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We study the exploration of an Erd\"os-R\'enyi random graph by arespondent-driven sampling method, where discovered vertices reveal theirneighbours. Some of them receive coupons to reveal in their turn their ownneighbourhood. This leads to the study of a Markov chain on the random graphthat we study. For sparse Erd\"os-R\'enyi graphs of large sizes, this processcorrectly renormalized converges to the solution of a deterministic curve,solution of a system of ODEs absorbed on the abscissa axis. The associatedfluctuation process is also studied, providing a functional central limittheorem, with a Gaussian limiting process. Simulations and numericalcomputation illustrate the study.

Respondent Driven Sampling on sparse Erd\"os-R\'enyi graphs