Comparative Bi-stochastizations and Associated Clusterings/Regionalizations of the 1995-2000 U. S. Intercounty Migration Network

TL;DR

This paper compares two bi-stochastization methods applied to the U.S. intercounty migration network, revealing distinct clustering structures and regionalizations through graph-theoretic analysis.

Contribution

It extends prior work by analyzing non-symmetric migration matrices with different divergence minimization methods, highlighting differences in resulting clusterings.

Findings

The Sinkhorn-Knopp method yields a matrix with a single unit entry, while the Wang-Li-Konig method produces many.

The directed graph has 2,352 strong components, including various regional groupings.

Hierarchical clustering reveals well-defined regional entities, including states and interstate regions.

Abstract

Wang, Li and Konig have recently compared the cluster-theoretic properties of bi-stochasticized symmetric data similarity (e. g. kernel) matrices, produced by minimizing two different forms of Bregman divergences. We extend their investigation to non-symmetric matrices, specifically studying the 1995-2000 U. S. 3,107 x 3,107 intercounty migration matrix. A particular bi-stochastized form of it had been obtained (arXiv:1207.0437), using the well-established Sinkhorn-Knopp (SK) (biproportional) algorithm--which minimizes the Kullback-Leibler form of the divergence. This matrix has but a single entry equal to (the maximal possible value of) 1. Highly contrastingly, the bi-stochastic matrix obtained here, implementing the Wang-Li-Konig-algorithm for the minimum of the alternative, squared-norm form of the divergence, has 2,707 such unit entries. The corresponding 3,107-vertex, 2,707-link directed graph has 2,352 strong components. These consist of 1,659 single/isolated counties, 654 doublets (thirty-one interstate in nature), 22 triplets (one being interstate), 13 quartets (one being interstate), three quintets and one septet. Not manifest in these graph-theoretic results, however, are the five-county states of Hawaii and Rhode Island and the eight-county state of Connecticut. These--among other regional configurations--appealingly emerged as well-defined entities in the SK-based strong-component hierarchical clustering.