AMS / MAA SPECTRUM VOL 76

SIX SOURCES OF COLLAPSE Charles R. Hadlock

A MATHEMATICIAN’S PERSPECTIVE ON HOW THINGS CAN FALL APART IN THE BLINK OF AN EYE i i “master” — 2012/10/11 — 22:40 — page i — #1 i i

10.1090/spec/076

Six Sources of Collapse

A Mathematician’s Perspective on How Things Can Fall Apart in the Blink of an Eye

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The photo of the Tacoma Narrows Bridge on the cover is used courtesy of UNIVERSITY OF WASHINGTON LIBRARIES SPECIAL COLLECTIONS Negative number: UW21418.

c 2012 by the Mathematical Association of America, Inc.

Library of Congress Catalog Card Number 2012950085 Print edition ISBN 978-0-88385-579-9 Electronic edition ISBN 978-1-61444-514-2 Printed in the United States of America Current Printing (last digit): 10987654321

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Six Sources of Collapse

A Mathematician’s Perspective on How Things Can Fall Apart in the Blink of an Eye

Charles R. Hadlock Bentley University

Published and Distributed by The Mathematical Association of America

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Council on Publications and Communications Frank Farris, Chair

Committee on Books Gerald Bryce, Chair

Spectrum Editorial Board Gerald L. Alexanderson, Co-Editor James J. Tattersall, Co-Editor Robert E. Bradley Susanna S. Epp RichardK.Guy KeithM.Kendig Shawnee L. McMurran Jeffrey L. Nunemacher Jean J. Pedersen Kenneth A. Ross Marvin Schaefer Franklin F. Sheehan

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SPECTRUM SERIES

The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publisha broad range of booksincluding biographies,accessible expositions of old or new mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers.

777 Mathematical Conversation Starters, by John de Pillis 99 Points of Intersection: Examples—Pictures—Proofs,by Hans Walser. Translated from the original German by Peter Hilton and Jean Pedersen Aha Gotcha and Aha Insight, by Martin Gardner All the Math That’s Fit to Print, by Keith Devlin Beautiful Mathematics, by Martin Erickson Calculus and Its Origins, by David Perkins Calculus Gems: Brief Lives and Memorable Mathematics, by George F. Simmons Carl Friedrich Gauss:Titan of Science, by G. Waldo Dunnington, with additional material by Jeremy Gray and Fritz-Egbert Dohse The Changing Space of Geometry, edited by Chris Pritchard Circles: A Mathematical View, by Dan Pedoe Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by Albrecht Beutelspacher The Early Mathematics of Leonhard Euler, by C. Edward Sandifer The Edge of the Universe: Celebrating 10 Years of Math Horizons, edited by Deanna Haunsperger and Stephen Kennedy Euler and Modern Science, edited by N. N. Bogolyubov, G. K. Mikhailov, and A. P. Yushkevich. Translated from Russian by Robert Burns. Euler at 300: An Appreciation, edited by Robert E. Bradley, Lawrence A. D’Antonio, and C. Edward Sandifer Expeditions in Mathematics, edited by Tatiana Shubin, David F. Hayes, and Gerald L. Alexanderson Five Hundred Mathematical Challenges, by Edward J. Barbeau,Murray S. Klamkin, and William O. J. Moser The Genius of Euler: Reflections on his Life and Work, edited by William Dunham The Golden Section, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Harmony of the World: 75 Years of Mathematics Magazine, edited by Gerald L. Alexanderson with the assistanceof Peter Ross A Historian Looks Back: The Calculus as Algebra and Selected Writings, by Judith Grabiner History of Mathematics: Highways and Byways, by Amy Dahan-Dalm´edico and Jeanne Peiffer, translated by Sanford Segal How Euler Did It, by C. Edward Sandifer In the Dark on the Sunny Side: A Memoir of an Out-of-Sight Mathematician, by Larry Baggett Is Mathematics Inevitable? A Miscellany, edited by Underwood Dudley I Want to Be a Mathematician, by Paul R. Halmos

