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Properties of latent variable network models

Published online by Cambridge University Press:  12 December 2016

RICCARDO RASTELLI Open the ORCID record for RICCARDO RASTELLI [Opens in a new window] ,
NIAL FRIEL  and
ADRIAN E. RAFTERY
RICCARDO RASTELLI
Affiliation:
School of Mathematics and Statistics, University College Dublin, Dublin, Ireland Insight: Centre for Data Analytics, Ireland (e-mail: riccardo.rastelli@ucdconnect.ie; nial.friel@ucd.ie)
NIAL FRIEL
Affiliation:
School of Mathematics and Statistics, University College Dublin, Dublin, Ireland Insight: Centre for Data Analytics, Ireland (e-mail: riccardo.rastelli@ucdconnect.ie; nial.friel@ucd.ie)
ADRIAN E. RAFTERY
Affiliation:
Department of Statistics and Sociology, University of Washington, Seattle, USA (e-mail: raftery@u.washington.edu)
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Abstract

We derive properties of latent variable models for networks, a broad class of models that includes the widely used latent position models. We characterize several features of interest, with particular focus on the degree distribution, clustering coefficient, average path length, and degree correlations. We introduce the Gaussian latent position model, and derive analytic expressions and asymptotic approximations for its network properties. We pay particular attention to one special case, the Gaussian latent position model with random effects, and show that it can represent the heavy-tailed degree distributions, positive asymptotic clustering coefficients, and small-world behaviors that often occur in observed social networks. Finally, we illustrate the ability of the models to capture important features of real networks through several well-known datasets.

Keywords

fitness modelslatent position modelslatent variable modelsrandom graphssocial networks
Type
Research Article
Information
Network Science , Volume 4 , Issue 4 , December 2016 , pp. 407 - 432
Copyright
Copyright © Cambridge University Press 2016 

