Stable homology-based cycle centrality measures

Volume 38, Issue 2, pp 179--200 https://dx.doi.org/10.22436/jmcs.038.02.03
Publication Date: December 13, 2024 Submission Date: April 25, 2024 Revision Date: July 25, 2024 Accteptance Date: October 16, 2024

Authors

J. R. D. Manzanares - Department of Mathematics and Computer Science, University of the Philippines Baguio, Baguio City, 2600, Philippines. P. S. P. Ignacio - Department of Mathematics and Computer Science, University of the Philippines Baguio, Baguio City, 2600, Philippines.


Abstract

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.


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ISRP Style

J. R. D. Manzanares, P. S. P. Ignacio, Stable homology-based cycle centrality measures, Journal of Mathematics and Computer Science, 38 (2025), no. 2, 179--200

AMA Style

Manzanares J. R. D., Ignacio P. S. P., Stable homology-based cycle centrality measures. J Math Comput SCI-JM. (2025); 38(2):179--200

Chicago/Turabian Style

Manzanares, J. R. D., Ignacio, P. S. P.. "Stable homology-based cycle centrality measures." Journal of Mathematics and Computer Science, 38, no. 2 (2025): 179--200


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