Approaches to Geo-mathematical Modelling: New Tools for Complexity Science(Wiley Series in Computational and Quantitative Social Science)

Geo-mathematical modelling: models from complexity science   Sir Alan Wilson, Centre for Advanced Spatial Analysis, University College London   Mathematical and computer models for a complexity science tool kit   Geographical systems are characterised by locations, activities at locations, interactions between them and the infrastructures that carry these activities and flows. They can be described at a great variety of scales, from individuals and organisations to countries. Our understanding, often partial, of these entities, and in many cases this understanding is represented in theories and associated mathematical models.   In this book, the main examples are models that represent elements of the global system covering such topics as trade, migration, security and development aid together with examples at finer scales. This provides an effective toolkit that can not only be applied to global systems, but more widely in the modelling of complex systems. All complex systems involve nonlinearities involving path dependence and the possibility of phase changes and this makes the mathematical aspects particularly interesting. It is through these mechanisms that new structures can be seen to ‘emerge’, and hence the current notion of ‘emergent behaviour’. The range of models demonstrated include account-based models and biproportional fitting, structural dynamics, space-time statistical analysis, real-time response models, Lotka-Volterra models representing ‘war’, agent-based models, epidemiology and reaction-diffusion approaches, game theory, network models and finally, integrated models.   Geo-mathematical modelling: Presents mathematical models with spatial dimensions. Provides representations of path dependence and phase changes. Illustrates complexity science using models of trade, migration, security and development aid. Demonstrates how generic models from the complexity science tool kit can each be applied in a variety of situations   This book is for practitioners and researchers in applied mathematics, geography, economics, and interdisciplinary fields such as regional science and complexity science. It can also be used as the basis of a modelling course for postgraduate students.

Table of Contents:
Notes on Contributors xv Acknowledgements xxi About the Companion Website xxiii Part I Approaches 1 The Toolkit 3 Alan G. Wilson Part II Estimating Missing Data: Bi-proportional Fitting and Principal Components Analysis 2 The Effects of Economic and Labour Market Inequalities on Interregional Migration in Europe 9 Adam Dennett 2.1 Introduction 9 2.2 The Approach 12 2.3 Data 12 2.4 Preliminary Analysis 13 2.5 Multinomial Logit Regression Analysis 15 2.6 Discussion 22 2.7 Conclusions 24 References 25 3 Test of Bi-Proportional Fitting Procedure Applied to International Trade 26 Simone Caschili and Alan G. Wilson 3.1 Introduction 26 3.2 Model 27 3.3 Notes of Implementation 28 3.4 Results 30 References 32 4 Estimating Services Flows 33 Robert G. Levy 4.1 Introduction 33 4.2 Estimation Via Iterative Proportional Fitting 34 4.2.1 The Method 34 4.2.2 With All Initial Values Equal 35 4.2.3 Equivalence to Entropy Maximisation 36 4.2.4 Estimation with Some Known Flows 37 4.2.5 Drawbacks to Estimating Services Flows with IPF 37 4.3 Estimating Services Flows Using Commodities Flows 37 4.3.1 The Gravity Model 37 4.3.2 Splitting Up Value Added 40 4.4 A Comparison of The Methods 40 4.4.1 Unbalanced Row and Column Margins 42 4.4.2 Iterative Proportional Fitting 42 4.4.3 Gravity Model 42 4.4.4 Gravity Model Followed by IPF 44 4.5 Results 45 4.5.1 Selecting a Representative Sector 45 4.5.2 Estimated