Network of Networks and the Climate System

Potsdam Institute for Climate Impact ResearchPotsdam Institute for Climate Impact Research&&

Institut of Physics, Humboldt-Universität zu Berlin Institut of Physics, Humboldt-Universität zu Berlin & &

King‘s College, University of AberdeenKing‘s College, University of Aberdeen

juergen.kurths@pik-potsdam.dejuergen.kurths@pik-potsdam.de

Jürgen KurthsJürgen Kurths

http://www.pik-potsdam.de/members/kurths/

Main Collaborators:Main Collaborators:

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

ZouZou

Networks with Complex Topology

Sociology, Economy, Biology, Engineering, Physics, Chemistry, Earth System,…

Biological Networks

Neural Networks

Genetic Networks

Protein interactionEcological 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 neighbourconnections

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) networksof 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.

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 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

Outer Synchronization:two coupled networks

Li, Sun, Kurths: Phys. Rev. E 76, 046204 (2007)

Li, Xu, Sun, J. Xu, Kurths, CHAOS 19, 013106 (2009)

Density of connections between the four com-munities

•Connections among the nodes: 2 … 35

•830 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