In this chapter we have discussed the idea of "structural equivalence" of actors, and seen some of the methodologies that are most commonly used to measure structural equivalence, find patterns in empirical data, and describe the sets of "substitutable" actors.
Structural equivalence of two actors is the degree to which the two actors have the same profile of relations across alters (all other actors in the network). Exact structural equivalence is rare in most social structures (one interpretation of exact structural equivalence is that it represents systematic redundancy of actors, which may be functional in some way to the network).
While it is sometimes possible to see patterns of structure equivalence "by eye" from an adjacency matrix or diagram, we almost always use numerical methods. Numerical methods allow us to deal with multiplex data, large numbers of actors, and valued data (as well as the binary type that we have examined here).
The first step in examining structural equivalence is to produce a "similarity" or a "distance" matrix for all pairs of actors. This matrix summarizes the overall similarity (or dissimilarity) of each pair of actors in terms of their ties to alters. While there are many ways of calculating such index numbers, the most common are the Pearson Correlation, the Euclidean Distance, the proportion of matches (for binary data), and the proportion of positive matches (Jaccard coefficient, also for binary data).
A number of methods may be used to identify patterns in the similarity or distance matrix, and to describe those patterns. Cluster analysis groups together the two most similar actors, recalculates similarities, and iterates until all actors are combined. What is produced is a "joining sequence" or map of which actors fall into a hierarchy of increasingly inclusive (and hence less exactly equivalent) groups. Multi-dimensional scaling and factor analysis can be used to identify what aspects of the tie profiles are most critical to making actors similar or different, and can also be used to identify groups. Groupings of structurally equivalent actors can also be identified by the divisive method of iterating the correlation matrix of actors (CONCOR), and by the direct method of permutation and search for perfect zero and one blocks in the adjacency matrix (Optimization by Tabu search).
Once the number of groupings that are useful has been determined, the data can be permuted and blocked, and images calculated. These techniques enable us to get a rather clear picture of how the actors in one set are "approximately equivalent" and why different sets of actors are different. That is, they enable us to describe the meaning of the groups, and the place of group members in the overall network in a general way.
Structural equivalence analysis often produces interesting and revealing findings about the patterns of ties and connections among the individual actors in a network. The structural equivalence concept aims to operationalize the notion that actors may have identical positions in a network - and hence be directly "substitutable" for one another. An alternative interpretation is that actors who are structurally equivalent face nearly the same matrix of constraints and opportunities in their social relationships.
Sociological analysis is not really about individual people. And, structural analysis is primarily concerned with the more general and abstract idea of the roles or positions that define the structure of the group - rather than the locations of specific actors with regard to specific others. For such analysis, we turn to a related set of tools for studying replicate sub-structures ("automorphic equivalence") and social roles ("regular equivalence").