in-Sample Flows 46 4.5.3 Estimated Export Totals 47 4.6 Conclusion 49 References 50 5 A Method for Estimating Unknown National Input–Output Tables Using Limited Data 51 Thomas P. Oléron Evans and Robert G. Levy 5.1 Motivation and Aims 51 5.2 Obstacles to The Estimation of National Input–Output Tables 52 5.3 Vector Representation of Input–Output Tables 53 5.4 Method 54 5.4.1 Concept 54 5.4.2 Estimation Procedure 55 5.4.3 Cross-Validation 57 5.5 In-Sample Assessment of The Estimates 58 5.5.1 Summary Statistics 58 5.5.2 Visual Comparison 61 5.6 Out-of-Sample Discussion of The Estimates 63 5.6.1 Final Demand Closeness 63 5.6.2 Technical Coefficient Clustering 65 5.7 Conclusion 67 References 68 Part III Dynamics in Account-based Models 6 A Dynamic Global Trade Model With Four Sectors: Food, Natural Resources, Manufactured Goods and Labour 71 Hannah M. Fry, Alan G. Wilson and Frank T. Smith 6.1 Introduction 71 6.2 Definition of Variables for System Description 73 6.3 The Pricing and Trade Flows Algorithm 73 6.4 Initial Setup 75 6.5 The Algorithm to Determine Farming Trade Flows 77 6.5.1 The Accounts for the Farming Industry 79 6.5.2 A Final Point on The Farming Flows 79 6.6 The Algorithm to Determine The Natural Resources Trade Flows 80 6.6.1 The Accounts for The Natural Resources Sector 80 6.7 The Algorithm to Determine Manufacturing Trade Flows 81 6.7.1 The Accounts for The Manufacturing Industry 82 6.8 The Dynamics 83 6.9 Experimental Results 84 6.9.1 Concluding Comments 88 References 90 7 Global Dynamical Input–Output Modelling 91 Anthony P. Korte and Alan G. Wilson 7.1 Towards a Fully Dynamic Inter-country Input–Output Model 91 7.2 National Accounts 92 7.2.1 Definitions 92 7.2.2 The Production Account 94 7.2.3 The Commodity Markets Account 94 7.2.4 The Household Account 94 7.2.5 The Capital Markets Account 94 7.2.6 The Rest of the World (RoW) Account 94 7.2.7 The Government Account 95 7.2.8 The Net Worth of an Economy and Revaluations 95 7.2.9 Overview of the National Accounts 95 7.2.10 Closing the Model: Making Final Demand Endogenous 96 7.3 The Dynamical International Model 97 7.3.1 Supply and Demand 97 7.3.2 The National Accounts Revisited 99 7.4 Investment: Modelling Production Capacity: The Capacity Planning Model 100 7.4.1 The Multi-region, Multi-sector Capacity Planning Model 100 7.5 Modelling Production Capacity: The Investment Growth Approach 103 7.5.1 Multi-region, multi-sector Investment Growth Models with Reversibility 103 7.5.2 One-country, One-sector Investment Growth Model with Reversibility 104 7.5.3 Two-country, Two-sector Investment Growth Model with Reversibility 106 7.5.4 A Multi-region, Multi-sector, Investment Growth Model without Reversibility 108 7.5.5 A Multi-region, Multi-sector, Investment Growth Model without Reversibility, with Variable Trade Coefficients 111 7.5.6 Dynamical Final Demand 114 7.5.7 Labour 115 7.5.8 The Price Model 118 7.6 Conclusions 121 References 122 Appendix 123 A.1 Proof of Linearity of the Static Model and the Equivalence of Two Modelling Approaches 123 Part IV Space–Time Statistical Analysis 8 Space–Time Analysis of Point Patterns in Crime and Security Events 127 Toby P. Davies, Shane D. Johnson, Alex Braithwaite and Elio Marchione 8.1 Introduction 127 8.1.1 Clustering 127 8.1.2 Clustering of Urban Crime 129 8.1.3 The Knox Test 130 8.2 Application in Novel Areas 132 8.2.1 Maritime Piracy 132 8.2.2 Space–Time Clustering of Piracy 134 8.2.3 Insurgency and Counterinsurgency in Iraq 136 8.3 Motif Analysis 138 8.3.1 Introduction 138 8.3.2 Event Networks 140 8.3.3 Network Motifs 140 8.3.4 Statistical Analysis 141 8.3.5 Random Network Generation 142 8.3.6 Results 143 8.4 Discussion 147 References 148 Part V Real-Time Response Models 9 The London Riots –1: Epidemiology, Spatial