Table of Contents
1 Perplexity of Complexity 1
1.1 A Compositional Containment Hierarchy of Complex Systems and Processes 1
1.2 Top-Down and Bottom-Up Processes Associated to Complex Systems and Processes 2
1.2.1 The Top-Down Process of Adaptation (Downward Causation) 3
1.2.2 The Bottom-Up Process of Speciation (Upward Causation) 3
1.3 Example: A Concept of Evolution by Natural Selection 5
1.4 Saltatory Temporal Evolution of Complex Systems 6
1.5 Prediction, Control and Uncertainty Relations 7
1.5.1 Physical Determinism and Probabilistic Causation 7
1.5.2 Rare and Extreme Events in Complex Systems 8
1.5.3 Uncertainty Relations 9
1.6 Uncertainty Relation for Survival Strategies 10
1.6.1 Situation of Adaptive Uncertainty 10
1.6.2 Coping with Growing Uncertainty 11
1.7 Resilient, Fragile and Ephemeral Complex Systems and Processes 12
1.7.1 Classification of Complex Systems and Processes According to the Prevalent Information Flows 13
1.8 Down the Rabbit-Hole: Simplicial Complexes as the Model for Complex Systems 16
1.8.1 Simplexes 16
1.8.2 Simplicial Complexes 18
1.8.3 Connectivity 18
1.9 Conclusion 20
2 Preliminaries: Permutations, Partitions, Probabilities and Information 23
2.1 Permutations and Their Matrix Representations 23
2.2 Permutation Orbits and Fixed Points 26
2.3 Fixed Points and the Inclusion-Exclusion Principle 28
2.4 Probability 30
2.5 Finite Markov Chains 31
2.6 Birkhoff-von Neumann Theorem 33
2.7 Generating Functions 34
2.8 Partitions 36
2.8.1 Compositions 36
2.8.2 Multi-Set Permutations 37
2.8.3 Weak Partitions 38
2.8.4 Integer Partitions 39
2.9 Information and Entropy 40
2.10 Conditional Information Measures for Complex Processes 42
2.11 Information Decomposition for Markov Chains 45
2.11.1 Conditional Information Measure for the Downward Causation Process 46
2.11.2 Conditional information Measure for the Upward Causation Process 47
2.11.3 Ephemeral information in Markov Chains 49
2.11.4 Graphic Representation of Information Decomposition for Markov Chains 50
2.12 Concluding Remarks and Further Reading 50
3 Theory of Extreme Events 53
3.1 Structure of Uncertainty 53
3.2 Model of Mass Extinction and Subsistence 54
3.3 Probability of Mass Extinction and Subsistence Under Uncertainty 57
3.4 Transitory Subsistence and Inevitable Mass Extinction Under Dual Uncertainty 59
3.5 Extraordinary Longevity is Possible Under Singular Uncertainty 60
3.6 Zipfian Longevity hi a Land of Plenty 62
3.7 A General Rule of Thumb for Subsistence Under Uncertainty 64
3.8 Exponentially Rapid Extinction after Removal of Austerity 65
3.9 On the Optimal Strategy of Subsistence Under Uncertainty 68
3.10 Entropy of Survival 70
3.11 Infinite Information Divergence Between Survival and Extinction 72
3.12 Principle of Maximum Entropy. Why is Zipf's Law so Ubiquitous in Nature? 73
3.13 Uncertainty Relation for Extreme Events 75
3.14 Fragility of Survival in the Model of Mass Extinction and Subsistence 76
3.15 Conclusion 78
4 Statistical Basis of Inequality and Discounting the Future and Inequality 79
4.1 Divide and Conquer Strategy for Managing Strategic Uncertainty 79
4.1.1 A Discrete Time Model of Survival with Reproduction 80
4.1.2 Cues to the 'Faster' Versus 'Slower' Behavioral Strategies 81
4.1.3 The Most Probable Partition Strategy 81
4.1.4 The Most Likely 'Rate' of Behavioral Strategy 83
4.1.5 Characteristic Time of Adaptation and Evolutionary Traps 84
4.2 The Use of Utility Functions for Managing Strategic Uncertainty 85
4.3 Logarithmic Utility of Time and Hyperbolic Discounting of the Future 86
4.3.1 The Arrow-Pratt Measure of Risk Aversion 88
4.3.2 Prudence 88
4.4 Would You Prefer a Dollar Today or Three Dollars Tomorrow? 89
4.5 Inequality Rising from Risk-Taking Under Uncertainty 90
4.6 Accumulated Advantage, Pareto Principle 92
4.6.1 A Stochastic Urn Process 92
4.6.2 Pareto Principle: 80-20 Rule 95
