Advertisement

Analysis of Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence

  • Woojin Kim
  • Facundo MémoliEmail author
  • Zane Smith
Conference paper
  • 74 Downloads
Part of the Abel Symposia book series (ABEL, volume 15)

Abstract

We overview recent work on obtaining persistent homology based summaries of time-dependent data. Given a finite dynamic graph (DG), one first constructs a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, it is possible to then obtain a persistence diagram or barcode from this zigzag persistence module. It turns out that these barcodes are stable under perturbations of the input DG under a certain suitable distance between DGs. We also overview how these results are also applicable in the setting of dynamic metric spaces, and describe a computational application to the analysis of flocking behavior.

Notes

Acknowledgements

We acknowledge funding from these sources: NSF-RI-1422400, NSF AF 1526513, NSF DMS 1723003, NSF CCF 1740761.

References

  1. 1.
  2. 2.
  3. 3.
    U. Bauer, C. Landi, and F. Memoli. The reeb graph edit distance is universal. arXiv preprint arXiv:1801.01866, 2018.Google Scholar
  4. 4.
    M. Benkert, J. Gudmundsson, F. Hübner, and T. Wolle. Reporting flock patterns. Computational Geometry, 41(3):111–125, 2008.MathSciNetCrossRefGoogle Scholar
  5. 5.
    H. B. Bjerkevik. Stability of higher-dimensional interval decomposable persistence modules. arXiv preprint arXiv:1609.02086, 2016.Google Scholar
  6. 6.
    H. B. Bjerkevik and M. B. Botnan. Computational complexity of the interleaving distance. In Proceedings of the thirty-fourth annual symposium on Computational geometry, 2018.Google Scholar
  7. 7.
    M. B. Botnan and M. Lesnick. Algebraic stability of persistence modules. arXiv preprint arXiv:1604.00655, 2016.Google Scholar
  8. 8.
    P. Bubenik and J. A. Scott. Categorification of persistent homology. Discrete & Computational Geometry, 51(3):600–627, 2014.MathSciNetCrossRefGoogle Scholar
  9. 9.
    G. Carlsson and V. De Silva. Zigzag persistence. Foundations of computational mathematics, 10(4):367–405, 2010.MathSciNetCrossRefGoogle Scholar
  10. 10.
    G. Carlsson, V. De Silva, and D. Morozov. Zigzag persistent homology and real-valued functions. In Proceedings of the twenty-fifth annual symposium on Computational geometry, pages 247–256. ACM, 2009.Google Scholar
  11. 11.
    G. Carlsson and F. Mémoli. Characterization, stability and convergence of hierarchical clustering methods. Journal of Machine Learning Research, 11:1425–1470, 2010.MathSciNetzbMATHGoogle Scholar
  12. 12.
    G. Carlsson and F. Mémoli. Classifying clustering schemes. Foundations of Computational Mathematics, 13(2):221–252, 2013.MathSciNetCrossRefGoogle Scholar
  13. 13.
    F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Oudot. Proximity of persistence modules and their diagrams. In Proc. 25th ACM Sympos. on Comput. Geom., pages 237–246, 2009.Google Scholar
  14. 14.
    F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. Mémoli, and S. Y. Oudot. Gromov-Hausdorff stable signatures for shapes using persistence. In Proc. of SGP, 2009.Google Scholar
  15. 15.
    D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103–120, 2007.MathSciNetCrossRefGoogle Scholar
  16. 16.
    N. Clause, and W. Kim. Spatiotemporal Persistent Homology computation tool. https://github.com/ndag/PHoDMSs, 2020.
  17. 17.
    V. De Silva, E. Munch, and A. Patel. Categorified reeb graphs. Discrete & Computational Geometry, 55(4):854–906, 2016.MathSciNetCrossRefGoogle Scholar
  18. 18.
    H. Edelsbrunner and J. Harer. Computational Topology - an Introduction. American Mathematical Society, 2010.zbMATHGoogle Scholar
  19. 19.
    J. Gudmundsson and M. van Kreveld. Computing longest duration flocks in trajectory data. In Proceedings of the 14th annual ACM international symposium on Advances in geographic information systems, pages 35–42. ACM, 2006.Google Scholar
  20. 20.