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Journey into Geometries, by Marta Sved JULIA: a life in mathematics, by Constance Reid The Lighter Side of Mathematics: Proceedings of the Eug`ene Strens Memorial Conferenceon Recre- ational Mathematics & Its History, edited by Richard K. Guy and Robert E. Woodrow Lure of the Integers, by Joe Roberts Magic Numbers of the Professor, by Owen O’Shea and Underwood Dudley Magic Tricks, Card Shuffling, and Dynamic Computer Memories: The Mathematics of the Perfect Shuffle, by S. Brent Morris Martin Gardner’s Mathematical Games: The entire collection of his Scientific American columns The Math Chat Book, by Frank Morgan Mathematical Adventures for Students and Amateurs, edited by David Hayes and Tatiana Shubin. With the assistance of Gerald L. Alexanderson and Peter Ross Mathematical Apocrypha, by Steven G. Krantz Mathematical Apocrypha Redux, by Steven G. Krantz Mathematical Carnival, by Martin Gardner Mathematical Circles Vol I: In Mathematical Circles Quadrants I, II, III, IV, by Howard W. Eves Mathematical Circles Vol II: Mathematical Circles Revisited and Mathematical Circles Squared, by Howard W. Eves Mathematical Circles Vol III: Mathematical Circles Adieu and Return to Mathematical Circles, by Howard W. Eves Mathematical Circus, by Martin Gardner Mathematical Cranks, by Underwood Dudley Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell Mathematical Fallacies, Flaws, and Flimflam, by Edward J. Barbeau Mathematical Magic Show, by Martin Gardner Mathematical Reminiscences, by Howard Eves Mathematical Treks: From Surreal Numbers to Magic Circles, by Ivars Peterson A Mathematician Comes of Age, by Steven G. Krantz Mathematics: Queen and Servant of Science, by E.T. Bell Mathematics in Historical Context,, by Jeff Suzuki Memorabilia Mathematica, by Robert Edouard Moritz Musings of the Masters: An Anthology of Mathematical Reflections, edited by Raymond G. Ayoub New Mathematical Diversions, by Martin Gardner Non-Euclidean Geometry, by H. S. M. Coxeter Numerical Methods That Work, by Forman Acton Numerology or What Pythagoras Wrought, by Underwood Dudley Out of the Mouths of Mathematicians, by Rosemary Schmalz Penrose Tiles to Trapdoor Ciphers ...and the Return of Dr. Matrix, by Martin Gardner Polyominoes, by George Martin Power Play, by Edward J. Barbeau Proof and Other Dilemmas: Mathematics and Philosophy, edited by Bonnie Gold and Roger Simons The Random Walks of George P´olya, by Gerald L. Alexanderson Remarkable Mathematicians, from Euler to von Neumann, by Ioan James

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The Search for E.T. Bell, also known as John Taine, by Constance Reid Shaping Space, edited by Marjorie Senechaland George Fleck Sherlock Holmes in Babylon and Other Tales of Mathematical History, edited by Marlow Anderson, Victor Katz, and Robin Wilson Six Sourcesof Collapse: A Mathematician’s Perspective on How Things Can Fall Apart in the Blink of an Eye, by Charles R. Hadlock Sophie’s Diary, Second Edition, by Dora Musielak Student Research Projects in Calculus, by Marcus Cohen, Arthur Knoebel, Edward D. Gaughan, Douglas S. Kurtz, and David Pengelley Symmetry, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Trisectors, by Underwood Dudley Twenty Years Before the Blackboard, by Michael Stueben with Diane Sandford Who Gave You the Epsilon? and Other Tales of Mathematical History, edited by Marlow Anderson, Victor Katz, and Robin Wilson The Words of Mathematics, by Steven Schwartzman

MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789

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This book is dedicated to my grandchildren, — Charles, Rob, Ben, Kent, Catherine, McKenzie, and Forrest — in the hope that they and their peers will successfully exert intellect, compassion, and leadership in taking on the new challenges of collapse that will surely confront their generation.

