List of Contributors XI Preface XV 1 Entropy and Renormalization in Chaotic Visibility Graphs 1 Bartolo Luque, Fernando Jesus Ballesteros, Alberto Robledo, and Lucas Lacasa 1.1 Mapping Time Series to Networks 2 1.1.1 Natural and Horizontal Visibility Algorithms 4 1.1.2 A Brief Overview of Some Initial Applications 8 1.1.2.1 Seismicity 8 1.1.2.2 Hurricanes 8 1.1.2.3 Turbulence 9 1.1.2.4 Financial Applications 9 1.1.2.5 Physiology 9 1.2 Visibility Graphs and Entropy 10 1.2.1 Definitions of Entropy in Visibility ...
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List of Contributors XI Preface XV 1 Entropy and Renormalization in Chaotic Visibility Graphs 1 Bartolo Luque, Fernando Jesus Ballesteros, Alberto Robledo, and Lucas Lacasa 1.1 Mapping Time Series to Networks 2 1.1.1 Natural and Horizontal Visibility Algorithms 4 1.1.2 A Brief Overview of Some Initial Applications 8 1.1.2.1 Seismicity 8 1.1.2.2 Hurricanes 8 1.1.2.3 Turbulence 9 1.1.2.4 Financial Applications 9 1.1.2.5 Physiology 9 1.2 Visibility Graphs and Entropy 10 1.2.1 Definitions of Entropy in Visibility Graphs 10 1.2.2 Pesin Theorem in Visibility Graphs 12 1.2.3 Graph Entropy Optimization and Critical Points 19 1.3 Renormalization Group Transformations of Horizontal Visibility Graphs 26 1.3.1 Tangent Bifurcation 29 1.3.2 Period-Doubling Accumulation Point 31 1.3.3 Quasi-Periodicity 32 1.3.4 Entropy Extrema and RG Transformation 34 1.3.4.1 Intermittency 35 1.3.4.2 Period Doubling 35 1.3.4.3 Quasi-periodicity 35 1.4 Summary 36 1.5 Acknowledgments 37 References 37 2 Generalized Entropies of Complex and Random Networks 41 Vladimir Gudkov 2.1 Introduction 41 2.2 Generalized Entropies 42 2.3 Entropy of Networks: Definition and Properties 43 2.4 Application of Generalized Entropy for Network Analysis 45 2.5 Open Networks 53 2.6 Summary 59 References 60 3 Information Flow and Entropy Production on Bayesian Networks 63 Sosuke Ito and Takahiro Sagawa 3.1 Introduction 63 3.1.1 Background 63 3.1.2 Basic Ideas of Information Thermodynamics 64 3.1.3 Outline of this Chapter 65 3.2 Brief Review of Information Contents 66 3.2.1 Shannon Entropy 66 3.2.2 Relative Entropy 67 3.2.3 Mutual Information 68 3.2.4 Transfer Entropy 69 3.3 StochasticThermodynamics for Markovian Dynamics 70 3.3.1 Setup 70 3.3.2 Energetics 72 3.3.3 Entropy Production and Fluctuation Theorem 73 3.4 Bayesian Networks 76 3.5 Information Thermodynamics on Bayesian Networks 79 3.5.1 Setup 79 3.5.2 Information Contents on Bayesian Networks 80 3.5.3 Entropy Production 83 3.5.4 Generalized Second Law 84 3.6 Examples 86 3.6.1 Example 1: Markov Chain 86 3.6.2 Example 2: Feedback Control with a Single Measurement 86 3.6.3 Example 3: Repeated Feedback Control with Multiple Measurements 89 3.6.4 Example 4: Markovian Information Exchanges 91 3.6.5 Example 5: Complex Dynamics 94 3.7 Summary and Prospects 95 References 96 4 Entropy, Counting, and Fractional Chromatic Number 101 Seyed Saeed Changiz Rezaei 4.1 Entropy of a Random Variable 102 4.2 Relative Entropy and Mutual Information 104 4.3 Entropy and Counting 104 4.4 Graph Entropy 107 4.5 Entropy of a Convex Corner 107 4.6 Entropy of a Graph 108 4.7 Basic Properties of Graph Entropy 110 4.8 Entropy of Some Special Graphs 112 4.9 Graph Entropy and Fractional Chromatic Number 116 4.10 Symmetric Graphs with respect to Graph Entropy 119 4.11 Conclusion 120 Appendix 4.A 121 References 130 5 Graph Entropy: Recent Results and Perspectives 133 Xueliang Li and MeiqinWei 5.1 Introduction 133 5.2 Inequalities and Extremal Properties on (Generalized) Graph Entropies 139 5.2.1 Inequalities for Classical Graph Entropies and Parametric Measures 139 5.2.2 Graph Entropy Inequalities with Information Functions f V , f P and f C 141 5.2.3 Information Theoretic Measures of UHG Graphs 143 5.2.4 Bounds for the Entropies of Rooted Trees and Generalized Trees 146 5.2.5 Information Inequalities for If (G) based on Different Information Functions 148 5.2.6 Extremal Properties of Degree- and Distance-Based Graph Entropies 153 5.2.7 Extremality of If (G), If 2 (G) If 3 (G) and Entropy Bounds for Dendrimers 157 5.2.8 Sphere-Regular Graphs and the Extremality Entropies If 2 (G) and If (G) 163 5.2.9 Information Inequalities for Generalized Graph Entropies 166 5.3 Relationships between Graph Structures, Graph Energies, Topological Indices, and Generalized Graph Entropies 171 5.4 Summary and Conclusion 179 References 180 6 Statistical Methods in Graphs: Parameter Estimati
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