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References

Airoldi, E. M., Blei, D. M., Fienberg, S. E., & Xing, E. P. (2008). Mixed membership stochastic blockmodels. Journal of Machine Learning Research, 9, 1981–2014.Google ScholarPubMed
Albert, R., Jeong, H., & Barabási, A. L. (1999). Internet: Diameter of the world-wide web. Nature, 401 (6749), 130–131.CrossRefGoogle Scholar
Albert, R., Jeong, H., & Barabási, A. L. (2000). Error and attack tolerance of complex networks. Nature, 406 (6794), 378–382.CrossRefGoogle ScholarPubMed
Amaral, L. A. N., Scala, A., Barthelemy, M., & Stanley, H. E. (2000). Classes of small-world networks. Proceedings of the National Academy of Sciences, 97 (21), 11149–11152.CrossRefGoogle ScholarPubMed
Ambroise, C., & Matias, C. (2012). New consistent and asymptotically normal parameter estimates for random-graph mixture models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 74 (1), 3–35.CrossRefGoogle Scholar
Barabási, A. L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286 (5439), 509–512.CrossRefGoogle ScholarPubMed
Batagelj, V., & Mrvar, A. (2006). Pajek datasets. http://vlado.fmf.uni-lj.si/pub/networks/data/ Google Scholar
Boguná, M. & Pastor-Satorras, R. (2003). Class of correlated random networks with hidden variables. Physical Review E, 68 (3), 036112.CrossRefGoogle ScholarPubMed
Caimo, A., & Friel, N. (2011). Bayesian inference for exponential random graph models. Social Networks, 33 (1), 41–55.CrossRefGoogle Scholar
Caldarelli, G., Capocci, A., De Los Rios, P., & Muñoz, M. A. (2002). Scale-free networks from varying vertex intrinsic fitness. Physical Review Letters, 89 (25), 258702.CrossRefGoogle ScholarPubMed
Cao, X., & Ward, M. D. (2014). Do democracies attract portfolio investment? Transnational portfolio investments modeled as dynamic network. International Interactions, 40 (2), 216–245.CrossRefGoogle Scholar
Carlson, R. O. (1965). Adoption of educational innovations. Eugene, OR: Center for the Advanced Study of Educational Administration, University of Oregon.Google Scholar
Channarond, A., Daudin, J. J., & Robin, S. (2012). Classification and estimation in the stochastic blockmodel based on the empirical degrees. Electronic Journal of Statistics, 6, 2574–2601.CrossRefGoogle Scholar
Chatterjee, S., & Diaconis, P. (2013). Estimating and understanding exponential random graph models. Annals of Statistics, 41 (5), 2428–2461.CrossRefGoogle Scholar
Chiu, G. S., & Westveld, A. H. (2011). A unifying approach for food webs, phylogeny, social networks, and statistics. Proceedings of the National Academy of Sciences, 108 (38), 15881–15886.CrossRefGoogle ScholarPubMed
Chiu, G. S., & Westveld, A. H. (2014). A statistical social network model for consumption data in trophic food webs. Statistical Methodology, 17 (4432), 139–160.CrossRefGoogle Scholar
Daudin, J. J., Picard, F., & Robin, S. (2008). A mixture model for random graphs. Statistics and Computing, 18 (2), 173–183.CrossRefGoogle Scholar
de Nooy, W., Mrvar, A., & Batgelj, V. (2011). Exploratory social network analysis with Pajek (2nd ed.). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Deprez, P. & Wüthrich, M. V. (2013). Scale-free percolation in continuum space. arxiv:1312.1948.Google Scholar
Dunbar, R. I. M. (1992). Neocortex size as a constraint on group size in primates. Journal of Human Evolution, 22 (6), 469–493.CrossRefGoogle Scholar
Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81 (395), 832–842.CrossRefGoogle Scholar
Fronczak, A., Fronczak, P. & Hołyst, J. A. (2004). Average path length in random networks. Physical Review E, 70 (5), 056110.CrossRefGoogle ScholarPubMed
Gollini, I., & Murphy, T. B. (2016). Joint modelling of multiple network views. Journal of Computational and Graphical Statistics, 25 (1), 246–265.CrossRefGoogle 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 (Statistics in Society), 170 (2), 301–354.CrossRefGoogle 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.CrossRefGoogle Scholar
Kiss, I. Z., & Green, D. M. (2008). Comment on “properties of highly clustered networks.” Physical Review E, 78 (4), 048101.CrossRefGoogle Scholar
Krackhardt, D. (1999). The ties that torture: Simmelian tie analysis in organizations. Research in the Sociology of Organizations, 16 (1), 183–210.Google Scholar
Krivitsky, P. N., & Handcock, M. S. (2014). A separable model for dynamic networks. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76 (1), 29–46.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle ScholarPubMed
Latouche, P., Birmelé, E., & Ambroise, C. (2011). Overlapping stochastic block models with application to the french political blogosphere. Annals of Applied Statistics, 5 (1), 309–336.CrossRefGoogle Scholar
Lusseau, D., Schneider, K., Boisseau, O. J., Haase, P., Slooten, E., & Dawson, S. M. (2003). The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting associations. Behavioral Ecology and Sociobiology, 54 (4), 396–405.CrossRefGoogle Scholar
MacRae, D. (1960). Direct factor analysis of sociometric data. Sociometry, 23 (4), 360–371.CrossRefGoogle Scholar
Mariadassou, M., & Matias, C. (2015). Convergence of the groups posterior distribution in latent or stochastic block models. Bernoulli, 21 (1), 537–573.CrossRefGoogle Scholar
Meester, R., & Roy, R. (1996). Continuum percolation. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Michael, J. H., & Massey, J. G. (1997). Modeling the communication network in a sawmill. Forest Products Journal, 47 (9), 25–30.Google Scholar
Newman, M. E. J. (2001). The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences, 98 (2), 404–409.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2002). Assortative mixing in networks. Physical Review Letters, 89 (20), 208701.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2003a). Properties of highly clustered networks. Physical Review E, 68 (2), 026121.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2003b). The structure and function of complex networks. SIAM Review, 45 (2), 167–256.CrossRefGoogle Scholar
Newman, M. E. J. (2003c). Random graphs as models of networks. In Bornholdt, S., & Schuster, H. G. (Eds.), Handbook of graphs and networks (35–68). Berlin: Wiley-VCH.Google Scholar
Newman, M. E. J. (2006). Finding community structure in networks using the eigenvectors of matrices. Physical Review E, 74 (3), 036104.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2009). Random graphs with clustering. Physical Review Letters, 103 (5), 058701.CrossRefGoogle ScholarPubMed
Newman, M. E. J., & Park, J. (2003). Why social networks are different from other types of networks. Physical Review E, 68 (3), 036122.CrossRefGoogle ScholarPubMed
Newman, M. E. J., Strogatz, S. H., & Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Physical Review E, 64 (2), 026118.CrossRefGoogle ScholarPubMed
Nowicki, K., & Snijders, T. A. B. (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96 (455), 1077–1087.CrossRefGoogle Scholar
Olhede, S. C., & Wolfe, P. J. Degree-based network models. arxiv:1211.6537.Google Scholar
Padgett, J. F., & Ansell, C. K. (1993). Robust Action and the Rise of the Medici, 1400–1434. American journal of sociology, 98 (6), 1259–1319.CrossRefGoogle Scholar
Penrose, M. D. (1991). On a continuum percolation model. Advances in Applied Probability, 23, 536–556.CrossRefGoogle Scholar
Perry, P. O., & Wolfe, P. J. (2013). Point process modelling for directed interaction networks. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75 (5), 821–849.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Sampson, S. F. (1968). A novitiate in a period of change: An experimental and case study of social relationships. Ph.D. thesis, Cornell University, September.Google Scholar
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
Shalizi, C. R., & Rinaldo, A. (2013). Consistency under sampling of exponential random graph models. Annals of Statistics, 41 (2), 508–535.CrossRefGoogle ScholarPubMed
Söderberg, B. (2002). General formalism for inhomogeneous random graphs. Physical Review E, 66 (6), 066121.CrossRefGoogle ScholarPubMed
Sweet, T. M., Thomas, A. C., & Junker, B. W. (2013). Hierarchical network models for education research: Hierarchical latent space models. Journal of Educational and Behavioral Statistics, 38 (3), 295–318.CrossRefGoogle Scholar
Wang, H., Tang, M., Park, Y., & Priebe, C. E. (2014). Locality statistics for anomaly detection in time series of graphs. IEEE Transactions on Signal Processing, 62, 703–717.CrossRefGoogle Scholar
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of small-world networks. Nature, 393 (6684), 440–442.CrossRefGoogle ScholarPubMed
Williams, R. J., & Martinez, N. D. (2000). Simple rules yield complex food webs. Nature, 404 (6774), 180–183.CrossRefGoogle ScholarPubMed
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