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Publisher:

Princeton University Press

Publication Date:

2012

Number of Pages:

240

Format:

Hardcover

Price:

45.00

ISBN:

9780691145495

Category:

Textbook

[Reviewed by , on ]

Peter Rabinovitch

08/8/2012

The first sentence in the first chapter of *Introduction to Mathematical Sociology *is “Mathematical sociology is not an oxymoron.” The authors then back up this statement with about 200 pages of ideas and examples, followed by an impassioned plea for students to accept that mathematics has role in sociology. I found it convincing.

This book has similar topics to Easley and Kleinberg’s *Networks, Crowds and Markets*, but at a much lower mathematical level. For example, eigenvalues are not even defined. This is partially made up for by having several interactive *Mathematica* demonstrations available, most with accompanying exercises. (Note that these demos do not require Mathematica, but rather the free CDF Player available from Wolfram Research.) Although playing with the demos is not “math,” it can certainly demonstrate to the target audience some interesting phenomena, as well as whet their appetites for a more thorough understanding.

After the introduction, some basic math (set theory, probability, relations, graph theory, and a little linear algebra) are presented before delving into the main topics of the book: weak ties, cliques, centrality, small worlds and scale free networks. This is followed by balance theory, demography, and two chapters on game theory, concluding with one chapter on complexity and chaos.

They book is easy bedtime reading for mathematicians, and a useful introduction for sociology students who are interested in how things like Facebook work. It could also be used for bright high school students who are interested in applications of mathematics to the softer sciences.

There are a few typos; none are showstoppers but I suspect they could confuse neophytes. The table of contents, first chapter and Mathematica demos are available at the publisher’s web site.

Peter Rabinovitch is a Systems Architect at Research in Motion. He recently defended his PhD thesis on Mallows permutations, and is still trying to condense it into something readable.

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

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