Maximally Persistent Cycles in Random Geometric Complexes
By Omer Bobrowski, Matthew Kahle, Primoz Skraba. 2015
77 citations recorded by the source corpus.
We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree-$k$ in persistent homology, for a either the \cech or the Vietoris--Rips filtration built on a uniform Poisson process of intensity $n$ in the unit cube $[0,1]^d$. This is a natural way of measuring the largest "$k$-dimensional hole" in a random point set. This problem is in the intersection of geometric probability and algebraic topology, and is naturally motivated by a probabilistic view of topological inference. We show that for all $d \ge 2$ and $1 \le k \le d-1$ the maximally persistent cycle has (multiplicative) persistence of order $$ \Theta \left(\left(\frac{\log n}{\log \log n} \right)^{1/k} \right),$$ with high probability, characterizing its rate of growth as $n \to \infty$. The implied constants depend on $k$, $d$, and on whether we consider the Vietoris--Rips or \cech filtration.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/f0e90f624073433f04fa1749ebc42c19c38109ab
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Citations in the corpus
- Cliques and cavities in the human connectome (2016, background): “An alternative measure is the death to birth ratio π = ρdeath/ρbirth which highlights topological cavities that survive exceptionally long in spite of being born early, a feature that is interesting in geometric random graphs (see Bobrowski et al. 2015 and Supporting Information).”
- Closures and Cavities in the Human Connectome (2016, methodology): “[7] Omer Bobrowski, Matthew Kahle, and Primoz Skraba.”