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Conditional estimation of exponential random graph models from snowball sampling designs. (English) Zbl 1281.62245

Summary: A complete survey of a network in a large population may be prohibitively difficult and costly. So it is important to estimate models for networks using data from various network sampling designs, such as link-tracing designs. We focus here on snowball sampling designs, designs in which the members of an initial sample of network members are asked to nominate their network partners, their network partners are then traced and asked to nominate their network partners, and so on. We assume an exponential random graph model (ERGM) of a particular parametric form and outline a conditional maximum likelihood estimation procedure for obtaining estimates of ERGM parameters. This procedure is intended to complement the likelihood approach developed by M. S. Handcock and K. J. Gile [Ann. Appl. Stat. 4, No. 1, 5–25 (2010; Zbl 1189.62187)] by providing a practical means of estimation when the size of the complete network is unknown and/or the complete network is very large. We report the outcome of a simulation study with a known model designed to assess the impact of initial sample size, population size, and number of sampling waves on properties of the estimates. We conclude with a discussion of the potential applications and further developments of the approach.

MSC:

62P25 Applications of statistics to social sciences
05C80 Random graphs (graph-theoretic aspects)
62F10 Point estimation
62D05 Sampling theory, sample surveys
91D30 Social networks; opinion dynamics
65C05 Monte Carlo methods

Citations:

Zbl 1189.62187

Software:

statnet; PNet
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Full Text: DOI

References:

