1995-2000年美国的比较双随机化和相关的区域化。S.县际移民网络
Comparative Bi-stochastizations and Associated Clusterings/Regionalizations of the 1995-2000 U. S. Intercounty Migration Network
摘要：
Wang，Li和Konig最近比较了双随机对称数据相似性的聚类理论性质（例如，G.核）矩阵，通过最小化两种不同形式的Bregman分歧产生。我们将他们的研究扩展到非对称矩阵，特别是研究1995-2000年的U。S. 3 107 x3 107个州际移徙矩阵。它的一个特殊的双随机形式已经获得（arXiv：1207.0437），使用完善的Sinkhorn-Knopp（SK）（双比例）算法-最小化Kullback-Leibler形式的分歧。这个矩阵只有一个元素等于1（的最大可能值）。高度对比，这里获得的双随机矩阵，实现了Wang-Li-Konig算法的最小值，平方范数形式的分歧，有2,707个这样的单位条目。对应的3，107个顶点，2，707个链接的有向图有2，352个强分支。其中包括1，659个单一/孤立的县，654个双胞胎（31个州际），22个三胞胎（一个是州际），13个四胞胎（一个是州际），三个五胞胎和一个七胞胎。然而，在这些图论的结果中并不明显的是夏威夷州和罗德岛州的五个县以及康涅狄格州的八个县。这些-在其他区域配置中-在基于SK的强成分层次聚类中作为定义良好的实体出现。
英文摘要：
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.
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