Different Classes of Model Specifications
A dependence assumption constrains the types of configurations among network tie-variables in the model. This point is argued more fully in Chapter 7, which presents technical details of dependence at greater length. For the moment, it suffices to know that a particular dependence assumption implies that the conditional probability of a tie is affected by certain network configurations. In other words, based on the nature of the dependence, a tie is more (or less) probable if it features in some of these configurations. Counts of these subgraphs in the observed graph then become the statistics in the model that permit us to move beyond logistic regression and take dependence into account. Different model specifications involve different combinations of statistics. We now describe a number of commonly used models based on different dependence assumptions, although this is not a complete list of possible models.
Undirected Graphs. The simplest ERGM is the Bernoulli model. As noted previously, for the homogeneous Bernoulli model, each possible tie can be modeled as the independent flip of a (p-) coin. The interpretation is that for each possible tie, we flip a (p-) coin, where the probability of heads is p: if this coin comes up heads, we deem a tie present; otherwise, it is absent. The conditional probability is then
The logit of the probability of a tie is given simply by в, the edge parameter. The joint ERGM probability mass function for the adjacency matrix according to the general formula (Equation (6.2)) is
The parameter eL is called the “edge parameter,” and the corresponding statistic for this is the number of edges L(x) = JT< j X/. There is only one network configuration relevant to this model: the single edge.
Directed Graphs. The homogeneous Bernoulli model for directed graphs follows the same principles as for undirected graphs with the difference that there are twice as many tie-variables.