Complex Network of sunspots

Document Type : Research Article

Authors

1 Department of Physics, Faculty of Science, University of Zanjan, Zanjan, Iran.

2 Laboratory of nanometer devices, Faculty of Electrical and Computer Engineering, Shahid Beheshti University, Tehran, Iran.

Abstract

The Sun is an external object that significantly impacts the earth's atmosphere and space weather conditions. Flares and coronal mass ejections are large-scale solar atmospheric features that mainly emerge at active regions above the sunspots and have influenced the earth. The sunspots, considered signatures of solar activity, are fascinating features related to the internal dynamics and activity of the Sun. The appearance of the sunspots in the photosphere shows the complexity of the magnetic field on the Sun. The frequency and size of sunspots change over time which show periods (e.g., eleven years periodicity) that may be a sign of the complex Sun. The time series of the sunspot numbers have been recorded for several centuries, and this time series is significantly changed over time. The complex network approach is a way to investigate the inherent property of complex time series, such as the sunspots time series.
In this study, the growing complex network with the visibility condition is constructed using the time series of the sunspots (time and numbers) for 1922 to 2016 collected by SILSO. We compute the complex network parameters such degree of nodes, shortest path length, and clustering coefficients. We examine the sunspot complex network's scale-free, small-world, and assortative properties.
We show that the degree distribution of the complex network for the time series of the sunspots obeys a power-law distribution function. We applied a method via maximum likelihood estimation in the Bayesian framework to obtain the power indicated. Therefore, the degree exponent is obtained larger than three, so the complex network for the time series of the sunspots is small-world and scale-free. The power-low behavior is an essential characteristic of self-organized or self-organized criticality systems. Limited productivity is a crucial property for these complex systems. The small word behavior indicates that the large sunspot numbers in the time series are clustered with several small values neighbors and linked with distinct large values in the time series.
The small-world network represents a small characteristic path length with a high clustering coefficient. The scale-free and small-word behavior for the network of the sunspots time series may imply that the sunspots and sunspot groups forming via complex non-linear dynamics. Changing the magnetic polarity of the sunspots during the solar cycle can be a characteristic of such complex systems. The limited predictability in sunspots' time series, e.g., the intensity of activity within a solar cycle, may also be another sign of the complex Sun.
The behavior of the degree of node distribution, clustering coefficient, and shortest path length indicates that the time series of sunspots is a non-random system. We showed that the degree of correlation is a function of the network size and can be considered as an assortative, dis-assortative, or neutral network.

