Characterizing Flows by Complex Network Methods

  • Reik V. Donner
  • Michael Lindner
  • Liubov Tupikina
  • Nora Molkenthin
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 22)


During the last years, complex network approaches have demonstrated their great potentials as versatile tools for exploring the structural as well as dynamical properties of complex systems from a variety of different fields. Among others, recent successful examples include their application to studying flow systems in both, abstract mathematical and real-world geophysical contexts. In this context, two recent developments are particularly notable: on the one hand, correlation-based functional network approaches allow inferring statistical interrelationships, for example between macroscopic regions of the Earth’s climate system, which are hidden to more classical statistical analysis techniques. On the other hand, Lagrangian flow networks provide a new tool to identify dynamically relevant structures in atmosphere, ocean or, more generally, the phase space of complex systems. This chapter summarizes these recent developments and provides some illustrative examples highlighting the application of both concepts to selected paradigmatic low-dimensional model systems.



This work has been financially supported by the BMBF Young Investigator’s Group CoSy-CC2 (grant no. 01LN1306A) and the International Research Training Group IRTG 1740/TRP 2014/50151-0, jointly funded by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) and the São Paulo Research Foundation (FAPESP, Fundação de Amparo à Pesquisa do Estado de São Paulo).


  1. 1.
    Newman, M. E. J. (2010). Networks: An introduction. Oxford: Oxford University Press.zbMATHCrossRefGoogle Scholar
  2. 2.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., & Hwang, D. U. (2006). Complex networks: Structure and dynamics. Physics Reports, 424, 175–308.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barrat, A., Barthélemy, M., & Vespignani, A. (2008). Dynamical processes on complex networks. Cambridge: Cambridge University Press.zbMATHCrossRefGoogle Scholar
  4. 4.
    Albert, R., & Barabási, A. L. (2002). Statistical mechanics of complex networks. Reviews of Modern Physics, 74(1), 47–98.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Review, 45, 167–256.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    da Fontoura Costa, L., Rodrigues, F. A., Travieso, G., & Villas Boas, P. R. (2007). Characterization of complex networks: A survey of measurements. Advances in Physics, 56(1), 167–242.CrossRefGoogle Scholar
  7. 7.
    Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486, 75–174.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., & Zhou, C. (2008). Synchronization in complex networks. Physics Reports, 469(3), 93–153.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Rossi, V., Ser-Giacomi, E., López, C., & Hernández-García, E. (2014). Hydrodynamic provinces and oceanic connectivity from a transport network help designing marine reserves. Geophysical Research Letters, 41(8), 2883–2891.CrossRefGoogle Scholar
  10. 10.
    Ser-Giacomi, E., Rossi, V., López, C., & Hernández-García, E. (2015). Flow networks: A characterization of geophysical fluid transport. Chaos, 25, 036404.CrossRefGoogle Scholar
  11. 11.
    Ser-Giacomi, E., Vasile, R., Hernández-García, E., & López, C. (2014). Most probable paths in temporal weighted networks: An application to ocean transport. Physical Review E, 92, 012818.CrossRefGoogle Scholar
  12. 12.
    Ser-Giacomi, E., Vasile, R., Recuerda, I., Hernández-García, E., & López, C. (2015). Dominant transport pathways in an atmospheric blocking event. Chaos, 25(8), 087413.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dods, J., Chapman, S. C., & Gjerloev, J. W. (2015). Network analysis of geomagnetic substorms using the SuperMAG database of ground-based magnetometer stations. Journal of Geophysical Research: Space Physics, 120(9), 7774.Google Scholar
  14. 14.
    Molkenthin, N., Rehfeld, K., Marwan, N., & Kurths, J. (2014). Networks from flows-from dynamics to topology. Scientific Reports, 4, 4119.CrossRefGoogle Scholar
  15. 15.
    Tupikina, L., Molkenthin, N., López, C., Hernández-García, E., Marwan, N., & Kurths, J. (2016). Correlation networks from flows. The case of forced and time-dependent advection-diffusion dynamics. PLoS One, 11(4), e0153703.CrossRefGoogle Scholar
  16. 16.
