A Complete Characterization of the One-Dimensional Intrinsic Čech Persistence Diagrams for Metric Graphs

Gasparovic, Ellen; Gommel, Maria; Purvine, Emilie; Sazdanovic, Radmila; Wang, Bei; Wang, Yusu; Ziegelmeier, Lori

doi:10.1007/978-3-319-89593-2_3

Cited by 35 publications

(22 citation statements)

References 15 publications

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“…Related papers include which studies the one‐dimensional persistence of Čech and Vietoris–Rips complexes of metric graphs, which extends this to geodesic spaces, which studies approximations of Vietoris–Rips complexes by finite samples even at higher scale parameters, and which applies Bestvina–Brady discrete Morse theory to Vietoris–Rips complexes.…”

Section: Preliminaries and Related Workmentioning

confidence: 99%

Metric Thickenings, Borsuk–ulam Theorems, and Orbitopes

2019

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Thickenings of a metric space capture local geometric properties of the space. Here, we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk–Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds of thickenings of the sphere allow us to prove that a weighted average of function values of odd maps Sn→double-struckRn+2 on a small diameter set is zero. We prove an additional generalization of the Borsuk–Ulam theorem for odd maps S2n−1→double-struckR2kn+2n−1. We prove such results for odd maps from the circle to any Euclidean space with optimal quantitative bounds. This in turn implies that any raked homogeneous trigonometric polynomial has a zero on a subset of the circle of a specific diameter; these results are optimal.

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Section: Preliminaries and Related Workmentioning

confidence: 99%

Metric Thickenings, Borsuk–ulam Theorems, and Orbitopes

2019

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“…Conversely, how to interpret elements of PH in terms of geometric properties? Some of the few settings in which such an interplay has been theoretically explained contain 1-dimensional PH of metric graphs [9], 1-dimensional PH and persistent fundamental group of geodesic spaces [14,15], the complete persistence of S 1 [1], and parts of PH of ellipses [3] and regular polygons [5].…”

Section: Introductionmentioning

confidence: 99%

Persistent Homology with Selective Rips complexes detects geodesic circles

Virk¹

2021

Preprint

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This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of S 1 ) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions we show that such a loop either terminates a 1-dimensional homology class or gives rise to a 2-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology in order to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes.

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“…This filtration is defined only for metric graphs in [26]. Let (G, d G ) be a metric graph with geometric realization |G|.…”

Section: Intrinsic čEch Filtration (Ic)mentioning

confidence: 99%

Persistence homology of networks: methods and applications

2019

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Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring similarity or distance between networks based on topology. However, classical graph-theoretic measures are usually local and mainly based on differences between either node or edge measurements or correlations without considering the topology of networks such as the connected components or holes. In recent years, mathematical tools and deep learning based methods have become popular to extract the topological features of networks. Persistent homology (PH) is a mathematical tool in computational topology that measures the topological features of data that persist across multiple scales with applications ranging from biological networks to social networks.In this paper, we provide a conceptual review of key advancements in this area of using PH on complex network science. We give a brief mathematical background on PH, review different methods (i.e. filtrations) to define PH on networks and highlight different algorithms and applications where PH is used in solving network mining problems. In doing so, we develop a unified framework to describe these recent approaches and emphasize major conceptual distinctions. We conclude with directions for future work. We focus our review on recent approaches that get significant attention in the mathematics and data mining communities working on network data. We believe our summary of the analysis of PH on networks will provide important insights to researchers in applied network science.

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A Complete Characterization of the One-Dimensional Intrinsic Čech Persistence Diagrams for Metric Graphs

Cited by 35 publications

References 15 publications

Metric Thickenings, Borsuk–ulam Theorems, and Orbitopes

Metric Thickenings, Borsuk–ulam Theorems, and Orbitopes

Persistent Homology with Selective Rips complexes detects geodesic circles

Persistence homology of networks: methods and applications

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