On the question of effective sample size in network modeling: an asymptotic inquiry

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 - natural analogues of standard questions already posed and addressed in more classical areas of statistics - that have yet to even be posed, much less addressed. 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 particular, adopting the (asymptotic) scaling of the variance of the maximum likelihood parameter estimates as a notion of effective sample size, say neff, 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 yield an order of magnitude difference in neff, from O(nV) to O(n2V). We then explore some practical implications of this result, using both simulation and data on food-sharing from Lamalera, Indonesia.