Multiscale Network Reduction Methodologies: Bistochastic and Disparity Filtering of Human Migration Flows between 3, 000+ U. S. Counties

To control for multiscale effects in networks, one can transform the matrix of (in general) weighted, directed internodal flows to bistochastic (doubly-stochastic) form, using the iterative proportional fitting (Sinkhorn-Knopp) procedure, which alternatively scales row and column sums to all equal 1. The dominant entries in the bistochasticized table can then be employed for network reduction, using strong component hierarchical clustering. We illustrate various facets of this wellestablished, widely-applied two-stage algorithm with the 3, 107 × 3, 107 (asymmetric) 1995-2000 intercounty migration flow table for the United States. We compare the results obtained with ones using the disparity filter, for ”extracting the ”multiscale backbone of complex weighted networks”, recently put forth by Serrano, Boguñá and Vespignani (SBV) (Proc. Natl. Acad. Sci. 106 [2009], 6483), upon which we have briefly commented (Proc. Natl. Acad. Sci. 106 [2009], E66). The performance of the bistochastic filter appears to be superior, in this specific case, in two respects: (1) it requires far fewer links to complete a strongly-connected network backbone; and (2) it ”belittles” small flows and nodes less–a principal desideratum of SBV–in the sense that the correlations of the nonzero raw flows are considerably weaker with the corresponding bistochastized links than with the significance levels yielded by the disparity filter. Further, the disparity filter, in general, relies upon a somewhat arbitrary choice of either AND or OR rules, while the bistochastic filter does not. Additional comparative studies–as called for by SBV–of these two filtering procedures, in particular as regards their topological properties, should be of considerable interest. Relatedly, in its many geographic applications, the two-stage procedure has–with rare exceptions–clustered contiguous areas, often reconstructing traditional regions (islands, for example), even though no contiguity constraints, at all, are imposed beforehand. PACS numbers: Valid PACS 02.10.Ox, 02.10.Yn, 89.65.Cd, 89.75.Hc ∗Electronic address: [email protected]