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Simultaneous and temporal autoregressive network models

Published online by Cambridge University Press:  12 February 2018

DANIEL K. SEWELL Open the ORCID record for DANIEL K. SEWELL [Opens in a new window]
DANIEL K. SEWELL*
Affiliation:
Department of Biostatistics, University of Iowa, Iowa City, IA 52242, USA (e-mail: daniel-sewell@uiowa.edu)
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Abstract

While logistic regression models are easily accessible to researchers, when applied to network data there are unrealistic assumptions made about the dependence structure of the data. For temporal networks measured in discrete time, recent work has made good advances (Almquist & Butts, 2014), but there is still the assumption that the dyads are conditionally independent given the edge histories. This assumption can be quite strong and is sometimes difficult to justify. If time steps are rather large, one would typically expect not only the existence of temporal dependencies among the dyads across observed time points but also the existence of simultaneous dependencies affecting how the dyads of the network co-evolve. We propose a general observation-driven model for dynamic networks that overcomes this problem by modeling both the mean and the covariance structures as functions of the edge histories using a flexible autoregressive approach. This approach can be shown to fit into a generalized linear mixed model framework. We propose a visualization method that provides evidence concerning the existence of simultaneous dependence. We describe a simulation study to determine the method's performance in the presence and absence of simultaneous dependence, and we analyze both a proximity network from conference attendees and a world trade network. We also use this last data set to illustrate how simultaneous dependencies become more prominent as the time intervals become coarser.

Keywords

dependence structuresdynamic networksgeneralized linear mixed modelsmultivariate probitobservation-driven model
Type
Research Article
Information
Network Science , Volume 6 , Issue 2 , June 2018 , pp. 204 - 231
Copyright
Copyright © Cambridge University Press 2018 

