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Network of Networks and the Climate System

Network of Networks and the Climate System. Jürgen Kurths. Potsdam Institute for Climate Impact Research & Institut of Physics, Humboldt-Universität zu Berlin & King‘s College, University of Aberdeen. juergen.kurths@pik-potsdam.de. http://www.pik-potsdam.de/members/kurths/.

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Network of Networks and the Climate System

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  1. Network of Networks and the Climate System Jürgen Kurths Potsdam Institute for Climate Impact Research & Institut of Physics, Humboldt-Universität zu Berlin & King‘s College, University of Aberdeen juergen.kurths@pik-potsdam.de http://www.pik-potsdam.de/members/kurths/

  2. Main Collaborators:B. Bookhagen, S. Breitenbach, J. Donges, R. Donner, B. Goswami, J. Heitzig, P. Menck, N. Malik, N. Marwan, K. Rehfeld, C. Zhou, Y. Zou

  3. Networks with Complex Topology Sociology, Economy, Biology, Engineering, Physics, Chemistry, Earth System,…

  4. Neural Networks Biological Networks Ecological Webs Protein interaction Genetic Networks Metabolic Networks

  5. Basic Problem: Structure vs. Functionality

  6. Dynamics on the nodes - synchronization

  7. Weighted Network of N Identical Oscillators F – dynamics of each oscillator H – output function G – coupling matrix combining adjacency A and weight W - intensity of node i (includes topology and weights)

  8. General Condition for Synchronizability Stability of synchronized state N eigenmodes of ith eigenvalue of G

  9. Main results Synchronizability universally determinedby: - mean degree K and - heterogeneity of the intensities or - minimum/ maximum intensities

  10. Synchronizability – Master Stability Formalism (Pecora&Carrol (1998) Synchronizability Ratio Stability Interval Synchronizability condition

  11. Stability/synchronizability in small-world (SW) networks Small-world (SW) networks (Watts, Strogatz, 1998 F. Karinthyhungarian writer – SW hypothesis, 1929)

  12. Small-world Networks Nearest neighbour and a few long-range connections Nearest neighbour connections Regular  Complex Topology

  13. MSF – local stability (Lyapunov stability)How to go beyond (not only small perturbations)?

  14. Basin Stabilitybasin volume of a state (regime) measures likelihood of arrival at this state (regime) NATURE PHYSICS (in press)

  15. Synchronizability and basin stability inWatts-Strogatz (WS) networks of chaotic oscillators. a: Expected synchronizability R versus the WS model's parameter p. The scale of the y-axis was reversed to indicate improvement upon increase in p. b: Expected basin stability S versus p. The grey shade indicates one standard deviation. The dashed line shows an exponential fitted to the ensemble results for p > 0.15. Solid lines are guides to the eye. The plots shown were obtained for N = 100 oscillators of Roessler type, each having on average k = 8 neighbours. Choices of larger N and different k produce results that are qualitatively the same.

  16. Topological comparison of ensemble results with real-world networks. - Circle represents the results for Watts-Strogatz networks with N = 100, k = 10 and rewiring probability p (increasing from left to right 0.05…1.0). - Circle's area proportional to expected basin stability S. - Circle's colour indicates ex- pected synchronizability R. - Squares represent real-world networks reported to display a small-world topology.

  17. Network of NetworksInterconnected NetworksInterdependent Networks

  18. Power grid in Japan

  19. Control of such networks?Pinning control (which nodes?)

  20. Papenburg: Monster Black-Out 06-11-2006 • Meyer Werft in Papenburg • Newly built ship Norwegian Pearl length: 294 m, width: 33 m • Cut one line of the power grid • Black-out in Germany ( > 10 Mio people) France (5 Mio people) Austria, Belgium, Italy, Spain

  21. OuterSynchronization:two coupled networks Li, Sun, Kurths: Phys. Rev. E 76, 046204 (2007) Li, Xu, Sun, J. Xu, Kurths, CHAOS 19, 013106 (2009)

  22. Cat Cerebal Cortex Density of connections between the four com-munities • Connections among the nodes: 2 … 35 • 830 connections • Mean degree: 15

  23. Pinning Control in Neuronal Networks • Pinning Control: apply control only to a few nodes, but reaching the control target for the whole network (some synchronization) • Problem: Identifying the controlling nodes PLoS ONE 7, e41375 (2012)

  24. Reference state Cortical network model Added feedback controller Control aim

  25. Multimodal Optimization Problem Identify the location of drivers satisfying Self-adaptive differential evolution method (JaDE)

  26. Main Results of JaDE Dependence on the number of driver nodes: • very small number (1-3): nodes with high degree and betweenness are best (hubs) • Intermediate number (4…15): nodes with small degree and betweenness best (!not hubs!) The auditory community is most prominent for driver node selection (although sparsely connected to the others)

  27. Technological Networks – Combined Design? World-Wide Web Internet Power Grid

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