Exponential random graph models and pendant-triangle statistics

The paper builds on the framework proposed by Pattison and Snijders (2012) for specifying exponential random graph models (ERGMs) for social networks. We briefly review the two-dimensional hierarchy of potential dependence structures for network tie variables that they outlined and provide proofs of the relationships among the model forms and of the nature of their sufficient statistics, noting that models in the hierarchy have the potential to reflect the outcome of processes of cohesion, closure, boundary and bridge formation and path creation over short or longer network distances. We then focus on the so-called partial inclusion dependence assumptions among network tie variables and the pendant-triangle, or paw, statistics to which they give rise, and illustrate their application in an empirical setting. We argue that the partial inclusion assumption leads to models that can reflect processes of boundary and bridge formation and that the model hierarchy provides a broad and useful framework for the statistical analysis of network data. We demonstrate in the chosen setting that pendant- triangle (or paw) effects, in particular, lead to a marked improvement in goodness-of-fit and hence add a potentially valuable capacity for modelling social networks.