Network centrality measures are concerned with evaluating the importance of nodes, paths, or cycles based on directed or reciprocal interactions inherent within graph structures encoded by vertices and edges. To accommodate higher-order connections between nodes, Estrada and Ross extended graph-based centrality measures to simplicial complexes by expanding node centrality to simplices. We follow this extension but digress in the approach in that we propose novel centrality measures by considering algebraically-computable topological signatures of cycles and their homological persistence. We apply tools from algebraic topology to extract multi-scale signatures within cycle spaces of weighted graphs by tracking homology generators that persist across a weight-induced filtration of simplicial complexes built over graphs. We take these persistent signatures, as well as the merge information of homology classes along the filtration to design centrality measures that quantify cycle importance not only via its geometric and topological significance, but also by its homological influence on other cycles. We also show that these measures are stable under small perturbations allowed by an appropriate metric.

翻译：网络中心化措施涉及评价节点、路径或周期的重要性,这些节点、路径或周期基于由脊椎和边缘编码的图形结构内固有的直接或对等互动的重要性; 将节点、埃斯特拉达和罗斯之间基于图表的延伸中心化措施与简化复合体之间的较高顺序连接,办法是扩大对模棱两可的中心点。 我们遵循这一延伸,但在提出新的中心点措施的方法中,我们提出新的中心点措施,方法是考虑周期的代数可计算表层特征及其同质性持久性。 我们从代数表层中运用各种工具,在加权图表的循环空间内提取多尺度的特征,方法是跟踪在图上构造的简化复合体的重源中持续存在的同系生成器。 我们采用这些持续特征,以及同系类的合并信息,以设计核心计量周期重要性的措施,不仅通过其几何和表层意义,而且通过其对其他周期的同质性影响。 我们还表明,这些措施在适当的指标允许的微孔孔下保持稳定。