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 Jürgen Kurths

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

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

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

Basic Problem: Structure vs. Functionality

Dynamics on the nodes - synchronization

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)

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

Main results Synchronizability universally determined by: - mean degree K and - heterogeneity of the intensities - minimum/ maximum intensities or

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

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

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

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

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

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 > 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.

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.

Network of Networks Interconnected Networks Interdependent Networks

Power grid in Japan

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

Papenburg: Monster Black-Out 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

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

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

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)

Reference state Cortical network model Added feedback controller Control aim

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

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)

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