Elsevier

Social Networks

Volume 29, Issue 2, May 2007, Pages 173-191
Social Networks

An introduction to exponential random graph (p*) models for social networks

https://doi.org/10.1016/j.socnet.2006.08.002Get rights and content

Abstract

This article provides an introductory summary to the formulation and application of exponential random graph models for social networks. The possible ties among nodes of a network are regarded as random variables, and assumptions about dependencies among these random tie variables determine the general form of the exponential random graph model for the network. Examples of different dependence assumptions and their associated models are given, including Bernoulli, dyad-independent and Markov random graph models. The incorporation of actor attributes in social selection models is also reviewed. Newer, more complex dependence assumptions are briefly outlined. Estimation procedures are discussed, including new methods for Monte Carlo maximum likelihood estimation. We foreshadow the discussion taken up in other papers in this special edition: that the homogeneous Markov random graph models of Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81, 832–842] are not appropriate for many observed networks, whereas the new model specifications of Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handock, M. New specifications for exponential random graph models. Sociological Methodology, in press] offer substantial improvement.

Section snippets

Why model social networks?

There are many well-known techniques that measure properties of a network, of the nodes, or of subsets of nodes (e.g., density, centrality and cohesive subsets). These techniques serve valuable purposes in describing and understanding network features that might bear on particular research questions. Why, then, might we want to go beyond these techniques and search for a well-fitting model of an observed social network, and in particular a statistical model? Reasons for doing so include the

The logic behind p* models for social networks—an outline1

We describe as the observed network the network data the researcher has collected and is interested in modeling. The observed network is regarded as one realization from a set of possible networks with similar important characteristics (at the very least, the same number of actors), that is, as the outcome of some (unknown) stochastic process. In other words, the observed network is seen as one particular pattern of ties out of a large set of possible patterns. In general, we do not know what

The general form of the exponential random graph model: dependence assumptions and parameter constraints

Exponential random graph models have the following form:Pr(Y=y)=1κexpAηAgA(y)where (i) the summation is over all configurations A; (ii) ηA is the parameter corresponding to the configuration A (and is non-zero only if all pairs of variables in A are assumed to be conditionally dependent);2 (iii) gA(y)=yijAyij is the network statistic corresponding to configuration A; gA(y) = 1 if the configuration is observed in the network y, and is 0 otherwise;3

Bernoulli graphs: the simplest dependence assumption

Bernoulli random graph distributions are generated when we assume that edges are independent, for instance if they occur randomly according to a fixed probability α (see Erdös and Renyi, 1959, Frank and Nowicki, 1993). The dependence assumption is simple in this case: all possible distinct ties are independent of one another. We noted above that the only configurations relevant to the model are those in which all possible ties in the configuration are conditionally dependent on each other. When

Estimation

Anderson et al. (1999) in their p* primer used pseudo-likelihood estimation introduced by Strauss and Ikeda (1990) in order to estimate the parameters of Markov models. We now know that, depending on the data, there may be serious problems with pseudo-likelihood estimates for these models. But for Markov random graph models, standard maximum likelihood estimation is not tractable for any but very small networks, because of the difficulties in calculating the normalizing constant in Eq. (1).

A short example: a Markov random graph model for Medici business network

Other papers in this special edition provide examples of fitting exponential random graph models to data, so here we present a very short example. We fit a Markov random graph model for the well-known non-directed network of business connections among 16 Florentine families, available in UCINET 5 (Borgatti et al., 1999). (For a fuller description of the context of the data, see Padgett and Ansell, 1993.) The model includes edge, two-star, three-star and triangle parameters as in Eq. (4). This

Conclusion

This article provides an introductory exposition of the formulation and application of exponential random graph models for social networks. We have concentrated on presenting the underlying logic and derivation of these models. Given the limitations of space, we have only given summary attention to more recent developments which will be discussed in other papers in this special edition.

Recent work on the Markov random graph models of Frank and Strauss (1986) shows that they may be inadequate

Acknowledgements

We thank an anonymous reviewer for helpful comments in improving earlier versions of the paper. This research was assisted by grants from the Australian Research Council.

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