  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.
