Exponential random graph models with big networks: Maximum pseudolikelihood estimation and the parametric bootstrap

With the growth of interest in network data across fields, the Exponential Random Graph Model (ERGM) has emerged as the leading approach to the statistical analysis of network data. ERGM parameter estimation requires the approximation of an intractable normalizing constant. Simulation methods represent the state-of-the-art approach to approximating the normalizing constant, leading to estimation by Monte Carlo maximum likelihood (MCMLE). MCMLE is accurate when a large sample of networks is used to approximate the normalizing constant. However, MCMLE is computationally expensive, and may be prohibitively so if the size of the network is on the order of 1,000 nodes (i.e., one million potential ties) or greater. When the network is large, one option is maximum pseudolikelihood estimation (MPLE). The standard MPLE is simple and fast, but generally underestimates standard errors. We show that a resampling method — the parametric bootstrap — results in accurate coverage probabilities for confidence intervals. We find that bootstrapped MPLE can be run in 1/5th the time of MCMLE. We study the relative performance of MCMLE and MPLE with simulation studies, and illustrate the two different approaches by applying them to a network of bills introduced in the United State Senate.

[1]  Harry Joe,et al.  Composite Likelihood Methods , 2012 .

[2]  D. J. Strauss,et al.  Pseudolikelihood Estimation for Social Networks , 1990 .

[3]  Pavel N Krivitsky,et al.  Computational Statistical Methods for Social Network Models , 2012, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[4]  Daniel A. McFarland,et al.  Ethnic Composition and Friendship Segregation: Differential Effects for Adolescent Natives and Immigrants1 , 2016, American Journal of Sociology.

[5]  Alessandro Lomi,et al.  Networks in markets and the propensity of companies to collaborate: An empirical test of three mechanisms , 2012 .

[6]  Tom A. B. Snijders,et al.  Markov Chain Monte Carlo Estimation of Exponential Random Graph Models , 2002, J. Soc. Struct..

[7]  S. Goodreau,et al.  Birds of a feather, or friend of a friend? using exponential random graph models to investigate adolescent social networks* , 2009, Demography.

[8]  A. Rinaldo,et al.  CONSISTENCY UNDER SAMPLING OF EXPONENTIAL RANDOM GRAPH MODELS. , 2011, Annals of statistics.

[9]  Wei Tang,et al.  Supervised Link Prediction Using Multiple Sources , 2010, 2010 IEEE International Conference on Data Mining.

[10]  P. Holland,et al.  An Exponential Family of Probability Distributions for Directed Graphs , 1981 .

[11]  M. Schweinberger.  Instability, Sensitivity, and Degeneracy of Discrete Exponential Families , 2011, Journal of the American Statistical Association.

[12]  Mark S. Handcock,et al.  A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models , 2009, Soc. Networks.

[13]  D. Lazer,et al.  The Coevolution of Networks and Political Attitudes , 2010 .

[14]  D. Hunter,et al.  Goodness of Fit of Social Network Models , 2008 .

[15]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[16]  P. Diaconis,et al.  Estimating and understanding exponential random graph models , 2011, 1102.2650.

[17]  Aapo Hyvärinen,et al.  Consistency of Pseudolikelihood Estimation of Fully Visible Boltzmann Machines , 2006, Neural Computation.

[18]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[19]  Bruce A. Desmarais,et al.  Inferential Network Analysis with Exponential Random Graph Models , 2011, Political Analysis.

[20]  Trish,et al.  Protecting adolescents from harm. Findings from the National Longitudinal Study on Adolescent Health. , 1997, JAMA.

[21]  Amanda L. Traud,et al.  Community Structure in Congressional Cosponsorship Networks , 2007, 0708.1191.

[22]  Mahzarin R. Banaji,et al.  Culture, Cognition, and Collaborative Networks in Organizations , 2011 .

[23]  Paul J. Laurienti,et al.  An exponential random graph modeling approach to creating group-based representative whole-brain connectivity networks , 2011, NeuroImage.

[24]  D. Hunter,et al.  Inference in Curved Exponential Family Models for Networks , 2006 .

[25]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[26]  S. Faraj,et al.  Electronic Knowledge Networks : Processes and Structure 1 Electronic Knowledge Networks : Processes and Structure , 2006 .

[27]  B. Desmarais.  Consistent Confidence Intervals for Maximum Pseudolikelihood Estimators , .

[28]  K. Axhausen,et al.  An agent model of social network and travel behavior interdependence , 2006 .

[29]  P. Pattison,et al.  New Specifications for Exponential Random Graph Models , 2006 .

[30]  James H. Fowler,et al.  Legislative cosponsorship networks in the US House and Senate , 2006, Soc. Networks.

[31]  Bruce A. Desmarais,et al.  Statistical Mechanics of Networks: Estimation and Uncertainty Forthcoming: Physica A , 2012 .

[32]  J. Fowler.  Connecting the Congress: A Study of Cosponsorship Networks , 2006, Political Analysis.

[33]  A. Bouskila,et al.  Similarity in sex and reproductive state, but not relatedness, influence the strength of association in the social network of feral horses in the Blauwe Kamer Nature Reserve , 2015 .

[34]  B. Efron.  The jackknife, the bootstrap, and other resampling plans , 1987 .

[35]  Paul J. Laurienti,et al.  Exponential Random Graph Modeling for Complex Brain Networks , 2010, PloS one.

[36]  Garry Robins,et al.  An introduction to exponential random graph (p*) models for social networks , 2007, Soc. Networks.

[37]  James S. Quinn,et al.  Individual attributes and self-organizational processes affect dominance network structure in pukeko , 2014 .

[38]  Martina Morris,et al.  ergm: A Package to Fit, Simulate and Diagnose Exponential-Family Models for Networks. , 2008, Journal of statistical software.

[39]  Stanley Wasserman,et al.  Statistical Models for Social Networks , 2000 .

[40]  J. Besag.  On the Statistical Analysis of Dirty Pictures , 1986 .