# Respondent driven sampling and sparse graph convergence

## TL;DR

This paper studies a respondent-driven sampling method modeled by a graphon, demonstrating that under certain conditions, the resulting sparse graphs converge to the graphon using advanced probabilistic tools.

## Contribution

It introduces a novel approach to analyze respondent-driven sampling via graphon convergence and develops a specific clumping procedure for sparse graph construction.

## Key findings

- Sparse graphs constructed via the method converge to the graphon in the cut-metric.
- Stationarity of the vertex-sets is key for convergence.
- Uses concentration inequalities and Stein-Chen method for analysis.

## Abstract

We consider a particular respondent-driven sampling procedure governed by a graphon. By a specific clumping procedure of the sampled vertices we construct a sequence of sparse graphs. If the sequence of the vertex-sets is stationary then the sequence of sparse graphs converge to the governing graphon in the cut-metric. The tools used are concentration inequality for Markov chains and the Stein-Chen method.