A generalized model of relational similarity

This paper introduces two principles for relational similarity, and based on these principles it proposes a novel geometric representation for similarity. The first principle generalizes earlier measures of similarity such as Pearson-correlation and structural equivalence: while correlation and structural equivalence measure similarity by the extent to which the actors have similar relationships to other actors or objects, the proposed model views two actors similar if they have similar relationships to similar actors or objects. The second principle emphasizes consistency among similarities: not only are actors similar if they have similar relationships to similar objects, but at the same time objects are similar if similar actors relate to them similarly. We examine the behavior of the proposed similarity model through simulations, and re-analyze two classic datasets: the Davis et al. (1941) data on club membership and the roll-call data of the U.S. Senate. We find that the generalized model of similarity is especially useful if (1) the dimensions of comparison are not independent, or (2) the data are sparse, or (3) the boundaries between clusters are not clear.