Complex Networks A Fashionable Topic or a Useful One

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Complex Networks A Fashionable Topic or a
Useful One?

• Jürgen Kurths¹, G. Osipov², G. Zamora¹, C. S.
  Zhou³
• ¹University Potsdam, Center for Dynamics of
  Complex Systems (DYCOS), Germany
  
• ² University Nizhny Novgorod, Russia
• 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

• Introduction
• Fashionable vs. 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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Networks with Complex Topology
Networks with complex topology

• Random graphs/networks (Erdös, Renyi, 1959)
• Small-world networks (Watts, Strogatz, 1998)
• Scale-free networks (Barabasi, Albert, 1999)
• Many participants (nodes) with complex
  interactions and complex dynamics at the nodes

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Complex networks a fashionable topic or a
useful one?
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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)
• 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 or air traffic etc.

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Scale-freee Networks

• Network resiliance
• Highly robust against random failure of a node
• Highly vulnerable to deliberate attacks on hubs
• Applications
• Immunization in networks of computers, humans, ...

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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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Universality in the synchronization of weighted
random networks
Our intention Include the influence of
weighted coupling for complete synchronization
Motter, Zhou, Kurths Phys. Rev. E
71, 016116 (2005) Phys. Rev. Lett. 96, 034101
(2006)
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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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Clusters of 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 Anatomic clusters

• Connections among the nodes 2-3 35
• 830 connections
• Mean degree 15

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Model for neuron i in area I
Fitz Hugh Nagumo model excitable system
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Transition to synchronized firing

• g coupling strength control parameter

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Network topology vs. Functional organization in
networks
Weak-coupling dynamics ? non-trivial
organization ? relationship to underlying
network topology
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Functional vs. Structural Coupling
Dynamic Clusters
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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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Initial brain states influence evoked activity
corr, significant
trial3
non significant
- corr, significant
trial13
On-going fluctuations single trial
EEG minus average ERP
trial15
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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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Summary

• Take home messages
• There are rich synchronization phenomena in
  complex networks (self-organized structure
  formation) hierarchical transitions
• The approach network of networks seems to be
  promising for understanding some aspects of
  structure formation in various fields
• 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) Phys. Rev. Lett. 96, 164102
(2006) 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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