Effective resistance (ER) is an attractive way to interrogate the structure of graphs. It is an alternative to computing the eigenvectors of the graph Laplacian. One attractive application of ER is to point clouds, i.e. graphs whose vertices correspond to IID samples from a distribution over a metric space. Unfortunately, it was shown that the ER between any two points converges to a trivial quantity that holds no information about the graph's structure as the size of the sample increases to infinity. In this study, we show that this trivial solution can be circumvented by considering a region-based ER between pairs of small regions rather than pairs of points and by scaling the edge weights appropriately with respect to the underlying density in each region. By keeping the regions fixed, we show analytically that the region-based ER converges to a non-trivial limit as the number of points increases to infinity. Namely the ER on a metric space. We support our theoretical findings with numerical experiments.

翻译：有效电阻（ER）是探究图结构的一种极具吸引力的方法，它是计算图拉普拉斯算子特征向量的替代方案。ER的一个吸引人应用在于点云——即顶点对应于度量空间上分布中独立同分布样本的图。然而，有研究表明，随着样本规模趋近于无穷大，任意两点间的有效电阻会收敛到一个不包含图结构信息的平凡值。本研究中，我们证明可以通过考虑小区域对（而非点对）的基于区域的ER，并根据每个区域的基础密度适当缩放边的权重，来规避这一平凡解。在保持区域固定的条件下，我们通过理论分析表明，随着点数增加至无穷大，基于区域的ER会收敛到一个非平凡极限——即度量空间上的有效电阻。我们通过数值实验支持了理论发现。