Complex Networks – a fashionable topic or a useful one? Jürgen Kurths¹ ², G. Zamora¹, L. Zemanova¹, C. S. Zhou ³ 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 Toolbox TOCSY

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, (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 … 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

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, (2007) Phys. Rev. E 71, (2005) Phys. Rev. E 76, (2007) CHAOS 16, (2006) New J. Physics 9, 178 (2007) Physica D 224, 202 (2006) Phys. Rev. E 77, (2008) Physica A 361, 24 (2006) Phys. Rev. E 77, (2008) Phys. Rev. E 74, (2006) Phys. Rev. E 77, (2008) Phys: Rev. Lett. 96, (2006) CHAOS 18, (2008) Phys. Rev. Lett. 96, (2006) J. Phys. A 41, (2008) Phys. Rev. Lett. 96, (2006) Phys. Rev. Lett. 97, (2006) Phys. Rev. E 76, (2007) Phys. Rev. E 76, (2007)