The Problem of Scaling in Exponential Random Graph Models

This study shows that residual variation can cause problems related to scaling in exponential random graph models (ERGM). Residual variation is likely to exist when there are unmeasured variables in a model-even those uncorrelated with other predictors-or when the logistic form of the model is inappropriate. As a consequence, coefficients cannot be interpreted as effect sizes or compared between models and homophily coefficients, as well as other interaction coefficients, cannot be interpreted as substantive effects in most ERGM applications. We conduct a series of simulations considering the substantive impact of these issues, revealing that realistic levels of residual variation can have large consequences for ERGM inference. A flexible methodological framework is introduced to overcome these problems. Formal tests of mediation and moderation are also proposed. These methods are applied to revisit the relationship between selective mixing and triadic closure in a large AddHealth school friendship network. Extensions to other classes of statistical work models are discussed.