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Preface

The world is a fascinating combination of fragility and resilience. In the midst of terrible wars and atrocities and countries being torn apart, we also see images of children playing soccer in the street and young couples starting futures together at wedding celebrations. The signs of past collapses are all around us, but we keep building new lives, structures, and institutionsto take their place. This is a book about collapse. It’s intended to help some of us manage collapse a little better, promoting it when we want to and protecting ourselves from it when we need to. The first step is to understand it better, using different tools and different points of view. Much of this book is about history and experience — the anatomies of past collapses in many different subject areas from finance to fighter jets, networks to nanostructures. In surveying these diverse cases, we find extraordinary commonalities. The same kinds of dynamics occur over and over again. Let me explain why my background as a mathematician gives me a particular perspec- tive on this subject. Some people think that math is all about solving equations, but they’re dead wrong. Math is much more about identifying common features and describing them in a way that captures their essence. Mathematical symbols and equations are basically a language that helps to clear away some of the detail so we can focus on those fundamental underlying features or principles. No matter what your math background, whether quite modest (say, comfort with the equations and graphs of high school pre-calculus) or highly advanced, I think you will be interested to see how using the mathematical language of probabilities, game theory, dynamical systems, networks, and related fields helps to or- ganize our accumulated experience about real world collapses. This can bring into much sharper focus the points of vulnerability in the systems of the present and future. That’s how we can bridge the gap, for example, from the amazing experiment of biological evo- lution to the design of organizations, or from the movements of a past civilization to the layout of fire exits. You can read this book looking for only what I have described above, skimming or skipping the rest. You’ll find the discussion of famous collapse events in language I think you will readily understand: what really happened at Chernobyl, why did the mortgage market crash, what was that “flash crash” you heard about, how could a falling branch knock out power to almost the entire Northeast, and more. This is serious material and the

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x Preface

book is not a bedtime novel, but I have tried to make these discussions simple, accurate, and interesting. The second way that math enters is by giving us tools that we can use to go deeper into the subject. Not only can we describe phenomena and find analogies, but we can analyze situations from the past, present, or future and make quantitative estimates of the importance of different factors. My goal is to give you some direct involvement with the use of these mathematical techniques. While this is not a textbook-like compendium of mathematical methods, each of the six technical chapters provides a serious engagement with representative methods that are encountered in that field. Use this to the extent you want or need it. The prerequisites are no more demanding than those listed earlier, but you will have to spend more time reading and thinking and reading again, as with almost any mathematical writing. We rarely get it the first time around (even those of us who are professional mathematicians). You can then turn to the references or to textbooks to learn more if you wish. A few paragraphs are flagged for readers who have a slightly more advanced back- ground, perhaps an elementary statistics course, a year of calculus, and occasionally more. But I believe that even for readers with a highly advanced mathematical background, there is much to stimulate thought here. I think it will place previous study in a new perspective. Because of the ready availability on personal computing devices of robust routines for actually solving equations, quantitative modeling has become a standard practice for many social scientists. The focus can thus stay more heavily on getting the models right and testing their predictions against real world experience, tasks that those who are experts in their applied fields are best suited for. Thus I hope that practicing social scientists, not just those from academia, can use this book to get new insights on collapse potential in many fields, from political polarization to nation building to the design of financial systems. I think we all recognize the vulnerabilities that exist all around us, and we need to make use of the best tools we can find to navigate our path. In this vein, perhaps I may at least contribute some small but useful insights to a fairly wide audience, including business managers and strategists, policy makers and their advisers, foundation managers who direct resources at important problems, teachers, writers, and journalists who help to educate us all, and others. I have the mental image of book writing as explaining something one-on-one across a table, rather than expounding on it from the lectern of an auditorium. Thus I have used a more informal and personal style than might be customary with many other books. I hope you enjoy it!

Charles R. Hadlock Lincoln, Massachusetts, and Silver Lake, New Hampshire 2012

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Acknowledgements

This book has been incubating for about ten years, and I cannot do justice to all the people who have shared key insightsthat helped me pick outwhat I feel are the importantcommon themes that permeate collapse. But I must single out my wife, Joanne, my children, Charles and Tessa, and my nephews, John Giasi and John Miscione, all experts in their own diverse fields, who brought insight and energy to many late night roundtable discussions of collapse processes, as well as strong encouragement on the value of organizing these ideas in book form. Friends and colleagues, including Norm Josephy, Rick Cleary, Vita Steblovskaya, Jack Knapp, Victor Donnay, Peter and Becky Mattison, Julia Sidorenko, Phil Knutel, and Gau- rav Shah, as well as others, made valuable contributionsand were a vital source of encour- agement. Several researchers were kind enough to correspond on particular matters or to review material for accuracy. These included Joshua Epstein, Lars-Erik Cederman, Juliet Johnson, Stuart Pimm, Michael Seiler, and Patrick McKenna. Richard Parris generously helped adapt his excellent Winplot freeware program to my needs in Chapters 5 and 6. I’ve been privileged at Bentley University to work with bright and highly motivated students, some of whom participated in an interdisciplinary senior honors seminar on this subject. The insights this group brought to the discussion table helped me identify my “six sources of collapse” and put me on the trail of examples I might never other- wise have found. They included Joseph Crossett, Michael DeLuca, Farris Jaber, Brian Jas- set, Raj Kochhar, Amanda Le, Rohith Lokareddy, Emily Nilson, Bryan Takvorian, Jillian Tourangeau, Sean Troy, and the late Patrick Kane, who had also kindly reviewed the entire manuscript before his tragic passing. In addition, Bentley University itself has provided outstanding support in many ways, such as a special interdisciplinary professorship, sab- batical leave, and many other resources. I’m grateful to Lee Bridges, Kevin “Kal” Kallaugher (Kaltoons), Lumeta Corporation, the University of Washington Special Collections, as well as several of the researchers mentioned above, for kindly allowing me to use some of their graphic material as figures. As usual, it’s a great pleasure to work with the Mathematical Association of America. Don Albers seized on the originalbook concept with great enthusiasm rightfrom the outset, Jerry Alexanderson led the review process, and Carol Baxter, Rebecca Elmo, and Beverly Ruedi made the entire production process a pleasure to participate in. I apologize for any errors or oversights and take full responsibility for them.

