Skip to main content
SpringerLink
Log in

Graph Dynamics of Solar Active Regions: Morse–Smale Complexes and Multiscale Graphs of Magnetograms

  • Published:
Astronomy Letters Aims and scope Submit manuscript

Abstract

We discuss the model for the evolution of an active region (AR) in which the graph constructed from the singular points of the magnetic field codes the AR magnetic patterns. The AR dynamic scenarios are mapped by the discrete structure of the network formed from the maxima, minima, and saddle points of the magnetic field. The AR dynamics leads to a rearrangement of the graph, graph dynamics. We discuss two graphs. The first of them is the graph of a Morse–Smale complex. It is a cellular gradient model of the magnetic field each cell of which contains a maximum, a minimum, and two saddle points. The Morse–Smale complex admits a simplification (editing) with prescribed detail that preserves the field topology. The second graph codes the AR dynamics simultaneously on different scales, in the so-called scale-space. This space is formed by a sequence of convolutions of the original magnetogram with a Gaussian kernel, so that the blurring scale is its additional coordinate. Non-Morse singular points, with a degenerate Hessian and Laplacian, are considered in each scale-space layer. The curves connecting these points in different scales form the critical paths whose vertices are called top points. The graph constructed from these points codes the structure of the degenerate AR field structures on different scales. We propose an efficient method of calculating the critical paths using a Jacobi set. The pre-flare regimes are believed to be associated with a significant change in the field topology, which controls the graph dynamics. Consequently, they must be accompanied by noticeable variations in the spectrum of eigenvalues for the discrete Laplacian of the graph (Kirchhoff–Laplace matrix). As an example, we present the evolution of the spectra for these graphs constructed from the magnetograms of the flaring AR 12673. The SDO/HMI magnetograms of the AR for the scalar line-of-sight (LOS) component served as the data. The possible connection of large variations in the spectrum with succeeding X flares is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. Thus, the ‘‘predictions’’ of the events that have already occurred in the AR history.

  2. Simplex stands for simplest. The 0-dimensional or 0-simplexes are simply points, 1-simplexes are edges.

  3. Homotopy is a family of mappings of two spaces that are continuous in parameter, but not necessarily unique. In our example these are the intersecting disks and the edge to which they shrink.

  4. A one-to-one differentiable mapping of two spaces, the inverse of which is also smooth, supplied with the structure of a one-parameter group relative to time.

  5. Interestingly, the complexes themselves may be considered as discrete multivalued dynamical systems and can be used to analyze the critical image regions (Allili et al. 2007).

  6. Imagine a horizontal plane crossing two hills separated by a saddle. There is only one connectivity component in the section below the saddle. There are already two when crossing the saddle and above.

  7. Another way to prove the equivalence of (1) and (2) is to write the solution (2) via the Green function using the Fourier transform.

REFERENCES

  1. M. A. Aizerman, L. A. Gusev, C. B. Petrov, and I. M. Smirnova, Avtom. Telemekh. 7, 135 (1997).

    Google Scholar 

  2. L. Allemand-Giorgis, G. P. Bonneau, and S. Hahmann, Topological Methods in Data Analysis and Visualization (Springer, Cham, 2015), p. 151.

    MATH  Google Scholar 

  3. M. Allili and D. Corriveau, Comput. Vision Image Understand. 105, 188 (2007).

    Article  Google Scholar 

  4. M. Allili, D. Corriveau, S. Derivière, T. Kaczynski, and A. Trahan, J. Math. Imaging Vision 28, 99 (2007).

    Article  MathSciNet  Google Scholar 

  5. M. J. Aschwanden, Solar Phys. 262, 235 (2010).

    Article  ADS  Google Scholar 

  6. M. G. Bobra, X. Sun, J. T. Hoeksema, M. Turmon, Y. Liu, K. Hayashi, G. Barnes, and K. D. Leka, Solar Phys. 289, 3549 (2014).

