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Relative interleavings and applications to sensor networks

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Abstract

In this paper, we prove several results on interleavings for persistent relative homology of sub-level sets, \({\check{\mathrm{C}}\hbox {ech}}\) complexes and Rips complexes. To prove the relative interleavings for \({\check{\mathrm{C}}\hbox {ech}}\) complexes and Rips complexes, we define a relative correspondence which is related to the Gromov-Hausdorff distance. We also apply the relative Rips interleaving to a coverage problem in sensor networks, and show that the interleaving captures some features about the sensors after perturbation from the information of the unperturbed system.

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Acknowledgments

The authors wish to express their sincere gratitude to Emerson Escolar for valuable discussions on this paper. This work is partially supported by JSPS 24684007.

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Authors and Affiliations

  1. WPI-Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai, 980-8577, Japan

    Yasuaki Hiraoka

  2. Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai, 980-8578, Japan

    Genki Kusano

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  1. Yasuaki Hiraoka
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  2. Genki Kusano
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Correspondence to Genki Kusano.

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Hiraoka, Y., Kusano, G. Relative interleavings and applications to sensor networks. Japan J. Indust. Appl. Math. 33, 99–120 (2016). https://doi.org/10.1007/s13160-016-0208-x

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  • DOI: https://doi.org/10.1007/s13160-016-0208-x

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