    Donges, J. F., Petrova, I., Loew, A., Marwan, N., & Kurths, J. (2015). How complex climate networks complement eigen techniques for the statistical analysis of climatological data. Climate Dynamics, 45(9), 2407–2424.CrossRefGoogle Scholar
  17. 17.
    Zhou, C., Zemanova, L., Zamora, G., Hilgetag, C. C., & Kurths, J. (2006). Hierarchical organization unveiled by functional connectivity in complex brain networks. Physical Review Letters, 97(23), 238103.CrossRefGoogle Scholar
  18. 18.
    Zhou, C., Zemanova, L., Zamora-Lopez, G., Hilgetag, C. C., & Kurths, J. (2007). Structure-function relationship in complex brain networks expressed by hierarchical synchronization. New Journal of Physics, 9(6), 178.CrossRefGoogle Scholar
  19. 19.
    Tsonis, A., & Roebber, P. (2004). The architecture of the climate network. Physica A, 333, 497–504.CrossRefGoogle Scholar
  20. 20.
    Yamasaki, K., Gozolchiani, A., & Havlin, S. (2008). Climate networks around the globe are significantly affected by El Niño. Physical Review Letters, 100(22), 228501.CrossRefGoogle Scholar
  21. 21.
    Donges, J. F., Zou, Y., Marwan, N., & Kurths, J. (2009). Complex networks in climate dynamics. European Physical Journal Special Topics, 174(1), 157.CrossRefGoogle Scholar
  22. 22.
    Barreiro, M., Marti, A. C., & Masoller, C. (2011). Inferring long memory processes in the climate network via ordinal pattern analysis. Chaos, 21(1), 013101.CrossRefGoogle Scholar
  23. 23.
    Mantegna, R. N. (1999). Hierarchical structure in financial markets. European Physical Journal B, 11(1), 193.CrossRefGoogle Scholar
  24. 24.
    Onnela, J. P., Kaski, K., & Kertész, J. (2004). Clustering and information in correlation based financial networks. European Physical Journal B, 38(2), 353.CrossRefGoogle Scholar
  25. 25.
    Liu, X. F., & Tse, C. K. (2012). A complex network perspective of world stock markets: synchronization and volatility. International Journal of Bifurcation and Chaos, 22(06), 1250142.CrossRefGoogle Scholar
  26. 26.
    Nicolis, G., García Cantú, A., & Nicolis, C. (2005). Dynamical aspects of interaction networks. International Journal of Bifurcation and Chaos, 15(11), 3467.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Sun, X., Small, M., Zhao, Y., & Xue, X. (2014). Characterizing system dynamics with a weighted and directed network constructed from time series data. Chaos, 24(2), 024402.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    McCullough, M., Small, M., Stemler, T., & Iu, H. H. C. (2015). Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems. Chaos, 25(5), 053101.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Donner, R. (2002). Dynamoeffekt in einem niedrig-dimensionalen Modell einer getriebenen Roberts-Strömung. Master’s thesis, University of Potsdam.Google Scholar
  30. 30.
    Donner, R., Seehafer, N., Sanjuán, M. A., & Feudel, F. (2006). Low-dimensional dynamo modelling and symmetry-breaking bifurcations. Physica D, 223(2), 151.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Donner, R., Feudel, F., Seehafer, N., & Sanjuán, M. A. F. (2007). Hierarchical modeling of a forced Roberts dynamo. International Journal of Bifurcation and Chaos, 17(10), 3589.zbMATHCrossRefGoogle Scholar
  32. 32.
    Barrat, A., & Weigt, M. (2000). On the properties of small-world network models. European Physical Journal B, 13, 547.CrossRefGoogle Scholar
  33. 33.
    Fagiolo, G. (2007). Clustering in complex directed networks. Physical Review E, 76, 026107.CrossRefGoogle Scholar
  34. 34.
    Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., & Alon, U. (2002). Network motifs: Simple building blocks of complex networks. Science, 298, 824.CrossRefGoogle Scholar
  35. 35.
    Latora, V., & Marchiori, M. (2001). Efficient behavior of small-world networks. Physical Review Letters, 87, 198701.CrossRefGoogle Scholar
  36. 36.
    Donges, J. F., Heitzig, J., Donner, R. V., & Kurths, J. (2012). Analytical framework for recurrence network analysis of time series. Physical Review E, 85, 046105.CrossRefGoogle Scholar
  37. 37.
