×

Adjusting for network size and composition effects in exponential-family random graph models. (English) Zbl 1215.91069

Summary: Exponential-family random graph models (ERGMs) provide a principled way to model and simulate features common in human social networks, such as propensities for homophily and friend-of-a-friend triad closure. We show that, without adjustment, ERGMs preserve density as network size increases. Density invariance is often not appropriate for social networks. We suggest a simple modification based on an offset which instead preserves the mean degree and accommodates changes in network composition asymptotically. We demonstrate that this approach allows ERGMs to be applied to the important situation of egocentrically sampled data. We analyze data from the National Health and Social Life Survey (NHSLS).

MSC:

91D30 Social networks; opinion dynamics
62P25 Applications of statistics to social sciences
05C80 Random graphs (graph-theoretic aspects)

Software:

bootlib; statnet; R
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] R. Admiraal, Dynamic network models based on revealed preference for observed relations and egocentric data, Ph.D. Thesis, University of Washington, Seattle, WA, 2009.; R. Admiraal, Dynamic network models based on revealed preference for observed relations and egocentric data, Ph.D. Thesis, University of Washington, Seattle, WA, 2009.
[2] Anderson, B. S.; Butts, C.; Carley, K., The interaction of size and density with graph-level indices, Social Networks, 21, 239-267 (1999)
[3] Davison, A. C.; Hinkley, D. V., Bootstrap Methods and their Application (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0886.62001
[4] Frank, O.; Strauss, D., Markov graphs, Journal of the American Statistical Association, 81, 832-842 (1986) · Zbl 0607.05057
[5] S.M. Goodreau, Results of analyses performed for Goodreau et al. (2008) that were not published in that article, Personal Communication, 2009.; S.M. Goodreau, Results of analyses performed for Goodreau et al. (2008) that were not published in that article, Personal Communication, 2009.
[6] Goodreau, S. M.; Kitts, J.; Morris, M., Birds of a feather, or friend of a friend? Using exponential random graph models to investigate adolescent social networks, Demography, 45, 103-125 (2008)
[7] Hamilton, D. T.; Handcock, M. S.; Morris, M., Degree distributions in sexual networks: a framework for evaluating evidence, Sexually Transmitted Diseases, 35, 30-40 (2008)
[8] Handcock, M. S.; Hunter, D. R.; Butts, C. T.; Goodreau, S. M.; Morris, M., Statnet: software tools for the representation, visualization, analysis and simulation of network data, Journal of Statistical Software, 24, 1-11 (2008)
[9] Helleringer, S.; Kohler, H.-P., Sexual network structure and the spread of HIV in Africa: evidence from Likoma Island, Malawi, AIDS, 21, 2323-2332 (2007)
[10] Holland, P. W.; Leinhardt, S., An exponential family of probability distributions for directed graphs, Journal of the American Statistical Association, 76, 33-65 (1981) · Zbl 0457.62090
[11] Hunter, D. R., Curved exponential family models for social networks, Social Networks, 29, 216-230 (2007)
[12] Hunter, D. R.; Handcock, M. S., Inference in curved exponential family models for networks, Journal of Computational & Graphical Statistics, 15, 565-583 (2006)
[13] Klovdahl, A. S.; Potterat, J. J.; Woodhouse, D. E.; B, M. J.; Muth, S. Q.; Darrow, W. W., Social networks and infectious disease: the colorado springs study, Social Science & Medicine, 38, 79 (1994)
[14] Koehly, L. M.; Goodreau, S. M.; Morris, M., Exponential family models for sampled and census network data, Sociological Methodology, 34, 241-270 (2004)
