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 Jkurths@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 Networks with complex 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 determinedby: - 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)
Mean-field approximation Each oscillator forced by a common signal Coupling strength ~ degree For nodes with rather large degree Scaling:
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
Connectivity Scannell et al., Cereb. Cort., 1999
Modelling • Intention: Macroscopic Mesoscopic Modelling
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)
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
Intermediate Coupling Intermediate Coupling: 3 main dynamical clusters
Correct words (Priester) Pseudowords (Priesper) Inferring networks from EEG during cognition Analysis and modeling of Complex Brain Networks underlyingCognitive (sub) ProcessesRelated to Reading, basing on single trial evoked-activity t2 t1 time Conventional ERP Analysis 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 • 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 greatones 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) )
Two coupled Funnel Roessler oscillators – Phase and General synchronized