    Pfeffer, J., & Carley, K. M. (2012). k-Centralities: Local approximations of global measures based on shortest paths. In Proceedings of the 21st International Conference Companion on World Wide Web (pp. 1043–1050). New York: ACMGoogle Scholar
  38. 38.
    Ercsey-Ravasz, M., & Toroczkai, Z. (2010). Centrality scaling in large networks. Physical Review Letters, 105(3), 038701.CrossRefGoogle Scholar
  39. 39.
    Ercsey-Ravasz, M., Lichtenwalter, R. N., Chawla, N. V., & Toroczkai, Z. (2012). Range-limited centrality measures in complex networks. Physical Review E, 85(6), 066103.CrossRefGoogle Scholar
  40. 40.
    Lindner, M., & Donner, R. V. (2017). Spatio-temporal organization of dynamics in a two-dimensional periodically driven vortex flow: A Lagrangian flow network perspective. Chaos, 27(3), 035806.MathSciNetCrossRefGoogle Scholar
  41. 41.
    Barthélemy, M. (2011). Spatial networks. Physics Reports, 499(1), 1.MathSciNetCrossRefGoogle Scholar
  42. 42.
    Gudmundsson, A., & Mohajeri, N. (2013). Entropy and order in urban street networks. Scientific Reports, 3, 3324.CrossRefGoogle Scholar
  43. 43.
    Mohajeri, N., French, J., & Gudmundsson, A. (2013). Entropy measures of street-network dispersion: Analysis of coastal cities in Brazil and Britain. Entropy, 15(9), 3340.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Mohajeri, N., & Gudmundsson, A. (2014). The evolution and complexity of urban street networks. Geographical Analysis, 46(4), 345.CrossRefGoogle Scholar
  45. 45.
    Molkenthin, N., Kutza, H., Tupikina, L., Marwan, N., Donges, J. F., Feudel, U., et al. (2017). Edge anisotropy and the geometric perspective on flow networks. Chaos, 27(3), 035802.MathSciNetCrossRefGoogle Scholar
  46. 46.
    Heitzig, J., Donges, J. F., Zou, Y., Marwan, N., & Kurths, J. (2012). Node-weighted measures for complex networks with spatially embedded, sampled, or differently sized nodes. European Physical Journal B, 85, 38.CrossRefGoogle Scholar
  47. 47.
    Molkenthin, N., Rehfeld, K., Stolbova, V., Tupikina, L., & Kurths, J. (2014). On the influence of spatial sampling on climate networks. Nonlinear Processes in Geophysics, 21, 651.CrossRefGoogle Scholar
  48. 48.
    Bialonski, S., Horstmann, M. T., & Lehnertz, K. (2010). From brain to earth and climate systems: Small-world interaction networks or not? Chaos, 20(1), 013134.MathSciNetCrossRefGoogle Scholar
  49. 49.
    Wiedermann, M., Donges, J. F., Kurths, J., & Donner, R. V. (2016). Spatial network surrogates for disentangling complex system structure from spatial embedding of nodes. Physical Review E, 93, 042308.CrossRefGoogle Scholar
  50. 50.
    Dellnitz, M., Hessel-von Molo, M., Metzner, P., Preis, R., & Schütte, C. (2006). Graph algorithms for dynamical systems. In A. Mielke (Ed.) Analysis, modeling and simulation of multiscale problems (pp. 619–645). Heidelberg: Springer.CrossRefGoogle Scholar
  51. 51.
    Santitissadeekorn, N., & Bollt, E. (2007). Identifying stochastic basin hopping by partitioning with graph modularity. Physica D, 231(2), 95.MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Bollt, E.M., & Santitissadeekorn, N. (2013). Applied and computational measurable dynamics. Philadelphia: SIAMzbMATHCrossRefGoogle Scholar
  53. 53.
    Froyland, G., & Dellnitz, M. (2003). Detecting and locating near-optimal almost-invariant sets and cycles. SIAM Journal on Scientific Computing, 24(6), 1839.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Froyland, G. (2005). Statistically optimal almost-invariant sets. Physica D, 200(3–4), 205.MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Froyland, G., Padberg, K., England, M. H., & Treguier, A. M. (2007). Detection of coherent oceanic structures via transfer operators. Physical Review Letters, 98(22), 224503.CrossRefGoogle Scholar
  56. 56.
