Introduction to the Theory of Complex Systems

Description

This book is a comprehensive introduction to quantitative approaches to complex adaptive systems. Practically all areas of life on this planet are constantly confronted with complex systems, be it ecosystems, societies, traffic, financial markets, opinion formation and spreading, or the internet and social media. Complex systems are systems composed of many elements that interact strongly with each other, which makes them extremely rich dynamical systems showing a huge range of phenomena. Properties of complex systems that are of particular importance are their efficiency, robustness, resilience, and proneness to collapse. The quantitative tools and concepts needed to understand the co-evolutionary nature of networked systems and their properties are challenging. The book gives a self-contained introduction to these concepts, so that the reader will be equipped with a toolset that allows them to engage in the science of complex systems. Topics covered include random processes of path-dependent processes, co-evolutionary dynamics, dynamics of networks, the theory of scaling, and approaches from statistical mechanics and information theory. The book extends beyond the early classical literature in the field of complex systems and summarizes the methodological progress made over the past 20 years in a clear, structured, and comprehensive way. Cover Introduction to the Theory of Complex Systems Copyright Preface Contents Chapter 1: Introduction to Complex Systems 1.1 Physics,biology,or social science? 1.2 Components from physics 1.2.1 The nature of the fundamental forces 1.2.2 What does predictive mean? 1.2.3 Statistical mechanics—predictability on stochastic grounds 1.2.4 The evolution of the concept of predictability in physics 1.2.5 Physics is analytic, complex systems are algorithmic 1.2.6 What are complex systems from a physics point of view? 1.2.7 A note on chemistry—the science of equilibria 1.3 Components from the life sciences 1.3.1 Chemistry of small systems 1.3.2 Biological interactions happen on networks—almost exclusively 1.3.3 What is evolution? 1.3.3.1 Evolution is not physics 1.3.3.2 The concept of the adjacent possible 1.3.3.3 Summary evolutionary processes 1.3.4 Adaptive and robust—the concept of the edge of chaos 1.3.4.1 How does nature find the edge of chaos? 1.3.4.2 Intuition behind self-organized critical systems—sand pile models 1.3.5 Components taken from the life sciences 1.4 Components from the social sciences 1.4.1 Social systems are continuously restructuring networks 1.5 What are Complex Systems? 1.5.1 What is co-evolution? 1.5.2 The role of the computer 1.6 The structure of the book 1.6.1 What has complexity science contributed to the history of science? Chapter 2: Probability and Random Processes 2.1 Overview 2.1.1 Basic concepts and notions 2.1.1.1 Trials, odds, chances, and probability 2.1.1.2 Experiments, random variables, and sample space 2.1.1.3 Systems and processes 2.1.1.4 Urns 2.1.1.5 Probability and distribution functions 2.1.2 Probability and information 2.1.2.1 Why are random processes important? 2.2 Probability 2.2.1 Basic probability measures and the Kolmogorov axioms 2.2.2 Histograms and relative frequencies 2.2.3 Mean, variance, and higher moments 2.2.4 More than one random variable 2.2.4.1 Comparing random variables 2.2.4.2 Joint and conditional probabilities 2.2.5 A note on Bayesian reasoning 2.2.6 Bayesian and frequentist thinking 2.2.6.1 Hypothesis testing in the frequentist approach 2.3 The law of large numbers—adding random numbers 2.3.1 The central limit theorem 2.3.1.1 Gaussian distribution function 2.3.1.2 The convolution product 2.3.1.3 Log-normal distribution function 2.3.2 Generalized limit theorems and α-stable processes 2.3.2.1 Distribution functions of α-stable processes 2.3.2.2 Gaussian distribution function 2.3.2.3 Cauchy distribution function 2.3.2.4 Lévy distribution function 2.3.2.5 Lévy flights 2.4 Fat-tailed distribution functions 2.4.1 Distribution functions that show power law tails 2.4.1.1 Pareto distribution 2.4.1.2 Zipf distribution 2.4.1.3 Zipf–Mandelbrot distribution 2.4.1.4 The q-exponential distribution function—Tsallis distribution 2.4.1.5 Student-t distribution 2.4.2 Other distribution functions 2.4.2.1 Exponential -or Boltzmann distribution 2.4.2.2 Stretched exponential distribution function 2.4.2.3 Gumbel distribution 2.4.2.4 Weibull distribution 2.4.2.5 Gamma distribution 2.4.2.6 Gompertz distribution 2.4.2.7 Generalized exponential distribution or Lambert-W exponential distribution 2.5 Stochastic processes 2.5.1 Simple stochastic processes 2.5.1.1 Bernoulli processes 2.5.1.2 Binomial distribution functions 2.5.1.3 Multinomial processes 2.5.1.4 Poisson processes 2.5.1.5 Markov processes 2.5.2 History- or path-dependent processes 2.5.3 Reinforcement processes 2.5.4 Driven dissipative systems 2.5.4.1 Processes with dynamical sample spaces 2.6 Summary 2.7 Problems Chapter 3: Scaling 3.1 Overview 3.1.1 Definition of scaling 3.2 Examples of scaling laws in statistical systems 3.2.1 A note on notation for distribution functions 3.2.1.1 Processes with natural order 3.2.1.2 Rank-ordered processes 3.2.1.3 Frequency distributions 3.3 Origins of scaling 3.3.1 Criticality 3.3.1.1 Critical exponents 3.3.1.2 Universality 3.3.1.3 Percolation 3.3.2 Self-organized criticality 3.3.3 Multiplicative processes 3.3.4 Preferential processes 3.3.5 Sample space reducing processes 3.3.5.1 Zipf’s law emerges 3.3.5.2 The influence of the driving rate 3.3.5.3 State-dependent driving rates and the emergence of statistics 3.3.5.4 Cascading SSR processes 3.3.5.5 Examples of SSR processes 3.3.6 Other mechanisms 3.3.6.1 Exponential growth with random exponentially distributed observation times 3.3.6.2 Random typewriting 3.3.6.3 Information-theoretic context 3.4 Power laws and how to measure them 3.4.1 Maximum likelihood estimator for power law exponents λ