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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Computational and Quantitative Social Science
- Sprache: Englisch
- Veröffentlichungsjahr: 2016

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.

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Cover

Title Page

Copyright

Notes on Contributors

Acknowledgements

About the Companion Website

Part One: Approaches

Chapter 1: The Toolkit

Part Two: Estimating Missing Data: Bi-proportional Fitting and Principal Components Analysis

Chapter 2: The Effects of Economic and Labour Market Inequalities on Interregional Migration in Europe

2.1 Introduction

2.2 The Approach

2.3 Data

2.4 Preliminary Analysis

2.5 Multinomial Logit Regression Analysis

2.6 Discussion

2.7 Conclusions

References

Chapter 3: Test of Bi-Proportional Fitting Procedure Applied to International Trade

3.1 Introduction

3.2 Model

3.3 Notes of Implementation

3.4 Results

References

Chapter 4: Estimating Services Flows

4.1 Introduction

4.2 Estimation Via Iterative Proportional Fitting

4.3 Estimating Services Flows Using Commodities Flows

4.4 A Comparison of The Methods

4.5 Results

4.6 Conclusion

References

Chapter 5: A Method for Estimating Unknown National Input–Output Tables Using Limited Data

5.1 Motivation and Aims

5.2 Obstacles to The Estimation of National Input–Output Tables

5.3 Vector Representation of Input–Output Tables

5.4 Method

5.5 In-Sample Assessment of The Estimates

5.6 Out-of-Sample Discussion of The Estimates

5.7 Conclusion

References

Part Three: Dynamics in Account-based Models

Chapter 6: A Dynamic Global Trade Model With Four Sectors: Food, Natural Resources, Manufactured Goods and Labour

6.1 Introduction

6.2 Definition of Variables for System Description

6.3 The Pricing and Trade Flows Algorithm

6.4 Initial Setup

6.5 The Algorithm to Determine Farming Trade Flows

6.6 The Algorithm to Determine The Natural Resources Trade Flows

6.7 The Algorithm to Determine Manufacturing Trade Flows

6.8 The Dynamics

6.9 Experimental Results

References

Chapter 7: Global Dynamical Input–Output Modelling

7.1 Towards a Fully Dynamic Inter-country Input–Output Model

7.2 National Accounts

7.3 The Dynamical International Model

7.4 Investment: Modelling Production Capacity: The Capacity Planning Model

7.5 Modelling Production Capacity: The Investment Growth Approach

7.6 Conclusions

References

Appendix

A.1 Proof of Linearity of the Static Model and the Equivalence of Two Modelling Approaches

Part Four: Space-Time Statistical Analysis

Chapter 8: Space–Time Analysis of Point Patterns in Crime and Security Events

8.1 Introduction

8.2 Application in Novel Areas

8.3 Motif Analysis

8.4 Discussion

References

Part Five: Real-Time Response Models

Chapter 9: The London Riots – 1: Epidemiology, Spatial Interaction and Probability of Arrest

9.1 Introduction

9.2 Characteristics of Disorder

9.3 The Model

9.4 Demonstration Case

9.5 Concluding Comments

References

Appendix

A.1 Note on Methods: Data

A.2 Numerical Simulations

Chapter 10: The London Riots – 2: A Discrete Choice Model

10.1 Introduction

10.2 Model Setup

10.3 Modelling the Observed Utility

10.4 Results

10.5 Simulating the 2011 London Riots: Towards a Policy Tool

10.6 Modelling Optimal Police Deployment

References

Part Six: The Mathematics of War

Chapter 11: Richardson Models with Space

11.1 Introduction

11.2 The Richardson Model

11.3 Empirical Applications of Richardson's Model

11.4 A Global Arms Race Model

11.5 Relationship to a Spatial Conflict Model

11.6 An Empirical Application

11.7 Conclusion

References

Part Seven: Agent-Based Models

Chapter 12: Agent-based Models of Piracy

12.1 Introduction

12.2 Data

12.3 An Agent-based Model

12.4 Model Calibration

12.5 Discussion

References

Chapter 13: A Simple Approach for the Prediction of Extinction Events in Multi-agent Models

13.1 Introduction

13.2 Key Concepts

13.3 The NANIA Predator–prey Model

13.4 Computer Simulation

13.5 Period Detection

13.6 A Monte Carlo Approach to Prediction

13.7 Conclusions

References

Part Eight: Diffusion Models

Chapter 14: Urban Agglomeration Through the Diffusion of Investment Impacts

14.1 Introduction

14.2 The Model

14.3 Mathematical Analysis for Agglomeration Conditions

14.4 Simulation Results

14.5 Conclusions

References

Part Nine: Game Theory

Chapter 15: From Colonel Blotto to Field Marshall Blotto

15.1 Introduction

15.2 The Colonel Blotto Game and its Extensions

15.3 Incorporating a Spatial Interaction Model of Threat

15.4 Two-front Battles

15.5 Comparing Even and Uneven Allocations in a Scenario with Five Fronts

15.6 Conclusion

References

Chapter 16: Modelling Strategic Interactions in a Global Context

16.1 Introduction

16.2 The Theoretical Model

16.3 Strategic Estimation

16.4 International Sources of Uncertainty in the Context of Repression and Rebellion

16.5 International Sources of Uncertainty Related to Outcomes

16.6 Empirical Analysis

16.7 Results

16.8 Additional Considerations Related to International Uncertainty

16.9 Conclusion

References

Chapter 17: A General Framework for Static, Spatially Explicit Games of Search and Concealment

