i.e., all edges are reciprocal. However, empirical social networks in general also include a substantial amount of non-reciprocal edges. In this paper, we develop an estimation method for RDS in populations connected by social networks that include reciprocal and non-reciprocal edges. We derive estimators of the selection probabilities of individuals as a function of the number of outgoing edges of sampled individuals. The proposed estimators are evaluated on articial and empirical networks and are shown to generally perform better than existing estimators. This is in particular the case when the fraction of directed edges in the network is large.