Ricard V. Sole is research professor and head of the Complex Systems Lab at Pompeu Fabra University and external professor at the Santa Fe Institute. He is the coauthor of "Signs of Life" (Basic) and "Self-Organization in Complex Ecosystems" (Princeton).

• Preface xi Chapter 1. Phase Changes 1 1.1 Complexity 1 1.2 Phase diagrams 4 1.3 Interactions make a difference 7 1.4 The Ising model: From micro to macro 11 1.5 Monte Carlo simulation 13 1.6 Scaling and universality 17 1.7 Mean field Ising model 18 1.8 Nonequilibrium transitions 23 Chapter 2. Stability and Instability 25 2.1 Dynamical systems 25 2.2 Attractors 27 2.3 Nonlinear dynamics 30 2.4 Linear stability analysis 33 Chapter 3. Bifurcations and Catastrophes 37 3.1 Qualitative changes 37 3.2 Symmetry breaking 38 3.3 Catastrophes and breakpoints 43 3.4 Critical slowing down 49 3.5 Multiple broken symmetries 51 Chapter 4. Percolation 53 4.1 Systems are connected 53 4.2 Percolation thresholds 54 4.3 Percolation is widespread 57 4.4 Bethe lattices 59 Chapter 5. Random Graphs 63 5.1 The Erdos-Renyi model 63 5.2 Statistical patterns 64 5.3 Percolation at critical connectivity 65 5.4 Real networks are heterogeneous 68 Chapter 6. Life Origins 70 6.1 Prebiotic evolution 70 6.2 Replicators and reproducers 71 6.3 Autocatalysis and cooperation 72 6.4 Prebiotic reaction networks 75 Chapter 7. Virus Dynamics 78 7.1 Molecular parasites 78 7.2 Exploring the hypercube 80 7.3 Swetina-Schuster quasispecies model 84 7.4 Critical genome size 87 7.5 Beyond the threshold 88 Chapter 8. Cell Structure 91 8.1 The architecture of cells 91 8.2 States and transitions 94 8.3 Dynamical instability model 95 8.4 Tensegrity 97 Chapter 9. Epidemic Spreading 99 9.1 Spreading diseases 99 9.2 SIS model 102 9.3 Vaccination thresholds 105 9.4 Pathogens and networks 107 Chapter 10. Gene Networks 109 10.1 Genes and cell types 109 10.2 Boolean dynamics 112 10.3 Percolation analysis 113 10.4 Is cell dynamics Boolean? 118 Chapter 11. Cancer Dynamics 120 11.1 Abnormal growth 120 11.2 Tumor decay under immune attack 122 11.3 Thresholds in cancer instability 128 11.4 Cancer as an evolving system 132 Chapter 12. Ecological Shifts 134 12.1 Change in ecology 134 12.2 Green-desert transitions 136 12.3 Continuous transitions 138 12.4 Sudden shifts 140 12.5 Metapopulation dynamics 143 12.6 Warning signals 146 Chapter 13. Information and Traffic Jams 148 13.1 Internet and computer networks 148 13.2 Mean field model of traffic flow 150 13.3 Fluid flow and congestion explosion 152 13.4 Self-regulated traffic 154 Chapter 14. Collective Intelligence 157 14.1 Swarm behavior 157 14.2 Foraging and broken symmetry 159 14.3 Order-disorder transitions in ant colonies 162 14.4 Structure and evolution in social insects 165 Chapter 15. Language 167 15.1 Lexical change 167 15.2 Words: Birth and death 169 15.3 String models 171 15.4 Symmetric model 174 15.5 Language thresholds 176 Chapter 16. Social Collapse 179 16.1 Rise and fall 179 16.2 Slow responses and sunk-cost effects 181 16.3 Ecological model of social collapse 182 16.4 Historical dynamics 186 References 189 Index 213

"Sole has compiled an interesting overview of the vast amount of real world systems in which phases play a role. It is a good introduction to the topic and the great variety of applications is inspirational. Phase Transitions is a good read for the JASSS audience interested in if, how, and when abrupt changes may occur--either as risks for collapse or as opportunities for salvation."--Emile Chappin, Journal of Artificial Societies and Social Simulation "Mean-field approaches constitute an elegant and considerable contribution to complex systems studies, and I commend Sole for his rigorous presentation."--Lael Parrott, BioScience