Equivalence in a social network

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Abstract
Equivalence has become a foundational concept in social network representations of social structure. Each use of an equivalence concept has two components: (i) the definition of equivalence and (ii) a computational algorithm for detecting equivalences, or the extent to which they exist. In all substantive contexts it is necessary to determine if a particular definition of equivalence is appropriate and the extent to which the computational algorithm faithfully mirrors the equivalence definition. An extensive comparison of structural equivalence, with two structural equivalence detectors, and regular equivalence, with one regular equivalence detector, is provided for a set of social structures. As blockmodels are, among other things, hypotheses about network structure, hypotheses about structure are used to obtain the networks used for these comparisons. For any network, there is, potentially, a multiplicity of equivalences that can be mobilized so as to complement each other in analyzing social structure.