Complexity Science: The Study of Emergence (Instructor Res. n. 1 of 2, Solution Manual, Solutions)

[At least 14 pages were not processed to pdf. 7 pages of the front matter including some of the contents. 7 pages of chapter 1. Maybe more.]Ecosystems, the human brain, ant colonies, and economic networks are all complex systems displaying collective behaviour, or emergence, beyond the sum of their parts. Complexity science is the systematic investigation of these emergent phenomena, and stretches across disciplines, from physics and mathematics, to biological and social sciences. This introductory textbook provides detailed coverage of this rapidly growing field, accommodating readers from a variety of backgrounds, and with varying levels of mathematical skill. Part I presents the underlying principles of complexity science, to ensure students have a solid understanding of the conceptual framework. The second part introduces the key mathematical tools central to complexity science, gradually developing the mathematical formalism, with more advanced material provided in boxes. A broad range of end of chapter problems and extended projects offer opportunities for homework assignments and student research projects, with solutions available to instructors online. Key terms are highlighted in bold and listed in a glossary for easy reference, while annotated reading lists offer the option for extended reading and research. Cover 1 Half-title 3 Title page 5 Copyright information 6 Dedication 7 Contents 9 Acknowledgements 13 Nomenclature 15 Preface 19 I Conceptual Foundation of Complexity Science 23 Introduction to Part I 25 1 The Science of Emergence 27 1.1 The Importance of Interaction 31 1.2 Past Views on Emergence 37 1.3 Further Reading 40 1.4 Exercises and Projects 41 2 Conceptual Framework of Emergence 43 2.1 Emergence of a Characteristic Scale or Lack of Scale 45 2.2 Emergence of Collective Robust Degrees of Freedom 48 2.3 Structural Coherence 50 2.4 Evolutionary Diffusion 53 2.5 Breaking of Symmetry 55 2.6 Emergence of Networks 57 2.7 Temporal Mode 59 2.8 Adaptive and Evolutionary Dynamics 61 2.9 Further Reading 62 2.10 Exercises and Projects 63 3 Specific Types of Emergent Behaviour 68 3.1 Ising-Type Models: Transitions and Criticality 70 3.2 Network Models and Scale vs. No Scale 74 3.3 Emergence of Coherence in Time: Synchronisation 79 3.4 Evolutionary Dynamics: Adaptation 82 3.5 Mean-Field Modelling: Dimensionality and Forecasting 86 3.6 Further Reading 91 3.7 Exercises and Projects 92 4 The Value of Prototypical Models of Emergence 97 4.1 The Need for Simplification of Models 98 4.2 O’Keeffe–Einstein Propositions at Work 100 4.3 Further Reading 104 4.4 Exercises and Projects 105 II Mathematical Tools of Complexity Science 109 Introduction to Part II 111 5 Branching Processes 115 5.1 Generator Functions: Sizes and Lifetimes 119 5.1.1 Size of the Progeny 121 5.1.2 Time to Extinction 124 5.2 Branching Trees and Random Walks 125 5.3 Further Reading 128 5.4 Exercises and Projects 129 6 Statistical Mechanics 132 6.1 Probabilities and Ensembles 132 6.2 The Ising Model 141 6.3 The Peculiar Nature of the Critical Point 147 6.4 Fluctuations, Response and Correlations 149 6.5 Examples of Correlation Functions: Brain, Flocks of Birds, Finance 154 6.6 Diverging Range of Correlations 155 6.6.1 Correlation Function – Exact Approach 156 6.6.2 Correlation Function – Intuitive Discussion 161 6.7 The Two-Dimensional XY Model 165 6.7.1 2d XY: Some Mathematical Details 170 6.7.2 Vortex Unbinding 175 6.7.3 The Vortex Unbinding Transition in Other Systems 176 6.8 Further Reading 178 6.9 Exercises and Projects 178 7 Synchronisation 185 7.1 The Kuramoto Model: The Onset of Synchronisation 186 7.2 Chimera States 192 7.3 Further Reading 196 7.4 Exercises and Projects 197 8 Network Theory 199 8.1 Basic Concepts 