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Contents

Preface ix Acknowledgements xi 1 Introduction 1 1.1 Whatisacollapse? ...... 1 1.2 ShadesofHitchcock,andothertales ...... 2 1.3 Whatmighttomorrowbring? ...... 6 1.4 Whatthisbookaimstodo...... 13 2 Predicting Unpredictable Events 15 2.1 Likeathiefinthenight?...... 15 2.2 Chanceandregularity ...... 17 2.3 Aquickstatisticsprimer...... 18 2.4 Normalregularity:thegood,thebad,andthemiraculous ...... 22 2.5 Abnormalregularity:extremevaluestatistics ...... 25 2.6 Gettingthingsrightwithheavy-taileddistributions ...... 31 2.7 Thedangersfromgettingyourprobabilitieswrong ...... 35 3 Group Behavior: Crowds, Herds, and Video Games 41 3.1 Fire! ...... 41 3.2 Birds,boids,andbicycles ...... 44 3.3 TheMonteCarloworld ...... 48 3.4 Modelswithprobabilities ...... 50 3.5 People,properties,andpoliticalsystems ...... 54 3.6 Connectionstootherchapters ...... 59 4 Evolutionand Collapse: Game Playing in a Changing World 61 4.1 MyNewHampshire...... 61 4.2 Strategiesandgames ...... 63 4.3 Iteratedandevolutionarygameplaying ...... 68 4.4 Modelingtheevolutionofspeciesandcultures ...... 74 4.5 Implicationsforunderstandingcollapse...... 80

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xiv Contents

5 Instability, Oscillation, and Feedback 85 5.1 Sharinganelectricblanketandotherchallenges ...... 85 5.2 Primerondifferentialequations...... 91 5.3 Stableandunstableequilibriumpointsandrelatedconcepts ...... 97 5.4 Thedynamicsofinteractingpopulations ...... 100 5.5 Structuralcollapsesandrelatedprocesses...... 106 5.6 Thescienceoftryingtomaintaincontrol ...... 112 5.7 TheChernobyldisaster ...... 115 6 Nonlinearity: Invitation to Chaos and Catastrophe 121 6.1 Theelephant’stoenail ...... 121 6.2 Locallinearity...... 122 6.3 Bifurcations,tippingpoints,andcatastrophes...... 127 6.4 Hysteresis:wheretheremaybenosimpleturningback ...... 134 6.5 Chaos:beginningwithabutterfly ...... 138 7 It’s All About Networks 145 7.1 How’syournetworking? ...... 145 7.2 Networkfundamentals...... 147 7.3 Importantvariationsinnetworkmacrostructure ...... 152 7.4 Unexpectednetworkcrashes ...... 157 7.5 Interactivedynamicsacrossnetworks ...... 161 7.6 Spreadingprocessesthroughnetworks ...... 165 7.7 Asurprisinggameonanetwork...... 167 7.8 Networksinanevolutionarycontext ...... 169 8 Putting It All Together: Looking at Collapse Phenomena in “6-D” 173 8.1 Aquickreview...... 173 8.2 The utilityof multipleperspectives in understanding the risk of collapse . 175 8.3 Wheretogofromhere:themodernfieldofcomplexitytheory ...... 186 References 189 Index 201 About the Author 207

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Index

[Note: only the first page of continuous discussionsis generally referenced.]

