Bayesian Analysis

Joint Modeling of Longitudinal Relational Data and Exogenous Variables

Rajarshi Guhaniyogi and Abel Rodriguez

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This article proposes a framework based on shared, time varying stochastic latent factor models for modeling relational data in which network and node-attributes co-evolve over time. Our proposed framework is flexible enough to handle both categorical and continuous attributes, allows us to estimate the dimension of the latent social space, and automatically yields Bayesian hypothesis tests for the association between network structure and nodal attributes. Additionally, the model is easy to compute and readily yields inference and prediction for missing link between nodes. We employ our model framework to study co-evolution of international relations between 22 countries and the country specific indicators over a period of 11 years.

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Bayesian Anal., Volume 15, Number 2 (2020), 477-503.

First available in Project Euclid: 8 June 2019

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latent factor model nodal attribute social network spike and slab prior systemic dimensions

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Guhaniyogi, Rajarshi; Rodriguez, Abel. Joint Modeling of Longitudinal Relational Data and Exogenous Variables. Bayesian Anal. 15 (2020), no. 2, 477--503. doi:10.1214/19-BA1160.

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  • Albert, J. H. and Chib, S. (1993). “Bayesian analysis of binary and polychotomous response data.” Journal of the American statistical Association, 88(422): 669–679.
  • Anderton, C. H., Beck, N., Carter, J. R., Dorussen, H., Gartzke, E., Gissinger, R., Gleditsch, N., Hegre, H., Levy, J. S., Li, Q., et al. (2003). Globalization and armed conflict. Rowman & Littlefield Publishers.
  • Brandes, U., Indlekofer, N., and Mader, M. (2012). “Visualization methods for longitudinal social networks and stochastic actor-oriented modeling.” Social Networks, 34(3): 291–308.
  • Butland, G., Peregrín-Alvarez, J. M., Li, J., Yang, W., Yang, X., Canadien, V., Starostine, A., Richards, D., Beattie, B., Krogan, N., et al. (2005). “Interaction network containing conserved and essential protein complexes in Escherichia coli.” Nature, 433(7025): 531–537.
  • Christakis, N. A. and Fowler, J. H. (2007). “The spread of obesity in a large social network over 32 years.” New England Journal of Medicine, 2007(357): 370–379.
  • De la Haye, K., Robins, G., Mohr, P., and Wilson, C. (2010). “Obesity-related behaviors in adolescent friendship networks.” Social Networks, 32(3): 161–167.
  • Desmarais, B. A. and Cranmer, S. J. (2010). “Consistent confidence intervals for maximum pseudolikelihood estimators.” In Proceedings of the Neural Information Processing Systems 2010 Workshop on Computational Social Science and the Wisdom of Crowds. Citeseer.
  • Desmarais, B. A. and Cranmer, S. J. (2012). “Statistical mechanics of networks: Estimation and uncertainty.” Physica A: Statistical Mechanics and its Applications, 391(4): 1865–1876.
  • Doreian, P. (2001). “Causality in social network analysis.” Sociological Methods & Research, 30(1): 81–114.
  • Durante, D. and Dunson, D. B. (2014). “Nonparametric Bayes dynamic modelling of relational data.” Biometrika, 101(4): 883–898.
  • Durante, D. and Dunson, D. B. (2017). “Bayesian inference and testing of group differences in brain networks.” Bayesian Analysis.
  • Durante, D., Mukherjee, N., and Steorts, R. C. (2017). “Bayesian learning of dynamic multilayer networks.” Journal of Machine Learning Research, 18(43): 1–29.
  • Fosdick, B. K. and Hoff, P. D. (2015). “Testing and modeling dependencies between a network and nodal attributes.” Journal of the American Statistical Association, 110(511): 1047–1056.
  • Fowler, J. H. and Christakis, N. A. (2008). “Dynamic spread of happiness in a large social network: longitudinal analysis over 20years in the Framingham Heart Study.” British Medical Journal, 337: a2338.
  • Frühwirth-Schnatter, S. (1994). “Data augmentation and dynamic linear models.” Journal of time series analysis, 15(2): 183–202.
  • George, E. I. and McCulloch, R. E. (1993). “Variable selection via Gibbs sampling.” Journal of the American Statistical Association, 88(423): 881–889.
  • Goldstein, J. S. (1992). “A conflict-cooperation scale for WEIS events data.” Journal of Conflict Resolution, 36(2): 369–385.
  • Guhaniyogi, R. and Banerjee, S. (2018). “Meta-kriging: Scalable Bayesian modeling and inference for massive spatial datasets.” Technometrics, (just-accepted).
  • Guhaniyogi, R., Li, C., Savitsky, T. D., and Srivastava, S. (2017). “A Divide-and-Conquer Bayesian Approach to Large-Scale Kriging.” arXiv preprint arXiv: 1712.09767.
  • Guhaniyogi, R. and Rodriguez, A. (2019). “Supplementary Material: Joint Modeling of Longitudinal Relational Data and Exogenous Variables.” Bayesian Analysis.
  • Handcock, M. S., Robins, G., Snijders, T., Moody, J., and Besag, J. (2003). “Assessing degeneracy in statistical models of social networks.” Technical report, Working paper.
  • Hanneke, S., Fu, W., Xing, E. P., et al. (2010). “Discrete temporal models of social networks.” Electronic Journal of Statistics, 4: 585–605.
  • Hanneke, S. and Xing, E. P. (2007). “Discrete temporal models of social networks.” In Statistical network analysis: Models, issues, and new directions, 115–125. Springer.