    Article  ADS  Google Scholar 

  7. C. Campi et al., Astrophys. J. 883, 150 (2019).

    Article  ADS  Google Scholar 

  8. F. R. Chung and F. C. Graham, Spectral Graph Theory, No. 92 of CBMS Regional Conference Series in Mathematics (Am. Math. Soc., Providence, RI, 1997).

  9. J. Damon, J. Differ. Equat. 115, 368 (1995).

    Article  ADS  Google Scholar 

  10. V. Deshmukh, T. E. Berger, E. Bradley, and J. D. Meiss, J. Space Weath. Space Climate 10, 13 (2020).

    Article  Google Scholar 

  11. H. Edelsbrunner and H. Harer, Foundations of Computational Mathematics (Cambridge Univ. Press, Cambridge, 2004), p. 37.

    MATH  Google Scholar 

  12. H. Edelsbrunner and J. Harer, Computational Topology: An Introduction (Am. Math. Soc., Providence, RI, 2010).

    MATH  Google Scholar 

  13. H. H. Edelsbrunner, D. Kirkpatrick, and R. Seidel, IEEE Trans. Inform. Theory 29, 551(1983).

    Article  MathSciNet  Google Scholar 

  14. J. Feldbrugge, M. van Engelen, R. van de Weygaert, P. Pranav, and G. Vegter, J. Cosmol. Astropart. Phys. 2019, 052 (2019).

  15. A. N. Gorban and I. Y. Tyukin, Philos. Trans. R. Soc., Ser. A 376, 20170237 (2018).

    Google Scholar 

  16. S. Gu, Y. Zheng, and C. Tomasi, in Proceedings of the European Conference on Computer Vision (Springer, Berlin, Heidelberg, 2010), p. 663.

  17. D. Günther, J. Reininghaus, H.-P. Seidel, and T. Weinkauf, Topological Methods in Data Analysis and Visualization III (Springer, Cham, 2014), p. 135.

    Google Scholar 

  18. B. Gutkin and U. Smilansky, J. Phys. A 34, 6061 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  19. M. Kac, Am. Math. Mon. 73 (4P2), 1 (1966).

  20. F. Kanters, M. Lillholm, R. Duits, B. Janssen, B. Platel, L. Florack, and B. ter Haar Romeny, in Proceedings of the International Conference on Scale-Space Theories in Computer Vision (Springer, Berlin, Heidelberg, 2005), p. 431.

  21. K. P. Knudson, Morse Theory: Smooth and Discrete (World Scientific, Singapore, 2015).

    Book  Google Scholar 

  22. I. S. Knyazeva, N. G. Makarenko, and F. A. Urt’ev, Geomagn. Aeron. 55, 1134 (2015).

    Article  ADS  Google Scholar 

  23. I. S. Knyazeva, N. G. Makarenko, I. Makarenko, A. Vdovina, and F. A. Urtiev, Phys. Proc. 738, 012070 (2016).

    Google Scholar 

  24. A. Kuijper, PhD Thesis (Utrecht Univ., Utrecht, 2002).

  25. K. D. Leka, S-H. Park, K. Kusano, J. Andries, G. Barnes, S. Bingham, D. Sh. Bloomfield, A. E. McCloskey, et al., Astrophys. J. 881, 101 (2019).

    Article  ADS  Google Scholar 

  26. D. Lim, Y. J. Moon, J. Park, E. Park, K. Lee, J. Y. Lee, and S. Jang, arXiv:1907.11373 (2019).

  27. T. Lindeberg, Scale-Space Theory in Computer Vision (Springer Science, New York, 2013).

    MATH  Google Scholar 

  28. H. Liu, C. Liu, J. T. Wang, and H. Wang, Astrophys. J. 877, 121 (2019).

    Article  ADS  Google Scholar 

  29. A. D. Lukyanov, N. G. Makarenko, and I. Knyazeva, Phys. Proc. 798, 012181 (2017).

    Google Scholar 

  30. N. G. Makarenko, D. B. Malkova, M. L. Myachin, I. S. Knyazeva, and I. N. Makarenko, Fundam. Prikl. Mat. 18, 79 (2013).