17.1 Introduction

17.2 Game Theoretic Concepts

17.3 Games of Search and Security: A Review

17.4 The Static Spatial Search Game (SSSG)

17.5 The Graph Search Game (GSG)

17.6 Summary and Conclusions

References

Part Ten: Networks

Chapter 18: Network Evolution: A Transport Example

18.1 Introduction

18.2 A Hierarchical Retail Structure Model as a Building Block

18.3 Extensions to Transport Networks

18.4 An Application in Transport Planning

18.5 A Case Study: Bagnoli in Naples

18.6 Conclusion

References

Chapter 19: The Structure of Global Transportation Networks

19.1 Introduction

19.2 Method

19.3 Analysis of the European Map

19.4 Towards a Global Spatial Economic Map: Economic Analysis by Country

19.5 An East-west Divide and Natural Economic Behaviour

19.6 Conclusion

References

Chapter 20: Trade Networks and Optimal Consumption

20.1 Introduction

20.2 The Global Economic Model

20.3 Perturbing Final Demand Vectors

20.4 Analysis

20.5 Conclusions

Acknowledgements

References

Appendix

Part Eleven: Integration

Chapter 21: Research Priorities

Index

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Cover

Table of Contents

Part One: Approaches

Begin Reading

Chapter 2: The Effects of Economic and Labour Market Inequalities on Interregional Migration in Europe

Figure 2.1 Average migration rate per 1000 population by ratio decile

Figure 2.2 Odds ratios for the effect of levels of PPS and over 25 unemployment on O/D migration flow volumes in Europe, 2002 (highest migration level reference)

Figure 2.3 Odds ratios for the effect of levels of PPS and over 25 unemployment on O/D migration flow volumes in Europe, 2002 (lowest migration-level reference)

Figure 2.4 Odds ratios for the effect of levels of PPS and over 25 unemployment on O/D migration flow volumes in Europe, 2004 (highest migration-level reference)

Figure 2.5 Odds ratios for the effect of levels of PPS and over 25 unemployment on highest migration flow volumes (decile 10) in Europe, 2004 (lowest migration-level reference)

Chapter 3:

Figure 3.1 Bilateral trade between countries and sectors. Each node represents a sector of a country. Each link has the colour of the exporter country, while link width is proportional to the value of bilateral trade between the two sectors. For ease of visualisation, we have pruned the graph from the links with bilateral trade lower than USD $100 million. Sectors: High Tech Manufacturing (MH), Low Tech Manufacturing (ML), Services (SE), Raw Materials (RM), Industrial (I), Retail Sale (SA), Government (G)

Figure 3.2 Inter-bilateral trade between Service and Government sectors. Width and colour of links are proportional to trade flows. For ease of visualisation, we have pruned the links with bilateral trade lower than USD $3000 million

Figure 3.3 Intra-bilateral trade in the Raw Material sector. Width and colour of links are proportional to trade flows. For ease of visualisation, we have pruned the links with bilateral trade lower than USD $2400 million

Figure 3.4 Intra-bilateral trade in the Manufacture High Tech sector. Width and colour of links are proportional to trade flows. For ease of visualisation, we have pruned the links with bilateral trade lower than USD $3000 million

Chapter 4: Estimating Services Flows

Figure 4.1 Density plot showing the results of multiple iterative proportional fitting runs. During each run, the initial value of every matrix element was tracked and paired with its value after the iteration completed. This graph shows these pairs of initial (-axis) and final values (-axis), having first binned the initial/final space into hexagonal-shaped regions for visual clarity. Darker hexagons represent more matrix elements occupying that space. The initial matrix was uniformly distributed between zero and one with shape 5 5. 10,000 runs were performed. The straight line shows

Chapter 5: A Method for Estimating Unknown National Input–Output Tables Using Limited Data

Figure 5.1 A density plot showing the relationship between the estimates of the 40 WIOD countries and the WIOD data itself. Darker shades indicate the presence of more data points. All data points are flows, measured in $US millions. A little over 100,000 flows are plotted.

Figure 5.2 Boxplots showing the estimation error of flows within the 10 largest countries and the 10 largest sectors by total flow magnitude. The dotted vertical lines show the 10th to the 90th percentiles. The box shows the interquartile range. The mean is represented as a small filled square, and the median by a horizontal line. Estimation errors are measured in millions of $US.

Figure 5.3 Visual comparison of WIOD data versus estimates for four countries

Figure 5.4 A co-occurrence matrix showing the result of 100 runs of the k-means clustering algorithm, set to split the technical coefficients of the countries shown into five clusters. Darker colours indicate a more frequent co-occurrence of the pair of countries in the same cluster

Chapter 6: A Dynamic Global Trade Model With Four Sectors: Food, Natural Resources, Manufactured Goods and Labour

Figure 6.1 A schematic diagram of how various elements of the model interact

Figure 6.2 A quantitative measure of benefit to a potential buyer against the parameter . See equation (6.59)

Figure 6.3 The demand on the natural resources products for various values of alpha, normalised by production capacity. In these plots, , ×

Figure 6.4 The actual sales price of natural resources for various values of

Figure 6.5 Some first results of simulation for the food flows. Here, the node size is the consumption per person, edge width is the size of the inter country flows and node colour is the percentage of consumption met locally (see scale)

Chapter 7: Global Dynamical Input–Output Modelling

Figure 7.1 National circular account flows and net worth of the economy

Figure 7.2 Production and its capacity, , , . (Left) 3rd country; (right) 5th country

Figure 7.3 Production and its capacity, , , . (Left) 3rd country; (right) 5th country

Figure 7.4 , , capacity planning model; production capacity (solid lines), production (dash-dot lines); same colours for as for in legend

Figure 7.5 Logistic, one-country, one-sector model with reversibility, . (a) ; (b) . Production capacity (black)/production (red) versus

Figure 7.6 Logistic, one-country, one-sector model, , , (a) Capacity versus ; (b) final demand and production capacity growth rate. Inset: period doubling contours

Figure 7.7 Logistic, one-country, one-sector non-reversible model,

a

=

b

=0.1,

p

=0.5,

f

=1, production capacity (solid lines), production (dash-dot lines) versus

Figure 7.8 Two-country, two-sector, reversible model, production capacity (blue)/production (black) versus . (a, b) Country 1; (c, d) country 2

