复杂性内在逻辑：从数学到可持续世界(英文版) Grammar of Complexity：From Mathematics to a Sustainable

前辅文

1 Perplexity of Complexity

1.1 A Compositional Containment Hierarchy of Complex Systems and Processes
1.2 Top-Down and Bottom-Up Processes Associated to Complex Systems and Processes
- 1.2.1 The Top-Down Process of Adaptation (Downward Causation)
- 1.2.2 The Bottom-Up Process of Speciation (Upward Causation)
1.3 Example: A Concept of Evolution by Natural Selection
1.4 Saltatory Temporal Evolution of Complex Systems
1.5 Prediction, Control and Uncertainty Relations
- 1.5.1 Physical Determinism and Probabilistic Causation
- 1.5.2 Rare and Extreme Events in Complex Systems
- 1.5.3 Uncertainty Relations
1.6 Uncertainty Relation for Survival Strategies
- 1.6.1 Situation of Adaptive Uncertainty
- 1.6.2 Coping with Growing Uncertainty
1.7 Resilient, Fragile and Ephemeral Complex Systems and Processes
- 1.7.1 Classification of Complex Systems and Processes According to the Prevalent Information Flows
1.8 Down the Rabbit-Hole: Simplicial Complexes as the Model for Complex Systems
- 1.8.1 Simplexes
- 1.8.2 Simplicial Complexes
- 1.8.3 Connectivity
1.9 Conclusion

2 Preliminaries: Permutations, Partitions, Probabilities and Information

2.1 Permutations and Their Matrix Representations
2.2 Permutation Orbits and Fixed Points
2.3 Fixed Points and the Inclusion-Exclusion Principle
2.4 Probability
2.5 Finite Markov Chains
2.6 Birkhoff–von Neumann Theorem
2.7 Generating Functions
2.8 Partitions
- 2.8.1 Compositions
- 2.8.2 Multi-Set Permutations
- 2.8.3 Weak Partitions
- 2.8.4 Integer Partitions
2.9 Information and Entropy
2.10 Conditional Information Measures for Complex Processes
- 2.11 Information Decomposition for Markov Chains
- 2.11.1 Conditional Information Measure for the Downward Causation Process
- 2.11.2 Conditional Information Measure for the Upward Causation Process
- 2.11.3 Ephemeral Information in Markov Chains
- 2.11.4 Graphic Representation of Information Decomposition for Markov Chains
2.12 Concluding Remarks and Further Reading

3 TheoryofExtremeEvents

3.1 Structure of Uncertainty
3.2 Model of Mass Extinction and Subsistence
3.3 Probability of Mass Extinction and Subsistence Under Uncertainty
3.4 Transitory Subsistence and Inevitable Mass Extinction Under Dual Uncertainty
3.5 Extraordinary Longevity is Possible Under Singular Uncertainty
3.6 Zipfian Longevity in a Land of Plenty
3.7 A General Rule of Thumb for Subsistence Under Uncertainty
3.8 Exponentially Rapid Extinction after Removal of Austerity
3.9 On the Optimal Strategy of Subsistence Under Uncertainty
3.10 Entropy of Survival
3.11 Infinite Information Divergence Between Survival and Extinction
3.12 Principle of Maximum Entropy. Why is Zipf’s Law so Ubiquitous in Nature?
3.13 Uncertainty Relation for Extreme Events
3.14 Fragility of Survival in the Model of Mass Extinction and Subsistence
3.15 Conclusion