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References

Airoldi, E. M., Fienberg, S. E., & Xing, E. P. (2008). Mixed membership stochastic blockmodels. Journal of Machine Learning Research, 9, 1981–2014.Google Scholar
Almquist, Z. W., & Butts, C. T. (2013). Dynamic network logistic regression: A logistic choice analysis of inter-and intra-group blog citation dynamics in the 2004 us presidential election. Political Analysis, 21 (4), 430–448.CrossRefGoogle ScholarPubMed
Almquist, Z. W., & Butts, C. T. (2014). Logistic network regression for scalable analysis of networks with joint edge/vertex dynamics. Sociological Methodology, 44 (1), 273–321.CrossRefGoogle ScholarPubMed
Anderson, T. W. (1973). Asymptotically efficient estimation of covariance matrices with linear structure. The Annals of Statistics, 1 (1), 135–141.Google Scholar
Barbieri, K., & Keshk, O. (2012). Correlates of war project trade data set codebook, version 3.0. Retrieved April 29, 2015, from http://correlatesofwar.org.Google Scholar
Barbieri, K., Keshk, O. M. G., & Pollins, B. M. (2009). Trading data: Evaluating our assumptions and coding rules. Conflict Management and Peace Science, 26 (5), 471–491.Google Scholar
Bharadwaj, H. M. (2016). Generalized linear mixed models in hearing science. The Journal of the Acoustical Society of America, 139 (4), 2101–2101.Google Scholar
Bickel, P., Choi, D., Chang, X., & Zhang, H. (2013). Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels. The Annals of Statistics, 41 (4), 1922–1943.Google Scholar
Bolker, B. M., Brooks, M. E., Clark, C. J., Geange, S. W., Poulsen, J. R., Stevens, M. H. H., & White, J.-S. S. (2009). Generalized linear mixed models: A practical guide for ecology and evolution. Trends in Ecology & Evolution, 24 (3), 127–135.Google Scholar
Chen, G., & Tsurumi, H. (2010). Probit and logit model selection. Communications in Statistics, 40 (1), 159–175.Google Scholar
Cox, D. R. (1981). Statistical analysis of time series: Some recent developments. Scandinavian Journal of Statistics, 8 (2), 93–115.Google Scholar
Demidenko, E. (2013). Mixed models: theory and applications with R. Hoboken, NJ: John Wiley & Sons.Google Scholar
Duijn, M. A. J., Snijders, T. A. B., & Zijlstra, B. J. H. (2004). p2: A random effects model with covariates for directed graphs. Statistica Neerlandica, 58 (2), 234–254.Google Scholar
Durante, D., & Dunson, D. B. (2014). Nonparametric bayes dynamic modelling of relational data. Biometrika, 101 (4), 125–138.Google Scholar
Eagle, N., & Pentland, A. (2006). Reality mining: Sensing complex social systems. Personal and Ubiquitous Computing, 10 (4), 255–268.CrossRefGoogle Scholar
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50 (4), 987–1007.Google Scholar
Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81 (395), 832–842.CrossRefGoogle Scholar
Gbur, E. (2012). Analysis of generalized linear mixed models in the agricultural and natural resources sciences. Madison, WI: Soil Science Society of America.Google Scholar
Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2004). Bayesian data analysis (3rd ed.). Boca Raton, USA: Chapman & Hall/CRC.Google Scholar
Gibler, D. M. (2009). International military alliances, 1648–2008. Washington, DC: CQ Press.Google Scholar
Handcock, M. S., Raftery, A. E., & Tantrum, J. M. (2007). Model-based clustering for social networks. Journal of the Royal Statistical Society, Series A, 170 (2), 301–354.Google Scholar
Hanneke, S., Fu, W., & Xing, E. P. (2010). Discrete temporal models of social networks. Electronic Journal of Statistics, 4, 585–605.Google Scholar
Hoff, P. D. (2005). Bilinear mixed-effects models for dyadic data. Journal of the American Statistical Association, 100 (469), 286–295.Google Scholar
Hoff, P. D., Raftery, A. E., & Handcock, M. S. (2002). Latent space approaches to social network analysis. Journal of the American Statistical Association, 97 (460), 1090–1098.Google Scholar
Holland, P. W., & Leinhardt, S. (1977). A dynamic model for social networks. Journal of Mathematical Sociology, 5 (1), 5–20.Google Scholar
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.Google Scholar
Holland, P. W., Laskey, K. B., & Leinhardt, S. (1983). Stochastic blockmodels: First steps. Social Networks, 5 (2), 109–137.Google 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.Google Scholar
Jin, I. H., & Liang, F. (2013). Fitting social network models using varying truncation stochastic approximation mcmc algorithm. Journal of Computational and Graphical Statistics, 22 (4), 927–952.Google Scholar