Figure 7.9 Two-country, two-sector non-reversible model, production capacity (blue)/production (black) versus

. (a, b) Country 1; (c, d) country 2

Figure 7.10 Two-country, two-sector non-reversible model: time series (period doubling) curves for production capacity (solid lines), production (dash-dot lines) for

(same colours for

as for

in legend)

Figure 7.11 Two-country, two-sector non-reversible model, with variable trade coefficients, production capacity (blue)/production (black) versus . (a, b) Country 1; (c, d) country 2

Figure 7.12 Two-country, two-sector model without reversibility: time series curves for (a) production capacity (solid lines), production (dash-dot lines) (same colours for as for in legend); (b) trade coefficients ,

Figure 7.13 Two-country, two-sector non-reversible model, trade coefficients versus

Figure 7.14 One-country, one-sector model, , , , , , (a) , , , , (b) , , , , including aid and remittances: production capacity (black)/production (red) versus

Figure 7.15 One-country, one-sector model, production capacity (black)/production (red) versus . (Left) ; (right)

Chapter 8: Space–Time Analysis of Point Patterns in Crime and Security Events

Figure 8.1 Kernel density map of incidents of piracy, January 1985 to January 2012

Figure 8.2 Monthly time series for two NGIA ocean subregions with high numbers of attacks (inset: mean monthly trends scaled by a factor of 2)

Figure 8.3 Knox ratios for attacks taking place in 2011 (shaded cells are statistically significant, )

Figure 8.4 Univariate and bivariate Knox analyses for insurgent and COIN activity (COIN events are cordon/searches, weapons cache discoveries and raids)

Figure 8.5 Close pair relationships for simple hypothetical sets of events. Vertex labels represent the time at which events occur, and their location is given by their position on the underlying grid, shown in grey. Red arrows represent close pairs, defined in this case as incidents occurring within two time units and one grid spacing

Figure 8.6 Motifs of three vertices and their real-world interpretation: (a) all three-vertex subgraphs which can arise in event networks, up to isomorphism; and (b) two example events, and , with a circular region of radius indicated for each

Figure 8.7 The process of randomised network generation involves iterative movement of points. The plots show the two possible outcomes at each iteration, for a hypothetical situation where four close pairs are required and it is assumed that all events are close temporally. The initial configuration, shown left, contains two close pairs. The point highlighted red is selected to move: one possible change gives three pairs and would be retained, and the other reduces the number to 1 and would be rejected

Figure 8.8 Statistical analysis of three-vertex motifs for a variety of spatio-temporal thresholds. Each cell corresponds to a particular combination of and , and the colours indicate the effect size

Figure 8.9 Statistical analysis of four-vertex motifs for a variety of spatio-temporal thresholds. Again, outlines and colours represent statistical significance and effect sizes, respectively

Chapter 9: The London Riots – 1: Epidemiology, Spatial Interaction and Probability of Arrest

Figure 9.1 Observations from arrest data: (a) log-linear plot of the complementary cumulative distribution function of , the distance between residential and offence locations. The straight line shows a hypothetical exponential distribution with parameter 0.274 ( for a confidence interval), for which the Kolmogorov–Smirnov distance statistic is 0.0246 (which compares with 0.332 for the equivalent fitted power-law). (b) Lorenz curve for the distribution of riot locations amongst Lower Super Output Areas (LSOAs; UK census units with an average population of approximately 1500) ranked according to deprivation (where 1 is the most deprived). The dashed line represents perfect equality. (c) Relationship between area-level deprivation and the proportion of residents involved in the disorder, where the horizontal axis represents a score derived from the UK Index of Multiple Deprivation (IMD) so that all values lie in and so that London's most deprived area is given a value of 1. (d) Temporal distribution of recorded crime

Figure 9.2 Borough-level choropleth of rioter residential locations from (a) data and (b) simulated results. Although the extreme dependence on initial conditions precludes our model from generating an exact replica of the observed incidents, the results show good qualitative agreement, with 26 of the 33 boroughs showing rioter percentages in the same or adjacent bands as the data. The remaining discrepancy may be accounted for by factors specific to the London disorder, such as communication between groups, other activity patterns occurring at the time, or social factors beyond the scope of this work. The labels 1, 2, 3 or 4 correspond to retail centres in Brixton, Croydon, Clapham Junction and Ealing, respectively, which are considered individually in our later simulations

Figure 9.3 Log-linear plot of the complementary cumulative distribution function for , the distances between residences and offence locations within the demonstration simulation

Figure 9.4 The susceptibility of retail sites. For each of the four centres worst affected in the riots – (a) Brixton, (b) Croydon, (c) Clapham Junction and (d) Ealing – we ran four separate simulations, pairing the site of interest with each of its closest geographical neighbours in turn. An initial disturbance of one rioter was included at both sites, and the model run to allow the incidents to evolve. Results shown for the sites of interest are the average of their four simulations, and in each case substantial growth is seen, particularly in comparison to the neighbouring centres

Figure 9.5 The effect on severity of modifying (a) the number of police officers, and (b) their response lag

Chapter 10: The London Riots – 2: A Discrete Choice Model

Figure 10.1 Odds ratios for the model without spillover effects

Figure 10.2 Odds ratios for the model with spillover effects

Figure 10.3 The 30 most targeted LSOAs sorted according to (a) the empirical data, and (b) the average of the simulations. The positive bar chart shows the empirical counts, and the negative bar chart shows the averaged simulated counts

Figure 10.4 Ratio of count to rank for each LSOA for both the empirical data and the simulation. The ranks are obtained by sorting the LSOAs according to their count. The plot is shown on a log-scale for clarity

Figure 10.5 Deployment utility (a) for the simulation and (b) according to the empirical data. Darker shades indicate higher levels of deployment utility. The values for deployment utility are plotted assuming that no police officers have been deployed

