Computing Pseudolikelihood Estimators for Exponential-Family Random Graph Models
Pub. online: 15 March 2023
Type: Computing In Data Science
Open Access
Open Access
Received
1 August 2022
1 August 2022
Accepted
24 February 2023
24 February 2023
Published
15 March 2023
15 March 2023
Abstract
The reputation of the maximum pseudolikelihood estimator (MPLE) for Exponential Random Graph Models (ERGM) has undergone a drastic change over the past 30 years. While first receiving broad support, mainly due to its computational feasibility and the lack of alternatives, general opinions started to change with the introduction of approximate maximum likelihood estimator (MLE) methods that became practicable due to increasing computing power and the introduction of MCMC methods. Previous comparison studies appear to yield contradicting results regarding the preference of these two point estimators; however, there is consensus that the prevailing method to obtain an MPLE’s standard error by the inverse Hessian matrix generally underestimates standard errors. We propose replacing the inverse Hessian matrix by an approximation of the Godambe matrix that results in confidence intervals with appropriate coverage rates and that, in addition, enables examining for model degeneracy. Our results also provide empirical evidence for the asymptotic normality of the MPLE under certain conditions.
Supplementary material
Supplementary MaterialThe R code file and the ergm package that implements the new methods.
References
Barnard GA, Jenkins GM, Winsten CB (1962). Likelihood inference and time series. Journal of the Royal Statistical Society. Series A. General, 125(3): 321–372. https://doi.org/10.2307/2982406
Besag J (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society, Series B, Methodological, 36(2): 192–236. https://doi.org/10.1111/j.2517-6161.1974.tb00999.x
Birnbaum A (1962). On the foundations of statistical inference. Journal of the American Statistical Association, 57(298): 269–306. https://doi.org/10.1080/01621459.1962.10480660
Desmarais BA, Cranmer SJ (2012). Statistical mechanics of networks: Estimation and uncertainty. Physica. A, 391(4): 1865–1876. https://doi.org/10.1016/j.physa.2011.10.018
Erdős P, Rényi A (1959). On random graphs. Publicationes Mathematicae Debrecen, 6: 290–297. https://doi.org/10.5486/PMD.1959.6.3-4.12
Frank O, Strauss D (1986). Markov graphs. Journal of the American Statistical Association, 81(395): 832–842. https://doi.org/10.1080/01621459.1986.10478342
Geyer CJ, Thompson EA (1992). Constrained Monte Carlo maximum likelihood for dependent data. Journal of the Royal Statistical Society, Series B, Methodological, 54(3): 657–699. https://doi.org/10.1111/j.2517-6161.1992.tb01443.x
Gilbert EN (1959). Random graphs. The Annals of Mathematical Statistics, 30(4): 1141–1144. https://doi.org/10.1214/aoms/1177706098
Godambe VP (1960). An optimum property of regular maximum likelihood estimation. The Annals of Mathematical Statistics, 31(4): 1208–1211. https://doi.org/10.1214/aoms/1177705693
Handcock MS, Hunter DR, Butts CT, Goodreau SM, Krivitsky PN, Morris M (2022). ergm: Fit, Simulate and Diagnose Exponential-Family Models for Networks. The Statnet Project (https://statnet.org). R package version 4.2.2.
Handcock MS, Hunter DR, Butts CT, Goodreau SM, Morris M (2008). statnet: Software tools for the representation, visualization, analysis and simulation of network data. Journal of Statistical Software, 24(1): 1–11. https://doi.org/10.18637/jss.v024.i01
Holland P, Leinhardt S (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 76(373): 33–50. https://doi.org/10.1080/01621459.1981.10477598
Hunter DR, Handcock MS (2006). Inference in curved exponential family models for networks. Journal of Computational and Graphical Statistics, 15(3): 565–583. https://doi.org/10.1198/106186006X133069
Krackhardt D (1987). Cognitive social structure. Social Networks, 9: 109–134. https://doi.org/10.1016/0378-8733(87)90009-8
Krivitsky PN, Handcock MS, Morris M (2011). Adjusting for network size and composition effects in exponential-family random graph models. Statistical Methodology, 8(4): 319–339. https://doi.org/10.1016/j.stamet.2011.01.005
Lubbers MJ, Snijders TAB (2007). A comparison of various approaches to the exponential random graph model: A reanalysis of 102 student networks in school classes. Social Networks, 29(2): 489–507. https://doi.org/10.1016/j.socnet.2007.03.002
Robins G, Snijders T, Wang P, Handcock M, Pattison P (2007). Recent developments in exponential random graph (${p^{\ast }}$) models for social networks. Social Networks, 29(2): 192–215. https://doi.org/10.1016/j.socnet.2006.08.003
Schweinberger M (2011). Instability, sensitivity, and degeneracy of discrete exponential families. Journal of the American Statistical Association, 106(496): 1361–1370. https://doi.org/10.1198/jasa.2011.tm10747
Strauss D, Ikeda M (1990). Pseudolikelihood estimation for social networks. Journal of the American Statistical Association, 85(409): 202–212. https://doi.org/10.1080/01621459.1990.10475327
van Duijn MA, Gile KJ, Handcock MS (2009). A framework for the comparison of maxmimum pseudo-likelikood and maximum likelihood estimation of exponential family random graph models. Social Networks, 31(1): 52–62. https://doi.org/10.1016/j.socnet.2008.10.003
White H (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50(1): 1–25. https://doi.org/10.2307/1912526
