9th International Summer School/Conference
LET'S FACE CHAOS THROUGH NONLINEAR DYNAMICS
at the University of Maribor, Slovenia
22 June - 6 July 2014
1. Introduction: The network approach to nature
2. The structure of complex networks
2.1. Definitions and notations
2.1.1. Node degree, degree distributions and correlations
2.1.2. Shortest path lengths, diameter and betweenness
2.1.5. Community structures
2.2. Topology of real networks
2.2.1. The small-world property
2.2.2. Scale-free degree distributions
2.3. Networks models
2.3.1. Random graphs
2.3.2. Generalized random graphs
2.3.3. Small-world networks
2.3.4. Static scale-free networks
2.3.5. Evolving scale-free networks
2.4. Weighted and spatial networks
3. Synchronization and collective dynamics
3.1. Introduction to synchronization
3.2. The Master stability function approach
3.3. Network propensity for synchronization
3.3.1. Synchronization in weighted networks: coupling matrices with real spectra
3.3.2. Synchronization in weighted networks: coupling matrices with complex spectra
4. Algorithms for finding community structures
4.1. The algorithm by Girvan and Newman
4.2. Other algorithms
These 5 lectures are meant to be introductory, with the main ideas presented in a pedagogical way by means of simple examples. They are aimed primarily at graduate students interested in complex phenomena occurring in various disciplines. In the first 2 lectures, we will be concerned with the emergence of collective behavior in the form of clustering, flocking, synchronization, etc. as they occur in dissipative systems of interest to Physics and Biology. Our approach will be to start from some crucial observation or experiment and seek to construct the appropriate mathematical model that captures the main features of the data. The remaining 3 lectures will focus on a class of conservative systems described by N-degree of freedom Hamiltonian functions, which are familiar to us from classical mechanics, astronomy and solid state physics. Our main point will be to show that despite a well established general theory, there are still many important local phenomena involving various degrees of order and chaos that need to be further understood, because of their global consequences regarding the physical properties of the system, especially for long times and large N. We will thus discover that to understand these complex aspects of Hamiltonian models, we need to combine the mathematical techniques of nonlinear dynamics with a statistical analysis of probability distributions of chaotic orbits in different regimes of the multi – dimensional phase space, where the motion of the system evolves.
The lectures (7h) are organized in two main parts: Part I will be devoted to the broad initiative 'FuturICT', Part II will be devoted to the applications of fractal geometry for the analysis of big-data sets of socio-economic-enviromental interest.
Part I: Global Community for our Complex connected World
FuturICT is a visionary project that will deliver new science and technology to explore, understand and manage our connected world. This will inspire new information and communication technologies (ICT) that are socially adaptive and socially interactive, supporting collective awareness.
Our increasingly dense interconnected world poses every day new challenges that need to be approached in several dimensions, at different temporal and spatial scales. In particular, given the scope and scale of the world's future Internet of everything, new technologies with the lowest energetic impact, unconventional computational schemes, novel phenomena and paradigm should be figure out for understanding and managing such increasing complexity. Revealing the hidden laws and processes underlying our complex, global, socially interactive systems constitutes one of the most pressing scientific challenges of the 21st Century. Integrating complexity science with ICT and the social sciences, will allow us to design novel robust, trustworthy and adaptive technologies based on socially inspired paradigms. Data from a variety of sources will help us to develop models of techno-socioeconomic systems. In turn, insights from these models will inspire a new generation of socially adaptive, self-organised ICT systems. This will create a paradigm shift and facilitate a symbiotic co-evolution of ICT and society. Further info at www.futurict.eu
Part II: A non-Random Walk through our complex connected world
Time series are a tool to describe biological, social and economic systems in one dimension, such as stock market indexes and genomic sequences. Extended systems evolving over space, such as urban textures, World Wide Web and firms are described in terms of high-dimensional random structures.
A short overview of the Detrending Moving Average (DMA) algorithm is presented. The DMA has the ability to quantify temporal and spatial long-range dependence of fractal sets with arbitrary dimension. Time series, profiles and surfaces can be characterized by the fractal dimension D, a measure of roughness, and by the Hurst exponent H, a measure of long-memory dependence. The method, in addition to accomplish accurate and fast estimates of the fractal dimension D and Hurst exponent H, can provide interesting clues between fractal properties, self-organized criticality and entropy of long-range correlated sequences. Further readings and tips about the DMA algorithm at www.polito.it/noiselab
We start with a recapitulation of basic notions of dynamics; flows, maps, local linear stability, heteroclinic connections, qualitative dynamics of stretching and mixing and symbolic dynamics.
