| Introduction to Permutations, Markov Chains, and Partitions | p. 1 |
| Permutations and Their Matrix Representations | p. 1 |
| Permutation Orbits and Fixed Points | p. 3 |
| Fixed Points and the Inclusion-Exclusion Principle | p. 5 |
| Finite Markov Chains | p. 8 |
| Birkhoff-von Neumann Theorem | p. 9 |
| Generating Functions | p. 10 |
| Partitions | p. 13 |
| Compositions | p. 13 |
| Multi-set Permutations | p. 14 |
| Weak Partitions | p. 15 |
| Integer Partitions | p. 16 |
| Concluding Remarks and Further Reading | p. 17 |
| Worth Another Binary Relation: Graphs | p. 19 |
| Binary Relations and Their Graphs | p. 19 |
| Representation of Graphs by Matrices | p. 20 |
| Algebraic Properties of Adjacency Operators | p. 23 |
| Perron-Frobenius Theory for Adjacency Matrices | p. 24 |
| Spectral Decomposition of Adjacency Operators | p. 26 |
| Adjacency and Walks on a Graph | p. 28 |
| Principal Invariants of the Graph Adjacency Matrix | p. 30 |
| Euler Characteristic and Genus of a Graph | p. 33 |
| Euler Characteristics and Genus of Complex Networks | p. 35 |
| Coloring a Graph | p. 36 |
| Shortest Paths in a Graph | p. 38 |
| Concluding Remarks and Further Reading | p. 41 |
| Permutations Sieved Through Adjacency: Graph Automorphisms | p. 43 |
| Graph Automorphisms | p. 43 |
| Nontrivial Graph Automorphisms and the Structure of Eigenvectors of the Adjacency Matrix | p. 45 |
| Automorphism Invariant Linear Functions of a Graph | p. 47 |
| Automorphism Invariant Stochastic Processes | p. 48 |
| Automorphism Invariant Harmonic Functions | p. 49 |
| Relations Between Eigenvalues of Automorphism Invariant Linear Functions | p. 51 |
| Summary | p. 54 |
| Exploring Undirected Graphs by Random Walks | p. 55 |
| Graphs as Discrete Time Dynamical Systems | p. 56 |
| Generating Functions of the Transition Probabilities | p. 57 |
| Cayley-Hamilton's Theorem for Random Walks | p. 58 |
| Stationary Distribution and Recurrence Time of Random Walks | p. 59 |
| Entropy of Random Walks Defined on a Graph | p. 61 |
| Hyperbolic Embeddings of Graphs by Transition Eigenvectors | p. 64 |
| Exploring the Shape of a Graph by Random Currents | p. 68 |
| Summary | p. 72 |
| Embedding of Graphs in Probabilistic Euclidean Space | p. 73 |
| Methods of Generalized Inverses in the Study of Graphs | p. 73 |
| Affine Probabilistic Geometry of Pseudo-inverses | p. 75 |
| Reduction to Euclidean Metric Geometry | p. 76 |
| Probabilistic Interpretation of Euclidean Geometry | p. 77 |
| Probabilistic Embedding of Simple Graphs | p. 79 |
| Group Generalized Inverse of the Laplace Operator for Directed Graphs | p. 81 |
| Summary | p. 83 |
| Random Walks and Electric Resistance Networks | p. 85 |
| Electric Resistance Network and its Probabilistic Interpretation | p. 85 |
| Dissipation and Effective Resistance in Electric Resistance Networks | p. 87 |
| Effective Resistance is Bounded Above by the Shortest Path Distance | p. 89 |
| Kirchhoff and Wiener Indexes of a Graph | p. 90 |
| Relation Between Effective Resistances and Commute Times | p. 90 |
| Summary | p. 91 |
| Random Walks and Diffusions on Directed Graphs and Interacting Networks | p. 93 |
| Random Walks on Directed Graphs | p. 93 |
| A Time Forward Random Walk | p. 94 |
| Backward Time Random Walks | p. 94 |
| Stationary Distributions of Random Walks on Directed Graphs | p. 95 |
| Laplace Operator Defined on Aperiodic Strongly Connected Directed Graphs | p. 96 |
| Bi-orthogonal Decomposition of Random Walks Defined on Strongly Connected Directed Graphs | p. 98 |
