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# 12.E: Positions and Roles - The Idea of Equivalence (Exercises)

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## Review Questions

1. How are network roles and social roles different from network "sub-structures" as ways of describing social networks?
2. Explain the differences among structural, automorphic, and regular equivalence.
3. Actors who are structurally equivalent have the same patterns of ties to the same other actors. How do correlation, distance, and match measures index this kind of equivalence or similarity?
4. If the adjacency matrix for a network can be blocked into perfect sets of structurally equivalent actors, all blocks will be filled with zeros or with ones. Why is this?
5. If two actors have identical geodesic distances to all other actors, they are (probably) automorphically equivalent. Why does having identical distances to all other actors make actors "substitutable" but not necessarily structurally equivalent?
6. Regularly equivalent actors have the same pattern of ties to the same kinds of other actors - but not necessarily the same distances to all other actors, or ties to the same other actors. Why is this kind of equivalence particularly important in sociological analysis?

## Application Questions

1. Think of the readings from the first part of the course. Did any studies use the idea of structural equivalence or network role? Did any studies use the idea of regular equivalence or social role?
2. Think about the star network. How many sets of structurally equivalent actors are there? What are the sets of automorphically equivalent actors? Regularly equivalent actors? What about the circle network?
3. Examine the line network carefully - this one's a little more tricky. Describe the structural equivalence and regular equivalence sets in a line network.
4. Consider our classical hierarchical bureaucracy, defined by a network of directed ties of "order giving" from the top to the bottom. Make an adjacency matrix for a simple bureaucracy like this. Block the matrix according to the regular equivalence sets; block the matrix according to structural equivalence sets. How (and why) do these blockings differ? How to the permuted matrices differ?
5. Think about some social role (e.g. "mother"). What would you say are the kinds of ties with other social roles that could be used to identify which persons in a population were "mothers" and which were not? Note the relational character of social roles - one social role can only be defined with respect to others. Provide some examples of social roles from an area of interest to you.