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

TL;DR

This paper compares bistochastic filtering and disparity filtering for reducing complex human migration networks, demonstrating that bistochastic filtering often produces more efficient and less biased network backbones in large-scale U.S. county migration data.

Contribution

It introduces a comparative analysis of bistochastic and disparity filtering methods for network reduction, highlighting the advantages of bistochastic filtering in large-scale migration networks.

Findings

Bistochastic filtering requires fewer links for a strongly-connected backbone.

It diminishes the influence of small flows and nodes more effectively.

The methods produce different topological properties, warranting further study.

Abstract

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 well-established, widely-applied two-stage algorithm with the 3, 107 x 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, 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 bistochastic filter appears to be superior-at least in this specific case-in two respects: (1) it requires far fewer links to complete a stongly-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. 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.