4 Statistical Basis of Inequality and Discounting the Future and Inequality

4.1 Divide and Conquer Strategy for Managing Strategic Uncertainty
- 4.1.1 A Discrete Time Model of Survival with Reproduction
- 4.1.2 Cues to the ‘Faster’ Versus ‘Slower’ Behavioral Strategies
- 4.1.3 The Most Probable Partition Strategy
- 4.1.4 The Most Likely ‘Rate’ of Behavioral Strategy
- 4.1.5 Characteristic Time of Adaptation and Evolutionary Traps
4.2 The Use of Utility Functions for Managing Strategic Uncertainty
4.3 Logarithmic Utility of Time and Hyperbolic Discounting of the Future
- 4.3.1 The Arrow-Pratt Measure of Risk Aversion
- 4.3.2 Prudence
4.4 Would You Prefer a Dollar Today or Three Dollars Tomorrow?
4.5 Inequality Rising from Risk-Taking Under Uncertainty
4.6 Accumulated Advantage, Pareto Principle
- 4.6.1 A Stochastic Urn Process
- 4.6.2 Pareto Principle: 80–20 Rule
- 4.6.3 Uncertainty Relation in the Process of Accumulated Advantage
4.7 Achieveing Success by Learning
4.8 Conclusion

5 Elements of Graph Theory. Adjacency, Walks, and Entropies

5.1 Binary Relations and Their Graphs
5.2 Background from Linear Algebra
5.3 Adjacency Operator and Adjacency Matrix
5.4 Adjacency and Walks
5.5 Determinant of Adjacency Matrix and Cycle Cover of a Graph
5.6 Principal Invariants of a Graph
5.7 Euler Characteristic and Genus of a Graph
5.8 Hyperbolicity of Scale-Free Graphs
5.9 Graph Automorphisms
5.10 Automorphism Invariant Linear Functions of a Graph
5.11 Relations Between Eigenvalues of Automorphism Invariant Linear Functions of a Graph
5.12 The Graph as a Dynamical System
5.13 Locally Anisotropic Random Walks on a Graph
5.14 Stationary Distributions of Locally Anisotropic Random Walks
5.15 Entropy of Anisotropic Random Walks
5.16 The Relative Entropy Rate for Locally Anisotropic Random Walks
5.17 Concluding Remarks and Further Reading

6 Exploring Graph Structures by Random Walks

6.1 Mixing Rates of Random Walks
6.2 Generating Functions of Random Walks
6.3 Cayley-Hamilton’s Theorem for Random Walks
6.4 Hyperbolic Embeddings of Graphs by Transition Eigenvectors
6.5 Exploring the Shape of a Graph by Random Currents
6.6 Exterior Algebra of Random Walks
6.7 Methods of Generalized Inverses in the Study of Graphs
6.8 Affine Probabilistic Geometry of Generzlied Inverses
6.9 Reduction of Graph Structures to Euclidean Metric Geometry
6.10 Probabilistic Interpretation of Euclidean Geometry by Random Walks
- 6.10.1 Norms of and Distances Between the Pointwise Distributions
- 6.10.2 Projections of the Pointwise Distributions onto Each Other
6.11 Group Generalized Inverses for Studying Directed Graphs
6.12 Electrical Resistance Networks
- 6.12.1 Probabilistic Interpretation of the Major Eigenvectors of the Kirchhoff Matrix
- 6.12.2 Probabilistic Interpretation of Voltages and Currents
6.13 Dissipation and Effective Resistance Distance
6.14 Effective Resistance Bounded by the Shortest Path Distance
6.15 Kirchhoff and Wiener Indexes of a Graph
6.16 Relation Between Effective Resistance and Commute Time Distances
6.17 Summary

7 We Shape Our Buildings

8 Complexity of Musical Harmony

8.1 Music as a Communication Process
8.2 Musical Dice Game as a Markov Chain
- 8.2.1 Musical Utility Function
- 8.2.2 Notes Provide Natural Discretization of Music
8.3 Encoding a Discrete Model of Music (MIDI) into a Markov Chain Transition Matrix
8.4 Musical Dice Game as a Generalized Communication Process
- 8.4.1 The Density and Recurrence Time to a Note in the MDG
- 8.4.2 Entropy and Redundancy in Musical Compositions
- 8.4.3 Downward Causation in Music: Long-Range Structural Correlations (Melody)
8.5 First-Passage Times to Notes Resolve Tonality of the Musical Score
8.6 Analysis of Selected Musical Compositions
8.7 First-Passage Times to Notes Feature a Composer
8.8 Conclusion

References

Index

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