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  • Filtration based on the entire sequence of weights satisfying discrete Morse function is equivalent to filtration based only on the subsequence of critical weights in terms of persistent homology. (a) Filtration of the network shown in Fig. 1 based on weights of the 4 critical simplices. There is a 0-hole (or connected component) that persists across the 4 stages of the filtration while another 0-hole is born at stage 2 on addition of critical vertex v2 but dies at the stage 3 which corresponds to the weight of the critical edge [v1,v7]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{v}_{1},{v}_{7}]$$\end{document}. Moreover, a 1-hole is born at the stage 4 on addition of the critical edge [v5,v7]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{v}_{5},{v}_{7}]$$\end{document}. (b) Five intermediate stages during the filtration between critical weights 1.1 (stage 2) and 2.35 (stage 3). (c) Four intermediate stages during the filtration between critical weights 2.35 (stage 3) and 3.48 (stage 4). It is seen that the homology of the clique complex remains unchanged during the intermediate stages of the filtration whereby the birth and death of holes occur only at stages which correspond to critical weights.
  • License:
  • CC-BY
  • Rights Holder:
  • Springer Nature
  • License Rights Holder:
  • © The Author(s) 2019
  • Asset Type:
  • Image
  • Asset Subtype:
  • Figure
  • Image Orientation:
  • Portrait
  • Image Dimensions:
  • 1713 x 2394
  • Image File Size:
  • 657 KB
  • Creator:
  • Harish Kannan, Emil Saucan, Indrava Roy, Areejit Samal
  • Credit:
  • Kannan, H., Saucan, E., Roy, I., & Samal, A. (2019). Persistent homology of unweighted complex networks via discrete Morse theory. Scientific Reports, 9(1), 13817. https://doi.org/10.1038/s41598-019-50202-3.
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  • Property Release:
  • No
  • Model Release:
  • No
  • Purchasable:
  • Yes
  • Sensitive Materials:
  • No
  • Article Authors:
  • Harish Kannan, Emil Saucan, Indrava Roy, Areejit Samal
  • Article Copyright Year:
  • 2019
  • Publication Volume:
  • 9
  • Publication Issue:
  • 1
  • Publication Date:
  • 09/25/2019
  • DOI:
  • https://doi.org/10.1038/s41598-019-50202-3

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