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Iterative estimation of mixed exponential random graph models with nodal random effects

Published online by Cambridge University Press:  25 January 2022

S. Kevork Open the ORCID record for S. Kevork [Opens in a new window]  and
G. Kauermann
S. Kevork*
Affiliation:
Institut für Statistik, Ludwig-Maximilians-Universität München, München, Germany.
G. Kauermann
Affiliation:
Institut für Statistik, Ludwig-Maximilians-Universität München, München, Germany.
*
*Corresponding author. Email: sevag.kevork@stat.uni-muenchen.de
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Abstract

The presence of unobserved node-specific heterogeneity in exponential random graph models (ERGM) is a general concern, both with respect to model validity as well as estimation instability. We, therefore, include node-specific random effects in the ERGM that account for unobserved heterogeneity in the network. This leads to a mixed model with parametric as well as random coefficients, labelled as mixed ERGM. Estimation is carried out by iterating between approximate pseudolikelihood estimation for the random effects and maximum likelihood estimation for the remaining parameters in the model. This approach provides a stable algorithm, which allows to fit nodal heterogeneity effects even for large scale networks. We also propose model selection based on the Akaike Information Criterion to check for node-specific heterogeneity.

Keywords

exponential random graph modelsrandom effectsgeneralized linear mixed modelsnetwork data analysis
Type
Research Article
Information
Network Science , Volume 9 , Issue 4 , December 2021 , pp. 478 - 498
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

