Hourglass Persistence interleaves sequences of graph inclusions and contractions to produce more expressive topological features than standard persistent homology for learning on graphs and higher-order complexes.
For completeness, we will also use a special case of this lemma to show how persistence modules can derive the usual interpretation of persistence pairs for a graph filtration
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Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells
Hourglass Persistence interleaves sequences of graph inclusions and contractions to produce more expressive topological features than standard persistent homology for learning on graphs and higher-order complexes.