On the Question of Effective Sample Size in Network Modeling

The modeling and analysis of networks and network data has seen an explosion of interest in recent years and represents an exciting direction for potential growth in statistics. Despite the already substantial amount of work done in this area to date by researchers from various disciplines, however, there remain many questions of a decidedly foundational nature — natu-ral analogues of standard questions already posed and addressed in more classical areas of statistics — that have yet to even be posed, much less ad-dressed. Here we raise and consider one such question in connection with network modeling. Specifically, we ask, “Given an observed network, what is the sample size? ” Using simple, illustrative examples from the class of exponential random graph models, we show that the answer to this question can very much depend on basic properties of the networks expected under the model, as the number of vertices Nv in the network grows. In particu-lar, we show that whether the networks are sparse or not under our model (i.e., having relatively few or many edges between vertices, respectively) is sufficient to change the asymptotic rates for maximum likelihood parameter estimation by an order of magnitude, from N1/2v to Nv. We then explor