Discussion on the paper by Handcock, Raftery and Tantrum.

Snijders, T. A. B. and Robinson, T. and Atkinson, A. C. and Riani, M. and Gormley, I. C. and Murphy, T. B. and Sweeting, T. and Leslie, D. S. and Longford, N. T. and Kent, J. T. and Lawrance, T. and Airoldi, E. M. and Besag, J. and Blei, D. and Fienberg, S. E. and Breiger, R. and Butts, C. T. and Doreian, P. and Batagelj, V. and Ferligoj, A. and Draper, D. and Van Duijn, M. A. J. and Faust, K. and Petrescu-Prahova, M. and Forster, J. J. and Gelman, A. and Goodreau, S. M. and Greenwood, P. E. and Gruenberg, Katharina Tatjana and Francis, Brian J. and Hennig, C. and Hoff, P. D. and Hunter, D. R. and Husmeier, D. and Glasbey, C. and Krackhardt, D. and Kuha, J. and Skrondal, A. and Lawson, A. and Liao, T. F. and Mendes, B. and Draper, D. and Reinert, G. and Richardson, S. and Lewin, A. and Titterington, D. M. and Wasserman, S. and Werhli, A. V. and Ghazal, P. (2007) Discussion on the paper by Handcock, Raftery and Tantrum. Journal of the Royal Statistical Society: Series A Statistics in Society, 170 (2). pp. 322-354. ISSN 0964-1998

Abstract

Network models are widely used to represent relations between interacting units or actors. Network data often exhibit transitivity, meaning that two actors that have ties to a third actor are more likely to be tied than actors that do not, homophily by attributes of the actors or dyads, and clustering. Interest often focuses on finding clusters of actors or ties, and the number of groups in the data is typically unknown. We propose a new model, the latent position cluster model, under which the probability of a tie between two actors depends on the distance between them in an unobserved Euclidean 'social space', and the actors' locations in the latent social space arise from a mixture of distributions, each corresponding to a cluster. We propose two estimation methods: a two-stage maximum likelihood method and a fully Bayesian method that uses Markov chain Monte Carlo sampling. The former is quicker and simpler, but the latter performs better. We also propose a Bayesian way of determining the number of clusters that are present by using approximate conditional Bayes factors. Our model represents transitivity, homophily by attributes and clustering simultaneously and does not require the number of clusters to be known. The model makes it easy to simulate realistic networks with clustering, which are potentially useful as inputs to models of more complex systems of which the network is part, such as epidemic models of infectious disease. We apply the model to two networks of social relations. A free software package in the R statistical language, latentnet, is available to analyse data by using the model.