Stochastic blockmodels with growing number of classes

Latent variable models are frequently used to identify structure in dichotomousnetwork data, in part because they give rise to a Bernoulli product likelihood thatis both well understood and consistent with the notion of exchangeable randomgraphs. In this article we propose conservative confidence sets that hold with re-spect to these underlying Bernoulli parameters as a function of any given partitionof network nodes, enabling us to assess estimates of residual network structure,that is, structure that cannot be explained by known covariates and thus cannot beeasily verified by manual inspection. We demonstrate the proposed methodologyby analyzing student friendship networks from the National Longitudinal Surveyof Adolescent Health that include race, gender, and school year as covariates. Weemploy a stochastic expectation-maximization algorithm to fit a logistic regres-sion model that includes these explanatory variables as well as a latent stochasticblockmodel component and additional node-specific effects. Although maximum-likelihood estimates do not appear consistent in this context, we are able to evalu-ate confidence sets as a function of different blockmodel partitions, which enablesus to qualitatively assess the significance of estimated residual network structurerelative to a baseline, which models covariates but lacks block structure.