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Immigration–emigration cluster-pairs with the countries rearranged by their cluster memberships. The frequency counts are represented with light to dark squares. (a) 2015 and (b) 2019.

Immigration–emigration cluster-pairs with the countries rearranged by their cluster memberships. The frequency counts are represented with light to dark squares. (a) 2015 and (b) 2019.

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We discuss spectral clustering from a variety of perspectives that include extending techniques to rectangular arrays, considering the problem of discrepancy minimization, and applying the methods to directed graphs. Near-optimal clusters can be obtained by singular value decomposition together with the weighted k-means algorithm. In the case of re...

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