On the Question of Effective Sample Size in Network Modeling: An Asymptotic Inquiry

TL;DR

This paper investigates the concept of effective sample size in network models, showing it varies significantly with network sparsity and structure, impacting statistical inference in network analysis.

Contribution

It introduces the notion of effective sample size in network modeling and demonstrates its dependence on network properties like sparsity and triadic effects.

Findings

Effective sample size can differ by an order of magnitude depending on network sparsity.

Networks with different properties exhibit distinct asymptotic behaviors of parameter estimates.

Simulation and real data illustrate the practical implications of these theoretical results.

Abstract

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 $n_V$ 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 ($n_{\mathrm{eff}}$), we show that when modeling the overall propensity to have ties and the propensity to reciprocate ties, whether the networks are sparse or not under the model (i.e., having a constant or an increasing number of ties per vertex, respectively) is sufficient to yield an order of magnitude difference in $n_{\mathrm{eff}}$, from $O(n_V)$ to $O(n^2_V)$. In addition, we report simulation study results that suggest similar properties for models for triadic (friend-of-a-friend) effects. We then explore some practical implications of this result, using both simulation and data on food-sharing from Lamalera, Indonesia.