Paul Samuel P. Ignacio

arXiv2024

Stable Homology-Based Cycle Centrality Measures

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.

cs.CG↗

arXiv2020

Intrinsic Hierarchical Clustering Behavior Recovers Higher Dimensional Shape Information

We show that specific higher dimensional shape information of point cloud data can be recovered by observing lower dimensional hierarchical clustering dynamics. We generate multiple point samples from point clouds and perform hierarchical clustering within each sample to produce dendrograms. From these dendrograms, we take cluster evolution and merging data that capture clustering behavior to construct simplified diagrams that record the lifetime of clusters akin to what zero dimensional persistence diagrams do in topological data analysis. We compare differences between these diagrams using the bottleneck metric, and examine the resulting distribution. Finally, we show that statistical features drawn from these bottleneck distance distributions detect artefacts of, and can be tapped to recover higher dimensional shape characteristics.

cs.CG↗