# Construct least-squares ultrametric (hierarchical clustering) fits to doubly-standardized “flow” tables and compare to single-linkage-type fits

Figure 1 of the paper, "Hierarchical Migration Regions of France" (IEEE Transactions on Systems, Man and Cybernetics, 4 (1976) 321-324) (https://www.researchgate.net/publication/224112435_Hierarchical_Internal_Migration_Regions_of_France) presents a hierarchical clustering of twenty-one regions of France (Table I). One immediately notes that the Paris Region is the most weakly integrated into the tree structure.

This dendrogram was produced by applying a directed-graph (strong component) version of single-linkage clustering (https://www.sciencedirect.com/science/article/pii/0020019082901363?via%3Dihub) to a (non-symmetric) table based on the recorded 1962-68 migration flows between the twenty-one regions--after the conversion ("double-standardization") of the table to one (Table II) in which all the rows and columns summed to 1000. (following Mosteller, Frederick. "Association and estimation in contingency tables." Journal of the American Statistical Association 63.321 (1968): 1-28.)

The relatively weak integration of the Paris Region into the tree clearly reflected the central ("cosmopolitan") nature of the region, in that it broadly interacted with the other regions, rather than its having a more narrow, "provincial" role. (The 1,000 "in-migrants" and "out-migrants" for Paris are more evenly distributed over the twenty other regions.)

This general procedure was subsequently applied to a wide variety of "transaction flow" tables. Of particular note, is its application to the 1995-2000 migration flows between the 3,000 + counties of the United States (https://arxiv.org/abs/1207.0437).

So, my question is can tree (ultrametric) structures be fitted to such doubly-standardized tables (even as large as $$3000 \times 3000$$) using least-squares, and if so what, in particular, will be the behavior of the cosmopolitan entities in such structures?

It is of interest to note that the double-standardization/single-linkage-type procedure yields fits in which the residuals from the fits are highly skewed negatively (since the larger values are used for the fitting), while, in least-squares, one anticipates that the residuals from the fits tend to be equally balanced between positive and negative.