"This book lays out a vision for a coherent framework for understanding complex systems'' (from the foreword by J. Doyne Farmer). By developing the genuine idea of Brownian agents, the author combines concepts from informatics, such as multiagent systems, with approaches of statistical many-particle physics. This way, an efficient method for computer simulations of complex systems is developed which is also accessible to analytical investigations and quantitative predictions. The book demonstrates that Brownian agent models can be successfully applied in many different contexts, ranging from physicochemical pattern formation, to active motion and swarming in biological systems, to self-assembling of networks, evolutionary optimization, urban growth, economic agglomeration and even social systems.
Keywords: Agent Models, Biological Systems, Brownian Agents, Complex Systems, Economic Agglomeration, Networks, Social Systems, Trail Formation
1. | Complex Systems and Agent Models | 1 |
1.1 | Introduction to Agent-Based Modeling | 1 |
1.1.1 | The Micro-Macro Link | 1 |
1.1.2 | The Role of Computer Simulations | 3 |
1.1.3 | Agents and Multi-Agent Systems | 6 |
1.1.4 | Complex versus minimalistic agents | 10 |
1.1.5 | Agent Ecology | 13 |
1.1.6 | Simulation Approaches | 17 |
1.2 | Brownian Agents | 22 |
1.2.1 | Outline of the Concept | 22 |
1.2.2 | Interaction as Communication | 28 |
1.2.3 | A short survey of the book | 32 |
1.3 | Brownian Motion | 39 |
1.3.1 | Observations | 39 |
1.3.2 | Langevin Equation of Brownian Motion | 42 |
1.3.3 | Probability Density and Fokker-Planck Equation | 46 |
2. | Active Particles | 51 |
2.1 | Active Motion and Energy Consumption | 51 |
2.1.1 | Storage of Energy in an Internal Depot | 51 |
2.1.2 | Velocity-Dependent Friction | 54 |
2.1.3 | Active Motion of Cells | 56 |
2.1.4 | Pumping By Space-Dependent Friction | 60 |
2.2 | Active Motion in One-Dimensional Systems | 65 |
2.2.1 | Adiabatic Approximations and Stationary Solutions | 65 |
2.2.2 | Stationary Velocities and Critical Parameters for U=const. | 67 |
2.2.3 | Stationary Solutions for a linear Potential U=ax | 70 |
2.2.4 | Deterministic Motion in a Ratchet Potential | 75 |
2.2.5 | Investigation of the Net Current | 82 |
2.2.6 | Stochastic Influences on the Net Current | 86 |
2.2.7 | Directed Motion in a Ratchet Potential | 92 |
2.3 | Active Motion in Two-Dimensional Systems | 95 |
2.3.1 | Distribution Function for U=const. | 95 |
2.3.2 | Deterministic Motion in a Parabolic Potential | 101 |
2.3.3 | Analytical Solutions for Deterministic Limit Cycle Motion | 103 |
2.3.4 | Deterministic Chaotic Motion in the Presence of Obstacles | 108 |
2.3.5 | Stochastic Motion in a Parabolic Potential | 109 |
2.3.6 | Stochastic Motion with Localized Energy Sources | 111 |
2.4 | Swarming of Active Particles | 114 |
2.4.1 | Canonical-Dissipative Dynamics of Swarms | 114 |
2.4.2 | Harmonic Swarms | 119 |
2.4.3 | Coupling via Mean Momentum and Mean Angular Momentum | 126 |
3. | Aggregation and Physico-Chemical Structure Formation | 133 |
3.1 | Indirect Agent Interaction | 133 |
3.1.1 | Response to External Stimulation | 133 |
3.1.2 | Generation of the Effective Potential Field | 136 |
3.1.3 | Master Equations and Density Equations | 138 |
3.1.4 | Stochastic Simulation Technique | 141 |
3.2 | Aggregation of Brownian Agents | 145 |
3.2.1 | Chemotactic Response | 145 |
3.2.2 | Stability Analysis for Homogeneous Distributions | 146 |
3.2.3 | Estimation of an Effective Diffusion Coefficient | 151 |
3.2.4 | Competition of Spikes | 153 |
3.2.5 | Derivation of a Selection Equation | 156 |
3.2.6 | Comparison to Biological Aggregation | 159 |
3.3 | Pattern Formation in Reaction-Diffusion Systems | 164 |
3.3.1 | Coexistence of Spikes | 164 |
3.3.2 | Spiral Waves and Travelling Spots | 169 |
3.3.3 | Travelling Waves | 171 |
4. | Self-Organization of Networks | 175 |
4.1 | Agent-Based Model of Network Formation | 175 |
4.1.1 | Basic Assumptions and Equations of Motion | 175 |
4.1.2 | Results of Computer Simulations | 179 |
4.2 | Estimation of the Network Connectivity | 182 |
4.2.1 | Critical Temperature | 182 |
