We introduce several geometric notions, including thick-thin decompositions and the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "size" or "persistence" of a homology class. As a case study, we analyze the power filtration on unweighted graphs, and provide explicit bounds for the life spans of homology classes in persistence diagrams in all dimensions.

翻译：我们引入了多种几何概念,包括厚深分解和同族元素的宽度等,以了解持久性同族元素理论。这些概念提供了持久性图表的几何解释。事实上,我们对同族元素类的“大小”或“持久性”进行定量和几何描述。作为案例研究,我们分析了未加权图表上的功率过滤,并在所有层面的持久性图表中为同族元素类别的生命范围提供了明确的界限。