A Sound of Thunder (short story) 144 Boston Red Sox 4 A&P 7 BP 4 ABM 45 Bradbury, Ray 144 Abnormal regularity 25 Braess paradox 167 Accidents, industrial 4, 36 Brand image 7 Agent-based modeling 42, 45 British Empire 6 AIG 38 Broomstick 87 Aircraft 112, 113, 125 Broughton suspension bridge 110 Albedo effect 88 Building collapses 8, 106 Alice in Wonderland 80 Burglaries 15 Alon, Uri 170 Butterfly effect 139 Anasazi simulation model 54 Arab Spring 9, 177, 184 Cape Ann earthquake 15 Arms races 81, 87 Car in snow 89, 107 Arthur Andersen 1 Catastrophe 127 Arthur D. Little 4 Cauchy distribution 21, 31 Artificial Anasazi project 54 CB radio 7 Asian financial crisis 37 CCDF 32 Asteroid impact 179 form for selected distributions 31 Attractor 141 Cederman, Lars Eric 57, 145, 177 Axelrod, Robert 69 Center, stable 99 Axtell, Robert 53 Central Limit Theorem 23 limitations 37 Balancing broomstick and pencil 87 Chaos 138 Bang or whimper 6 Charting 43 Behavioral finance and economics 43 Chernobyl nuclear accident 115 Betweenness centrality 151 Chromosome 74 Bhopal 4 synthetic 77 Bicycle game 44 Civilizations and empires 5, 176 Bifurcations 127 Clustering coefficient 150 Black swans 25 Coca-Cola 7 Boeing 737 wing cracks 125 Cockroaches 81 Book flipping 86 Coconut Grove Nightclub 41 Boom and bust cycles 8 Code Red virus 167

201

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202 Index

Cold War 81 Equilibrium, unstable 98 Collapse, definition 1 Eurofighter 113 Company-related collapse issues 180 EVI 29 Complexity levels 177 values for selected distributions 31 Complexity theory 186 Evolution 61 Computer viruses and malware 166 aimless 80 Control theory 113 industrial 81 Controllability 114 of ecosystems71 Cooperation vs. competition 63 of species and cultures 74 Copula, Gaussian 39 synthetic 75 Corporate collapse 5 Evolutionary games 68 Cost projections 126 Exponential distribution 31 Crossover 75 Extinction 3, 6, 10, 62, 144, 166, 179 Cycle 100 Extreme storms 26 Extreme value index 29 Dark networks 169 Extreme Value Theorem 25 Deepwater Horizon 4 Deer on power line 40 Fat tail 22 Democracy vs. autocracy 57 Feedback 85 Dependence (statistical) among failures 39 negative 89 Dependence (mathematical) on initial condi- positive 88 tions 143 Financial collapse 4 Derivative (mathematics) 91 Fire disasters 41 Derivative (financial) 39 Fisheries 3, 93, 103, 128 Descartes 80 Fitness (evolutionary) 59, 62, 77, 169, 176 Diamond Crystal 36 Fitting a distribution to data 20 Dice, maximum of multiple throws 27 Flash crash 161 Differential equation 93 Flocking behavior 46 Dinosaurs 1, 6, 81, 179 Floods 16 Direction field 95 Flu, simulating the spread of 51 Disease, collapse issues 5, 10, 184 Folkman, Judah 170 eradication 133 Fortune 500 7 model for spreading 131 Fractals 144 simulation modeling 50 Fracture models 125 DNA 74 Frechet distribution 29, 31 Domain of attraction 97 Fukushima nuclear accident 115 Dow Jones data 17 Fundamental Theorem of Game Theory 66 Drilling into salt mine 35 Future collapse possibilities 6

Earthquakes 15 Galileo 122 Ecosystems 62 Galloping Gertie 106 Edge (graph) 148 Game theory 63 Efficient market hypothesis 42 lessons learned 71, 73 Eisenhower, President Dwight 87 on a network 169 Elastic models, linear and nonlinear 125 Gaussian copula 39 Eldredge, Niles 144 Gaussian distribution 17 Electric blanket 85 General circulation models (GCMs) 88 Emergency exits 41 General Public Utilities 4 Enron 1 Generalized Extreme Value Theorem(GEV) 28 Environmental Protection Agency (EPA) 35 Genetic algorithm 77 Epidemics 131 Genetics 62 Epstein, Joshua 53 Geodesic 150 Equilibrium points, classification 97 GEV 28

