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.h
  tml
• Toolbox TOCSY
• Jkurths_at_gmx.de

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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
Networks with complex topology
A Fashionable Topic or a Useful One?
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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
Local Transportation
Road Maps
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Synchronization in such networks

• Synchronization properties strongly influenced by
  the networks 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
or
- minimum/ maximum intensities
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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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Mean-field approximation
Each oscillator forced by a common
signal Coupling strength degree
For nodes with rather large degree
? 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

• Connection matrix of the cortical network of the
  cat brain (anatomical)
• 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

t2
t1
time
Dynamical Network Approach
Conventional ERP Analysis
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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 (nm
  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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Poincarés Recurrence
Arnolds 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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