Consistency under sampling of exponential random graph models

TL;DR

This paper investigates whether exponential random graph models (ERGMs) are consistent under sampling, revealing that many popular models violate this property and discussing implications for statistical inference in network analysis.

Contribution

The paper demonstrates that many ERGMs are not consistent under sampling and provides conditions for their consistency, extending to general exponential families of dependent variables.

Findings

Many ERGMs violate the sampling consistency condition.

Satisfying the consistency condition limits ERGM's expressive power.

Provides conditions for maximum likelihood estimation consistency in ERGMs.

Abstract

The growing availability of network data and of scientific interest in distributed systems has led to the rapid development of statistical models of network structure. Typically, however, these are models for the entire network, while the data consists only of a sampled sub-network. Parameters for the whole network, which is what is of interest, are estimated by applying the model to the sub-network. This assumes that the model is consistent under sampling, or, in terms of the theory of stochastic processes, that it defines a projective family. Focusing on the popular class of exponential random graph models (ERGMs), we show that this apparently trivial condition is in fact violated by many popular and scientifically appealing models, and that satisfying it drastically limits ERGM's expressive power. These results are actually special cases of more general results about exponential families of dependent random variables, which we also prove. Using such results, we offer easily checked conditions for the consistency of maximum likelihood estimation in ERGMs, and discuss some possible constructive responses.