Curved exponential family models for social networks

Abstract

Curved exponential family models are a useful generalization of exponential random graph models (ERGMs). In particular, models involving the alternating k-star, alternating k-triangle, and alternating k-twopath statistics of Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins, G.L., Handcock, M.S., in press. New specifications for exponential random graph models. Sociological Methodology] may be viewed as curved exponential family models. This article unifies recent material in the literature regarding curved exponential family models for networks in general and models involving these alternating statistics in particular. It also discusses the intuition behind rewriting the three alternating statistics in terms of the degree distribution and the recently introduced shared partner distributions. This intuition suggests a redefinition of the alternating k-star statistic. Finally, this article demonstrates the use of the statnet package in R for fitting models of this sort, comparing new results on an oft-studied network dataset with results found in the literature.

Introduction

For a fixed set of n actors, or nodes, and a network on those nodes, assume that Y denotes the $n\times n$ adjacency matrix for the network; that is

$$Y_{ij}=\left\{\begin{array}{cc}1\hfill & \text{if an edge exists from}i\text{to}j\text{,}\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\right.$$

In some social networks applications, the goal is to produce a probabilistic model for Y based on an observed network dataset. It is the goal of this article to explain a particular class of models, called curved exponential family models, for achieving this end. We assume here that the reader is at least somewhat conversant in certain basic techniques of statistical modelling such as logistic regression, though not necessarily familiar with the intricacies of statistical modelling of networks.

Implicit in definition (1) is the fact that we assume no valued or multiple edges are allowed; furthermore, we disallow self-edges, so $Y_{ii}=0$ for all i. Finally, we will treat only undirected networks in this article, which implies that $Y_{ij}=Y_{ji}$, though we make this choice only to simplify some of the arguments; there is little difficulty in extending all of the results here to the case of directed networks.

Beyond the network information contained in Y, there are often additional data, such as a set of measured characteristics for each node in the network. For instance, when the nodes are people, we may know each person’s age and sex. Throughout this article, we let X denote the additional data.

We assume throughout this article that the probability of observing a particular network is a function of statistics that may depend on the network itself as well as covariates measured on the nodes. For the particular class of models known as exponential random graph models (ERGMs), the relationship between a particular graph y and its probability of occurrence conditional on the additional data X is generally expressed as

$$P_{\eta }(\text{Y}=\text{y})=\frac{\exp \{\sum _{i=1}^p{\eta }_i{g}_i(\text{y}\text{,}\text{X})\}}{\kappa (\eta )}=\frac{\exp \{{\eta }^t\text{g}(\text{y}\text{,}\text{X})\}}{\kappa (\eta )}\text{,}$$

where $\text{g}(\text{y}\text{,}\text{X})$ is a user-defined p-vector of statistics and $\eta \in R^p$ denotes the statistical parameter governing the probabilistic formation of the network. The denominator, $\kappa (\eta )$, is a normalizing constant that ensures that the sum of (2) over all possible y equals 1.

ERGMs are sometimes known in the social networks literature as p-star models (Wasserman and Pattison, 1996). We use “ERGM” instead of “p-star” here due to the vast statistical literature covering exponential family models (e.g., Barndorff-Nielsen, 1978, Brown, 1986). Nevertheless, we consider “p-star” to be synonymous with “ERGM” in this article, with one caveat: Wasserman and Pattison (1996) used a method of parameter estimation, maximum pseudo-likelihood estimation, that has come to be closely associated with the p-star models themselves. However, in this article we separate the name of the models (ERGMs or p-star models) from the method of estimating their parameters. In particular, we do not discuss maximum pseudo-likelihood estimation here, focusing instead on the better-understood method of maximum likelihood estimation. ERGMs are discussed in detail by Robins et al. (2007a).

The remainder of this article concerns generalizations of (2) known as curved exponential family models. Section 2 discusses these models in general terms, while Sections 3 Rewriting alternating, 4 Shared partner statistics focus on specific examples—namely, models involving the alternating k-star, alternating k-triangle, and alternating k-twopath statistics developed by Snijders et al. (in press). These statistics have recently shown great promise in producing parsimonious models that fit certain social network datasets well; they are discussed further by Robins et al. (2007b). Much of Sections 3 Rewriting alternating, 4 Shared partner statistics is devoted to a reformulation of these statistics in terms of the degree statistics and the recently developed shared partner statistics, as well as an attempt to reveal how this reformulation aids in interpreting these statistics. Finally, Section 5 demonstrates the use of these models on real data. The analysis, which recreates and extends some earlier work using the same dataset, is carried out using the statnet package for R, available for use at csde.washington.edu/statnet. The computer code used in Section 5 may be found in Appendix A.