The Reeb graph has been utilized in various applications including the analysis of scalar fields. Recently, research has been focused on using topological signatures such as the Reeb graph to compare multiple scalar fields by defining distance metrics on the topological signatures themselves. Here we survey five existing metrics that have been defined on Reeb graphs: the bottleneck distance, the interleaving distance, functional distortion distance, the Reeb graph edit distance, and the universal edit distance. Our goal is to (1) provide definitions and concrete examples of these distances in order to develop the intuition of the reader, (2) visit previously proven results of stability, universality, and discriminativity, (3) identify and complete any remaining properties which have only been proven (or disproven) for a subset of these metrics, (4) expand the taxonomy of the bottleneck distance to better distinguish between variations which have been commonly miscited, and (5) reconcile the various definitions and requirements on the underlying spaces for these metrics to be defined and properties to be proven.

翻译：Reeb 图形用于各种应用,包括分析 scalar 字段。 最近, 研究的重点是使用 Reeb 图形等地形特征,通过界定地形特征本身的距离度量来比较多个标度域。 我们在这里调查Reeb 图形上定义的五个现有指标: 瓶颈距离、 间断距离、 功能扭曲距离、 Reeb 图形编辑距离和通用编辑距离。 我们的目标是(1) 提供这些距离的定义和具体例子, 以发展读者的直觉;(2) 访问先前证明的稳定性、 普遍性和歧视性结果;(3) 查明并完成这些指标组中仅被证明( 或无法证明) 的任何剩余属性;(4) 扩大瓶颈距离的分类,以更好地区分通常错误的变异;(5) 调和关于这些计量所要界定和所要证明的属性的基本空间的各种定义和要求。