Random spanning trees of a graph $G$ are governed by a corresponding probability mass distribution (or “law”), $\mathrm{\mu }$, defined on the set of all spanning trees of $G$. This paper addresses the problem of choosing $\mathrm{\mu }$ in order to utilize the edges as “fairly” as possible. This turns out to be equivalent to minimizing, with respect to $\mathrm{\mu }$, the expected overlap of two independent random spanning trees sampled with law $\mathrm{\mu }$. In the process, we introduce the notion of homogeneous graphs. These are graphs for which it is possible to choose a random spanning tree so that all edges have equal usage probability. The main result is a deflation process that identifies a hierarchical structure of arbitrary graphs in terms of homogeneous subgraphs, which we call homogeneous cores. A key tool in the analysis is the spanning tree modulus, for which there exists an algorithm based on minimum spanning tree algorithms, such as Kruskal’s or Prim’s.

图的随机生成树 $G$ 由相应的概率质量分布（或“定律”）控制， $\mathrm{\mu }$，定义在的所有生成树的集合上 $G$。本文解决选择的问题$\mathrm{\mu }$为了尽可能“公平地”利用边缘。事实证明，相对于$\mathrm{\mu }$，两个依法采样的独立随机生成树的预期重叠 $\mathrm{\mu }$。在此过程中，我们介绍了齐次图的概念。这些图可以选择一个随机生成树，以便所有边具有相等的使用概率。主要结果是一个放气过程，该过程根据齐次子图（我们称为齐次核心）识别任意图的层次结构。分析中的关键工具是生成树模量，为此，存在一种基于最小生成树算法的算法，例如Kruskal或Prim。