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To control for multiscale effects in networks, one can transform the matrixof (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 toall equal 1. The dominant entries in the bistochasticized table can then beemployed for network reduction, using strong component hierarchical clustering.We illustrate various facets of this well-established, widely-applied two-stagealgorithm with the 3, 107 x 3, 107 (asymmetric) 1995-2000 intercounty migrationflow table for the United States. We compare the results obtained with onesusing the disparity filter, for "extracting the "multiscale backbone of complexweighted networks", recently put forth by Serrano, Boguna 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 bistochasticfilter appears to be superior-at least in this specific case-in two respects:(1) it requires far fewer links to complete a stongly-connected networkbackbone; and (2) it "belittles" small flows and nodes less-a principaldesideratum of SBV-in the sense that the correlations of the nonzero raw flowsare considerably weaker with the corresponding bistochastized links than withthe significance levels yielded by the disparity filter. Additional comparativestudies--as called for by SBV-of these two filtering procedures, in particularas regards their topological properties, should be of considerable interest.Relatedly, in its many geographic applications, the two-stage procedurehas--with rare exceptions-clustered contiguous areas, often reconstructingtraditional regions (islands, for example), even though no contiguityconstraints, at all, are imposed beforehand.

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