Complex Networks – a fashionable topic or a useful one?
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Complex Networks – a fashionable topic or a
useful one?Jürgen Kurths¹ ², G. Zamora¹,
L. Zemanova¹, C. S. Zhou³
¹University Potsdam, Center for Dynamics of Complex Systems (DYCOS), Germany
² Humboldt University Berlin and Potsdam Institute
for Climate Impact Research, Germany
³ Baptist University, Hong Kong
http://www.agnld.uni-potsdam.de/~juergen/juergen.htmlToolbox [email protected]
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Outline
• Complex Networks Studies: Fashionable or Useful?
• Synchronization in complex networks via hierarchical (clustered) transitions
• Application: structure vs. functionality in complex brain networks – network of networks
• Retrieval of direct vs. indirect connections in networks (inverse problem)
• Conclusions
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Ensembles: Social Systems
• Rituals during pregnancy: man and woman isolated from community; both have to follow the same tabus (e.g. Lovedu, South Africa)
• Communities of consciousness and crises
• football (mexican wave: la ola, ...)• Rhythmic applause
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Networks with complex topology
A Fashionable Topic or a Useful One?
Networks with Complex Topology
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Inferring Scale-free Networks
What does it mean: the power-law behavior is clear?
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Hype: studies on complex networks
• Scale-free networks – thousands of examples in the recent literature
• log-log plots (frequency of a minimum number of connections nodes in the network have): find „some plateau“ Scale-Free Network - similar to dimension estimates in the 80ies…)
!!! What about statistical significance? Test statistics to apply!
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Hype
• Application to huge networks (e.g. number of different sexual partners in one country SF) – What to learn from this?
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Useful approaches with networks
• Many promising approaches leading to useful applications, e.g.
• immunization problems (spreading of diseases)
• functioning of biological/physiological processes as protein networks, brain dynamics, colonies of thermites
• functioning of social networks as network of vehicle traffic in a region, air traffic, or opinion formation etc.
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Transportation Networks
Airport Networks
Road Maps
Local Transportation
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Synchronization in such networks
• Synchronization properties strongly influenced by the network´s structure (Jost/Joy, Barahona/Pecora, Nishikawa/Lai, Timme et al., Hasler/Belykh(s), Boccaletti et al., etc.)
• Self-organized synchronized clusters can be formed (Jalan/Amritkar)
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Universality in the synchronization of weighted
random networks
Our intention:
Include the influence of weighted coupling for complete synchronization
(Motter, Zhou, Kurths; Boccaletti et al.; Hasler et al….)
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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)
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Main results
Synchronizability universally determined by:
- mean degree K and
- heterogeneity of the intensities
- minimum/ maximum intensities
or
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Hierarchical Organization of Synchronization in Complex
Networks
Homogeneous (constant number of connections in each node)
vs.
Scale-free networks
Zhou, Kurths: CHAOS 16, 015104 (2006)
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Identical oscillators
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Transition to synchronization
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Each oscillator forced by a common signal
Coupling strength ~ degree
For nodes with rather large degree
Mean-field approximation
Scaling:
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Clusters of synchronization
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Non-identical oscillators
phase synchronization
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Transition to synchronization in complex networks
• Hierarchical transition to synchronization via clustering
• Hubs are the „engines“ in cluster formation AND they become synchronized first among themselves
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Cat Cerebal Cortex
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Connectivity
Scannell et al.,
Cereb. Cort., 1999
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Modelling
• Intention:
Macroscopic Mesoscopic Modelling
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Network of Networks
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Hierarchical organization in complex brain networks
a) Connection matrix of the cortical network of the cat brain (anatomical)
b) Small world sub-network to model each node in the network (200 nodes each, FitzHugh Nagumo neuron models - excitable)
Network of networks
Phys Rev Lett 97 (2006), Physica D 224 (2006)
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Density of connections between the four com-munities
•Connections among the nodes: 2-3 … 35
•830 connections
•Mean degree: 15
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Model for neuron i in area I
FitzHugh Nagumo model
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Transition to synchronized firing
g – coupling strength – control parameter
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Functional vs. Structural Coupling
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Intermediate Coupling
Intermediate Coupling:
3 main dynamical clusters
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Strong Coupling
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Inferring networks from EEG during cognition
Analysis and modeling of Complex Brain Networks
underlying Cognitive (sub) Processes Related to Reading, basing on single trial evoked-activity
time
Dynamical Network Approach
Correct words (Priester)Pseudowords (Priesper)
Conventional ERP Analysis
t1 t2
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Identification of connections – How to avoid spurious ones?
Problem of multivariate statistics: distinguish direct and indirect interactions
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Linear Processes
• Case: multivariate system of linear stochastic processes
• Concept of Graphical Models (R. Dahlhaus, Metrika 51, 157 (2000))
• Application of partial spectral coherence
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Extension to Phase Synchronization Analysis
• Bivariate phase synchronization index (n:m synchronization)
• Measures sharpness of peak in histogram of
Schelter, Dahlhaus, Timmer, Kurths: Phys. Rev. Lett. 2006
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Partial Phase Synchronization
Synchronization Matrix
with elements
Partial Phase Synchronization Index
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Example
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Example
• Three Rössler oscillators (chaotic regime) with additive noise; non-identical
• Only bidirectional coupling 1 – 2; 1 - 3
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Extension to more complex phase dynamics
• Concept of recurrence
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H. Poincare
If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at the succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws.
But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.
(1903 essay: Science and Method)
Weak Causality
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Concept of Recurrence
Recurrence theorem:
Suppose that a point P in phase space is covered by a conservative system. Then there will be trajectories which traverse a small surrounding of P infinitely often.That is to say, in some future time the system will return arbitrarily close to its initial situation and will do so infinitely often. (Poincare, 1885)
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Poincaré‘s Recurrence
Arnold‘s cat map
Crutchfield 1986, Scientific American
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Probability of recurrence after a certain time
• Generalized auto (cross) correlation function
(Romano, Thiel, Kurths, Kiss, Hudson Europhys. Lett. 71, 466 (2005) )
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Roessler Funnel – Non-Phase coherent
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Two coupled Funnel Roessler oscillators - Non-synchronized
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Two coupled Funnel Roessler oscillators – Phase and General synchronized
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Phase Synchronization in time delay systems
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Generalized Correlation Function
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Phase and Generalized Synchronization
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Summary
Take home messages:
• There are rich synchronization phenomena in complex networks (self-organized structure formation) – hierarchical transitions
• This approach seems to be promising for understanding some aspects in cognitive and neuroscience
• The identification of direct connections among nodes is non-trivial
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Our papers on complex networks
Europhys. Lett. 69, 334 (2005) Phys. Rev. Lett. 98, 108101 (2007)Phys. Rev. E 71, 016116 (2005) Phys. Rev. E 76, 027203 (2007)CHAOS 16, 015104 (2006) New J. Physics 9, 178 (2007)Physica D 224, 202 (2006) Phys. Rev. E 77, 016106 (2008) Physica A 361, 24 (2006) Phys. Rev. E 77, 026205 (2008)Phys. Rev. E 74, 016102 (2006) Phys. Rev. E 77, 027101 (2008)Phys: Rev. Lett. 96, 034101 (2006) CHAOS 18, 023102 (2008)Phys. Rev. Lett. 96, 164102 (2006) J. Phys. A 41, 224006 (2008)Phys. Rev. Lett. 96, 208103 (2006)Phys. Rev. Lett. 97, 238103 (2006)Phys. Rev. E 76, 036211 (2007)Phys. Rev. E 76, 046204 (2007)
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