# Network driven sampling; a critical threshold for design effects

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Web crawling and respondent-driven sampling (RDS) are two types of network driven sampling techniques that are popular when it is difficult to contact individuals in the population of interest. This paper studies network driven sampling as a Markov process on the social network that is indexed by a tree. Each node in this tree corresponds to an observation and each edge in the tree corresponds to a referral. Indexing with a tree, instead of a chain, allows for the sampled units to refer multiple future units into the sample. In survey sampling, the design effect characterizes the additional variance induced by a novel sampling strategy. If the design effect is $D$, then constructing an estimator from the novel design makes the variance of the estimator $D$ times greater than it would be under a simple random sample. Under certain assumptions on the referral tree, the design effect of network driven sampling has a critical threshold that is a function of the referral rate $m$ and the clustering structure in the social network, represented by the second eigenvalue of the Markov transition matrix $\lambda_2$. If $m < 1/\lambda_2^2$, then the design effect is finite (i.e. the standard estimator is $\sqrt{n}$-consistent). However, if $m > 1/\lambda_2^2$, then the design effect grows with $n$ (i.e. the standard estimator is no longer $\sqrt{n}$-consistent; it converges at the slower rate of $\log_m \lambda_2$).