发表机构
Department of Mathematics and Statistics, University of North Carolina at Greensboro（数学与统计学系，北卡罗来纳州格林斯堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究提出用Metropolis-Hastings扩散距离衡量图上概率分布差异，当第二个分布均匀时可测空间聚类，扩展了Moran's \(I\)。建立相关理论并给出统计检验方法，实验表明其在合成数据和实际数据上比Moran's \(I\)更具优势。
AI 中文摘要
我们提出了一种衡量图上两个概率分布\(f\)和\(g\)差异的新方法——扩散距离，它测量在具有平稳分布\(g\)的图约束马尔可夫链下\(f\)收敛到\(g\)的速率。默认使用以\(g\)为目标的Metropolis-Hastings转移矩阵，提议由图上随机游走给出。主要关注\(g\)为均匀分布时，此时扩散距离成为\(f\)中空间聚类的度量。它扩展了Moran's \(I\)型空间自相关度量，纳入了全局图几何。建立了理论边界和稳定性结果，还概述了基于扩散距离的空间聚类统计检验。通过实验比较了扩散距离与Moran's \(I\)，结果表明扩散距离在合成数据上有更高功效，在对美国100个城市黑人人口分布的实证分析中能检测到Moran's \(I\)未发现的城市隔离模式细微差异。
英文摘要
We propose a novel measure of the discrepancy between two probability distributions $f$ and $g$ on a graph - which we call the diffusion distance - that measures the rate of convergence of $f$ to $g$ under a graph-constrained Markov chain with stationary distribution $g$. As a default choice for this Markov chain, we use the Metropolis-Hastings transition matrix targeting $g$ with proposals given by a random walk on the graph. Our primary case of interest is when the second distribution $g$ is uniform, in which case the diffusion distance becomes a measure of spatial clustering in $f$. Used in this way, (Metropolis-Hastings) diffusion distance to uniformity extends Moran's $I$-type measures of spatial autocorrelation by incorporating global graph geometry rather than just local patterns. Indeed, Moran's $I$, the most well-known measure of spatial autocorrelation, can be viewed as a one-step heuristic for diffusion distance, so long as specific spatial weights are used. We establish theoretical bounds and a stability result for our measure, connecting it to graph spectra and optimal transport. We then turn our attention to outlining a statistical test for spatial clustering using diffusion distance. Under permutation null models, we derive high-probability bounds on diffusion distance underpinned by exact spectral formulas for convergence of distributions, enabling an efficient statistical test for spatial clustering on large datasets. We empirically compare diffusion distance to Moran's $I$ both as a numerical measure and as a statistical test. We show that diffusion distance exhibits higher power on synthetic data using a stochastic block model. Empirical analysis of Black population distributions for 100 U.S. cities shows that diffusion distance detects subtle differences in urban segregation patterns that Moran's $I$ does not.
