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

Models for Directed Networks

For directed networks, the direction of an arc permits more dyadic effects. In Table 8.2, lines 1 to 3 present single arc effects that include both an activity and a popularity effect, in addition to a homophily parameter. Investigation of homophily for a binary attribute in a directed network should include at least these three parameters. Lines 4 to 6 present counterpart configurations for continuous attributes. Once again, homophily for the binary interaction effect is indicated by a positive parameter value, and a continuous difference effect is indicated by a negative value. Sometimes there may also be interest in studying attribute effects in relation to reciprocated ties in directed networks, in which case the mutual activity and homophily effects for binary attributes (lines 7 and 8) could also be included (and counterpart effects for continuous attribute variables are possible but not presented here).

Conditional Odds Ratios

Just as with a logistic regression, it can sometimes be helpful for interpretation of selection effects to calculate a (conditional) odds ratio. Suppose we have an undirected network with a single binary attribute on the nodes, and we present a model with activity and homophily selection effects drawn from Table 8.1. Equation 8.1 then becomes

where Qa and are activity and homophily parameters, respectively. Just as described in Chapter 6, there is an equivalent form of Equation (8.2) in a conditional odds form:

where QT5+(x) is the weighted sum of change statistics for endogenous network effects described in Chapter 6 for the conditional logit. Note how the activity statistic becomes a sum of the attribute values for i and j because in predicting the conditional odds of Xij in an undirected network, there are no grounds to privilege either i or j.

For the purposes of interpretation, suppose that we have two nodes, i and j, in a fixed neighborhood of network ties so that 8+(x) is always the same in what follows. Then let us assign different attribute values to the nodes. Of course, this is purely an abstract thought experiment, but it does help us interpret attribute effects. So, suppose the tie is between nodes without the attribute (yi = yj = 0), then the conditional odds of the tie are simply exp{QT5+(x)}; if the tie is between one node with the attribute and one without (yi = 1, yj = 0), then the conditional odds of the tie are exp(QT8+(x) + Qa); and if the tie is between nodes with the attribute (yi = yj = 1), then the conditional odds of the tie is exp(QT8+j(x) + 2Qa + Qy). The sum inside the exponential function in each case is called the “conditional log-odds.”

Let us choose the case of two nodes without attributes (i.e., yi = yj = 0) as the baseline case. Then the ratio of the odds for one of the other two cases to the odds for the baseline case is called the “odds ratio.” Thus, the odds ratio for the case where one node has the attribute is exp(ea), and for both nodes with the attribute, the odds ratio is exp(2ea + 6h). The interpretation is then straightforward and is analogous to the interpretation of odds ratios in logistic regression. The odds of the tie occurring for the pattern of attributes in the dyad are increased by a factor of the odds ratio over the baseline.

As an example, suppose in a study of gender homophily in schools where female is coded as “1” and male “0”, the parameter estimate for activity is —1 and for homophily is 2. Then the odds ratio for a tie between a boy and a girl is e—1 = 0.37, and between two girls is e—2+2 = e0 = 1.0. Remember that these odds ratios use the baseline of a tie between two boys. So, all other things being equal, the odds of a tie between two boys and two girls is the same (odds ratio = 1), whereas the odds of a tie between a boy and a girl is only 37% of that of a within-sex tie. The phrase “all other things being equal,” of course, supposes that the two ties being compared are in an identical neighborhood of network ties, a situation that may not even occur in the data. The odds ratio analysis does, however, permit us to understand the effect of attribute variables over and above the endogenous processes in the data.

There are equivalent odds ratios calculations for directed models, but in these we need to distinguish between a sender and a receiver having the attribute. Suppose the model includes the three binary attribute single arc effects at the top of Table 8.2, with вs, вr, and вh being the estimates for sender, receiver, and homophily parameters, respectively. Again, let the case of yi = yj = 0 be the baseline. Then the odds ratio for a node with the attribute sending a tie to a node without the attribute is exp(es); the odds ratio for a node without the attribute sending a tie to a node with the attribute is exp(er); and the odds ratio for a tie between both nodes with the attribute is exp(es + er + eh).

For continuous attributes, as with logistic regression, it is possible to make an interpretive decision on low and high scores on the attribute (e.g., 1 standard deviation below and above the mean, respectively, or the minimum and maximum values), and then select two low scoring nodes as a baseline and compare that with other patterns of attributes using odds ratios.

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