11 - Relational Algebras

Summary

In this chapter we turn from methods for analyzing properties of actors and social positions to methods for analyzing properties of relations and the associations among relations. The methods we discuss in this chapter are concerned with the theoretical notion of social role, where social role is conceptualized as regular patterns in the relations between social positions. Our focus is on formalization of social role in network terms. Following the scheme that we presented in Chapter 9 (Figure 9.1) as an overview of positional and role analysis, this chapter considers methods for “grouping” relations. We will be traversing the horizontal paths, both on the top and bottom of the figure. The methods in this chapter depend on the notions of social position and the mathematical property of structural equivalence that we discussed in Chapters 9 and 10. However, they take a different perspective by focusing on relations, rather than on actors or subsets of actors.

As we saw in the previous chapter, interpreting the results of a positional analysis can be quite complicated when the analysis includes more than one relation. Distinct interpretations for separate relations become tedious and at times ad hoc. It is useful to have a unified and consistent approach for describing and modeling multiple relations and the associations among these relations. Association among relations means that some relations tend to link the same actors, or that the presence of one relation implies the presence of a second relation.