network of networks and the climate system potsdam institute for climate impact research &...
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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
[email protected]@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
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
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.
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)
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)