14.S: Automorphic Equivalence (Summary)
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The kind of equivalence expressed by the notion of automorphism falls between structural and regular equivalence, in a sense. Structural equivalence means that individual actors can be substituted one for another. Automorphic equivalence means that sub-structures of graphs can can be substituted for one another. As we will see next, regular equivalence goes further still, and seeks to deal with classes or types of actors--where each member of any class has similar relations with some member of each other.
The notion of structural equivalence corresponds well to analyses focusing on how individuals are embedded in networks -- or network positional analysis. The notion of regular equivalence focuses our attention on classes of actors, or "roles" rather than individuals or groups. Automorphic equivalence analysis falls between these two more conventional foci, and has not received as much attention in empirical research. Still, the search for multiple substitutable sub-structures in graphs (particularly in large and complicated ones) may reveal that the complexity of very large structures is more apparent than real; sometimes very large structures are decomposable (or partially so) into multiple similar smaller ones.