Abstract

Random-Effects Models for Network Dependence

Peter Hoff, (University of Washington), ` hoff@stat.washington.edu`

**Abstract**

One impediment to the statistical analysis of network data has been the difficulty in modeling the dependence among the observations. In the very simple case of binary network data, some researchers have parameterized network dependence in terms of exponential family representations. Accurate parameter estimation in this setting can be difficult, and the most commonly used models often display a significant lack of fit. Additionally, such models are generally limited to binary data. In contrast, random-effects models have been a widely successful tool in capturing statistical dependence for a variety of data types, and allow for prediction, imputation, and hypothesis testing within a general regression context. However, their application to network data has been limited.

We propose the use of novel random-effects structures for the statistical modeling of dependent network data. Such an approach typically proceeds by fitting a standard regression model, except that the error term is decomposed into a set of simple random effects which induce statistical dependence. We decompose the error term into sender-specific, receiver-specific, and dyad-specific random effects. The sender-specific effects, for example, can capture the positive dependence among observations having a common sender, and similarly for the receiver-specific effects. More difficult is the modeling of the dyad-specific effects, which ideally are able to capture more complicated forms of network dependence such as reciprocity, transitivity, and balance. We take a "latent similarity" approach to modeling dyadic dependence, in which the dyad-specific random effect is a simple function measuring the similarity of additional node-specific random effects. Such an approach provides a flexible strategy for the statistical modeling of network data using well known statistical tools such as regression and generalized linear models. The method also provides a model-based graphical representation of network data structure.

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