200 8.2 Measures of the Importance of Nodes 201 8.2.1 Degree Centrality 201 8.2.2 Eigenvector Centrality 206 8.2.3 Closeness Centrality 209 8.2.4 Betweenness Centrality 209 8.2.5 How Well Does it Work? 210 8.3 Community Detection 210 8.4 Spreading on Networks – Giant Cluster 218 8.5 Analysis of Dynamics of and on Networks 225 8.5.1 Generating Networks 226 8.5.2 Random Walk on Networks 234 8.5.3 Synchronisation on Networks 238 8.6 Further Reading 246 8.7 Exercises and Projects 247 9 Information Theory and Entropy 252 9.1 Information Theory and Interdependence 254 9.2 Entropy and Estimates of Causal Relations 259 9.3 From Time Series to Networks 263 9.4 From Entropy to Probability Distribution 267 9.5 Measures of Degrees of Complexity 278 9.5.1 Lempel–Ziv Complexity Measure 278 9.5.2 Information-Theoretic Approach to Emergence 281 9.5.3 Group Entropy Measure of Complexity 294 9.6 Further Reading 296 9.7 Exercises and Projects 297 10 Stochastic Dynamics and Equations for the Probabilities 301 10.1 Random Walk and Diffusion 302 10.2 First Passage and First Return Times 315 10.3 Correlations in Time 319 10.4 Random Walk with Persistence or Anti-persistence: Hurst Exponent 324 10.5 Stationary Diffusion: Ornstein–Uhlenbeck Process 329 10.6 Evolutionary Dynamics and Clustering 331 10.7 Master Equation, Coarse Graining and Free Energy 335 10.8 Further Reading 340 10.9 Exercises and Projects 341 11 Agent-Based Modelling 346 11.1 Flocks of Birds or Schools of Fish 347 11.2 Models of Segregation 350 11.3 The Tangled Nature Model 359 11.4 Further Reading 371 11.5 Exercises and Projects 372 12 Intermittency 378 12.1 Self-Organised Criticality 379 12.1.1 Sandpile Models 380 12.1.2 Mean-Field Analysis 383 12.1.3 Lessons from Sandpile Models 386 12.1.4 Forest Fire Model 389 12.2 Record Dynamics 392 12.2.1 Statistics of Records 393 12.2.2 Spin Glasses, Superconductors, Ants and Evolution 397 12.3 Tangent Map Intermittency 401 12.4 Further Reading 404 12.5 Exercises and Projects 405 13 Tipping Points, Transitions and Forecasting 409 13.1 Externally Induced Transitions 409 13.2 Intrinsic Instability 411 13.3 Further Reading 417 13.4 Exercises and Projects 417 14 Concluding Comments and a Look to the Future 419 14.1 Further Reading 421 Glossary 423 References 433 Index 458 "Ecosystems, the human brain, ant colonies, and economic networks are all complex systems displaying collective behaviour, or emergence, beyond the sum of their parts. Complexity science is the systematic investigation of these emergent phenomena, and stretches across disciplines, from physics and mathematics, to biological and social sciences. This introductory textbook provides detailed coverage of this rapidly growing field, accommodating readers from a variety of backgrounds, and with varying levels of mathematical skill. Part I presents the underlying principles of complexity science, to ensure students have a solid understanding of the conceptual framework. The second part introduces the key mathematical tools central to complexity science, gradually developing the mathematical formalism, with more advanced material provided in boxes. A broad range of end of chapter problems and extended projects offer opportunities for homework assignments and student research projects, with solutions available to instructors online. Key terms are highlighted in bold and listed in a glossary for easy reference, while annotated reading lists offer the option for extended reading and research"-- Provided by publisher This introductory textbook provides detailed coverage of the rapidly growing field of complexity science and accommodates readers from a wide variety of backgrounds, and with varying levels of mathematical skill. The book contains a broad range of end of chapter problems and extended projects, with solutions available to instructors online.