The Relationship Between the Intrinsic Cech and Persistence Distortion Distances for Metric Graphs
The Relationship Between the Intrinsic Cech and Persistence Distortion Distances for Metric Graphs
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DOI：
10.20382/jocg.v10i1a16
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发表时间：
2018-12
期刊：
影响因子：
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通讯作者：
Ellen Gasparovic;Maria Gommel;Emilie Purvine;R. Sazdanovic;Bei Wang;Yusu Wang;Lori Ziegelmeier
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文献类型：
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作者：
Ellen Gasparovic;Maria Gommel;Emilie Purvine;R. Sazdanovic;Bei Wang;Yusu Wang;Lori Ziegelmeier
度量图是用于建模复杂结构的有意义的对象，这些复杂结构出现在许多现实世界的应用中，例如道路网络、河流系统、地震断层、血管和星系中的附属结构。为了研究度量图的比较的背景下，我们有兴趣在确定两个基于拓扑的距离之间的任意有限度量图的相对判别能力：持久性失真距离和内在的切赫距离。我们明确地展示了如何计算两个度量图之间的内在切赫距离的基础上的知识，最短的循环系统的图形。我们的主要定理建立了一个不等式的内在Cech和持久性失真距离的情况下，当一个图是花束图和其他是任意的。当这两个图都是通过圈和边的楔形和构造时，这种关系也成立。
Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric graphs in the context of comparison, we are interested in determining the relative discriminative capabilities of two topology-based distances between a pair of arbitrary finite metric graphs: the persistence distortion distance and the intrinsic Cech distance. We explicitly show how to compute the intrinsic Cech distance between two metric graphs based solely on knowledge of the shortest systems of loops for the graphs. Our main theorem establishes an inequality between the intrinsic Cech and persistence distortion distances in the case when one of the graphs is a bouquet graph and the other is arbitrary. The relationship also holds when both graphs are constructed via wedge sums of cycles and edges.