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Index 203

Gladwell, Malcolm 179 Link (network) 147 Golden Gate bridge 112 Location factor 29 Goldman Sachs 7 Logistic distribution 31 Goodwin, Richard 106 Logistic model 101 Gould, Stephen Jay 144 London riots 184 Government 9, 183 Long House Valley Anasazi 54 Greenland 77, 82 Long-Term Capital Management 16, 36 Group behavior 41 Lorenz system 141 Grumman X-29 112 Lorenz, Edward 139 Gulf oil spill 4 Lotka-Volterra equations 103 Gumbel distribution 29 Low probability events 35 Guns 87 LTCM 16, 36 Gypsy moths 144 Macondo well blowout 4 Halliday, T. R. 179 Malthusian population model 93, 100 Han Dynasty 6 Mandelbrot, Benoit 144 Hartford Civic Center 8 Markets 5 Heavy tail 22, 154 collapse issues 8, 183 underestimation 34 Martin Luther King 83 Hedge 37 Mass extinctions 6 Hitchcock movie 2 Max Planck 83 Holland, John 77 Maximum rainfall 25 Hooke’s law 108, 124 Maximum sustainable yield 105, 129 Hotel New World (Singapore) 8 McKenna, Joseph 111, 125 Housing bubble 39 Mean 20 Hundred-year wind 29 Meiosis 74 Hysteresis 134 Merrill Lynch 5 examples related to collapse 136 Milgram, Stanley 150 Military-industrial complex 87 Independence assumptions 30 Minsky model 106 Industrial accidents 4, 36 Mixed strategy 66 Influenza 51 Models, simple vs. complex 121 Inkombank 90 Monte Carlo simulation 48 Instability 85 Mortgage crisis 37 Internet structure 154, 157 Mortgage-backed securities 38 Interstate Highway system 147 Movement diagram 65 Isle Royale 106 Murphy’s law 40, 75 Iterated games 68 Mutation 75, 185 Jefferson Island 36 Joint distribution function, approximation of 39 Nash equilibrium 66 Nash, John 66 Kahneman, Daniel 43 Natural gas pipeline network 147 King Oscar II 138, 141 Neckties 7 NetLogo 46 Laplace transform 114 Network motifs 170 Lattice network 147, 152 Networks 5, 11, 145, 185 Law of Large Numbers 24 crashes of 157 Lazer, Alan 111 evolving 169 Lehman Brothers 5 in medicine 170 Leveraging 36 New Hampshire 61 Liberty ship failures 124 Newton’s law 97, 108 Linear approximation 122 Node (network) 147 Linear thinking 126 Node, stable 98

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204 Index

Node, unstable 99 Risk, hidden 39 Nonlinearity, global 126 management in business 175 Normal distribution 17, 20, 31 RNA 75 why common 22 Roman Empire 6, 177 Normalization of pdfs 28 Running to stay in place 81 Northeast power blackout 158 Russian sovereign debt crisis 37, 89 Nuclear power 115 Saddle point 98 Observability 114 Salt mine 35 Ockham’s razor principle 121 Santa Fe Institute 77, 186 Oscillation 85 Sasser worm 166 Overfishing 128 Scale factor 29 Scale-free network 153 Parameter estimation 30 Securitization 38 Pareto distribution 31 Seismic risk 15 Pareto type 33 Shape factor 29 Passengerpigeon 2, 82, 178 Sharks 81 Pastor-Satorras, Romualdo 167 Ship failures 125 Patches 54 SIR model 131, 166 pdf 18 diverse applications 133 form for selected distributions 31 SIS model 166 Pendulum 122 Six sources of collapse 13, 174 Poincar´e-Bendixen Theorem 141 Slope field 94 Polar bear 82 Slowly varying function 33 Political gridlock 6 Small-world network 150, 152 Politics and government 5, 9, 183 Smith, Vernon 43 Power law 155 Social order 5, 9, 183 distribution 31 Solar system, stability question 138, 141 tail 33 Sony Betamax 7, 73 Predators and prey 73, 102 Sovereign debt 6 Preisach model of hysteresis 135 Soviet Union 1, 81, 119, 177 Prisoner’s dilemma 63 Special purpose vehicle 38 evolutionary model 77 Species and ecosystems5, 178 tournament 69 Stability, asymptotic 98 Probabilities, simulation of 48 Stable center 99 Probability, density function 18 Stable distribution 34 distribution 18 Stable node 97 estimation 35 Stable vortex 100 Products and fads 5, 179 Stampedes 41 Punctuated equilibrium 144 Standard deviation 20 Pure strategy 66 Station Nightclub 41 Stock market cartoon 42 Radioactive waste 35 Strange attractor 141 Random variable 18 Strategy (game theory) 63 Rats 4, 61 invasive 73 Real estate foreclosure contagion 55 Stress-strain relationships 125 Red noise 81 Structural collapse 5, 106, 181 Red Queen 80 Student’s t-distribution 31 Red River Valley flood 143 Stuxnet virus 167 Regular dependence on initial conditions 140 Subprime mortgage crisis 37 Replicator dynamics 75 Sugarscape model 53 Reproduction 62 Swarming behavior 46 Resonance 110 System diagram 114