Chapter 11: Richardson Models with Space

Figure 11.1 The trace–determinant diagram for linear planar systems. The location of the matrix on this diagram determines the dynamics near to the equilibrium. The points (a–f) correspond to the subFigure in Figure 11.2, which show the qualitative dynamics for each case

Figure 11.2 The dynamics around the equilibrium value of the Richardson model in equation 11.1 for different parameter values. The parameters are chosen to coincide with each point in Figure 11.1

Figure 11.3 Unilateral policy options available to nation i. (a) The change in the system according to the trace–determinant diagram from Figure 11.1 when is decreased; and (b) the change when is increased. Note that the half plane with is shaded since if , as Richardson hypothesised, the system will not lie in this portion of the plane. (c) The impact of nation reducing external grievances when the system is a saddle. The equilibrium point will move towards the positive quadrant, meaning that for initial conditions given by , the system will tend towards greater cooperation, rather than greater threats

Figure 11.4 The change in the sum of defence budgets against the sum of defence budgets for four nations during the four years prior to the First World War. The four nations are Russia, Germany, France and Austria-Hungary, and the values plotted represent the sum of defence budgets over these nations. Defence expenditure data were gathered from various sources by Richardson, and the line represents the best fit that would be expected from the model in equation 11.1, assuming that and . This Figure is reproduced from Richardson (1960). The gradient is given by and is estimated by Richardson to be 0.73. An ordinary least squares regression produces the same output

Figure 11.5 The increase in from the minimum of those tested, multiplied by , for different parameters and

Chapter 12: Agent-based Models of Piracy

Figure 12.1 Pirate attacks and vessel routes in 2010

Figure 12.5 Visualisation of the simulation (red dots, pirates; blue dots, vessels; green dots, naval units)

Figure 12.2 Density map of observed pirate attacks in 2010

Figure 12.3 Vessel route map

Figure 12.4 An example of the pirates' range of action

Figure 12.6 Probability of pirates attacks around the Gulf of Aden for the period 1999–2009. Source: National Geospatial Intelligence Agency's Anti-Shipping Activity Messages database

Figure 12.7 (a) The observed map and (b) a map generated during one run of the model that used the best parameter settings.

Chapter 13: A Simple Approach for the Prediction of Extinction Events in Multi-agent Models

Figure 13.1 Comparison of the Von Neumann and Moore adjacency schemes in graphs over rectangular lattices. Source: Hogeweg 1988. Reproduced with permission of Elsevier

Figure 13.2 Visualisation of the Twin Ribbon Initialisation

Figure 13.3 Sample dynamics of the predator and prey populations. The horizontal axis displays the number of steps since a particular chosen observation (). This starting point was chosen since it provides a good example of the population cycles of the system, with no extinctions observed in the following 500 steps

Figure 13.4 (a) Autocorrelations of the predator and prey population density time series calculated from the full set of 400,000 observations. Note that the number of paired observations used in each calculation varies, because fewer high-lag autocorrelations may be calculated from a time series of given length than low-lag autocorrelations. For all lags in the range 0 to 500, the number of pairs used exceeds 100,000. (b) The number of pairs used in each of the 501 calculations

Figure 13.5 399,979 observations of the NANIA model using the specifications of Section 13.4.1. Observations are classified by extinction of the foxes with a lead time of . Observations for which the fox population was extinct within 21 steps are coloured red; otherwise, they are coloured green. Note that the red points are plotted over the green points, increasing their apparent prominence.

Figure 13.6 The observations of Figure 13.5 partitioned into bins of the form for . Observations for which the fox population was extinct are excluded. Brighter coloured cells indicate higher numbers of observations.

Figure 13.7 Estimated probabilities of fox extinction for observations in each of the bins of Figure 13.6 for a lead time of 21. Paler coloured cells indicate lower estimated probabilities. Cells marked N/A contain no observations, so no extinction probabilities could be estimated for these cells.

Figure 13.8 Size of the Clopper–Pearson confidence intervals for each of the estimated probabilities of Figure 13.7 for a confidence level of 95%. Paler coloured cells indicate smaller intervals and therefore less uncertainty over the true extinction probability for observations in that cell. Cells in which no observations were recorded necessarily have a confidence interval of size 1. Such cells are distinguished by their white text.

Figure 13.9 Upper bounds of the Clopper–Pearson confidence intervals visualised in Figure 13.8. Paler coloured cells indicate lower upper bounds on the estimated probability. Cells in which no observations were recorded necessarily have an upper bound of 1. Such cells are distinguished by their white text. The area enclosed by the orange (yellow) boundary indicates the region of phase space in which we may be ‘confident’ that the probability of extinction within 21 steps does not exceed 5% (50%).

Figure 13.10 ROC curves for three families of classifiers , generated from binned training data, where varies over . Each family of classifiers is generated using bins of different sizes, and performances are measured on the unseen test data. The dotted line represents performances that could be achieved by classifiers making random predictions according to a Bernoulli variable, independently of the data.

Figure 13.11 Precision-recall curves for three families of classifiers , generated from binned training data, where varies over . Each family of classifiers is generated using bins of different size, and performances are measured on the unseen test data. Note that the term

recall

is synonymous with

true positive rate

. The dotted line at (the proportion of positively categorised points in the data) represents performances that could be achieved by classifiers making random predictions according to a Bernoulli variable, independently of the data.

Chapter 14: Urban Agglomeration Through the Diffusion of Investment Impacts

Figure 14.1 Development of population agglomeration zones

Figure 14.2 Utility, rent and transport curve

Figure 14.3 Scatterplot of urban agglomeration parameters

Figure 14.4 Scatterplot of agglomeration with two migration factors (on the left side the migration factor is equal to 0.1, and on the right side the migration factor is equal to 0.01)

Chapter 15: From Colonel Blotto to Field Marshall Blotto

Figure 15.1 The payoff function to strategy against . The distance metric is given in equation 15.23

Figure 15.2 The payoff function to strategy against . The distance metric is given in equation 15.24

Chapter 16: Modelling Strategic Interactions in a Global Context

Figure 16.1 The strategic situation.

