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Home arrow Sociology arrow Exponential Random Graph Models for Social Networks Theory, Methods, and Applications

Models for Undirected Networks

For undirected networks (Table 8.1), dyadic attribute effects are relatively simple. For binary attribute variables, line 1 of Table 8.1 presents a configuration of a node with the attribute sending a tie to another node (irrespective of its attribute status). The statistic is the count of this configuration in the graph. This is an activity effect associated with the attribute. A positive parameter indicates that nodes with the attribute tend to have higher network activity (i.e., more ties) than nodes without the attribute (this may also be called the “main effect” of an attribute). Line 2, however, represents a homophily effect, whereby nodes with the attribute tend to have ties with each other. There are, of course, various ways to implement this effect: in line 2, the statistic indicates an interaction (i.e., a product) between the two attribute variables for nodes i and j, so that it comes into effect when both variables equal 1 (i.e., when both nodes have the attribute).

It is usually helpful to have both activity and homophily effects in the one model. If one type of node (e.g., females) is more active, such nodes will form more ties among themselves as a simple consequence of greater activity, without the necessity of an independent process whereby females specifically select females. With both parameters in the model, the activity effect is controlled and a significant homophily parameter then indicates an independent effect for females selecting females. It is certainly desirable to have both parameters in the model if the research question relates to possible homophily effects.

For categorical attribute variables (e.g., different departments in an organization), homophily could be implemented with a “same category” effect, with statistic JL< jxijI{yi = yj}, where I{a} is an indicator that is 1 when the statement is true (0, otherwise), when a positive parameter would indicate homophily (i.e., two nodes in the same category tend to be associated with the presence of a network tie. This effect is referred to as nodematch in statnet; statnet offers a variety of ways of parametrizing different covariate effects, such as nodecov, nodefactor, and nodemix, the latter being a stochastic a priori blockmodel).

Counterpart effects for continuous attributes are in lines 3 and 4 of Table 8.1. Here, the homophily statistic indicates an absolute difference effect, in which case a negative parameter indicates homophily (i.e., a smaller difference in attribute between the two nodes is associated with the presence of a tie). Of course, other parameterizations of homophily are also possible (e.g., Euclidean instead of absolute distance or interaction). Interaction for continuous covariates captures a different aspect of homophily, namely, where the homophily operates on the basis of extremes - the further the two actors are away from zero (or the mean if the scores are standardized), the more likely they are to choose each other.

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