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.html
Toolbox TOCSY
[email protected]
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
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
Networks
with
Complex
Networks with
complex topology
Topology
A Fashionable Topic or a Useful
One?
Inferring Scale-free Networks
What does it mean: the power-law behavior is clear?
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!
Hype
• Application to huge networks
(e.g. number of different sexual partners in
one country SF) – What to learn from
this?
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.
Transportation Networks
Airport Networks
Local Transportation
Road Maps
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)
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….)
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)
Main results
Synchronizability universally determined by:
- mean degree K and
- heterogeneity of the intensities
or
- minimum/ maximum intensities
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)
Identical oscillators
Transition to synchronization
Mean-field approximation
Each oscillator forced by a common signal
Coupling strength ~ degree
For nodes with rather large degree
 Scaling:
Clusters of
synchronization
Non-identical oscillators
 phase synchronization
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
Cat Cerebal Cortex
Connectivity
Scannell et al.,
Cereb. Cort., 1999
Modelling
• Intention:
Macroscopic  Mesoscopic Modelling
Network of Networks
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)
Density of connections
between the four communities
•Connections among
the nodes: 2-3 … 35
•830 connections
•Mean degree: 15
Model for neuron i in area I
FitzHugh Nagumo model
Transition to synchronized firing
g – coupling strength – control parameter
Functional vs. Structural Coupling
Intermediate Coupling
Intermediate Coupling:
3 main dynamical clusters
Strong Coupling
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
Correct words (Priester)
Pseudowords (Priesper)
Conventional ERP Analysis
time
Dynamical Network Approach
Identification of connections – How to avoid
spurious ones?
Problem of multivariate statistics: distinguish
direct and indirect interactions
Linear Processes
• Case: multivariate system of linear stochastic
processes
• Concept of Graphical Models (R. Dahlhaus,
Metrika 51, 157 (2000))
• Application of partial spectral coherence
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
Partial Phase Synchronization
Synchronization Matrix
with elements
Partial Phase Synchronization Index
Example
Example
• Three Rössler oscillators (chaotic regime)
with additive noise;
non-identical
• Only bidirectional coupling 1 – 2; 1 - 3
Extension to more complex phase dynamics
• Concept of recurrence
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
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)
Poincaré‘s Recurrence
Arnold‘s cat map
Crutchfield 1986,
Scientific American
Probability of recurrence after a certain time
• Generalized auto (cross) correlation function
(Romano, Thiel, Kurths, Kiss, Hudson
Europhys. Lett. 71, 466 (2005) )
Roessler Funnel – Non-Phase coherent
Two coupled Funnel Roessler oscillators - Non-synchronized
Two coupled Funnel Roessler oscillators – Phase and General
synchronized
Phase Synchronization in time delay systems
Generalized Correlation Function
Phase and Generalized Synchronization
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
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)