Computing continuous core/periphery structures for social relations data with MINRES/SVD

When diagonal values are missing or excluded, MINRES is a natural continuous model for the core/periphery structure of a symmetric social network matrix. Symmetric models, however, are not so useful when dealing with asymmetric data. Singular value decomposition (SVD) is a natural choice to model asymmetry, but this method also requires the presence of diagonal values. In this paper we offer an alternative, more general, approach to continuous core/periphery structures, the minimum residual singular value decomposition (MINRES/SVD), where each node in the network receives two indices, an “in-coreness” and an “out-coreness.” The algorithm for computing these coreness vectors is a least squares computation similar to, but distinct from the SVD, again because of the missing diagonal values. And in contrast to the standard, symmetric MINRES algorithm, we can more accurately model asymmetric matrices. This allows us to distinguish, for example, countries in the world economy that are more in the exporting core than they are in the importing core. We propose two nested PRE (proportional reduction of error) measures of fit: (1) the PRE from the MINRES vector with respect to the data and (2) the PRE of the product of the two MINRES/SVD vectors. Applying the resulting method to citations between journals and to international trade in clothing, we illustrate insights gained from being able to model asymmetrical flow patterns. Finally, two permutation tests are introduced to test independently for the MINRES and MINRES/SVD results.