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Estimation and prediction for stochastic blockmodels for graphs with latent block structure. (English) Zbl 0896.62063
Summary: A statistical approach to a posteriori blockmodeling for graphs is proposed. The model assumes that the vertices of the graph are partitioned into two unknown blocks and that the probability of an edge between two vertices depends only on the blocks to which they belong. Statistical procedures are derived for estimating the probabilities of edges and for predicting the block structure from observations of the edge patterns only. ML estimators can be computed using the EM algorithm, but this strategy is practical only for small graphs. A Bayesian estimator, based on Gibbs sampling, is proposed. This estimator is practical also for large graphs. When ML estimators are used, the block structure can be predicted based on predictive likelihood. When Gibbs sampling is used, the block structure can be predicted from posterior predictive probabilities.
A side result is that when the number of vertices tends to infinity while the probabilities remain constant, the block structure can be recovered correctly with probability tending to 1.

62H99 Multivariate analysis
05C80 Random graphs (graph-theoretic aspects)
62P25 Applications of statistics to social sciences
05C90 Applications of graph theory
62F15 Bayesian inference
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