  • Heaton, M. J., Datta, A., Finley, A., Furrer, R., Guhaniyogi, R., Gerber, F., Gramacy, R. B., Hammerling, D., Katzfuss, M., Lindgren, F., et al. (2017). “Methods for analyzing large spatial data: A review and comparison.” arXiv preprint arXiv:1710.05013.
  • Hoff, P. D. (2005). “Bilinear mixed-effects models for dyadic data.” Journal of the american Statistical association, 100(469): 286–295.
  • Hoff, P. D. (2009). “Multiplicative latent factor models for description and prediction of social networks.” Computational and mathematical organization theory, 15(4): 261.
  • Hoff, P. D. (2015). “Multilinear tensor regression for longitudinal relational data.” The annals of applied statistics, 9(3): 1169.
  • Hoff, P. D., Raftery, A. E., and Handcock, M. S. (2002). “Latent space approaches to social network analysis.” Journal of the american Statistical association, 97(460): 1090–1098.
  • Holland, P. W., Laskey, K. B., and Leinhardt, S. (1983). “Stochastic blockmodels: First steps.” Social networks, 5(2): 109–137.
  • Holland, P. W. and Leinhardt, S. (1981). “An exponential family of probability distributions for directed graphs.” Journal of the american Statistical association, 76(373): 33–50.
  • Ishwaran, H., Rao, J. S., et al. (2005). “Spike and slab variable selection: frequentist and Bayesian strategies.” The Annals of Statistics, 33(2): 730–773.
  • Kalyagin, V., Koldanov, A., Koldanov, P., Pardalos, P., and Zamaraev, V. (2014). “Measures of uncertainty in market network analysis.” Physica A: Statistical Mechanics and its Applications, 413: 59–70.
  • Lin, X. (2010). “Identifying peer effects in student academic achievement by spatial autoregressive models with group unobservables.” Journal of Labor Economics, 28(4): 825–860.
  • Miguel, E., Satyanath, S., and Sergenti, E. (2004). “Economic shocks and civil conflict: An instrumental variables approach.” Journal of political Economy, 112(4): 725–753.
  • Morrow, J. D. (1999). “How could trade affect conflict?” Journal of Peace Research, 36(4): 481–489.
  • Niezink, N. M., Snijders, T. A., et al. (2017). “Co-evolution of social networks and continuous actor attributes.” The Annals of Applied Statistics, 11(4): 1948–1973.
  • Nishihara, R., Murray, I., and Adams, R. P. (2014). “Parallel MCMC with generalized elliptical slice sampling.” The Journal of Machine Learning Research, 15(1): 2087–2112.
  • Polachek, S. W. (1980). “Conflict and trade.” Journal of Conflict resolution, 24(1): 55–78.
  • Polachek, S. W. and Sevastianova, D. (2012). “Does conflict disrupt growth? Evidence of the relationship between political instability and national economic performance.” The Journal of International Trade & Economic Development, 21(3): 361–388.
  • Reuveny, R. (2000). “The trade and conflict debate: A survey of theory, evidence and future research.” Peace Economics, Peace Science and Public Policy, 6(1).
  • Reuveny, R. and Kang, H. (1996). “International trade, political conflict/cooperation, and Granger causality.” American Journal of Political Science, 943–970.
  • Robins, G. and Pattison, P. (2001). “Random graph models for temporal processes in social networks.” Journal of Mathematical Sociology, 25(1): 5–41.
  • Robins, G., Pattison, P., Kalish, Y., and Lusher, D. (2007). “An introduction to exponential random graph (p∗) models for social networks.” Social networks, 29(2): 173–191.
  • Rodríguez, A. and Moser, S. (2015). “Measuring and accounting for strategic abstentions in the US Senate, 1989–2012.” Journal of the Royal Statistical Society: Series C (Applied Statistics), 64(5): 779–797.
  • Rodrik, D. (1999). “Where did all the growth go? External shocks, social conflict, and growth collapses.” Journal of economic growth, 4(4): 385–412.
  • Sarkar, P. and Moore, A. W. (2006). “Dynamic social network analysis using latent space models.” In Advances in Neural Information Processing Systems, 1145–1152.
  • Scott, J. G., Berger, J. O., et al. (2010). “Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem.” The Annals of Statistics, 38(5): 2587–2619.
  • Sewell, D. K. and Chen, Y. (2015). “Latent space models for dynamic networks.” Journal of the American Statistical Association, 110(512): 1646–1657.
  • Shoham, D. A., Hammond, R., Rahmandad, H., Wang, Y., and Hovmand, P. (2015). “Modeling social norms and social influence in obesity.” Current epidemiology reports, 2(1): 71–79.
  • Snijders, T. A. (2002). “Markov chain Monte Carlo estimation of exponential random graph models.” Journal of Social Structure, 3(2): 1–40.
  • Snijders, T. A. (2013). “Stochastic actor-oriented models for network change.” In Evolution of social networks, 193–216. Routledge.
  • Snijders, T. A., Van de Bunt, G. G., and Steglich, C. E. (2010). “Introduction to stochastic actor-based models for network dynamics.” Social networks, 32(1): 44–60.
  • Wasserman, S. and Anderson, C. (1987). “Stochastic a posteriori blockmodels: Construction and assessment.” Social Networks, 9(1): 1–36.
  • Watts, D. J. and Dodds, P. (2009). “Threshold models of social influence.” The Oxford handbook of analytical sociology, 475–497.

Supplemental materials

  • Supplementary Material: Joint Modeling of Longitudinal Relational Data and Exogenous Variables. Supplementary material consists of three sections. Section 1 presents the full conditional posteriors for the MCMC algorithm. Section 2 states and proves theorems related to large support of the prior. Section 3 argues identifiability of $q_{S}$, $q_{A}$, $q_{N}$ and $R^{*}$.