Chapter 17: A General Framework for Static, Spatially Explicit Games of Search and Concealment

Figure 17.1 The Static Spatial Search Game (SSSG). Players deploy simultaneously at points in some metric space . Suppose Player deploys at and can search all points within a radius . Player loses the game if deployed at , but wins if deployed at . A win results in a payoff of 1; a loss results in a payoff of 0

Figure 17.2 The Graph Search Game (GSG). Players deploy simultaneously at vertices of the graph. Suppose that Player deploys at the vertex marked . If , Player loses the game if she deploys at any of the vertices marked or , but wins if she deploys at any of the vertices marked or . If , Player loses the game if she deploys at any of the vertices marked , or , but wins if she deploys at either of the vertices marked . A win results in a payoff of 1; a loss results in a payoff of 0

Figure 17.3 A representation of a hypothetical street network, in which high-risk locations (the vertices) must be protected from criminals by a single unmarked police unit

Figure 17.4 The graph

for a GSG with

, played over the graph

that was depicted in Figure 17.3

Figure 17.5 Four simple graphs. Solutions of GSGs played over these graphs and the values of the four bounds , , and are summarised in Table 17.1

Figure 17.6 The vertex subset

(whose members are highlighted in this figure) is an element of the set

(see Proposition 17.13), because the union of the closed neighbourhoods of these vertices (shaded) completely covers the vertices of the graph

Figure 17.7 The vertex subset

(whose members are highlighted in this figure) is an element of the set

(see Proposition 17.13), because the distance between any pair of these vertices is always at least 3. This is demonstrated in the figure by shading the region

around each vertex in

. Note that no shaded region contains more than one highlighted vertex

Chapter 18: Network Evolution: A Transport Example

Figure 18.1 Arterial roads classification in urban and suburban areas. Source: Federal Highway Administration, U.S Department of Transportation

Figure 18.2 The test site on Google Earth (top) and the related modelling of the area (bottom) with urban arterials as a continuous black line and the urban highway Tangenziale di Napoli in black dots

Figure 18.3 Network links with allowed or not allowed upgrading

Figure 18.4 Actual scenario facility type

Figure 18.5 Facility types at the third step of the incremental demand scenario

Figure 18.6 Facility types at the fifth step of the incremental demand scenario with budget constraints

Figure 18.7 Facility types at the sixth step of the incremental demand scenario with budget constraints. (This configuration can be reached without budget constraints at the fifth step.)

Figure 18.8 (a) Facility types at the 11th step of the incremental demand scenario without budget constraints. (b) Facility types at the 11th step of the incremental demand scenario with budget constraints

Figure 18.9 Facility types at the 12th step of the incremental demand scenario with budget constraints

Figure 18.10 Network evolution through facility type upgrading (dotted lines for budget constraints scenario)

Figure 18.11 Average travel times for different facility types

Chapter 19: The Structure of Global Transportation Networks

Figure 19.1 Spatial distribution of choice values using 50 km radii for road and rail networks. Five link classes have been calculated using the natural breaks (Jenks) technique, and zero values have been excluded from the visualisation

Figure 19.2 Spatial distribution of choice values using 1000 km radii for road and rail networks. Five link classes have been calculated using the natural breaks (Jenks) technique, and zero values have been excluded from the visualisation

Figure 19.3 Comparison between the (log) sum of choice measures for road and rail networks against (log) total GDP for different radii

Figure 19.4 Comparison between the (log) total choice measures for road and rail networks against (log) total GDP, both normalised by unit of area, for different radii

Figure 19.5 Comparison between the sum of the top 10% (log) choice measures for road and rail networks against (log) total GDP for different radii

Figure 19.6 Map layout showing how we have defined countries belonging to Eastern (purple) and Western (green) Europe

Figure 19.7 Comparison between the average choice measures for road and rail networks against GDP per capita for Eastern (red) and Western (blue) Europe countries and for different radii

Chapter 20: Trade Networks and Optimal Consumption

Figure 20.1 Perturbation distance (rotation angle) against % change in GDP for Great Britain. All trajectories originate at Great Britain's consumption pattern (). Destinations are the consumption patterns of each target country in the feasible set

Figure 20.2 In-degree, a count of the number of countries who would benefit from consuming like the country in question, plotted against out-degree, a count of the number of countries the country in question would benefit from consuming like. An ordinary least squares (OLS) regression suggests with a -value of the order

Figure 20.3 Figure 20.3a shows in-degree plotted against the M1 metric of sectoral diversity; Figure 20.3b shows out-degree plotted against the M1 metric of sectoral diversity. An OLS regression line has been added to each plot. The -value in both cases is of the order

Figure 20.4 A network of improvements with countries as nodes and defining edges; see equation (20.5). Outgoing links lie counter clockwise of the direct line between two nodes and are limited in this diagram to the five largest. Node size is proportional to population; colour is defined by GDP per capita set along a scale where red and blue indicate lowest and highest GDP per capita, respectively

Figure 20.5 Co-occurrence matrix of the optimal clustering ordered similarity. White areas are very dissimilar; dark areas are highly similar. We see two well-defined clusters, along with a small number of countries without clear cluster membership

Figure 20.6 A network of improvements showing the two clusters of Figure 20.5. Edges between the clusters have been removed, with the exception of the three transition countries Korea (KOR), Indonesia (IDN) and Greece (GRC). Only improvements producing changes greater than 0.1% of GDP are shown. Node colouring reflects underlying GDP per capita as before; edge colouring reflects the averaged GDP per capita of source and destination country

Chapter 4: Estimating Services Flows

Table 4.1 Results of using equation 4.4

Table 4.2 Regression similar in structure to Table 4.1 but with divided into its constituent sectors

Table 4.3 The root mean squared error (RMSE) for each services sector when point-to-point flows are estimated using the method associated with each column. RMSE is calculated as . The specification number refers to the column of Table 4.1 used as the gravity model coefficients

