Goodness of Fit Test for Network Models

We study networks from a point of view of exponential random graph models. This flexible and versatile statistical family comes equipped with tools to perform inferential statistics, however the standard toolbox assumes large sample sizes and the use of asymptotic theory. In networks, this assumption is not met, as we only observe a sample of size 1. Moreover, due to their complex nature, model specification is challenging, and many networks exhibit poor model fit. This motivates the study of exact measures of model fit. We explore the inner workings of regular exponential families which allows us to develop a geometric interpretation of model fit. Specifically, we develop a new framework for carrying out statistical inference. We propose two statistics - the MLE and the Error operators - both are exact statistics to be used to compute the Maximum Likelihood Estimator, and the model misspecification. The proposed methodology can be used for any regular exponential family model and is a natural generalization of existing goodness of fit tests developed in literature in the last century. We showcase the methodology on three stochastic blockmodels: the Erdos-Renyi, the additive, and the degeneracy-restricted beta model.