Stable Homology-Based Cycle Centrality Measures for Weighted Graphs
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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.
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