Interaction and Probability of Arrest 153 Toby P. Davies, Hannah M. Fry, Alan G. Wilson and Steven R. Bishop 9.1 Introduction 153 9.2 Characteristics of Disorder 156 9.3 The Model 158 9.3.1 Outline 158 9.3.2 General Concepts 158 9.3.3 Riot Participation 159 9.3.4 Spatial Assignment 160 9.3.5 Interaction between Police and Rioters 162 9.4 Demonstration Case 162 9.5 Concluding Comments 166 References 166 Appendix 168 A.1 Note on Methods: Data 168 A.2 Numerical Simulations 169 10 The London Riots –2: A Discrete Choice Model 170 Peter Baudains, Alex Braithwaite and Shane D. Johnson 10.1 Introduction 170 10.2 Model Setup 170 10.3 Modelling the Observed Utility 172 10.4 Results 176 10.5 Simulating the 2011 London Riots: Towards a Policy Tool 181 10.6 Modelling Optimal Police Deployment 187 References 190 Part VI The Mathematics of War 11 Richardson Models with Space 195 Peter Baudains 11.1 Introduction 195 11.2 The Richardson Model 196 11.3 Empirical Applications of Richardson’s Model 202 11.4 A Global Arms Race Model 204 11.5 Relationship to a Spatial Conflict Model 206 11.6 An Empirical Application 207 11.6.1 Two Models of Global Military Expenditure 207 11.6.2 The Alliance Measure C ij 208 11.6.3 A Spatial Richardson Model of Global Military Expenditure 210 11.6.4 Results 211 11.7 Conclusion 212 References 213 Part VII Agent-based Models 12 Agent-based Models of Piracy 217 Elio Marchione, Shane D. Johnson and Alan G. Wilson 12.1 Introduction 217 12.2 Data 219 12.3 An Agent-based Model 221 12.3.1 Defining Maritime Piracy Maps 221 12.3.2 Defining Vessel Route Maps 222 12.3.3 Defining Pirates’, Naval Units’ and Vessels’ Behaviours 224 12.3.4 Comparing Risk Maps 227 12.4 Model Calibration 232 12.5 Discussion 232 References 235 13 A Simple Approach for the Prediction of Extinction Events in Multi-agent Models 237 Thomas P. Oléron Evans, Steven R. Bishop and Frank T. Smith 13.1 Introduction 237 13.2 Key Concepts 238 13.2.1 Binary Classification 238 13.2.2 Measures of Classifier Performance 238 13.2.3 Stochastic Processes 240 13.3 The NANIA Predator–prey Model 241 13.3.1 Background 241 13.3.2 An ODD Description of the NANIA Model 241 13.3.3 Behaviour of the NANIA Model 245 13.3.4 Extinctions in the NANIA Model 246 13.4 Computer Simulation 247 13.4.1 Data Generation 247 13.4.2 Categorisation of the Data 249 13.5 Period Detection 249 13.6 A Monte Carlo Approach to Prediction 252 13.6.1 Binned Data 252 13.6.2 Confidence Intervals 257 13.6.3 Predicting Extinctions using Binned Population Data 257 13.6.4 ROC and Precision-recall Curves for Monte Carlo Prediction of Predator Extinctions 260 13.7 Conclusions 263 References 264 Part VIII Diffusion Models 14 Urban Agglomeration Through the Diffusion of Investment Impacts 269 Minette D’Lima, Francesca R. Medda and Alan G. Wilson 14.1 Introduction 269 14.2 The Model 270 14.3 Mathematical Analysis for Agglomeration Conditions 272 14.3.1 Introduction 272 14.3.2 Case: r < c 274 14.3.3 Case: r ≥ c 274 14.4 Simulation Results 275 14.5 Conclusions 279 References 279 Part IX Game Theory 15 From Colonel Blotto to Field Marshall Blotto 283 Peter Baudains, Toby P. Davies, Hannah M. Fry and Alan G. Wilson 15.1 Introduction 283 15.2 The Colonel Blotto Game and its Extensions 285 15.3 Incorporating a Spatial Interaction Model of Threat 286 15.4 Two-front Battles 288 15.5 Comparing Even and Uneven Allocations in a Scenario with Five Fronts 289 15.6 Conclusion 292 References 292 16 Modelling Strategic Interactions in a Global Context 293 Janina Beiser 16.1 Introduction 293 16.2 The Theoretical Model 294 16.3 Strategic Estimation 295 16.4 International Sources of Uncertainty in the Context of Repression and Rebellion 297 16.4.1 International Sources of Uncertainty Related to Actions 297 16.5 International