ERGM Social Selection Models

An ERGM social selection model involves various forms of dependence between attribute and network tie-variables. The precise details are beyond the scope of this chapter, but, broadly, attribute variables are taken as exogenous predictors of network tie-variables at the same time that there are dependencies among the network variables. Interested readers should consult Robins, Elliott, and Pattison (2001), who set out dependencies for ERGM selection models within the framework

94 Exponential Random Graph Models for Social Networks

Table 8.1. Some social selection configurations for undirected networks

Configuration	Statistic	Parameter
Binary attributes
		Attribute-based activity
		Homophily (interaction)
Continuous attributes
		Attribute-based activity
		Homophily (difference)

of Markov endogenous network processes. Each attribute effect in an ERGM selection model is a statistical interaction between at least one attribute and one tie-variable, so that configurations involve not just patterns of ties but also colors on the nodes (for binary and categorical attribute variables), or size of nodes (for continuous attribute variables).

The general form of an ERGM social selection model is as follows:

where Q and z are parameters and statistics for endogenous network effects as discussed in previous chapters, and Q_a and z_a are parameters and statistics for social selection configurations involving an interaction of network (x) and attribute (y) variables.

For binary and continuous attribute variables, the most important social selection configurations are presented in Tables 8.1 and 8.2. In these tables, a filled circle represents a node with attribute value y_i = 1. For instance, if the attribute variable gender is scored 0 for male and 1 for female, the filled circle represents “female.” (For simplicity, we just say that the node “has the attribute”; thus, in the example of gender, the attribute can be understood as “being female,” given the scoring.) For continuous attribute variables, a larger score on the attribute is represented by a node of larger size. An unfilled circle represents a node, irrespective of attribute status. So, for instance, line 1 of Table 8.1 presents a configuration with a tie from a node with the attribute to another node; that is, the configuration represents the network activity of nodes with the attribute. (In the statistics columns of Tables 8.1 and 8.2, summations are across all nodes i and j, but with the proviso in some cases that i < j to avoid double counting of configurations.)

Table 8.2. Some social selection configurations for directed networks

Single arc effects Binary attributes
		Sender effect (attribute-based activity)
		Receiver effect (attribute-based popularity)
		Homophily (interaction)
Continuous attributes
		Sender effect (attribute-based activity)
		Receiver effect (attribute-based popularity)
		Homophily (single arc difference)
Mutual effects Binary attributes
		Mutual activity
		Mutual homophily (mutual difference)

Tables 8.1 and 8.2 present only selection effects related to dyads. It is also possible to have effects relating to stars, triads, and other configurations, with one or more colored (or sized) nodes occupying different positions within the configuration. See Robins, Elliott, and Pattison (2001) for examples.