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 TOCSYJkurths@gmx.de

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 topology

A Fashionable Topic or a Useful One?

Networks with Complex Topology

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

Road Maps

Local Transportation

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

- minimum/ maximum intensities

or

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

Each oscillator forced by a common signal

Coupling strength ~ degree

For nodes with rather large degree

Mean-field approximation

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 com-munities

•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

time

Dynamical Network Approach

Correct words (Priester)Pseudowords (Priesper)

Conventional ERP Analysis

t1 t2

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)