[1] Baddeley, A.; Möller, J., Nearest-neighbour Markov point processes and random sets, International Statistical Review, 57, 89-121 (1989) · Zbl 0721.60010
[2] Barton, A. H., Paul Lazarsfeld as institutional inventor, International Journal of Public Opinion Research, 13, 245-269 (2001)
[3] Besag, J. E., Spatial interaction and the statistical analysis of lattice systems, Journal of the Royal Statistical Society: Series B, 36, 96-127 (1974), (with discussion)
[4] Bieleman, B.; Diaz, A.; Merlo, G.; Kaplan, C., Lines across Europe: nature and extent of cocaine use in Barcelona, Rotterdam and Turin (1993), Swets & Zeitlinger: Swets & Zeitlinger Amsterdam
[5] Bott, E., Family and social network (1957), Tavistock Publications: Tavistock Publications London
[6] Coleman, J. S., Relational analysis: the study of social organizations with survey methods, Human Organization, 16, 28-36 (1958)
[7] Dávid, B.; Snijders, T. A.B., Estimating the size of the homeless population in Budapest, Hungary, Quality and Quantity, 36, 291-303 (2002)
[8] Erdös, P.; Rényi, A., On random graphs I, Publicationes Mathematicae Debrecen, 6, 290-297 (1959) · Zbl 0092.15705
[9] Frank, O., Network sampling and model fitting, (Carrington, P. J.; Scott, J.; Wasserman, S., Models and methods in social network analysis (2005), Cambridge University Press: Cambridge University Press New York), 31-56
[10] Frank, O.; Snijders, T. A.B., Estimating hidden populations using snowball sampling, Journal of Official Statistics, 10, 53-67 (1994)
[11] Frank, O.; Strauss, D., Markov graphs, Journal of the American Statistical Association, 81, 832-842 (1986) · Zbl 0607.05057
[12] Goodman, L. A., Snowball sampling, The Annals of Mathematical Statistics, 32, 148-170 (1961) · Zbl 0099.14203
[13] Goodreau, S., Advances in exponential random graph \((p^\ast )\) models applied to a large social network, Social Networks, 29, 231-248 (2007)
[14] Handcock, M. S., Statistical models for social networks, (Breiger, R. L.; Carley, K. M.; Pattison, P. E., Dynamic social network modeling and analysis (2003), National Academies Press: National Academies Press Washington, DC)
[17] Handcock, M. S.; Gile, K. J., Modelling networks from sampled data, Annals of Applied Statistics, 4, 5-25 (2010) · Zbl 1189.62187
[18] Handcock, M. S.; Hunter, D.; Butts, C.; Goodreau, S.; Morris, M., Statnet: software tools for representation, visualization, analysis and simulation of network data, Journal of Statistical Software, 24, 1 (2008), URL: http://www.jtatsoft.org/v24/i01
[19] Heckathorn, D. D., Respondent-driven sampling: a new approach to the study of hidden populations, Social Problems, 44, 174-199 (1997)
[20] Holland, P.; Leinhardt, S., An exponential family of probability distributions for directed graphs, Journal of the American Statistical Association, 76, 33-50 (1981) · Zbl 0457.62090
[21] Hunter, D. R., Curved exponential family models for social networks, Social Networks, 29, 216-230 (2007)
[22] Hunter, D. R.; Handcock, M. S., Inference in curved exponential family models for networks, Journal of Computational and Graphical Statistics, 15, 565-583 (2006)
[23] Katz, E.; Lazarsfeld, P., Personal influence (1955), The Free press: The Free press Glencoe, IL
[24] Lazarsfeld, P.; Berelson, B.; Gaudet, H., The people’s choice: how the voter makes up his mind in a presidential election (1944), Duell, Sloan & Pearce: Duell, Sloan & Pearce New York
[25] Lubbers, M. J., Group composition and network structure in school classes: a multilevel application of the \(p^\ast\) model, Social Networks, 25, 309-332 (2003)
[26] Merton, R. K., Social theory and social structure (1957), The Free Press: The Free Press Glencoe, IL
[27] Milgram, S., The small world problem, Psychology Today, 1, 60-67 (1967)
[28] Pattison, P. E.; Robins, G. L., Neighbourhood-based models for social networks, Sociological Methodology, 32, 301-337 (2002)
[29] Pattison, P.; Snijders, T. A.B., Modelling social networks: next steps, (Lusher, D.; Koskinen, J.; Robins, G., Exponential Random Graph Models for Social Networks: Theory, Method and Applications (2013), Cambridge University Press), 287-302
[30] Robins, G. L.; Pattison, P. E.; Wang, P., Closure, connectivity and degree distributions: exponential random graph \((p^\ast )\) models for directed social networks, Social Networks, 31, 105-117 (2009)
[31] Robins, G. L.; Pattison, P. E.; Woolcock, J., Small and other worlds: global network structures from local processes, American Journal of Sociology, 110, 894-936 (2005)
[32] Robins, G. L.; Snijders, T. A.B.; Wang, P.; Handcock, M. S.; Pattison, P. E., Recent developments in exponential random graph \((p^\ast )\) models, Social Networks, 29, 216-230 (2007)
[33] Snijders, T. A.B., Markov chain Monte Carlo estimation of exponential random graph models, Journal of Social Structure, 3, 2 (2002)
[34] Snijders, T. A.B., Conditional marginalization for exponential random graph models, Journal of Mathematical Sociology, 34, 239-252 (2010) · Zbl 1201.91170
[35] Snijders, T. A.B.; Pattison, P.; Robins, G. L.; Handcock, M., New specifications for exponential random graph models, Sociological Methodology, 36, 99-153 (2006)
[36] Thompson, S. K.; Collins, L. M., Adaptive sampling in research on risk-related behaviors, Drug and Alcohol Dependence, 68, S57-S67 (2002) · Zbl 1294.93019
[37] Thompson, S. K.; Frank, O., Model-based estimation with link-tracing designs, Survey Methodology, 26, 87-98 (2000)
[38] Wang, P.; Pattison, P. E.; Robins, G. L., Exponential random graph model specifications for bipartite networks—a dependence hierarchy, Social Networks, 35, 2, 211-222 (2013)
[40] Wang, P.; Sharpe, K.; Robins, G.; Pattison, P., Exponential random graph \((p^\ast )\) models for affiliation networks, Social Networks, 31, 12-23 (2009)
[41] Wasserman, S.; Pattison, P. E., Logit models and logistic regressions for social networks, I. An introduction to Markov graphs and \(p^\ast \), Psychometrika, 61, 401-425 (1996) · Zbl 0866.92029
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