The Annals of Statistics
- Ann. Statist.
- Volume 41, Number 2 (2013), 508-535.
Consistency under sampling of exponential random graph models
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.
Ann. Statist. Volume 41, Number 2 (2013), 508-535.
First available in Project Euclid: 26 April 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 91D30: Social networks 62B05: Sufficient statistics and fields
Secondary: 60G51: Processes with independent increments; Lévy processes 62M99: None of the above, but in this section 62M09: Non-Markovian processes: estimation
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.
- 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.