Network centrality measures play a crucial role in understanding graph structures, assessing the importance of nodes, paths, or cycles based on directed or reciprocal interactions encoded by vertices and edges. Estrada and Ross extended these measures to simplicial complexes to account for higher-order connections. In this work, we introduce novel centrality measures by leveraging algebraically-computable topological signatures of cycles and their homological persistence. We apply tools from algebraic topology to extract multiscale signatures within cycle spaces of weighted graphs, tracking homology generators persisting across a weight-induced filtration of simplicial complexes built over point clouds. This approach incorporates persistent signatures and merge information of homology classes along the filtration, quantifying cycle importance not only by geometric and topological significance but also by homological influence on other cycles. We demonstrate the stability of these measures under small perturbations using an appropriate metric to ensure robustness in practical applications. Finally, we apply these measures to fractal-like point clouds, revealing their capability to detect information consistent with, and possibly overlooked by, common topological summaries.

翻译：网络中心性度量在理解图结构、评估节点、路径或环基于顶点和边编码的有向或互惠交互的重要性方面起着关键作用。Estrada与Ross将这些度量扩展到单纯复形以考虑高阶连接。在本工作中，我们通过利用循环及其同调持续性的代数可计算拓扑特征，引入新的中心性度量。我们应用代数拓扑中的工具，在加权图的循环空间中提取多尺度特征，追踪在点云上构建的单纯复形的权重诱导滤过中持续存在的同调生成元。该方法沿滤过整合持续特征与同调类的合并信息，不仅通过几何与拓扑显著性，还通过同调对其他循环的影响来量化循环重要性。我们通过合适的度量证明了这些度量在小扰动下的稳定性，以确保实际应用中的鲁棒性。最后，我们将这些度量应用于分形点云，揭示了其检测与常见拓扑摘要一致且可能被其忽略的信息的能力。