## The Annals of Statistics

### Consistency under sampling of exponential random graph models

#### 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.

#### Article information

Source
Ann. Statist. Volume 41, Number 2 (2013), 508-535.

Dates
First available in Project Euclid: 26 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1366980556

Digital Object Identifier
doi:10.1214/12-AOS1044

Mathematical Reviews number (MathSciNet)
MR3099112

Zentralblatt MATH identifier
1269.91066

#### Citation

Shalizi, Cosma Rohilla; Rinaldo, Alessandro. Consistency under sampling of exponential random graph models. Ann. Statist. 41 (2013), no. 2, 508--535. doi:10.1214/12-AOS1044. https://projecteuclid.org/euclid.aos/1366980556.

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#### Supplemental materials

• Supplementary material: Non-uniform base measures and conditional projectibility. In the supplementary material we consider the case of nonuniform base measures and also study a more general form of conditional projectibility, which implies, in particular, that stochastic block models are projective.