Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-28T20:53:31.743Z Has data issue: false hasContentIssue false
  • English
  • Français

DERGMs: Degeneracy-restricted exponential family random graph models

Published online by Cambridge University Press:  31 March 2022

Vishesh Karwa ,
Sonja Petrović Open the ORCID record for Sonja Petrović [Opens in a new window]  and
Denis Bajić
Vishesh Karwa
Affiliation:
Temple University, Philadelphia, PA 19122, USA
Sonja Petrović*
Affiliation:
Illinois Institute of Technology, Chicago, IL 60616, USA
Denis Bajić
Affiliation:
Illinois Institute of Technology, Chicago, IL 60616, USA
*
*Corresponding author. Email: sonja.petrovic@iit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Exponential random graph models, or ERGMs, are a flexible and general class of models for modeling dependent data. While the early literature has shown them to be powerful in capturing many network features of interest, recent work highlights difficulties related to the models’ ill behavior, such as most of the probability mass being concentrated on a very small subset of the parameter space. This behavior limits both the applicability of an ERGM as a model for real data and inference and parameter estimation via the usual Markov chain Monte Carlo algorithms. To address this problem, we propose a new exponential family of models for random graphs that build on the standard ERGM framework. Specifically, we solve the problem of computational intractability and “degenerate” model behavior by an interpretable support restriction. We introduce a new parameter based on the graph-theoretic notion of degeneracy, a measure of sparsity whose value is commonly low in real-world networks. The new model family is supported on the sample space of graphs with bounded degeneracy and is called degeneracy-restricted ERGMs, or DERGMs for short. Since DERGMs generalize ERGMs—the latter is obtained from the former by setting the degeneracy parameter to be maximal—they inherit good theoretical properties, while at the same time place their mass more uniformly over realistic graphs. The support restriction allows the use of new (and fast) Monte Carlo methods for inference, thus making the models scalable and computationally tractable. We study various theoretical properties of DERGMs and illustrate how the support restriction improves the model behavior. We also present a fast Monte Carlo algorithm for parameter estimation that avoids many issues faced by Markov Chain Monte Carlo algorithms used for inference in ERGMs.

Keywords

networkERGMdegeneracystabilitysufficiency
Type
Research Article
Information
Network Science , Volume 10 , Issue 1 , March 2022 , pp. 82 - 110
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Karwa was partially supported by NSF TRIPODS+X grant number 1947919.

‡

SP is partially supported by the Simons Foundation’s the Collaboration Grant forMathematicians 854770. This work was initially supported by U.S. Air Force Office of Scientific Research Grant #FA9550-14-1-0141 to Illinois Tech. A small subset of the simulations for this work were completed on Illinois Tech’s Karlin cluster.

