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Properties of latent variable network models


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.

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Airoldi E. M., Blei D. M., Fienberg S. E., & Xing E. P. (2008). Mixed membership stochastic blockmodels. Journal of Machine Learning Research, 9, 19812014.
Albert R., Jeong H., & Barabási A. L. (1999). Internet: Diameter of the world-wide web. Nature, 401 (6749), 130131.
Albert R., Jeong H., & Barabási A. L. (2000). Error and attack tolerance of complex networks. Nature, 406 (6794), 378382.
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), 1114911152.
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), 335.
Barabási A. L., & Albert R. (1999). Emergence of scaling in random networks. Science, 286 (5439), 509512.
Batagelj V., & Mrvar A. (2006). Pajek datasets.
Boguná M. & Pastor-Satorras R. (2003). Class of correlated random networks with hidden variables. Physical Review E, 68 (3), 036112.
Caimo A., & Friel N. (2011). Bayesian inference for exponential random graph models. Social Networks, 33 (1), 4155.
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.
Cao X., & Ward M. D. (2014). Do democracies attract portfolio investment? Transnational portfolio investments modeled as dynamic network. International Interactions, 40 (2), 216245.
Carlson R. O. (1965). Adoption of educational innovations. Eugene, OR: Center for the Advanced Study of Educational Administration, University of Oregon.
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, 25742601.
Chatterjee S., & Diaconis P. (2013). Estimating and understanding exponential random graph models. Annals of Statistics, 41 (5), 24282461.
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), 1588115886.
Chiu G. S., & Westveld A. H. (2014). A statistical social network model for consumption data in trophic food webs. Statistical Methodology, 17 (4432), 139160.
Daudin J. J., Picard F., & Robin S. (2008). A mixture model for random graphs. Statistics and Computing, 18 (2), 173183.
de Nooy W., Mrvar A., & Batgelj V. (2011). Exploratory social network analysis with Pajek (2nd ed.). Cambridge, UK: Cambridge University Press.
Deprez P. & Wüthrich M. V. (2013). Scale-free percolation in continuum space. arxiv:1312.1948.
Dunbar R. I. M. (1992). Neocortex size as a constraint on group size in primates. Journal of Human Evolution, 22 (6), 469493.
Frank O., & Strauss D. (1986). Markov graphs. Journal of the American Statistical Association, 81 (395), 832842.
Fronczak A., Fronczak P. & Hołyst J. A. (2004). Average path length in random networks. Physical Review E, 70 (5), 056110.
Gollini I., & Murphy T. B. (2016). Joint modelling of multiple network views. Journal of Computational and Graphical Statistics, 25 (1), 246265.
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), 301354.
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), 10901098.
Kiss I. Z., & Green D. M. (2008). Comment on “properties of highly clustered networks.” Physical Review E, 78 (4), 048101.
Krackhardt D. (1999). The ties that torture: Simmelian tie analysis in organizations. Research in the Sociology of Organizations, 16 (1), 183210.
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), 2946.
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), 204213.
Latouche P., Birmelé E., & Ambroise C. (2011). Overlapping stochastic block models with application to the french political blogosphere. Annals of Applied Statistics, 5 (1), 309336.
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), 396405.
MacRae D. (1960). Direct factor analysis of sociometric data. Sociometry, 23 (4), 360371.
Mariadassou M., & Matias C. (2015). Convergence of the groups posterior distribution in latent or stochastic block models. Bernoulli, 21 (1), 537573.
Meester R., & Roy R. (1996). Continuum percolation. Cambridge, UK: Cambridge University Press.
Michael J. H., & Massey J. G. (1997). Modeling the communication network in a sawmill. Forest Products Journal, 47 (9), 2530.
Newman M. E. J. (2001). The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences, 98 (2), 404409.
Newman M. E. J. (2002). Assortative mixing in networks. Physical Review Letters, 89 (20), 208701.
Newman M. E. J. (2003a). Properties of highly clustered networks. Physical Review E, 68 (2), 026121.
Newman M. E. J. (2003b). The structure and function of complex networks. SIAM Review, 45 (2), 167256.
Newman M. E. J. (2003c). Random graphs as models of networks. In Bornholdt S., & Schuster H. G. (Eds.), Handbook of graphs and networks (3568). Berlin: Wiley-VCH.
Newman M. E. J. (2006). Finding community structure in networks using the eigenvectors of matrices. Physical Review E, 74 (3), 036104.
Newman M. E. J. (2009). Random graphs with clustering. Physical Review Letters, 103 (5), 058701.
Newman M. E. J., & Park J. (2003). Why social networks are different from other types of networks. Physical Review E, 68 (3), 036122.
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.
Nowicki K., & Snijders T. A. B. (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96 (455), 10771087.
Olhede S. C., & Wolfe P. J. Degree-based network models. arxiv:1211.6537.
Padgett J. F., & Ansell C. K. (1993). Robust Action and the Rise of the Medici, 1400–1434. American journal of sociology, 98 (6), 12591319.
Penrose M. D. (1991). On a continuum percolation model. Advances in Applied Probability, 23, 536556.
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), 821849.
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), 901919.
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.
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), 647676.
Shalizi C. R., & Rinaldo A. (2013). Consistency under sampling of exponential random graph models. Annals of Statistics, 41 (2), 508535.
Söderberg B. (2002). General formalism for inhomogeneous random graphs. Physical Review E, 66 (6), 066121.
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), 295318.
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, 703717.
Watts D. J., & Strogatz S. H. (1998). Collective dynamics of small-world networks. Nature, 393 (6684), 440442.
Williams R. J., & Martinez N. D. (2000). Simple rules yield complex food webs. Nature, 404 (6774), 180183.
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Network Science
  • ISSN: 2050-1242
  • EISSN: 2050-1250
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