生成树重组的紧性统计
Compactness statistics for spanning tree recombination
摘要：
包围分析已成为量化不公正划分选区的重要工具;其主要思想是生成一个大的、随机的分区计划样本（一个“集合”），任何拟议的计划都可以与之进行比较。如果一个拟议的计划是一个极端的离群相比，总体关于各种重划标准，这可能表明，该计划是故意设计，以产生一个特定的结果。许多方法已经被用来构建集合，一个基本的问题是：给定一个构建计划的方法，我们能否在计划空间上确定一个概率分布，来描述通过该方法构建任何特定计划的概率？近年来，MCMC方法已成为构造系综的主要工具。在这里，我们重点介绍MCMC方法，称为“ReCom”，该方法由MGGG Redistricting Lab于2018年推出。与其他一些方法相比，ReCom倾向于产生具有更紧凑区域的计划，我们试图更好地理解这种现象。我们采用了一个离散的模拟区周长称为“切边”作为一个定量的措施，区紧凑性，这种措施是由杜钦和特纳，它避免了一些困难与紧凑性措施的基础上，地理周长，如Polsby-Popper得分。为了对基本的ReCom步骤进行建模，我们为两个网格图和博尔德县的选区图构建了2区计划的集合，我们发现，对任何特定计划进行抽样的概率--大致与两个地区的生成树数量的乘积成正比--也近似地与平面中切割边缘的数量的指数衰减函数成比例。这是一个重要的一步，了解紧凑性的分区计划由ReCom方法。
英文摘要：
Ensemble analysis has become an important tool for quantifying gerrymandering; the main idea is to generate a large, random sample of districting plans (an "ensemble") to which any proposed plan may be compared. If a proposed plan is an extreme outlier compared to the ensemble with regard to various redistricting criteria, this may indicate that the plan was deliberately engineered to produce a specific outcome. Many methods have been used to construct ensembles, and a fundamental question that arises is: Given a method for constructing plans, can we identify a probability distribution on the space of plans that describes the probability of constructing any particular plan by that method? Recently, MCMC methods have become a predominant tool for constructing ensembles. Here we focus on the MCMC method known as "ReCom," which was introduced in 2018 by the MGGG Redistricting Lab. ReCom tends to produce plans with more compact districts than some other methods, and we sought to better understand this phenomenon. We adopted a discrete analog of district perimeter called "cut edges" as a quantitative measure for district compactness; this measure was proposed by Duchin and Tenner, and it avoids some of the difficulties associated with compactness measures based on geographic perimeter, such as the Polsby-Popper score. To model the basic ReCom step, we constructed ensembles of 2-district plans for two grid graphs and for the precinct graph of Boulder County, CO. We found that the probability of sampling any particular plan -- which is roughly proportional to the product of the numbers of spanning trees for each of the two districts -- is also approximately proportional to an exponentially decaying function of the number of cut edges in the plan. This is an important step towards understanding compactness properties for districting plans produced by the ReCom method.
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