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

TL;DR

This paper analyzes a respondent-driven sampling process on large sparse Erd"os-Rényi graphs, showing convergence to deterministic ODEs and establishing a Gaussian fluctuation limit, supported by simulations.

Contribution

It introduces a new Markov chain model for respondent-driven sampling on sparse Erd"os-Rényi graphs and characterizes its asymptotic behavior through ODEs and a CLT.

Findings

Convergence of the sampling process to a deterministic ODE system.

Derivation of a Gaussian fluctuation process with a CLT.

Validation of theoretical results through simulations.

Abstract

We study the exploration of an Erd\"os-R\'enyi random graph by a respondent-driven sampling method, where discovered vertices reveal their neighbours. Some of them receive coupons to reveal in their turn their own neighbourhood. This leads to the study of a Markov chain on the random graph that we study. For sparse Erd\"os-R\'enyi graphs of large sizes, this process correctly renormalized converges to the solution of a deterministic curve, solution of a system of ODEs absorbed on the abscissa axis. The associated fluctuation process is also studied, providing a functional central limit theorem, with a Gaussian limiting process. Simulations and numerical computation illustrate the study.