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 1．5．1Physical 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， Probabilitiesand 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 Conditionallnformation 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 oflnformation Decomposition for Markov Chains 2．12 Concluding Remarks and Further Reading 3 Theory of Extreme Events 3．1 Structure of Uncertainty 3．2 Model of Mass Extinction and Subsistence 3．3 Probability of Mass Extinction and Subsistence UnderUncertainty 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 UnderUncertainty 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 …… 4 Statistical Basis oflnequality and Discounting the Futureand Inequality 5 Elements of Graph Theory． Adjacency， Walks， andEntropies 6 Exploring Graph Structures by Random Walks 7 We Shape Our Buildings; Thereafter They Shape Us 8 Complexity of Musical Harmony References Index
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