| Spectral Analysis of Self-adjoint Operators Defined on Directed Graphs | p. 101 |
| Self-adjoint Operators Defined on Interacting Networks | p. 103 |
| Summary | p. 105 |
| Structural Analysis of Networks and Databases | p. 107 |
| Structure and Function in Complex Networks and Databases | p. 108 |
| Graph Cut Problems | p. 109 |
| Weakly Connected Graph Components | p. 110 |
| Graph Partitioning Objectives as Trace Optimization Problems | p. 112 |
| Markov Chains Estimate Land Value in Cities | p. 116 |
| Spatial Networks of Urban Environments | p. 117 |
| Spectra of Cities | p. 118 |
| First-passage Times to Ghettos | p. 120 |
| Random Walks Estimate Land Value in Manhattan | p. 121 |
| Unraveling the Tangles of Language Evolution | p. 123 |
| Applying Phylogenetic Methods to Language Taxonomies | p. 124 |
| The Data Set We Have Used | p. 125 |
| The Relations Among Languages Encoded in the Matrix of Lexical Distances | p. 126 |
| The Structural Component Analysis on Language Data | p. 128 |
| Principal Structural Components of the Lexical Distance Data | p. 131 |
| Geometric Representation of the Indo-European Family | p. 132 |
| In Search of Lost Time | p. 135 |
| Evidence for Proto-Indo-Europeans | p. 137 |
| In Search of Polynesian Origins | p. 140 |
| Geometric Representation of Malagasy Dialects | p. 144 |
| Austronesian Languages Riding an Express Train | p. 148 |
| Markov Chain Analysis of Musical Dice Games | p. 152 |
| Musical Dice Game as a Markov Chain | p. 153 |
| Encoding of a Discrete Model of Music (MIDI) into a Transition Matrix | p. 156 |
| Musical Dice Game as a Generalized Communication Process | p. 160 |
| First Passage Times to Notes Resolve Tonality of Musical Dice Games | p. 164 |
| First Passage Times to Notes Feature a Composer | p. 167 |
| Summary | p. 170 |
| When Feedbacks Matter: Epidemics, Synchronization, and Self-regulation in Complex Networks | p. 171 |
| Susceptible-Infected-Susceptible Models in Epidemics | p. 172 |
| Dynamical Equation of the Epidemic Spreading in Scale Free Networks | p. 172 |
| Simplified Equation for Low Infection Rates | p. 174 |
| Stationary Solution of the Epidemic Equation for Low Infection Rates | p. 175 |
| Dynamical Solution of the Evolution Equation for Low Infection Rates | p. 178 |
| Epidemic Spreading in Evolutionary Scale Free Networks | p. 180 |
| Transitions to Intermittency and Collective Behavior in Randomly Coupled Map Networks | p. 183 |
| The Model of Random Networks of Coupled Maps | p. 185 |
| Spatiotemporal Intermittency and Collective Behavior | p. 186 |
| The Evolution of G(N,k) with k | p. 193 |
| Thermodynamics of Random Networks of Coupled Maps | p. 196 |
| Large Gene Expression Regulatory Networks | p. 202 |
| A Model of a Large Gene Expression Regulatory Networks | p. 203 |
| Numerical Analysis of Large Gene Expression Regulatory Networks | p. 206 |
| Mean Field Approach to the Large Transcription Regulatory Networks | p. 213 |
| Summary | p. 217 |
| Critical Phenomena on Large Graphs with Regular Subgraphs | p. 219 |
| Description of the Model and the Results | p. 221 |
| The Regular Subgraphs Viewed as Riemann Surfaces | p. 222 |
| Nonlinear Diffusions Through Complex Networks | p. 224 |
| Diffusion as a Generalized Brownian Motion | p. 229 |
| Scaling of a Scalar Field Coupled to a Complex Network | p. 233 |
| Summary | p. 235 |
| References | p. 237 |
| Glossary of Graph Theory | p. 258 |
| Index | p. 259 |
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