Pattison and Robins (2002) suggested that in some circumstances, two

tie-variables, X_{i}j and Xh_{m}, may be conditionally dependent given the

presence of other network tie-variables even when they do not share a

node. This is a “partial conditional independence assumption” that is a

generalization of the more familiar notion of conditional independence,

which refers to the situation of statistical independence of two variables

given the state of a third variable (Dawid, 1979, 1980). In the case of

partial conditional independence for tie-variables, two tie-variables are

statistically independent if and only if a third tie-variable is in a particular 2

state.^{2}

An example of a partial conditional independence assumption is the “social circuit dependence assumption,” where two tie-variables, Xij and

This approach is similar in form to that used by Baddeley and Moller (1989) for spatial models, which is why Pattison and Robins (2002) adopted the same term referring to these models as “realization-dependent models.”

Figure 6.2. Social circuit dependence.

Xh_{m}, that do not share a vertex are conditionally dependent if ties exists between i and h and between j and m. In this case, if the two-variables are observed (i.e., x_{i}hxj_{m} = 1), then a 4-cycle is created as in Figure 6.2. This dependence may be typified in collaborative ties, when i works with h and j works with m. Then the presence of collaboration between i and j is likely to affect whether h and m also collaborate.

This assumption in combination with the Markov dependence assumption gives rise to a set of additional configurations in the network model, including sets of 2-paths with common starting and ending nodes, and sets of triangles with a common base.