    J. Gudmundsson, M. van Kreveld, and B. Speckmann. Efficient detection of patterns in 2d trajectories of moving points. Geoinformatica, 11(2):195–215, 2007.CrossRefGoogle Scholar
  21. 21.
    Y. Huang, C. Chen, and P. Dong. Modeling herds and their evolvements from trajectory data. In International Conference on Geographic Information Science, pages 90–105. Springer, 2008.Google Scholar
  22. 22.
    S.-Y. Hwang, Y.-H. Liu, J.-K. Chiu, and E.-P. Lim. Mining mobile group patterns: A trajectory-based approach. In PAKDD, volume 3518, pages 713–718. Springer, 2005.Google Scholar
  23. 23.
    N. Jardine and R. Sibson. Mathematical taxonomy. John Wiley & Sons Ltd., London, 1971. Wiley Series in Probability and Mathematical Statistics.Google Scholar
  24. 24.
    H. Jeung, M. L. Yiu, X. Zhou, C. S. Jensen, and H. T. Shen. Discovery of convoys in trajectory databases. Proceedings of the VLDB Endowment, 1(1):1068–1080, 2008.CrossRefGoogle Scholar
  25. 25.
    P. Kalnis, N. Mamoulis, and S. Bakiras. On discovering moving clusters in spatio-temporal data. In SSTD, volume 3633, pages 364–381. Springer, 2005.Google Scholar
  26. 26.
    W. Kim and F. Memoli. Stable signatures for dynamic graphs and dynamic metric spaces via zigzag persistence. arXiv preprint arXiv:1712.04064, 2017.Google Scholar
  27. 27.
    W. Kim and F. Mémoli. Formigrams: Clustering summaries of dynamic data. In Proceedings of 30th Canadian Conference on Computational Geometry (CCCG18), 2018.Google Scholar
  28. 28.
    W. Kim, and F. Mémoli. Spatiotemporal persistent homology for dynamic metric spaces. Discrete & Computational Geometry. https://doi.org/10.1007/s00454-019-00168-w, 2020.
  29. 29.
    W. Kim, F. Mémoli, and Z. Smith. https://research.math.osu.edu/networks/formigrams, 2017.
  30. 30.
    W. Kim, F. Mémoli, and D. Verano. Formigramator: Formigram computation tool, https://research.math.osu.edu/networks/formigramator, 2020.
  31. 31.
    Z. Li, B. Ding, J. Han, and R. Kays. Swarm: Mining relaxed temporal moving object clusters. Proceedings of the VLDB Endowment, 3(1–2):723–734, 2010.CrossRefGoogle Scholar
  32. 32.
    F. Mémoli. A distance between filtered spaces via tripods. arXiv preprint arXiv:1704.03965, 2017.Google Scholar
  33. 33.
    N. Milosavljević, D. Morozov, and P. Skraba. Zigzag persistent homology in matrix multiplication time. In Proceedings of the Twenty-seventh Annual Symposium on Computational Geometry, SoCG ’11, pages 216–225, New York, NY, USA, 2011. ACM.Google Scholar
  34. 34.
    J. K. Parrish and W. M. Hamner. Animal groups in three dimensions: how species aggregate. Cambridge University Press, 1997.CrossRefGoogle Scholar
  35. 35.
    C. W. Reynolds. Flocks, herds and schools: A distributed behavioral model. In ACM SIGGRAPH computer graphics, volume 21, pages 25–34. ACM, 1987.Google Scholar
  36. 36.
    Z. Smith, S. Chowdhury, and F. Mémoli. Hierarchical representations of network data with optimal distortion bounds. In Signals, Systems and Computers, 2016 50th Asilomar Conference on, pages 1834–1838. IEEE, 2016.Google Scholar
  37. 37.
    D. J. Sumpter. Collective animal behavior. Princeton University Press, 2010.CrossRefGoogle Scholar
  38. 38.
    M. R. Vieira, P. Bakalov, and V. J. Tsotras. On-line discovery of flock patterns in spatio-temporal data. In Proceedings of the 17th ACM SIGSPATIAL international conference on advances in geographic information systems, pages 286–295. ACM, 2009.Google Scholar
  39. 39.
    Y. Wang, E.-P. Lim, and S.-Y. Hwang. Efficient algorithms for mining maximal valid groups. The VLDB Journal – The International Journal on Very Large Data Bases, 17(3):515–535, 2008.CrossRefGoogle Scholar
  40. 40.
    Wikipedia. Formicarium — Wikipedia, the free encyclopedia, 2017. [Online; accessed 03-June-2017].Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Department of Computer Science and EngineeringThe Ohio State UniversityColumbusUSA
  3. 3.Department of Computer Science and EngineeringUniversity of MinnesotaMinneapolisUSA

Personalised recommendations