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Index 205

Tacoma Narrows Bridge 106, 111, 124 Unstable equilibrium 98 Tainter, Joseph 177 Unstable node 99 Takeover, corporate 4 Technical analysis 43 Value at risk 38 Ten Commandments 73 Variance 21 Tennis racquet theorem 86 Vertex 148 Texaco 36 Vespignani, Alessandro 167 Thermostats 134 Virus (computer) 166 Three Mile Island 4 Vortex, stable 100 Ticks (time steps) 47 Tipping point 6, 127 Walmart 7 Tit-for-tat strategy 70 Weather, extreme 26 Tomcod, Hudson River 74 Weibull distribution 29 Turbulence (fluid flow) 127 Whales 6 Turchin, Peter 106, 178 Wilder, J. W. 144 Twin Towers 8 World Trade Center 8, 112, 182 Tyndallization 82 World War III 6 World Wide Web structure 157 Uniform distribution 31 Union Carbide 4 X-29 aircraft 112

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About the Author

Charles Hadlock received his Ph.D. in mathematics from the University of Illinois in 1970. He has studied collapse processes from many points of view: from one career as a world traveling consultant with Arthur D. Little, Inc., working to head off catastrophic risks in the chemical, power, transportation, and mining industries; to another as the Dean of a business school witnessing collapses of corporations, currencies, and markets. Add to this his broad mathematical and scientific background; collaborative work with political scientists, engineers, and others; and extensive experience with management challenges at the top levels of corporations and governments. He is an award winning author with a Carus Monograph on field theory and an acclaimed text on mathematical modeling in environmental management. He has served on the mathematics faculties of Amherst and Bowdoin Colleges, as a Visiting Professor of Earth, Atmospheric, and Planetary Sciences at MIT, and is currently Trustee Professor of Technology, Policy, and Decision Making at Bentley University.

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i i AMS / MAA SPECTRUM

SIX SOURCES OF COLLAPSE A MATHEMATICIAN’S PERSPECTIVE ON HOW THINGS CAN FALL APART IN THE BLINK OF AN EYE Charles R. Hadlock

Beginning with one of the most remarkable ecological collapses of re- cent time, that of the passenger pigeon, Hadlock goes on to survey col- lapse processes across the entire spectrum of the natural and man-made world. He takes us through extreme weather events, technological disas- ters, evolutionary processes, crashing markets and companies, the cha- otic nature of Earth’s orbit, revolutionary political change, the spread and elimination of disease, and many other fascinating cases. His key thesis is that one or more of six fundamental dynamics consistently show up across this wide range. These “six sources of collapse” can all be best de- scribed and investigated using fundamental mathematical concepts. They include low probability events, group dynamics, evolutionary games, in- stability, nonlinearity, and network effects, all of which are explained in readily understandable terms. Almost the entirety of the book can be un- derstood by readers with a minimal mathematical background, but even professional mathematicians are likely to get rich insights from the range of examples.

The author tells his story with a warmly personal tone and weaves in many of his own experiences, whether from his consulting career of rac- ing around the world trying to head off industrial disasters to his story of watching collapse after collapse in the evolution of an ecosystem on his New Hampshire farm. Creative teachers could use this book for anything from a liberal arts math course to a senior capstone seminar, and one re- viewer suggested that it should be required reading for any mathematics graduate student heading off into a teaching career.

This book will also be of interest to readers in the fields under discussion, such as business, engineering, ecology, political science, and others.