Table 4.5 The 10 best estimated and most poorly estimated Financial Services flows using gravity model specification (4) followed by IPF. Error shown is simply the difference between the estimated flow and the flow in data. Only flows greater than $US 10 million are included

Chapter 5: A Method for Estimating Unknown National Input–Output Tables Using Limited Data

Table 5.1 The seven “super-sectors” into which each of the 35 WIOD sectors has been aggregated in order to facilitate visual analysis of the estimates

Table 5.2 The 52 non-WIOD countries with a population of greater than 10 million, grouped by the WIOD country to which the final demand vector is closest in angle, defined by equation (5.2)

Chapter 6: A Dynamic Global Trade Model With Four Sectors: Food, Natural Resources, Manufactured Goods and Labour

Table 6.1 A summary of the inputs and outputs of the model

Table 6.2 Parameters used in base case simulation

Chapter 7: Global Dynamical Input–Output Modelling

Table 7.1 Variables used to define the national accounts at a country level

Table 7.2 Indexed variables for the international system

Chapter 9: The London Riots – 1: Epidemiology, Spatial Interaction and Probability of Arrest

Table 9.1 Parameters used in base case simulation

Chapter 10: The London Riots – 2: A Discrete Choice Model

Table 10.1 The variables used to estimate the observed utility of each target associated with crime pattern theory

Table 10.2 The number of offences and the number of LSOAs affected by day of rioting. The total number number of LSOAs affected is the total number of LSOAs that experienced rioting over the 5 days

Chapter 11: Richardson Models with Space

Table 11.1 Regression results for three model specifications. Model 1 has no lagged dependent variable, model 2 includes this variable and model 3 incorporates clustered standard errors on each country. All parameters except the intercept in models 2 and 3 have

Chapter 12: Agent-based Models of Piracy

Table 12.1 Vessels transiting the Gulf of Aden

Table 12.3 Vessel flag passing through the Gulf of Aden in 2010. Source: Canal Suez Traffic Statistics

Table 12.2 Vessel type passing through the Gulf of Aden in 2010. Source: Canal Suez Traffic Statistics

Table 12.4 Model calibration

Chapter 14: Urban Agglomeration Through the Diffusion of Investment Impacts

Table 14.1 Parameter space classification for urban agglomeration

Table 14.2 Parameter space classification for urban agglomeration

Chapter 16: Modelling Strategic Interactions in a Global Context

Table 16.1 The effect of variables on actor utilities. Errors are assumed to be related to actions

Chapter 17: A General Framework for Static, Spatially Explicit Games of Search and Concealment

Table 17.1 Summary of the values of bounds , , and ; the true values to Player ; and examples of OMSs for each player for GSGs played over the four graphs shown in Figure 17.5, with and . Shaded cells indicate that the relevant bounds are attained. All figures are rounded to two decimal places

Chapter 18: Network Evolution: A Transport Example

Table 18.1

Highway Capacity Manual

(Transportation Research Board, 2010) characteristics for different facility types, and the facility types used for the case study described in Section 18.5

Table 18.2 OD matrix of actual scenario, with centroid 41 not yet active (in brackets, values of the centroid 41 incremental demand at first step)

Table 18.3 Incremental demand assignment results without budget constraints

Table 18.4 Incremental demand assignment results with budget constraints

Table 18.5 Network link costs

Table 18.6 Network costs at milestones

Chapter 20: Trade Networks and Optimal Consumption

Table 20.1 Results of consumption pattern perturbation for all base-target country pairs. In Table 20.1a, target countries are ranked by the number of base countries they improve (in-degree). Conversely, Table 20.1b shows base countries ranked by the number of improving target countries they have (out-degree)

Table 20.2 OLS regressions. Dependent variable is in all specifications. Standard errors are shown in brackets below each point estimate. ; ;

Table 20.3 Eigenvector centrality by country for the network defined in Section 20.4.3. Only the 10 largest are shown

Wiley Series in Computational and Quantitative Social Science

Computational Social Science is an interdisciplinary field undergoing rapid growth due to the availability of ever increasing computational power leading to new areas of research.

Embracing a spectrum from theoretical foundations to real world applications, the Wiley Series in Computational and Quantitative Social Science is a series of titles ranging from high level student texts, explanation and dissemination of technology and good practice, through to interesting and important research that is immediately relevant to social / scientific development or practice. Books within the series will be of interest to senior undergraduate and graduate students, researchers and practitioners within statistics and social science.

Behavioral Computational Social Science

Riccardo Boero

Tipping Points: Modelling Social Problems and Health

John Bissell (Editor), Camila Caiado (Editor), Sarah Curtis (Editor), Michael Goldstein (Editor), Brian Straughan (Editor)

Understanding Large Temporal Networks and Spatial Networks: Exploration, Pattern Searching, Visualization and Network Evolution

Vladimir Batagelj, Patrick Doreian, Anuska Ferligoj, Natasa Kejzar

Analytical Sociology: Actions and Networks

Gianluca Manzo (Editor)

Computational Approaches to Studying the Co-evolution of Networks and Behavior in Social Dilemmas

Rense Corten

The Visualisation of Spatial Social Structure

Danny Dorling

Edited by

Alan G.Wilson

Centre for Advanced Spatial Analysis, University College London, London, UK

This edition first published 2016

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Peter Baudains is a Research Associate at the Department of Security and Crime Science at University College London. He obtained his PhD in Mathematics from UCL in 2015 and worked for five years on the EPSRC-funded ENFOLDing project, contributing to a wide range of research projects. His research interests are in the development and application of novel analytical techniques for studying complex social systems, with a particular attention on crime, rioting and terrorism. He has authored research articles appearing in journals such as Criminology, Applied Geography, Policing and the European Journal of Applied Mathematics.

