## The Annals of Applied Probability

### Dense graph limits under respondent-driven sampling

#### Abstract

We consider certain respondent-driven sampling procedures on dense graphs. We show that if the sequence of the vertex-sets is ergodic then the limiting graph can be expressed in terms of the original dense graph via a transformation related to the invariant measure of the ergodic sequence. For specific sampling procedures, we describe the transformation explicitly.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 4 (2016), 2193-2210.

Dates
Revised: September 2015
First available in Project Euclid: 1 September 2016

https://projecteuclid.org/euclid.aoap/1472745456

Digital Object Identifier
doi:10.1214/15-AAP1144

Mathematical Reviews number (MathSciNet)
MR3543894

Zentralblatt MATH identifier
1350.05151

#### Citation

Athreya, Siva; Röllin, Adrian. Dense graph limits under respondent-driven sampling. Ann. Appl. Probab. 26 (2016), no. 4, 2193--2210. doi:10.1214/15-AAP1144. https://projecteuclid.org/euclid.aoap/1472745456

#### References

• [1] Aaronson, J., Burton, R., Dehling, H., Gilat, D., Hill, T. and Weiss, B. (1996). Strong laws for $L$- and $U$-statistics. Trans. Amer. Math. Soc. 348 2845–2866.
• [2] Aldous, D. J. (1981). Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11 581–598.
• [3] Bollobás, B. and Riordan, O. (2009). Metrics for sparse graphs. In Surveys in Combinatorics 2009. London Mathematical Society Lecture Note Series 365 211–287. Cambridge Univ. Press, Cambridge.
• [4] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008). Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.
• [5] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2012). Convergent sequences of dense graphs II. Multiway cuts and statistical physics. Ann. of Math. (2) 176 151–219.
• [6] Diaconis, P. and Janson, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. Appl. (7) 28 33–61.
• [7] Gile, K. J. and Handcock, M. S. (2010). Respondent-driven sampling: An assessment of current methodology. Sociological Methodology 40 285–327.
• [8] Goel, S. and Salganik, M. J. (2010). Assessing respondent-driven sampling. Proc. Natl. Acad. Sci. USA 107 6743–6747.
• [9] Heckathorn, D. D. (1997). Respondent-driven sampling: A new approach to the study of hidden populations. Social Problems 44 174–199.
• [10] Heckathorn, D. D. (2002). Respondent-driven sampling II: Deriving valid population estimates from chain-referral samples of hidden populations. Social Problems 49 11–34.
• [11] Hernández-Lerma, O. and Lasserre, J. B. (1998). Ergodic theorems and ergodic decomposition for Markov chains. Acta Appl. Math. 54 99–119.
• [12] Hoover, D. N. (1979). Relations on probability spaces and arrays of random variables. Preprint, Institute for Advanced Studies. Princeton, NJ.
• [13] Jagers, P. (1989). General branching processes as Markov fields. Stochastic Process. Appl. 32 183–212.
• [14] Jagers, P. and Nerman, O. (1996). The asymptotic composition of supercritical multi-type branching populations. In Séminaire de Probabilités, XXX. Lecture Notes in Math. 1626 40–54. Springer, Berlin.
• [15] Janson, S. (2008). Connectedness in graph limits. Technical Report 2008:8, Uppsala.
• [16] Lovász, L. (2012). Large Networks and Graph Limits. American Mathematical Society Colloquium Publications 60. Amer. Math. Soc., Providence, RI.
• [17] Lovász, L. and Szegedy, B. (2006). Limits of dense graph sequences. J. Combin. Theory Ser. B 96 933–957.
• [18] McDiarmid, C. (1989). On the method of bounded differences. In Surveys in Combinatorics, 1989 (Norwich, 1989). London Mathematical Society Lecture Note Series 141 148–188. Cambridge Univ. Press, Cambridge.
• [19] Ney, P. and Nummelin, E. (1987). Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Probab. 15 561–592.
• [20] Niemi, S. and Nummelin, E. (1986). On nonsingular renewal kernels with an application to a semigroup of transition kernels. Stochastic Process. Appl. 22 177–202.
• [21] Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics 83. Cambridge Univ. Press, Cambridge.
• [22] Volz, E. and Heckathorn, D. D. (2008). Probability based estimation theory for respondent driven sampling. Journal of Official Statistics 24 79–97.
• [23] Yosida, K. (1980). Functional Analysis, 6th ed. Grundlehren der Mathematischen Wissenschaften 123. Springer, Berlin.