    Dellnitz, M., Froyland, G., Horenkamp, C., Padberg-Gehle, K., & Sen Gupta, A. (2009). Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators. Nonlinear Processes in Geophysics, 16(6), 655.CrossRefGoogle Scholar
  57. 57.
    Froyland, G., Santitissadeekorn, N., & Monahan, A. (2010). Transport in time-dependent dynamical systems: Finite-time coherent sets. Chaos, 20(4), 043116.MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Santitissadeekorn, N., Froyland, G., & Monahan, A. (2010). Optimally coherent sets in geophysical flows: A transfer-operator approach to delimiting the stratospheric polar vortex. Physical Review E, 82(5), 056311.CrossRefGoogle Scholar
  59. 59.
    Froyland, G., Horenkamp, C., Rossi, V., Santitissadeekorn, N. & Gupta, A. S. (2012). Three-dimensional characterization and tracking of an Agulhas Ring. Ocean Modelling, 52, 69.CrossRefGoogle Scholar
  60. 60.
    Banisch, R., & Koltai, P. (2017). Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets. Chaos, 27(3), 035804.MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Lindner, M., & Hellmann, F. (2018). Stochastic basins of attraction and generalized committor functions. Preprint. arXiv:1803.06372.Google Scholar
  62. 62.
    Jacobi, M. N., André, C., Döös, K., & Jonsson, P. R. (2012). Identification of subpopulations from connectivity matrices. Ecography, 35(11), 1004.CrossRefGoogle Scholar
  63. 63.
    Froyland, G. (2005). Statistically optimal almost-invariant sets. Physica D, 200(3–4), 205.MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Froyland, G., & Padberg, K. (2009). Almost-invariant sets and invariant manifolds-connecting probabilistic and geometric descriptions of coherent structures in flows. Physica D, 238(16), 1507.MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Froyland, G. 2001. Extracting dynamical behavior via Markov models. In Nonlinear dynamics and statistics (pp. 281–321). Berlin: SpringerCrossRefGoogle Scholar
  66. 66.
    Klus, S., Koltai, P., & Schütte, C. (2016). On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 3(1), 51–79.MathSciNetzbMATHGoogle Scholar
  67. 67.
    Froyland, G., Junge, O., & Koltai, P. (2013). Estimating long-term behavior of flows without trajectory integration: the infinitesimal generator approach. SIAM Journal on Numerical Analysis, 51(1), 223–247MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Cencini, M., Lacorata, G., Vulpiani, A., & Zambianchi, E. (1999). Mixing in a meandering jet: a Markovian approximation. Journal of Physical Oceanography, 29(10), 2578.CrossRefGoogle Scholar
  69. 69.
    Bower, A. S. (1991). A simple kinematic mechanism for mixing fluid parcels across a meandering jet. Journal of Physical Oceanography, 21(1), 173.CrossRefGoogle Scholar
  70. 70.
    Samelson, R. M. (1992). Fluid exchange across a meandering jet. Journal of Physical Oceanography, 22, 431.CrossRefGoogle Scholar
  71. 71.
    Raynal, F., & Wiggins, S. (2006). Lobe dynamics in a kinematic model of a meandering jet. I. Geometry and statistics of transport and lobe dynamics with accelerated convergence. Physica D, 223(1), 7.MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Peikert, R., Pobitzer, A., Sadlo, F., & Schindler, B. (2014). A Comparison of Finite-Time and Finite-Size Lyapunov Exponents (pp. 187–200). Cham: Springer International Publishing.zbMATHGoogle Scholar
  73. 73.
    Shadden, S. C., Lekien, F., & Marsden, J. E. (2005). Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D, 212(3–4), 271.MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Rodríguez-Méndez, V., Ser-Giacomi, E., & Hernández-García, E. (2017). Clustering coefficient and periodic orbits in flow networks. Chaos, 27(3), 035803.MathSciNetCrossRefGoogle Scholar
  75. 75.
    Tabeling, P., Perrin, B., & Fauve, S. (1987). Instability of a linear array of forced vortices. Europhysics Letters, 3(4), 459.CrossRefGoogle Scholar
  76. 76.