Janina Beiser obtained her PhD in the Department of Political Science at University College London. During her PhD, she was part of the security workstream of the ENFOLDing project at UCL's Centre for Advanced Spatial Analysis for three years. Her research is concerned with the contagion of armed civil conflict as well as with government repression. She is now a Research Fellow in the Department of Government at the University of Essex.

Steven R. Bishop is a Professor of Mathematics at University College London, where he has been since arriving in 1984 as a postdoctoral researcher. He published over 150 academic papers, edited books and has had appearances on television and radio. Historically, his research investigated topics such as chaos theory, reducing vibrations of engineering structures and how sand dunes are formed, but he has more recently worked on ‘big data’ and the modelling of social systems. Steven held a prestigious ‘Dream’ Fellowship funded by the UK Research Council (EPSRC) until December 2013, allowing him to consider creative ways to arrive at scientific narratives. He was influential in the formation of a European network of physical and social scientists in order to investigate how decision support systems can be developed to assist policy makers and, to drive this, has organised conferences in the UK and European Parliaments. He has been involved in several European Commission–funded projects and has helped to forge a research agenda which looks at the behaviour of systems that cross policy domains and country borders.

Alex Braithwaite is an Associate Professor in the School of Government and Public Policy at the University of Arizona, as well as a Senior Research Associate in the School of Public Policy at University College London. He obtained a PhD in Political Science from the Pennsylvania State University in 2006 and has since held academic positions at Colorado State University, UCL, and the University of Arizona. He was a co-investigator on the EPSRC-funded ENFOLDing project between 2010 and 2013, contributing to a wide range of projects under the “security” umbrella. His research interests lie in the causes and geography of violent and nonviolent forms of political conflict and has been published in journals such as Journal of Politics, International Studies Quarterly, British Journal of Political Science, Journal of Peace Research, Criminology and Journal of Quantitative Criminology.

Simone Caschili has a PhD in Land Engineering and Urban Planning, and after being a Research Associate at the Centre for Advanced Spatial Analysis (at University College London) and Senior Fellow of the UCL QASER Lab, is currently an Associate at LaSalle Investment Management, London. His research interests cover the modelling of urban and regional systems, property markets, spatial-temporal and economic networks and policy evaluation for planning in both transport and environmental governance.

Minette D'Lima is a researcher in the QASER (Quantitative and Applied Spatial Economics Research) Laboratory at University College London. She was trained as a pure mathematician with bachelors' degrees in Mathematics and Computer Technology, followed by a PhD in Algebraic Geometry. She works in a multidisciplinary group of mathematicians, physicists and economists providing innovative solutions to financial and economic problems. Her research covers a broad range of projects from complexity analysis and stochastic modelling to structuring portfolios in urban investments. She has been a researcher on an EPSRC Programme Grant, “SCALE: Small Changes Lead to Large Effects,” and developed a discrete spatial interaction model to study the effect of transport investments on urban space. She has developed a stochastic model for quantifying resilience on a FuturICT-sponsored project, “ANTS: Adaptive Networks for Complex Transport Systems.” She has also worked on developing a mathematical model using portfolio theory and agent-based modelling to simulate the agricultural supply chain in Uganda for a World Bank project, “Rethinking Logistics in Lagging Regions.” She is currently working in the EPSRC Programme Grant “Liveable Cities,” structuring and optimising portfolios for urban investments, and taking into account the socio-environmental impacts of such investments and their interactions.

Toby P. Davies is a Research Associate working on the Crime, Policing and Citizenship project at University College London, having previously been a member of the UCL SECReT Doctoral Training Centre. His background is in mathematics, and his work concerns the application of mathematical techniques in the analysis and modelling of crime and other security issues. His main area of interest is the spatio-temporal distribution of crime, in particular its relationship with urban form and its analysis using network-based methods.

Valerio de Martinis is Scientific Assistant at the Institute of Transport Planning and System (ETH Zurich). He is part of SCCER Mobility (Swiss Competence Center on Energy Research), and his research activities focus on energy efficiency and railway systems. He received his PhD in Transportation Systems in 2008 at the University of Naples Federico II.

Adam Dennett is a Lecturer in Urban Analytics in the Centre for Advanced Spatial Analysis at University College London. He is a geographer and fellow of the Royal Geographical Society and has worked for a number of years in the broad area of population geography, applying quantitative techniques to the understanding of human populations; much of this involves the use of spatial interaction models to understand the migration flows of people around the UK, Europe and the world. A former secondary school teacher, Adam arrived at UCL in 2010 after completing a PhD at the University of Leeds.

Robert J. Downes is a MacArthur Fellow in Nuclear Security working at the Centre for Science and Security Studies at the Department of War Studies, King's College London. Trained as a mathematician, Rob received his PhD in mathematics from University College London in 2014; he studied the interplay between geometry and spectral theory with applications to physical systems and gravitation. He also holds an MSci in Mathematics with Theoretical Physics awarded by UCL. As a Postdoctoral Research Associate on the ENFOLDing project at The Bartlett Centre for Advanced Spatial Analysis, Rob studied the structure and dynamics of global socio-economic systems using ideas from complexity science, with particular emphasis on national economic structure and development aid.

Hannah M. Fry is a Lecturer in the Mathematics of Cities at the Centre for Advanced Spatial Analysis (CASA). She was trained as a mathematician with a first degree in mathematics and theoretical physics, followed by a PhD in fluid dynamics. This technical background is now applied in the mathematical modelling of complex social and economic systems, her main research interest. These systems can take a variety of forms, from retail to riots and terrorism, and exist at various scales, from the individual to the urban, regional and global, but – more generally – they deal with the patterns that emerge in space and time.

Sean Hanna is Reader in Space and Adaptive Architectures at University College London, Director of the Bartlett Faculty of the Built Environment's MSc/MRes programmes in Adaptive Architecture and Computation, and Academic Director of UCL's Doctoral Training Centre in Virtual Environments, Imaging and Visualisation. He is a member of the UCL Space Group. His research is primarily in developing computational methods for dealing with complexity in design and the built environment, including the comparative modelling of space, and the use of machine learning and optimisation techniques for the design and fabrication of structures. He maintains close design industry collaboration with world-leading architects and engineers.

