Maximally Persistent Cycles in Random Geometric Complexes

@article{Bobrowski2015MaximallyPC,
  title={Maximally Persistent Cycles in Random Geometric Complexes},
  author={O. Bobrowski and M. Kahle and P. Skraba},
  journal={arXiv: Probability},
  year={2015}
}
  • O. Bobrowski, M. Kahle, P. Skraba
  • Published 2015
  • Mathematics
  • arXiv: Probability
  • 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… CONTINUE READING
    48 Citations

    Figures from this paper.

    Limit theorems for process-level Betti numbers for sparse, critical, and Poisson regimes
    • 6
    • PDF
    Homotopy Types of Random Cubical Complexes
    • 2
    • PDF
    A Fractal Dimension for Measures via Persistent Homology
    • 7
    • PDF
    Persistent Betti numbers of random Čech complexes
    • 4
    • PDF
    The density of expected persistence diagrams and its kernel based estimation
    • 17
    • PDF

    References

    SHOWING 1-10 OF 61 REFERENCES
    Homological Connectivity Of Random 2-Complexes
    • 215
    • PDF
    Topology of random clique complexes
    • M. Kahle
    • Mathematics, Computer Science
    • Discret. Math.
    • 2009
    • 128
    • PDF
    On the vanishing of homology in random Čech complexes
    • 23
    • PDF
    Random geometric complexes in the thermodynamic regime
    • 53
    • PDF
    Crackle: The Homology of Noise
    • 35
    • PDF
    When does the top homology of a random simplicial complex vanish?
    • 28
    • PDF
    Linear-size approximations to the vietoris-rips filtration
    • D. Sheehy
    • Mathematics, Computer Science
    • SoCG '12
    • 2012
    • 87
    • PDF