1. Network of Networks and the Climate System Potsdam Institute for Climate Impact Research & Institut of Physics, Humboldt-Universität zu Berlin & King‘s College, University of Aberdeen juergen.kurths@pik-potsdam.de Jürgen Kurths 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. Biological Networks Neural Networks Genetic Networks Protein interaction Ecological Webs 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 determined by: - mean degree K and - heterogeneity of the intensities - minimum/ maximum intensities or

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

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

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

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

16. Basin Stability basin volume of a state (regime) measures likelihood of arrival at this state (regime) NATURE PHYSICS (in press)

20. 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.

21. 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.

22. Network of Networks Interconnected Networks Interdependent Networks

23. Power grid in Japan

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

25. 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

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

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

28. 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)

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

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

31. 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)

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