Krackhardt, D., & Handcock, M. S. (2007). Heider vs simmel: Emergent features in dynamic structures. In Airoldi, E. M., Blei, D. M., Fienberg, S. E., Goldenberg, A., Xing, E. P., & Zheng, A. X. (Eds.), Statistical network analysis: Models, issues, and new directions, ICML 2006 Workshop on Statistical Network Analysis (pp. 14–27). Pittsburgh, PA: Springer.Google Scholar
Krivitsky, P. N., & Handcock, M. S. (2014). A separable model for dynamic networks. Journal of the Royal Statistical Society, Series B, 76 (1), 29–46.Google Scholar
Krivitsky, P. N., Handcock, M. S., Raftery, A. E., & Hoff, P. D. (2009). Representing degree distributions, clustering, and homophily in social networks with latent cluster random effects models. Social Networks, 31 (3), 204–213.Google Scholar
Krueger, D. C., & Montgomery, D. C. (2014). Modeling and analyzing semiconductor yield with generalized linear mixed models. Applied Stochastic Models in Business & Industry, 30 (6), 691–707.Google Scholar
Leenders, R. Th., A. J. (1995). Models for network dynamics: A markovian framework. The Journal of Mathematical Sociology, 20 (1), 1–21.Google Scholar
Lerner, J., Indlekofer, N., Nick, B., & Brandes, U. (2013). Conditional independence in dynamic networks. Journal of Mathematical Psychology, 57 (6), 275–283.CrossRefGoogle Scholar
Maoz, Z., & Henderson, E. A. (2013). The world religion dataset, 1945–2010: Logic, estimates, and trends. International Interactions, 39 (3), 265–291.CrossRefGoogle Scholar
Okabayashi, S. (2011). Parameter estimation in social network models. Ph.D. thesis. University of Minnesota, Minneapolis, MN.Google Scholar
OpenAMD. (2008). AMD hope RFID data. Retrieved September 21, 2015, from http://networkdata.ics.uci.edu/data.php?d=amdhope.Google Scholar
Raftery, A. E., Niu, X., Hoff, P. D., & Yeung, K. Y. (2012). Fast inference for the latent space network model using a case-control approximate likelihood. Journal of Computational and Graphical Statistics, 21 (4), 901–919.Google Scholar
Ripley, R., Boitmanis, K., & Snijders, T. A. B. (2013). Rsiena: Siena – Simulation investigation for empirical network analysis. R package version 1.1-232.Google Scholar
Robins, G., & Pattison, P. (2001). Random graph models for temporal processes in social networks. Journal of Mathematical Sociology, 25 (1), 5–41.Google Scholar
Salter-Townshend, M., & Murphy, T. B. (2013). Variational bayesian inference for the latent position cluster model for network data. Computational Statistics & Data Analysis, 57 (1), 661–671.Google Scholar
Sarkar, P., & Moore, A. W. (2005). Dynamic social network analysis using latent space models. ACM SIGKDD Explorations Newsletter, 7 (2), 31–40.Google Scholar
Sewell, D. K., & Chen, Y. (2015). Latent space models for dynamic networks. Journal of the American Statistical Association, 110 (512), 1646–1657.CrossRefGoogle Scholar
Shephard, N. (1995). Generalized linear autoregressions. Economics Papers 8. Economics Group, Nuffield College, University of Oxford.Google Scholar
Simmel, G., & Wolff, K. H. (1950). The sociology of Georg Simmel. vol. 92892. New York, NY: Simon and Schuster.Google Scholar
Snijders, T. A. B. (1996). Stochastic actor-oriented models for network change. Journal of Mathematical Sociology, 21 (1–2), 149–172.Google Scholar
Snijders, T. A. B., & Nowicki, K. (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure. Journal of Classification, 14 (1), 75–100.Google Scholar
Vanhems, P, Barrat, A., Cattuto, C., Pinton, J.-F., Khanafer, N., Régis, C., . . . Voirin, N. (2013). Estimating potential infection transmission routes in hospital wards using wearable proximity sensors. PLoS One, 8 (9), e73970.Google Scholar
Wang, Y. J., & Wong, G. Y. (1987). Stochastic blockmodels for directed graphs. Journal of the American Statistical Association, 82, 8–19.CrossRefGoogle Scholar
Warner, R. M., Kenny, D. A., & Stoto, M. (1979). A new round robin analysis of variance for social interaction data. Journal of Personality and Social Psychology, 37 (10), 1742–1757.Google Scholar
Wasserman, S. (1980). Analyzing social networks as stochastic processes. Journal of the American Statistical Association, 75 (370), 280–294.Google Scholar
Wasserman, S., & Faust, K. (1994). Social network analysis: methods and applications. New York, NY: Cambridge University Press.Google Scholar
Westveld, A. H., & Hoff, P. D. (2011). A mixed effects model for longitudinal relational and network data, with applications to international trade and conflict. The Annals of Applied Statistics, 5 (2A), 843–872.Google Scholar
Xing, E. P., Fu, W., & Song, L. (2010). A state-space mixed membership blockmodel for dynamic network tomography. The Annals of Applied Statistics, 4 (2), 535–566.Google Scholar
Zeger, S. L., & Qaqish, B. (1988). Markov regression models for time series: A quasi-likelihood approach. Biometrics, 44 (4), 1019–1031.Google Scholar