Respondent Driven Sampling and Random Graph Convergence

We consider certain respondent-driven sampling procedures on dense and sparse graphs. Under the assumption that the sequence of the vertex-sets is ergodic we understand possible limiting graphs. In the dense regime they can be expressed in terms of the original dense graph via a transformation related to the invariant measure of the ergodic sequence. In the sparse regime by a specific clumping procedure of the sampled vertices we construct a sequence of sparse graphs which converge to the governing graphon in the cut-metric.