    Tabeling, P., Cardoso, O., & Perrin, B. (1990). Chaos in a linear array of vortices. Journal of Fluid Mechanics, 213, 511.CrossRefGoogle Scholar
  77. 77.
    Witt, A., Braun, R., Feudel, F., Grebogi, C., & Kurths, J. (1999). Tracer dynamics in a flow of driven vortices. Physical Review E, 59(2), 1605.MathSciNetCrossRefGoogle Scholar
  78. 78.
    Feudel, F., Witt, A., Gellert, M., Kurths, J., Grebogi, C., & Sanjuán, M. (2005). Intersections of stable and unstable manifolds: The skeleton of Lagrangian chaos. Chaos, Solitons and Fractals, 24(4), 947.MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Hadjighasem, A., Karrasch, D., Teramoto, H., & Haller, G. (2016). Spectral-clustering approach to Lagrangian vortex detection. Physical Review E, 93, 063107.CrossRefGoogle Scholar
  80. 80.
    Padberg-Gehle, K., & Schneide, C. (2017). Network-based study of Lagrangian transport and mixing. Nonlinear Processes in Geophysics, 24(4), 661.CrossRefGoogle Scholar
  81. 81.
    Froyland, G. (2013). An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems. Physica D, 250, 1.MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Froyland, G., Stuart, R. M., & van Sebille, E. (2014). How well-connected is the surface of the global ocean? Chaos, 24(3), 033126.MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Van Sebille, E., England, M. H., & Froyland, G. (2012). Origin, dynamics and evolution of ocean garbage patches from observed surface drifters. Environmental Research Letters, 7(4), 044040.CrossRefGoogle Scholar
  84. 84.
    Froyland, G., & Padberg-Gehle, K. (2012). Finite-time entropy: A probabilistic approach for measuring nonlinear stretching. Physica D, 241(19), 1612.zbMATHCrossRefGoogle Scholar
  85. 85.
    Fujiwara, N., Kirchen, K., Donges, J. F., & Donner, R. V. (2017). A perturbation-theoretic approach to Lagrangian flow networks. Chaos, 27(3), 035813.MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Rheinwalt, A., Marwan, N., Kurths, J., Werner, P. C., & Gerstengarbe, F. W. (2012). Boundary effects in network measures of spatially embedded networks. Europhysics Letters, 100(2), 28002.CrossRefGoogle Scholar
  87. 87.
    Rehfeld, K., Marwan, N., Breitenbach, S. F. M., & Kurths, J. (2013). Late Holocene Asian summer monsoon dynamics from small but complex networks of paleoclimate data. Climate Dynamics, 41(1), 3.CrossRefGoogle Scholar
  88. 88.
    Gelbrecht, M., Boers, N., & Kurths, J. (2017). A complex network representation of wind flows. Chaos, 27, 035808.CrossRefGoogle Scholar
  89. 89.
    Van Der Mheen, M., Dijkstra, H.A., Gozolchiani, A., Den Toom, M., Feng, Q., Kurths, J., et al. (2013). Interaction network based early warning indicators for the Atlantic MOC collapse. Geophysical Research Letters, 40(11), 2714.CrossRefGoogle Scholar
  90. 90.
    Tirabassi, G., Viebahn, J., Dakos, V., Dijkstra, H., Masoller, C., Rietkerk, M., et al. (2014). Interaction network based early-warning indicators of vegetation transitions. Ecological Complexity, 19, 148.CrossRefGoogle Scholar
  91. 91.
    Feng, Q. Y., Viebahn, J. P., & Dijkstra, H. A. (2014). Deep ocean early warning signals of an Atlantic MOC collapse. Geophysical Research Letters, 41(16), 6009.CrossRefGoogle Scholar
  92. 92.
    Rodríguez-Méndez, V., Eguíluz, V. M., Hernández-García, E., & Ramasco, J. J. (2016). Percolation-based precursors of transitions in extended systems. Scientific Reports, 6, 29552.CrossRefGoogle Scholar
  93. 93.
    Tsonis, A., & Swanson, K. (2008). Topology and predictability of El Niño and La Niña networks. Physical Review Letters, 100(22), 1.CrossRefGoogle Scholar
  94. 94.
