We now turn our attention to stochastic models for social network data. The methodology described here continues the development of statistical methods for network data begun in Chapter 13. We begin in Chapter 15 by considering a (very special) class of statistical distributions for random directed graphs, which, as we will show, is a special case of the uniform random directed graph distributions presented in Chapter 13. This class is more interesting than the distributions of Chapter 13, and contains substantively meaningful parameters which reflect a wide variety of graph properties. Further, the parameters can actually be estimated from data. The basic model has many generalizations and extensions, some of which are described in Chapter 16.

In Chapter 16 we turn to the last question raised in Chapter 9 concerning methodology for studying a positional analysis. We want to measure the adequacy of a representation of a positional analysis. We stated that there are four tasks that have to be undertaken in a positional analysis:

1. (i) Define equivalence

2. (ii) Measure how closely the actors adhere to this definition

3. (iii) Represent the equivalences of the actors

4. (iv) Measure the adequacy of this representation

Two of the necessary tasks are measurement-oriented. These tasks are the second and fourth. The second task requires the analyst to determine how equivalent the actors are, for a given set of relations; that is, one must find which actors are equivalent, and which ones are not, using some measurement device(s).