    Google Scholar 

  31. N. G. Makarenko, D. O. Park, and V. V. Alexeev, Phys. Proc. 675, 032026 (2016).

    Google Scholar 

  32. N. G. Makarenko, L. Karimova, A. Terekhov, and M. Novak, Phys. A (Amsterdam, Neth.) 289, 278 (2001).

  33. J. C. Maxwell, Philos. Mag., Ser. 4 40, 421 (1870).

    Article  Google Scholar 

  34. J. J. Molitierno, Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs (Chapman and Hall/CRC, New York, 2016).

    Book  Google Scholar 

  35. S.-H. Park, K. D. Leka, K. Kusano, J. Andries, G. Barnes, S. Bingham, D. Sh. Bloomfield, A. E. McCloskey, et al., Astrophys. J. 890, 124 (2020).

    Article  ADS  Google Scholar 

  36. P. Pranav, H. Edelsbrunner, R. van de Weygaert, G. Vegter, M. Kerber, B. J. Jones, M. Wintraecken, Mon. Not. R. Astron. Soc. 465, 4281 (2017).

    Article  ADS  Google Scholar 

  37. F. P. Preparata and M. I. Shamos, Computational Geometry: An Introduction (Springer, New York, 1985).

    Book  Google Scholar 

  38. V. Robins, P. J. Wood, and A. P. Sheppard, IEEE Trans. Pattern Anal. Mach. Intell. 33, 1646 (2011).

    Article  Google Scholar 

  39. S. Rosenberg, The Laplacian on a Riemannian Manifold, Vol. 31 of London Math. Soc. Student Texts (Cambridge Univ. Press, Cambridge, 1997).

  40. A. Samal, R. P. Sreejith, J. Gu, S. Liu, E. Saucan, and J. Jost, Sci. Rep. 8, 8650 (2018).

    Article  ADS  Google Scholar 

  41. C. J. Schrijver, A. A. van Ballegooijen, H. J. Hagenaar, and R. A. Shine, Astrophys. J. 487, 424 (1997).

    Article  ADS  Google Scholar 

  42. T. Sousbie, Mon. Not. R. Astron. Soc. 414, 350 (2011).

    Article  ADS  Google Scholar 

  43. X. Sun and A. A. Norton, Res. Not. AAS 1, 24 (2017).

    Article  ADS  Google Scholar 

  44. M. Turmon and S. Mukhtar, in Proceedings of the IEEE International Conference on Image Processing, CL 97-1147 (1997), Vol. 3, p. 320; in Proceedings of the Proc. Compstat-98, Bristol, UK, CL 98-0761 (1998), p. 473.

  45. H. N. Wang and J. Wang, Astron. Astrophys. 313, 285 (1996).

    ADS  Google Scholar 

  46. E. V. Zhuzhoma, V. S. Medvedev, and N. V. Isaenkova, Nelin. Din. 13, 399 (2017).

    Article  Google Scholar 

  47. V. A. Zorich, Teor. Veroyatn. Primen. 59, 436 (2014).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150000, Russia

    V. V. Alekseev

  2. Pulkovo Astronomical Observatory, Russian Academy of Sciences, , Pulkovskoe sh. 65, , St. Petersburg, 196140, Russia

    V. V. Alekseev, N. G. Makarenko & I. S. Knyazeva

  3. Institute of Information and Computing Technologies, Almaty, Kazakhstan

    N. G. Makarenko

  4. Institute of Information and Computing Technologies, Ministry of Education and Science of Kazakhstan, Almaty, Kazakhstan

    I. S. Knyazeva

  5. St. Petersburg State University, St. Petersburg, Russia

    I. S. Knyazeva

Authors
  1. V. V. Alekseev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. G. Makarenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. S. Knyazeva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. G. Makarenko.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alekseev, V.V., Makarenko, N.G. & Knyazeva, I.S. Graph Dynamics of Solar Active Regions: Morse–Smale Complexes and Multiscale Graphs of Magnetograms. Astron. Lett. 46, 488–500 (2020). https://doi.org/10.1134/S1063773720070014

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063773720070014

Keywords:

Search

Navigation