Action Editor: Stanley Wasserman

References

Akaike, H. (1974). A new look at the statistical model identification. In Selected Papers of Hirotugu Akaike (pp. 215—222). Springer.Google Scholar
Box-Steffensmeier, J. M., Christenson, D. P., & Morgan, J. W. (2018). Modeling unobserved heterogeneity in social networks with the frailty exponential random graph model. Political Analysis, 26(1), 3–19.CrossRefGoogle Scholar
Breslow, N. E., & Clayton, D. G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88(421), 9–25.Google Scholar
Caimo, A., & Friel, N. (2011). Bayesian inference for exponential random graph models. Social Networks, 33(1), 41–55.CrossRefGoogle Scholar
Cranmer, S. J., & Desmarais, B. A. (2011). Inferential network analysis with exponential random graph models. Political Analysis, 19(1), 66–86.CrossRefGoogle Scholar
Desmarais, B., & Cranmer, S. (2012). Statistical mechanics of networks: Estimation and uncertainty. Physica A: Statistical Mechanics and its Applications, 391(4), 1865–1876.CrossRefGoogle Scholar
Duijn, M. A. J., Snijders, T. A., & Zijlstra, B. J. H. (2004). $ p_2 $: A random effects model with covariates for directed graphs. Statistica Neerlandica, 58(2), 234–254.CrossRefGoogle Scholar
Faraway, J. J. (2016). Extending the linear model with R: generalized linear, mixed effects and nonparametric regression models. CRC Press.Google Scholar
Fellows, I., & Handcock, M. S. (2012). Exponential-family random network models. Retrieved from arXiv preprint arXiv:1208.0121.Google Scholar
Fienberg, S. E. (2012). A Brief History of Statistical Models for Network Analysis and Open Challenges. Journal of Computational and Graphical Statistics, 21(4), 825–839.CrossRefGoogle Scholar
Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81(395), 832–842.CrossRefGoogle Scholar
Goldenberg, A., Zheng, A. X., Fienberg, S. E., & Airoldi, E. M. (2010). A survey of statistical network models. Foundations and Trends in Machine Learning, 2(2), 129–233.CrossRefGoogle Scholar
Greven, S., & Kneib, T. (2010). On the behaviour of marginal and conditional AIC in linear mixed models. Biometrika, 97(4), 773–789.CrossRefGoogle Scholar
Henry, T. R., Gates, K. M., Prinstein, M. J., & Steinley, D. (2020). Modeling heterogeneous peer assortment effects using finite mixture exponential random graph models. Psychometrika, 85(1), 8–34.CrossRefGoogle ScholarPubMed
Holland, P. W., & Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 76(373), 33–50.CrossRefGoogle Scholar
Hummel, R. M., Hunter, D. R., & Handcock, M. S. (2012). Improving simulation-based algorithms for fitting ergms. Journal of Computational and Graphical Statistics, 21(4), 920–939.CrossRefGoogle ScholarPubMed
Hunter, D. R. & Handcock, M. S. (2006). Inference in curved exponential family models for networks. Journal of Computational and Graphical Statistics, 15(3), 565–583.CrossRefGoogle Scholar
Hunter, D. R., Handcock, M. S., Butts, C. T., Goodreau, S. M., & Morris, M. (2008). ergm: A package to fit, simulate and diagnose exponential-family models for networks. Journal of Statistical Software, 24(3), 1–29.CrossRefGoogle ScholarPubMed
Hunter, D. R., Krivitsky, P. N., & Schweinberger, M. (2012). Computational statistical methods for social network analysis. Journal of Computational and Graphical Statistics, 21(4), 856–882.CrossRefGoogle Scholar
Kolaczyk, E. D. (2009). Statistical Analysis of Network Models. New York: Springer.CrossRefGoogle Scholar
Kolaczyk, E. D. & CsÁrdi, G. (2014). Statistical analysis of network data with R. New York: Springer.CrossRefGoogle Scholar
Koskinen, J. (2009). Using latent variables to account for heterogeneity in exponential family random graph models.Google Scholar
Lusher, D., Koskinen, J., & Robins, G. (2013). Exponential random graph models for social networks. Cambridge: Cambridge University Press.Google Scholar
Mastrandrea, R., Fournet, J., & Barrat, A. (2015). Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys. PloS one, 10(9), e0136497.CrossRefGoogle Scholar
McAuley, J. J., & Leskovec, J. (2012). In NIPS, volume 2012. Citeseer (pp. 548–656).Google Scholar
Morris, M., Handcock, M. S., & Hunter, D. R. (2008). Specification of exponential-family random graph models: terms and computational aspects. Journal of statistical software, 24(4), 1548.CrossRefGoogle ScholarPubMed
Robins, G. L., Elliot, P., & Pattison, P. (2001). Network models for social selection processes. Social Networks, 23(1), 1–30.CrossRefGoogle Scholar
Schmid, C. S. & Desmarais, B. A. (2017). Exponential random graph models with big networks: Maximum pseudolikelihood estimation and the parametric bootstrap. In 2017 IEEE International Conference on Big Data (Big Data). IEEE (pp. 116—121).Google Scholar
Schweinberger, M. (2011). Instability, sensitivity, and degeneracy of discrete exponential families. Journal of the American Statistical Association, 106(496), 1361–1370.CrossRefGoogle ScholarPubMed
Schweinberger, M. and Handcock, M. S. (2015). Local dependence in random graph models: characterization, properties and statistical inference. Journal of the American Statistical Association, 77(3), 647.Google ScholarPubMed
Schweinberger, M., Krivitsky, P. N., & Butts, C. T. (2017). Foundations of finite-, super-, and infinite-population random graph inference. arXiv preprint arXiv:1707.04800.Google Scholar
Snijders, T. A. B. (2002). Markov chain monte carlo estimation of exponential random graph models. Journal of Social Structure, 3(2), 1–40.Google Scholar
Snijders, T. A. B., Pattison, P. E., Robins, G. L., & Handcock, M. S. (2006). New specifications for exponential random graph models. Sociological Methodology, 36(1), 99–153.CrossRefGoogle Scholar
Strauss, D. & Ikeda, M. (1990). Pseudolikelihood estimation for social networks. Journal of the American Statistical Association, 85(409), 204–212.CrossRefGoogle Scholar
Thiemichen, S., Friel, N., Caimo, A., & Kauermann, G. (2016). Bayesian exponential random graph models with nodal random effects. Social Networks, 46, 11–28.CrossRefGoogle Scholar
Vaida, F. & Blanchard, S. (2005). Conditional akaike information for mixed-effects models. Biometrika, 92(2), 351–370.CrossRefGoogle Scholar
van Duijn, M. A., Gile, K. J., & Handcock, M. S. (2009). A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models. Social Networks, 31(1), 52–62.CrossRefGoogle ScholarPubMed
Wood, S. N. (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Journal of the Royal Statistical Society (B), 73(1), 3–36.CrossRefGoogle Scholar
Zachary, W. W. (1977). An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33(4), 452–473.CrossRefGoogle Scholar
Zijlstra, B. J. H., Duijn, M. A. J., & Snijders, T. A. B. (2006). The multilevel $p_2$ model: A random effects model for the analysis of multiple social networks. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, 2(1), 42–47.CrossRefGoogle Scholar
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