Sources of Uncertainty Related to Outcomes 299 16.6 Empirical Analysis 301 16.6.1 Data and Operationalisation 301 16.7 Results 303 16.8 Additional Considerations Related to International Uncertainty 304 16.9 Conclusion 304 References 305 17 A General Framework for Static, Spatially Explicit Games of Search and Concealment 306 Thomas P. Oléron Evans, Steven R. Bishop and Frank T. Smith 17.1 Introduction 306 17.2 Game Theoretic Concepts 307 17.3 Games of Search and Security: A Review 310 17.3.1 Simple Search Games 310 17.3.2 Search Games with Immobile Targets 311 17.3.3 Accumulation Games 311 17.3.4 Search Games with Mobile Targets 311 17.3.5 Allocation Games 312 17.3.6 Rendez-vous Games 312 17.3.7 Security Games 313 17.3.8 Geometric Games 313 17.3.9 Motivation for Defining a New Spatial Game 314 17.4 The Static Spatial Search Game (SSSG) 314 17.4.1 Definition of the SSSG 314 17.4.2 The SSSG and other Games 316 17.4.3 The SSSG with Finite Strategy Sets 317 17.4.4 Dominance and Equivalence in the SSSG 318 17.4.5 Iterated Elimination of Dominated Strategies 323 17.5 The Graph Search Game (GSG) 324 17.5.1 Definition of the GSG 324 17.5.2 The GSG with r ≠ 1 326 17.5.3 Preliminary Observations 327 17.5.4 Bounds on the Value of the GSG 330 17.6 Summary and Conclusions 335 References 336 Part X Networks 18 Network Evolution: A Transport Example 343 Francesca Pagliara, Alan G. Wilson and Valerio de Martinis 18.1 Introduction 343 18.2 A Hierarchical Retail Structure Model as a Building Block 344 18.3 Extensions to Transport Networks 345 18.4 An Application in Transport Planning 347 18.5 A Case Study: Bagnoli in Naples 350 18.6 Conclusion 360 References 361 19 The Structure of Global Transportation Networks 363 Sean Hanna, Joan Serras and Tasos Varoudis 19.1 Introduction 363 19.2 Method 364 19.3 Analysis of the European Map 366 19.4 Towards a Global Spatial Economic Map: Economic Analysis by Country 368 19.5 An East-west Divide and Natural Economic Behaviour 373 19.6 Conclusion 376 References 377 20 Trade Networks and Optimal Consumption 378 Robert J. Downes and Robert G. Levy 20.1 Introduction 378 20.2 The Global Economic Model 379 20.2.1 Introduction 379 20.2.2 Data Sources 380 20.2.3 Model Overview 380 20.3 Perturbing Final Demand Vectors 380 20.3.1 Introduction 380 20.3.2 Perturbation Process 382 20.4 Analysis 384 20.4.1 Introduction 384 20.4.2 A Directed Network Representation 384 20.4.3 A Weighted Directed Network Representation 389 20.4.4 Communities in the Network of Improvements 390 20.5 Conclusions 393 Acknowledgements 394 References 394 Appendix 396 Part XI Integration 21 Research Priorities 399 Alan G. Wilson Index 403

About the Author :
Alan Geoffrey Wilson, Centre for Advanced Spatial Analysis, University College London, UK. His research interests have been concerned with many aspects of mathematical modelling and the use of models in planning in relation to all aspects of cities and regions - including demography, economic input-output modelling, transport and locational structures. He was responsible for the introduction of a number of model building techniques which are now in common use internationally. These models have been widely used in areas such as transport planning. He made important contributions through the rigorous deployment of accounts' concepts in demography and economic modelling. In recent years he has been particularly concerned with applications of dynamical systems theory in relation to the task of modelling the evolution of urban structure, initially described in Catastrophe theory and bifurcation: applications to urban and regional systems. His current research, supported by ESRC and EPSRC grants of around ?3M, is on the evolution of cities and the dynamics of global trade and migration.