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حامدی‌وفا، ه. (1383). مطالعه‌ای در ساختارهای ریز لکه‌های خورشیدی. رساله دکتری، دانشگاه صنعتی شریف، ایران.
صفری، ح.؛ قنادی، ر.؛ علیپور راد، ن. و فرهنگ، ن. (1397). مغناطوهیدرودینامیک خورشید. زنجان: مؤسسه انتشارات دانشگاه زنجان.
لطفی، ن. (1390). شبکه پیچیده زلزله‌های ایران. پایان‌نامه کارشناسی‌ارشد، دانشگاه زنجان، ایران.
محمدی‌گونه، پ. (1396). ساختار شبکه سری زمانی لک‌های خورشیدی. پایان‌نامه کارشناسی‌ارشد، دانشگاه زنجان، ایران.
Alipour, N., & Safari, H. (2015). Statistical properties of solar coronal bright points. The Astrophysical Journal, 807(2), 175.
Aschwanden, M. (2006). Physics of the solar corona: an introduction with problems and solutions. Springer Science & Business Media.
Aschwanden, M. J. (2015). Thresholded power law size distributions of instabilities in astrophysics. The Astrophysical Journal, 814(1), 19.
Bar-Yam, Y., McKay, S. R., & Christian, W. (1998). Dynamics of complex systems (Studies in nonlinearity). Computers in Physics, 12(4), 335-336.
Barabási, A. L. (2003). Linked: The new science of networks.
Bazargan, S., Safari, H., & Kaashisaaz, H. (2019). Classification of mini-dimmings associated with extreme ultraviolet eruptions by using graph theory. Iranian Journal of Physics Research, 16(2), 217-223.
Bondy, J. A., & Murty, U. S. R. (1976). Graph theory with applications (Vol. 290). London: Macmillan.
Newman, M. E. (2000). Models of the small world. Journal of Statistical Physics, 101(3), 819-841.
Bornholdt, S., & Schuster, H. G. (2001). Handbook of graphs and networks. From Genome to the Internet, Willey-VCH (2003 Weinheim).
Braun, H., Christl, M., Rahmstorf, S., Ganopolski, A., Mangini, A., Kubatzki, C., Roth, K., & Kromer, B. (2005). Possible solar origin of the 1,470-year glacial climate cycle demonstrated in a coupled model. Nature, 438(7065), 208-211.
Caldarelli, G. (2007). Large Scale Structure and Dynamics of Complex Networks, World Scientific.
Daei, F., Safari, H., & Dadashi, N. (2017). Complex network for solar active regions. The Astrophysical Journal, 845(1), 36.
Dekking, F. M., Kraaikamp, C., Lopuhaä, H. P., & Meester, L. E. (2005). A Modern Introduction to Probability and Statistics: Understanding why and how (Vol. 488). London: Springer.
Dorogovtsev, S. N., & Mendes, J. F. (2003). Evolution of networks: From biological nets to the Internet and WWW. Oxford university press.
Farhang, N., Safari, H., & Wheatland, M. S. (2018). Principle of minimum energy in magnetic reconnection in a self-organized critical model for solar flares. The Astrophysical Journal, 859(1), 41.
Fortunato, S., Mangioni, G., Menezes, R., & Nicosia, V. (Eds.). (2009). Complex Networks: Results of the 2009 International Workshop on Complex Networks (CompleNet 2009). Springer Berlin Heidelberg.
Gheibi, A., Safari, H., & Javaherian, M. (2017). The solar flare complex network. The Astrophysical Journal, 847(2), 115.
Humphries, M. D., & Gurney, K. (2008). Network ‘small-world-ness’: a quantitative method for determining canonical network equivalence. PloS one, 3(4), e0002051.
Kaki, B., Farhang, N., & Safari, H. (2022). Evidence of Self-Organised Criticality in Time Series by the Horizontal Visibility Graph Approach.
Koutchmy, S., Filippov, B., Tavabi, E., Noëns, J. C., & Wurmser, O. (2022). Polar regions activity and the prediction of the height of the solar cycle 25. arXiv preprint arXiv:2205.09089.
Lacasa, L., Luque, B., Ballesteros, F., Luque, J., & Nuno, J. C. (2008). From time series to complex networks: The visibility graph. Proceedings of the National Academy of Sciences, 105(13), 4972-4975.
Mohammadi, Z., Alipour, N., Safari, H., & Zamani, F. (2021). Complex network for solar protons and correlations with flares. Journal of Geophysical Research: Space Physics, 126(7), e2020JA028868.
Newman, M. E. (2003). The structure and function of complex networks. SIAM review, 45(2), 167-256.
Rezaei, S., Darooneh, A. H., Lotfi, N., & Asaadi, N. (2017). The earthquakes network: Retrieving the empirical seismological laws. Physica A: Statistical Mechanics and its Applications, 471, 80-87.
Rubinov, M., & Sporns, O. (2010). Complex network measures of brain connectivity: uses and interpretations. Neuroimage, 52(3), 1059-1069.
Lotfi, N., & Darooneh, A. H. (2012). The earthquakes network: the role of cell size. The European Physical Journal B, 85(1), 1-4.
Lotfi, N., Javaherian, M., Kaki, B., Darooneh, A. H., & Safari, H. (2020). Ultraviolet solar flare signatures in the framework of complex network. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(4), 043124.
Pastor-Satorras, R., & Vespignani, A. (2001). Epidemic spreading in scale-free networks. Physical review letters, 86(14), 3200
Priest, E. (2014). Magnetohydrodynamics of the Sun. Cambridge University Press.
Solanki, S. K., Usoskin, I. G., Kromer, B., Schüssler, M., & Beer, J. (2004). Unusual activity of the Sun during recent decades compared to the previous 11,000 years. Nature, 431(7012), 1084-1087.
Sonett, C. P., & Finney, S. A. (1990). The spectrum of radiocarbon. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 330(1615), 413-426.
Taran, S., Khodakarami, E., & Safari, H. (2022). Complex network view to solar flare asymmetric activity. Advances in Space Research, 70(8), 2541-2550.
Usoskin, I. G., Solanki, S. K., Schüssler, M., Mursula, K., & Alanko, K. (2003). Millennium-scale sunspot number reconstruction: Evidence for an unusually active Sun since the 1940s. Physical Review Letters, 91(21), 211101.
Usoskin, I. G., Solanki, S. K., & Kovaltsov, G. A. (2007). Grand minima and maxima of solar activity: new observational constraints. Astronomy & Astrophysics, 471(1), 301-309.
Vázquez, A., Pastor-Satorras, R., & Vespignani, A. (2002). Large-scale topological and dynamical properties of the Internet. Physical Review E, 65(6), 066130.
Van Steen, M. (2010). Graph theory and complex networks. An introduction, 144.
Vespignani, A., & Caldarelli, G. (Eds.). (2007). Large scale structure and dynamics of complex networks: from information technology to finance and natural science (Vol. 2). World scientific.
Zou, Y., Small, M., Liu, Z., & Kurths, J. (2014). Complex network approach to characterize the statistical features of the sunspot series. New Journal of Physics, 16(1), 01305.