4.6.3 Uncertainty Relation in the Process of Accumulated Advantage 96
4.7 Achicveing Success by Learning 97
4.8 Conclusion 102
5 Elements of Graph Theory. Adjacency, Walks, and Entropies 103
5.1 Binary Relations and Their Graphs 103
5.2 Background from Linear Algebra 104
5.3 Adjacency Operator and Adjacency Matrix 105
5.4 Adjacency and Walks 106
5.5 Determinant of Adjacency Matrix and Cycle Cover of a Graph 107
5.6 Principal Invariants of a Graph 108
5.7 Euler Characteristic and Genus of a Graph 111
5.8 Hyperbolicity of Scale-Free Graphs 113
5.9 Graph Automorphisms 114
5.10 Automorphism Invariant Linear Functions of a Graph 115
5.11 Relations Between Eigenvalues of Automorphism Invariant Linear Functions of a Graph 118
5.12 The Graph as a Dynamical System 120
5.13 Locally Anisotropic Random Walks on a Graph 121
5.14 Stationary Distributions of Locally Anisotropic Random Walks 123
5.15 Entropy of Anisotropic Random Walks 126
5.16 The Relative Entropy Rate for Locally Anisotropic Random Walks 128
5.17 Concluding Remarks and Further Reading 130
6 Exploring Graph Structures by Random Walks 131
6.1 Mixing Rates of Random Walks 131
6.2 Generating Functions of Random Walks 132
6.3 Cayley-Hamilton's Theorem for Random Walks 134
6.4 Hyperbolic Embeddings of Graphs by Transition Eigenvectors 135
6.5 Exploring the Shape of a Graph by Random Currents 139
6.6 Exterior Algebra of Random Walks 141
6.7 Methods of Generalized Inverses in the Study of Graphs 142
6.8 Affine Probabilistic Geometry of Generzlied Inverses 144
6.9 Reduction of Graph Structures to Euclidean Metric Geometry 145
6.10 Probabilistic Interpretation of Euclidean Geometry by Random Walks 146
6.10.1 Norms of and Distances Between the Pointwise Distributions 146
6.10.2 Projections of the Pointwise Distributions onto Each Other 147
6.11 Group Generalized Inverses for Studying Directed Graphs 149
6.12 Electrical Resistance Networks 151
6.12.1 Probabilistic Interpretation of the Major Eigenvectors of the Kirchhoff Matrix 152
6.12.2 Probabilistic Interpretation of Voltages and Currents 153
6.13 Dissipation and Effective Resistance Distance 153
6.14 Effective Resistance Bounded by the Shortest Path Distance 155
6.15 Kirchhoff and Wiener Indexes of a Graph 156
6.16 Relation Between Effective Resistance and Commute Time Distances 157
6.17 Summary 157
7 We Shape Our Buildings; Thereafter They Shape Us 159
7.1 The City as the Major Editor of Human Interactions 160
7.2 Build Environments Organizing Spatial Experience in Humans 160
7.3 Spatial Graphs of Urban Environments 162
7.4 How a City Should Look? 163
7.4.1 Labyrinths 164
7.4.2 Manhattan's Grid 168
7.4.3 German Organic Cities 170
7.4.4 The Diamond Shaped Canal Network of Amsterdam 172
7.4.5 The Canal Network of Venice 174
7.4.6 A Regional Railway Junction 177
7.5 First-Passage Times to Ghettos 178
7.6 Why is Manhattan so Expensive? 179
7.7 First-Passage Times and the Tax Assessment Rate of Land 182
7.8 Mosque and Church in Dialog 183
7.9 Which Place is the Ideal Crime Scene? 185
7.10 To Act Now to Sustain Our Common Future 188
7.11 Conclusion 190
8 Complexity of Musical Harmony 191
8.1 Music as a Communication Process 191
8.2 Musical Dice Game as a Markov Chain 193
8.2.1 Musical Utility Function 193
8.2.2 Notes Provide Natural Discretization of Music 194
8.3 Encoding a Discrete Model of Music (MIDI) into a Markov Chain Transition Matrix 196
8.4 Musical Dice Game as a Generalized Communication Process 200
8.4.1 The Density and Recurrence Time to a Note in the MDG 200
8.4.2 Entropy and Redundancy in Musical Compositions 201
8.4.3 Downward Causation in Music: Long-Range Structural Correlations (Melody) 203
8.5 First-Passage Times to Notes Resolve Tonality of the Musical Score 204
8.6 Analysis of Selected Musical Compositions 207
8.7 First-Passage Times to Notes Feature a Composer 247
8.8 Conclusion 249
References 251
Index 267