[15] E.O. Laumann, J.H. Gagnon, R.T. Michael, S. Michaels, National health and social life survey, Chicago, IL, USA: University of Chicago and National Opinion Research Center [producer], 1995, Ann Arbor, MI, USA: Inter-University Consortium for Political and Social Research [distributor], 2008-04-17, Computer File, 1992.; E.O. Laumann, J.H. Gagnon, R.T. Michael, S. Michaels, National health and social life survey, Chicago, IL, USA: University of Chicago and National Opinion Research Center [producer], 1995, Ann Arbor, MI, USA: Inter-University Consortium for Political and Social Research [distributor], 2008-04-17, Computer File, 1992.
[16] Laumann, E. O.; Gagnon, J. H.; Michael, R. T.; Michaels, S., The Social Organization of Sexuality (1994), University of Chicago Press: University of Chicago Press Chicago
[17] Leskovec, J.; Kleinberg, J.; Faloutsos, C., Graph evolution: densification and shrinking diameters, (ACM Transactions on Knowledge Discovery from Data (TKDD) (2007), ACM Press: ACM Press New York, NY, USA), 1-41
[18] McCullagh, P.; Nelder, J. A., (Generalized Linear Models. Generalized Linear Models, Monographs on Statistics and Applied Probability, vol. 37 (1989), Chapman & Hall/CRC) · Zbl 0744.62098
[19] McPherson, M.; Smith-Lovin, L.; Cook, J. M., Birds of a feather: homophily in social networks, Annual Review of Sociology, 27, 415-444 (2001)
[20] Morris, M., A log-linear modeling framework for selective mixing, Mathematical Biosciences, 107, 349-377 (1991)
[21] Morris, M.; Kretzschmar, M., Concurrent partnerships and the spread of HIV, AIDS, 11, 641-648 (1997)
[22] Pattison, P.; Robins, G. L., Neighborhood-based models for social networks, Sociological Methodology, 32, 301-337 (2002)
[23] Pattison, P.; Wasserman, S., Logit models and logistic regressions for social networks: II. Multivariate relations, British Journal of Mathematical and Statistical Psychology, 52, 169-193 (1999)
[24] R Development Core Team, R: a language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, version 2.6.1, 2009.; R Development Core Team, R: a language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, version 2.6.1, 2009.
[25] Robins, G.; Pattison, P., Random graph models for temporal processes in social networks, Journal of Mathematical Sociology, 25, 5-41 (2001) · Zbl 0986.91048
[26] Robins, G.; Pattison, P.; Wang, P., Closure, connectivity and degree distributions: exponential random graph \((p^\ast )\) models for directed social networks, Social Networks, 31, 105-117 (2009)
[27] Snijders, T. A.B.; Pattison, P. E.; Robins, G. L.; Handcock, M. S., New specifications for exponential random graph models, Sociological Methodology, 36, 99-153 (2006)
[28] Strauss, D.; Ikeda, M., Pseudolikelihood estimation for social networks, Journal of the American Statistical Association, 85, 204-212 (1990)
[29] Strichartz, R. S., The Way of Analysis (2000), Jones and Bartlett Publishers
[30] J.R. Udry, The national longitudinal study of adolescent health (add health), waves I & II, 1994-1996; wave III, 2001-2002, Technical Report, Carolina Population Center, University of North Carolina at Chapel Hill, 2003.; J.R. Udry, The national longitudinal study of adolescent health (add health), waves I & II, 1994-1996; wave III, 2001-2002, Technical Report, Carolina Population Center, University of North Carolina at Chapel Hill, 2003.
[31] van Duijn, M. A.J.; Gile, K. J.; Handcock, M. S., A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models, Social Networks, 31, 52-62 (2009)
[32] Wasserman, S. S.; Pattison, P., 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
[33] Woodhouse, D. E.; Rothenberg, R. B.; Potterat, J. J.; Darrow, W. W.; Muth, S. Q.; Klovdahl, A. S.; Zimmerman, H. P.; Rogers, H. L.; Maldonado, T. S.; Muth, J. B.; Reynolds, J. U., Mapping a social network of heterosexuals at high risk for HIV infection, AIDS, 8, 1331-1336 (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.