Shane D. Johnson is a Professor in the Department of Security and Crime Science at University College London. He has worked within the fields of criminology and forensic psychology for over 15 years, and has particular interests in complex systems, patterns of crime and insurgent activity, event forecasting and design against crime. He has published over 100 articles and book chapters.

Anthony Korte is a Research Associate in the Centre for Advanced Spatial Analysis at University College London, where he works on spatial interaction and input–output models relevant to the mathematical modelling of global trade dynamics.

Robert G. Levy is a researcher at the Centre for Advanced Spatial Analysis at University College London. He has a background in quantitative economics, database administration, coding and visualisation. His first love was Visual Basic but now writes Python and Javascript, with some R when there's no way to avoid it.

Elio Marchione is a Consultant for Ab Intio Software Corporation. Elio was Research Associate at the Centre for Advanced Spatial Analysis at University College London. He obtained his PhD at the University of Surrey at the Centre for Research in Social Simulation, MSc in Applied Mathematics at the University of Essex and MEng at the University of Naples. His current role consists, among others, in designing and building scalable architectures addressing parallelism, data integration, data repositories and analytics, while developing heavily parallel CPU-bound applications in a dynamic, high-volume environment. Elio's academic interests are in designing and/or modelling artificial societies or distributed intelligent systems enabled to produce novelty or emergent behaviour.

Francesca R. Medda is a Professor in Applied Economics and Finance at University College London. She is the Director of the UCL QASER (Quantitative and Applied Spatial Economics Research) Laboratory. Her research focusses on project finance, financial engineering and risk evaluation in different infrastructure sectors such as the maritime industry, energy innovation and new technologies, urban investments (smart cities), supply chain provision and optimisation and airport efficiency.

Thomas P. Oléron Evans is a Research Associate in the Centre for Advanced Spatial Analysis at University College London, where he has been working on the ENFOLDing project since 2011. In 2015, he completed a PhD in Mathematics, on the subject of individual-based modelling and game theory. He attained a Master's degree in Mathematics from Imperial College London in 2007, including one year studying at the École Normale Supérieure in Lyon, France. He is also an ambassador for the educational charity Teach First, having spent two years teaching mathematics at Bow School in East London, gaining a Postgraduate Certificate in Education from Canterbury Christ Church University in 2010.

Francesca Pagliara is Assistant Professor of Transport at the Polytechnic School and of the Basic Sciences of the University of Naples Federico II. During her PhD course, in 2000 she worked at David Simmonds Consultancy in Cambridge. In 2002 she worked at the Transport Studies Unit of the University of Oxford, and in 2006 she worked at the Institute for Transport Planning and Systems of ETH in Zurich. She was visiting professor at Transportation Research Group of the University of Southampton (2007 and 2009) and at TRANSyt of the University of Madrid (2007 and 2010). She had further research experience in 2013 in France, where she worked at the LVMT of the University of Paris-Est. She is author of academic books, both in Italian and in English, and of almost 100 papers. She has participated at several research projects.

Joan Serras is a Senior Research Associate in the Centre for Advanced Spatial Analysis (CASA) at University College London. He received his PhD in Engineering Design for Complex Transportation Systems from the Open University in 2007. While at CASA, he has been involved in three research grants: “SCALE: Small Changes Lead to Large Effects” (EPSRC), ENFOLDing (EPSRC) and EUNOIA (EU FP7). His research focusses on the development of tools to support decision making in urban planning, mainly in the transport sector.

Frank T. Smith FRS does research on social-interaction, industrial, biochemical and biomedical modelling, as Goldsmid Chair of applied mathematics at University College London. Author of over 300 refereed papers, he collaborates internationally, nationally and within London, and has taken part in many research programmes. Frank has contacts with government organisations, industry, commerce and NHS hospitals, with real-world applications ranging very widely and including consumer choice, social issues and city growth. Recent support has come from international and national bodies and companies. His applications-driven work deals with social applications to help understanding of crime, opinion dynamics, security strategies and hub development, as well as biomedical, biochemical and industrial applications. Frank tends to use modelling combined with analysis, computations and experimental or observational links throughout. He has been on many peer review panels, has contributed to several books is a long-standing Fellow of the Royal Society and is Director of the London Taught Course Centre for doctoral studies in the Mathematical Sciences.

Tasos Varoudis is a Senior Research Associate in the Bartlett School of Architecture at UCL. He is a registered architect, computer scientist, designer and technologist. He studied Architectural Engineering at the National Technical University of Athens and took his doctorate in Computing Engineering at Imperial College, London. His research ranges from architectural computation and the analysis of hybrid architectural spaces to architecture and human–computer interaction.

Sir Alan Wilson FBA FAcSS FRS is Professor of Urban and Regional Systems in the Centre for Advanced Spatial Analysis at University College London. He is Chair of the Home Office Science Advisory Council and of the Lead Expert Group for the GO-Science Foresight Project on the Future of Cities. He was responsible for the introduction of a number of model-building techniques which are now in common use – including ‘entropy’ in building spatial interaction models. His current research, supported by ESRC and EPSRC grants, is on the evolution of cities and global dynamics. He was one of two founding directors of GMAP Ltd in the 1990s – a successful university spin-out company. He was Vice-Chancellor of the University of Leeds from 1991 to 2004, when he became Director-General for Higher Education in the then DfES. From 2007 to 2013, he was Chair of the Arts and Humanities Research Council. He is a Fellow of the British Academy, the Academy of Social Sciences and the Royal Society. He was knighted in 2001 for services to higher education. His recent books include Knowledge power (2010), The Science of Cities and Regions, his five-volume (edited) Urban Modelling (both 2013) and (with Joel Dearden) Explorations in Urban and Regional Dynamics (2015).

I am grateful to the following publishers for permission to use material.