4.2.2 | Network Connectivity and Threshold | 186 |
4.2.3 | Numerical Results | 190 |
4.3 | Construction of a Dynamic Switch | 192 |
4.3.1 | Setup for the Switch | 192 |
4.3.2 | Simulations of the Dynamic Switch | 194 |
4.3.3 | Estimation of the Switch Delay | 198 |
5. | Tracks and Trail Formation in Biological Systems | 203 |
5.1 | Active Walker Models | 203 |
5.1.1 | Master Equation Approach to Active Walkers | 203 |
5.1.2 | Active Walker Models of Fractal Growth Patterns | 206 |
5.1.3 | Active Walker Models of Bacterial Growth | 208 |
5.2 | Discrete Model of Track Formation | 212 |
5.2.1 | Biased Random Walks | 212 |
5.2.2 | Reinforced Biased Random Walks | 217 |
5.2.3 | Formation of Tracks | 221 |
5.3 | Track Formation and Aggregation in Myxo-bacteria | 225 |
5.3.1 | Modification of the Active Walker Model | 225 |
5.3.2 | Simulation of Myxobacteria Aggregation | 228 |
5.4 | Trunk Trail Formation of Ants | 232 |
5.4.1 | Biological Observations | 232 |
5.4.2 | Active Walker Model of Trail Formation in Ants | 235 |
5.4.3 | Simulation of Trunk Trail Formation in Ants | 240 |
6. | Movement and Trail Formation of Pedestrians | 247 |
6.1 | Movement of Pedestrians | 247 |
6.1.1 | The Social Force Model | 247 |
6.1.2 | Simulation of Pedestrian Motion | 249 |
6.2 | Trail Formation of Pedestrians | 251 |
6.2.1 | Model of Trail Formation | 251 |
6.2.2 | Human Trail Formation | 255 |
6.2.3 | Simulation of Pedestrian Trail Systems | 258 |
6.2.4 | Macroscopic Equations of Trail Formation | 261 |
7. | Evolutionary Optimization Using Brownian Searchers | 267 |
7.1 | Evolutionary Optimization Strategies | 267 |
7.1.1 | Ensemble Search with Brownian Agents | 267 |
7.1.2 | Boltzmann Strategy and Darwin Strategy | 270 |
7.1.3 | Mixed Boltzmann--Darwin Strategy | 275 |
7.2 | Evaluation and Optimization of Road Networks | 279 |
7.2.1 | Road Networks | 279 |
7.2.2 | The Evaluation Function | 281 |
7.2.3 | Results of Computer Simulations | 284 |
7.3 | Asymptotic Results on the Optimization Landscape | 289 |
7.3.1 | Optimization Values in the Asymptotic Limit | 289 |
7.3.2 | Density of States in the Asymptotic Limit | 291 |
8. | Analysis and Simulation of Urban Aggregation | 295 |
8.1 | Spatial Structure of Urban Aggregates | 295 |
8.1.1 | Urban Growth and Population Distribution | 295 |
8.1.2 | Mass Distribution of Urban Aggregates: Berlin | 300 |
8.1.3 | Fractal Properties of Urban Aggregates | 304 |
8.2 | Rank-Size Distribution of Urban Aggregates | 307 |
8.2.1 | Analysis of the Rank-Size Distribution | 307 |
8.2.2 | Master Equation Approach to Urban Growth | 309 |
8.2.3 | Simulation of the Rank-Size Distribution: Berlin | 312 |
8.2.4 | Forecast of the Future Evolution: Daegu | 315 |
8.3 | Kinetic Models of Urban Growth | 318 |
8.3.1 | Fractal Growth and Correlated Growth Models | 318 |
8.3.2 | Shift of Growth Zones | 322 |
8.3.3 | Simulating Urban Growth with Brownian Agents | 326 |
8.3.4 | Results of Computer Simulations: Berlin | 329 |
9. | Economic Agglomeration | 335 |
9.1 | Migration and Agglomeration of Workers | 335 |
9.1.1 | Spatial Economic Patterns | 335 |
9.1.2 | Model Equations for Migration and Employment | 337 |
9.1.3 | Derivation of a Competition Dynamics | 341 |
9.2 | Dynamic Model of Economic Concentration | 345 |
9.2.1 | Production Function and Transition Rates | 345 |
9.2.2 | Simulation of Spatial Economic Agglomeration | 350 |
10. | Spatial Opinion Structures in Social Systems | 357 |
10.1 | Quantitative Sociodynamics | 357 |
10.1.1 | Socioconfiguration | 357 |
10.1.2 | Stochastic Changes and Transition Rates | 359 |
10.2 | Collective Opinion Formation of Brownian Agents | 363 |
10.2.1 | Dynamic Equations | 363 |
10.2.2 | Subpopulation Sizes in a System with Fast Communication | 366 |
10.2.3 | Influence of External Support | 369 |
10.2.4 | Critical Conditions for Spatial Opinion Separation | 371 |
10.3 | Spatial Opinion Patterns in a Model of Direct Interactions | 375 |
10.3.1 | Transition Rates and Mean Value Equations | 375 |
10.3.2 | Stationary Solutions for a Single Box | 379 |
10.3.3 | Results of Computer Simulations | 382 |
Bibliography | 387 | |
Index | 415 |