The lecture notes and videos are available online, as parts of the advanced nonlinear dynamics course, ChaosBook.org/version15/Maribor13.shtml
A motion on a strange attractor can be approximated by shadowing the orbit by a sequence of nearby periodic orbits of finite length. This notion is here made precise by approximating orbits by primitive cycles, and evaluating associated curvatures. A curvature measures the deviation of a longer cycle from its approximation by shorter cycles; the smoothness of the dynamical system implies exponential (or faster) fall-off for (almost) all curvatures. The technical prerequisite for implementing this shadowing is a good understanding of the symbolic dynamics of the classical dynamical system. The resulting cycle expansions offer an efficient method for evaluating classical and quantum periodic orbit sums; accurate estimates can be obtained by using as input the lengths and eigenvalues of a few prime cycles.
All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. Our goal is to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise. That is achieved by computing the local eigenfunctions of the Fokker-Planck evolution operator in linearized neighborhoods of the periodic orbits of the corresponding deterministic system, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables.
Dynamical systems often come equipped with symmetries, such as the reflection symmetries of various potentials. Symmetries simplify the dynamics in a rather beautiful way:
If dynamics is invariant under a set of discrete symmetries G, the state space M is tiled by a set of symmetry-related tiles, and the dynamics can be reduced to dynamics within one such tile, the fundamental domain M/G. If the symmetry is continuous, the dynamics is reduced to a lower-dimensional desymmetrized system M/G, with ``ignorable" coordinates eliminated (but not forgotten). We reduce a continuous symmetry by slicing the state space in such a way that an entire class of symmetry-equivalent points is represented by a single point.
In either case, families of symmetry-related full state space cycles are replaced by fewer and often much shorter ``relative'' cycles. In presence of a symmetry the notion of a prime periodic orbit has to be reexamined: it is replaced by the notion of a relative periodic orbit, the shortest segment of the full state space cycle which tiles the cycle under the action of the group. Furthermore, the group operations that relate distinct tiles do double duty as letters of an alphabet which assigns symbolic itineraries to trajectories.
As a turbulent flow evolves, every so often we catch a glimpse of a familiar pattern. For any finite spatial resolution, the system follows approximately for a finite time a pattern belonging to a finite alphabet of admissible patterns. In ``Hopf's vision of turbulence,'' the long term turbulent dynamics is a walk through the space of such unstable patterns.
At first sight, it seems a bit far-fetched that physicists work on economics problems. A closer look, however, reveals that the connection between physics and economics is rather natural -- and not even new! Many physicists are surprised to hear that the mathematician Bachelier developed a theory of stochastic processes very similar to the theory of Brownian motion which Einstein put forward in 1905. Bachelier did it in the context of financial instruments, and he was even a bit earlier than Einstein. Moreover, not all physicists know that financial time series were a major motivation for Mandelbrot when he started his work on fractals. Mathematical modeling in physics and economics, in particular finance, is similar!
In the last 15 or 20 years, the physicists' interest in economic issues grew ever faster, and the term ``econophysics'' was coined. Econophysics developed into a recognized subject. The crucial reason for this was the dramatic improvement of the data situation, a wealth of data became available and (electronically) accessible. Moreover, complex systems moved into the focus of physics research. The economy certainly qualifies as a complex system and poses serious challenges for basic research. Simultaneously, economics started to develop into a more quantitative science. From a more practical viewpoint, the need to quantitatively improve risk management in economics is a driving force in econophysics.
The presentation starts from scratch, no background in economics is needed, it consists of five lectures: (1) Basic Concepts, (2) Detailed Look at Stock Markets and Trading, (3) Financial Correlations and Portfolio Optimization, (4) Quantitative Identification of Market States, (5) Credit Risk.
The field develops quickly, implying that not all of the topics in the course can be found in text books appropriate for a physics audience. Some good text books written by physicists are listed below, further literature will be given in the course.
Mantegna R.N and Stanley H.E. 2000
An Introduction to Econophysics,
Cambridge University Press, Cambridge
Bouchaud J.P.and Potters M. 2003 Theory of Financial Risk and Derivative Pricing, Cambridge University Press, Cambridge
Voit J. 2001 The Statistical Mechanics of Financial Markets, Springer, Heidelberg
Last modified: 26 December 2013