Mathematical models and computer simulations of complex social systems have become everyday tools in sociology. Yet until now, students had no up-to-date textbook from which to learn these techniques. Introduction to Mathematical Sociology fills this gap, providing undergraduates with a comprehensive, self-contained primer on the mathematical tools and applications that sociologists use to understand social behavior. Phillip Bonacich and Philip Lu cover all the essential mathematics, including linear algebra, graph theory, set theory, game theory, and probability. They show how to apply these mathematical tools to demography; patterns of power, influence, and friendship in social networks; Markov chains; the evolution and stability of cooperation in human groups; chaotic and complex systems; and more. Introduction to Mathematical Sociology also features numerous exercises throughout, and is accompanied by easy-to-use Mathematica-based computer simulations that students can use to examine the effects of changing parameters on model behavior.
* Provides an up-to-date and self-contained introduction to mathematical sociology * Explains essential mathematical tools and their applications * Includes numerous exercises throughout * Features easy-to-use computer simulations to help students master concepts
Phillip Bonacich is professor emeritus of sociology at the University of California, Los Angeles. Philip Lu is a PhD candidate in sociology at UCLA.
List of Figures ix List of Tables xiii Preface xv Chapter 1. Introduction 1 Epidemics 2 Residential Segregation 6 Exercises 11 Chapter 2. Set Theory and Mathematical Truth 12 Boolean Algebra and Overlapping Groups 19 Truth and Falsity in Mathematics 21 Exercises 23 Chapter 3. Probability: Pure and Applied 25 Example: Gambling 28 Two or More Events: Conditional Probabilities 29 Two or More Events: Independence 30 A Counting Rule: Permutations and Combinations 31 The Binomial Distribution 32 Exercises 36 Chapter 4. Relations and Functions 38 Symmetry 41 Reflexivity 43 Transitivity 44 Weak Orders-Power and Hierarchy 45 Equivalence Relations 46 Structural Equivalence 47 Transitive Closure: The Spread of Rumors and Diseases 49 Exercises 51 Chapter 5. Networks and Graphs 53 Exercises 59 Chapter 6. Weak Ties 61 Bridges 61 The Strength of Weak Ties 62 Exercises 66 Chapter 7. Vectors and Matrices 67 Sociometric Matrices 69 Probability Matrices 71 The Matrix, Transposed 72 Exercises 72 Chapter 8. Adding and Multiplying Matrices 74 Multiplication of Matrices 75 Multiplication of Adjacency Matrices 77 Locating Cliques 79 Exercises 82 Chapter 9. Cliques and Other Groups 84 Blocks 86 Exercises 87 Chapter 10. Centrality 89 Degree Centrality 93 Graph Center 93 Closeness Centrality 94 Eigenvector Centrality 95 Betweenness Centrality 96 Centralization 99 Exercises 101 Chapter 11. Small-World Networks 102 Short Network Distances 103 Social Clustering 105 The Small-World Network Model 111 Exercises 116 Chapter 12. Scale-Free Networks 117 Power-Law Distribution 118 Preferential Attachment 121 Network Damage and Scale-Free Networks 129 Disease Spread in Scale-Free Networks 134 Exercises 136 Chapter 13. Balance Theory 137 Classic Balance Theory 137 Structural Balance 145 Exercises 148 The Markov Assumption: History Does Not Matter 156 Transition Matrices and Equilibrium 157 Exercises 158 Chapter 15. Demography 161 Mortality 162 Life Expectancy 167 Fertility 171 Population Projection 173 Exercises 179 Chapter 16. Evolutionary Game Theory 180 Iterated Prisoner's Dilemma 184 Evolutionary Stability 185 Exercises 188 Chapter 17. Power and Cooperative Games 190 The Kernel 195 The Core 199 Exercises 200 Chapter 18. Complexity and Chaos 202 Chaos 202 Complexity 206 Exercises 212 Afterword: "Resistance Is Futile" 213 Bibliography 217 Index 219