Action Editor: Stanley Wasserman

References

Bajić, D. (2016). Dergms: Supplementary material on GitHub. https://github.com/dbajic/degen.Google Scholar
Bannister, M. J., Devanny, W. E., & Eppstein, D. (2014). ERGMs are hard. Preprint arXiv:1412.1787 [cs.DS].Google Scholar
Barndorff-Nielsen, O. (2014). Information and exponential families in statistical theory. John Wiley & Sons.CrossRefGoogle Scholar
Batagelj, V., & Mrvar, A. (2006). Pajek datasets. http://vlado.fmf.uni-lj.si/pub/networks/data/.Google Scholar
Batagelj, V., & Zaversnik, M. (2003). An o (m) algorithm for cores decomposition of networks. arxiv preprint cs/0310049.Google Scholar
Bauer, R., Krug, M., & Wagner, D. (2010). Enumerating and generating labeled k-degenerate graphs. In Proceedings of the seventh workshop on analytic algorithmics and combinatorics (analco).CrossRefGoogle Scholar
Caimo, A., & Friel, N. (2011). Bayesian inference for exponential random graph models. Social Networks, 33(1), 41–55.CrossRefGoogle Scholar
Chatterjee, S., & Diaconis, P. (2013). Estimating and understanding exponential random graph models. Annals of Statistics, 41(5), 2428–2461.CrossRefGoogle Scholar
Engström, A., & Norén, P. (2011). Polytopes from subgraph statistics. Discrete Mathematics and Theoretical Computer Science.CrossRefGoogle Scholar
Fellows, I., & Handcock, M. (2017). Removing phase transitions from gibbs measures. In Artificial intelligence and statistics (pp. 289–297).Google Scholar
Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81(395), 832–842.CrossRefGoogle Scholar
Geyer, C. J., & Thompson, E. A. (1992). Constrained monte carlo maximum likelihood for dependent data. Journal of the Royal Statistical Society. Series B (Methodological), 657–699.CrossRefGoogle Scholar
Giatsidis, C., Thilikos, D. M., & Vazirgiannis, M. (2011). D-cores: Measuring collaboration of directed graphs based on degeneracy. In IEEE 11th international conference on data mining.CrossRefGoogle Scholar
Goldenberg, A., Zheng, A. X., Fienberg, S. E., & Airoldi, E. M. (2009). A survey of statistical network models. Foundations and Trends in Machine Learning, 2(2), 129–233.CrossRefGoogle Scholar
Goodreau, S. M., Kitts, J. A., & Morris, M. (2009). Birds of a feather, or friend of a friend? using exponential random graph models to investigate adolescent social networks*. Demography, 46(1), 103–125.Google ScholarPubMed
Handcock, M. S. (2003). Assessing degeneracy in statistical models of social networks. Center for Statistics and the Social Sciences, University of Washington, Working Paper No. 39.Google Scholar
Horvát, S., Czabarka, É., & Toroczkai, Z. (2015). Reducing degeneracy in maximum entropy models of networks. Physical Review Letters, 114(15), 158701.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. (2008a). ergm: A package to fit, simulate and diagnose exponential-family models for networks. Journal of Statistical Software, 24(3). http://www.jstatsoft.org/v24/i03.CrossRefGoogle Scholar
Hunter, D. R., Goodreau, S. M., & Handcock, M. S. (2008b). Goodness of fit of social network models. Journal of the American Statistical Association, 103(481), 248–258.CrossRefGoogle Scholar
Hunter, D. R., Krivitsky, P. N., & Schweinberger, M. (2012). Computational statistical methods for social network models. Journal of Computational and Graphical Statistics, 21(4), 856–882.CrossRefGoogle ScholarPubMed
Karwa, V., & Slavković, A. (2016). Inference using noisy degrees: Differentially private $\beta$ -model and synthetic graphs. The Annals of Statistics, 44(1), 87–112.CrossRefGoogle Scholar
Karwa, V., Pelsmajer, M. J., Petrović, S., Stasi, D., & Wilburne, D. (2017). Statistical models for cores decomposition of an undirected random graph. Electronic Journal of Statistics, 11(1), 1949–1982. Preprint, arXiv:1410.7357 [math.ST].CrossRefGoogle Scholar
Kolaczyk, E. D., & Krivitsky, P. N. (2015). On the question of effective sample size in network modeling: An asymptotic inquiry. Statistical Science: A Review Journal of the Institute of Mathematical Statistics, 30(2), 184.Google ScholarPubMed
Krivitsky, P. N. (2012). Exponential-family random graph models for valued networks. Electronic Journal of Statistics, 6(none), 1100–1128.CrossRefGoogle ScholarPubMed
Krivitsky, P. N., Handcock, M. S., & Morris, M. (2011). Adjusting for network size and composition effects in exponential-family random graph models. Statistical Methodology, 8(4), 319–339.CrossRefGoogle Scholar
Lauritzen, S., Rinaldo, A., & Sadeghi, K. (2018). Random networks, graphical models and exchangeability. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(3), 481–508.CrossRefGoogle Scholar
Lick, D. R., & White, A. T. (1970). k-degenerate graphs. Canadian Journal of Mathematics, 22, 1082–1096.CrossRefGoogle Scholar
Rinaldo, A., Fienberg, S. E., & Zhou, Y. (2009). On the geometry of discrete exponential families with application to exponential random graph models. Electronic Journal of Statistics, 3, 446–484.CrossRefGoogle Scholar
Rinaldo, A., Petrović, S., & Fienberg, S. E. (2013). Maximum lilkelihood estimation in the $\beta$ -model. The Annals of Statistics, 41(3), 1085–1110.CrossRefGoogle Scholar
Robbins, H., & Monro, S. (1985). A stochastic approximation method. In Herbert robbins selected papers (pp. 102–109). Springer.CrossRefGoogle Scholar
Sampson, S. F. (1968). A novitiate in a period of change: An experimental and case study of relationships. Ph.D. thesis, Department of Sociology, Cornell University.Google Scholar
Saul, Z. M., & Filkov, V. (2007). Exploring biological network structure using exponential random graph models. Bioinformatics, 23(19), 2604–2611.CrossRefGoogle ScholarPubMed
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., & Handcock, M. S. (2015). Local dependence in random graph models: Characterization, properties and statistical inference. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(3), 647–676.CrossRefGoogle ScholarPubMed
Seidman, S. B. (1983). Network structure and minimum degree. Social Networks, 5(3), 269–287.CrossRefGoogle 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
Thiemichen, S., & Kauermann, G. (2017). Stable exponential random graph models with non-parametric components for large dense networks. Social Networks, 49, 67–80.CrossRefGoogle Scholar
Yin, M., Rinaldo, A., Fadnavis, S., et al. (2016). Asymptotic quantization of exponential random graphs. The Annals of Applied Probability, 26(6), 3251–3285.CrossRefGoogle Scholar
Supplementary material: PDF

Karwa et al. supplementary material

Karwa et al. supplementary material

Download Karwa et al. supplementary material(PDF)
PDF 624 KB