    Gozolchiani, A., Yamasaki, K., Gazit, O. & Havlin, S. (2008). Pattern of climate network blinking links follows El Niño events. Europhysics Letters, 83(2), 28005.CrossRefGoogle Scholar
  95. 95.
    Martin, E., Paczuski, M., & Davidsen, J. (2013). Interpretation of link fluctuations in climate networks during El Niño periods. Europhysics Letters, 102(4), 48003.CrossRefGoogle Scholar
  96. 96.
    Radebach, A., Donner, R. V., Runge, J., Donges, J. F., & Kurths, J. (2013). Disentangling different types of El Niño episodes by evolving climate network analysis. Physical Review E, 88(5), 052807.CrossRefGoogle Scholar
  97. 97.
    Wiedermann, M., Radebach, A., Donges, J. F., Kurths, J., & Donner, R. V. (2016). A climate network-based index to discriminate different types of El Niño and La Niña. Geophysical Research Letters, 43(13), 7176.CrossRefGoogle Scholar
  98. 98.
    Ludescher, J., Gozolchiani, A., Bogachev, M. I., Bunde, A., Havlin, S., & Schellnhuber, H. J. (2013). Improved El Niño forecasting by cooperativity detection. Proceedings of the National Academy of Sciences, 110(29), 11742.CrossRefGoogle Scholar
  99. 99.
    Ludescher, J., Gozolchiani, A., Bogachev, M. I., Bunde, A., Havlin, S., & Schellnhuber, H. J. (2014). Very early warning of next El Niño. Proceedings of the National Academy of Sciences, 111(6), 2064.Google Scholar
  100. 100.
    Quian Quiroga, R., Kreuz, T., & Grassberger, P. (2002). Event synchronization: A simple and fast method to measure synchronicity and time delay patterns. Physical Review E, 66, 041904.MathSciNetCrossRefGoogle Scholar
  101. 101.
    Malik, N., Marwan, N., & Kurths, J. (2010). Spatial structures and directionalities in Monsoonal precipitation over South Asia. Nonlinear Processes in Geophysics, 17(5), 371.CrossRefGoogle Scholar
  102. 102.
    Malik, N., Bookhagen, B., Marwan, N., & Kurths, J. (2011). Analysis of spatial and temporal extreme monsoonal rainfall over South Asia using complex networks. Climate Dynamics, 39(3–4), 971.Google Scholar
  103. 103.
    Stolbova, V., Martin, P., Bookhagen, B., Marwan, N., & Kurths, J. (2014). Topology and seasonal evolution of the network of extreme precipitation over the Indian subcontinent and Sri Lanka. Nonlinear Processes in Geophysics, 21(4), 901.CrossRefGoogle Scholar
  104. 104.
    Boers, N., Bookhagen, B., Marwan, N., Kurths, J., & Marengo, J. (2013). Complex networks identify spatial patterns of extreme rainfall events of the South American Monsoon System. Geophysical Research Letters, 40(16), 4386.CrossRefGoogle Scholar
  105. 105.
    Boers, N., Bookhagen, B., Barbosa, H. M. J., Marwan, N., Kurths, J., & Marengo, J. A. (2014). Prediction of extreme floods in the eastern Central Andes based on a complex networks approach. Nature Communications, 5, 5199.CrossRefGoogle Scholar
  106. 106.
    Pöschke, P., Sokolov, I. M., Nepomnyashchy, A. A., & Zaks, M. A. (2016). Anomalous transport in cellular flows: The role of initial conditions and aging. Physical Review E, 94, 032128.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Reik V. Donner
    • 1
  • Michael Lindner
    • 2
    • 3
  • Liubov Tupikina
    • 4
  • Nora Molkenthin
    • 5
  1. 1.Potsdam Institute for Climate Impact ResearchPotsdamGermany
  2. 2.Potsdam Institute for Climate Impact ResearchPotsdamGermany
  3. 3.Department of MathematicsHumboldt UniversityBerlinGermany
  4. 4.Laboratoire de Physique de la Matiere CondenseeEcole PolytechniquePalaiseauFrance
  5. 5.Max Planck Institute for Dynamics and Self-OrganizationGöttingenGermany

Personalised recommendations