L – Modeling and Simulating Social Systems with MATLAB

851-0585-04L – Modeling and Simulating Social Systems with MATLAB
L – Modeling and Simulating Social Systems with MATLAB Lecture 4 – Cellular Automata Karsten Donnay and Stefano Balietti Chair of Sociology, in particular of Modeling and Simulation © ETH Zürich | © ETH Zürich |

Schedule of the course Introduction to MATLAB
Schedule of the course Introduction to MATLAB 26.09. 03.10. 10.10. 17.10. 24.10. 31.10. 07.11. 14.11. 21.11. 28.11. 05.12. 12.12. 19.12. Create and Submit a Research Plan DEADLINE: Introduction to social-science modeling and simulations Working on projects (seminar thesis) Handing in seminar thesis and giving a presentation

Projects – Suggested Topics
Projects – Suggested Topics 1 Traffic Dynamics 9 Evacuation Bottleneck 17 Self-organized Criticality 25 Facebook 2 Civil Violence 10 Friendship Network Formation 18 Social Networks Evolution 26 Sequential Investment Game 3 Collective Behavior 11 Innovation Diffusion 19 Task Allocation & Division of Labor 27 Modeling the Peer Review System 4 Disaster Spreading 12 Interstate Conflict 20 Artificial Financial Markets 28 Modeling Science 5 Emergence of Conventions 13 Language Formation 21 Desert Ant Behavior 29 Simulation of Networks in Science 6 Emergence of Cooperation 14 Learning 22 Trail Formation 30 Opinion Formation in Science 7 Emergence of Culture 15 Opinion Formation 23 Wikipedia 31 Organizational Learning 8 Emergence of Values 16 Pedestrian Dynamics 24 Social Contagion 3 3

Goals of Lecture 4: students will
Consolidate their knowledge of dynamical systems, through brief repetition of the main concepts and revision of the exercises. Get familiar with how and why simulation models may contribute to our understanding of complex socio-dynamic processes. Understand the important concept of Cellular Automata as discrete representations of interactions on an abstract grid (or configuration space). Implement simple Cellular Automata in MATLAB (Game of Life, Highway Simulation, Epidemics: Kermack- McKendrick model revisited)

Repetition dynamical systems described by a set of differential equations (example: Lotka-Volterra) numerical solutions iteratively for instances using 1st Euler’s Method (example: Kermack-McKendrick) the values and ranges of parameters critically matter; they determine which dynamics the model represents (Ex. 2, the ratio of recuperation to infection parameter determines the epidemiological threshold) time resolution in Euler Method must be sufficiently high to capture (‘fast’) system dynamics (Ex. 3) not all MATLAB-own ODE solvers work equally well for every dynamical system under consideration (Ex. 3)

Simulation Models What do we mean when we speak of (simulation) models? Formalized (computational) representation of social, economic etc. dynamics Implies reduction of complexity, i.e. usually making (strong) simplifying assumptions Not trying to reproduce reality but rather systematize specific interdependencies of “real” systems Formal framework to test and evaluate causal hypotheses against empirical data (or stylized facts)

Simulation Models Strength of (simulation) models?
Simulation Models Strength of (simulation) models? Computational laboratory: test how micro dynamics lead to macro patters There are usually no experiments in social sciences but computer models can be a “testing ground” Particularly suitable where formal models fail (or where dynamics are too complex for formal modeling) Possible to combine empirical input with quantitative validation of both the results and mechanisms

Simulation Models Weakness of (simulation) models?
Simulation Models Weakness of (simulation) models? The choice of model parameters, implementation etc. may strongly influence the simulation outcome We can only model aspects of a system, i.e. the models are necessarily incomplete & reductionist More complex models are NOT necessarily better: dynamics between model components often poorly understood (a known problem in climate modeling) Can construct a computational model of virtually anything but hard to verify if you are actually studying a realistic empirical question!

Simulation Models How to we deal with known limitations?
Simulation Models How to we deal with known limitations? Use empirical input and formal optimization to rule out arbitrariness of the model Test for implementation- and specification- dependence of simulations Validate the model mechanism both with observations and causal theory Use empirical data to evaluate the predictive power of the simulation model

Cellular Automaton (plural: Automata)
Cellular Automaton (plural: Automata) A cellular automaton is a rule, defining how the state of a cell in a grid is updated, depending on the states of its neighbor cells. May be represented as grids with arbitrary dimension. Cellular-automata simulations are discrete both in time and space.

Cellular Automaton (plural: Automata)
Cellular Automaton (plural: Automata) Simplest example for a (reduced) representation of an interaction model Automata dimensions represent an abstract “distance” that could be spatial, social relations etc. Natural micro-to-macro link from interactions between cells to patterns visible in the automaton

Cellular Automaton The grid can have an arbitrary number of dimensions: 1-dimensional cellular automaton 2-dimensional cellular automaton

Moore Neighborhood The cells are interacting with each neighbor cells, and the neighborhood can be defined in different ways, e.g. the Moore neighborhood: 1st order Moore neighborhood 2nd order Moore neighborhood

Von-Neumann Neighborhood
Von-Neumann Neighborhood The cells are interacting with each neighbor cells, and the neighborhood can be defined in different ways, e.g. the Von-Neumann neighborhood: 1st order Von-Neumann neighborhood 2nd order Von-Neumann neighborhood

Game of Life Ni = Number of 1st order Moore neighbors to cell i that are activated. For each cell i: Deactivate: If Ni <2 or Ni >3. Activate: if cell i is deactivated and Ni =3

Highway Simulation Simple example of a 1-dimensional cellular automaton Rules for each car at cell i: Stay: If the cell directly to the right is occupied. Move: Otherwise, move one step to the right, with probability p Move to the next cell, with the probability p

Highway Simulation We have prepared some files for the highway simulations: draw_car.m : Draws a car, with the function draw_car(x0, y0, w, h) simulate_cars.m: Runs the simulation, with the function simulate_cars(moveProb, inFlow, withGraphics)

Highway Simulation Running the simulation is done like this: simulate_cars(0.9, 0.2, true)

Kermack-McKendrick Model
Kermack-McKendrick Model In lecture 3, we introduced the Kermack- McKendrick model, used for simulating disease spreading. We will now implement the model again, but this time instead of using differential equations we define it within the cellular-automata framework.

Kermack-McKendrick Model The Kermack-McKendrick model is specified as: S: Susceptible persons I: Infected persons R: Removed (immune) persons β: Infection rate γ: Immunity rate S β transmission R I γ recovery

Kermack-McKendrick Model The Kermack-McKendrick model is specified as: S: Susceptible persons I: Infected persons R: Removed (immune) persons β: Infection rate γ: Immunity rate

Kermack-McKendrick Model The Kermack-McKendrick model is specified as: For the MATLAB implementation, we need to decode the states {S, I, R}={0, 1, 2} in a matrix x. S I R 1 2

Kermack-McKendrick Model The Kermack-McKendrick model is specified as: We now define a 2-dimensional cellular- automaton, by defining a grid (matrix) x, where each of the cells is in one of the states: 0: Susceptible 1: Infected 2: Recovered

Kermack-McKendrick Model Define microscopic rules for Kermack- McKendrick model: In every time step, the cells can change states according to: A Susceptible individual can be infected by an Infected neighbor with probability β, i.e. State 0 -> 1, with probability β. An individual can recover from an infection with probability γ, i.e. State 1 -> 2, with probability γ.

Cellular-Automaton Implementation
Cellular-Automaton Implementation Implementation of a 2-dimensional cellular- automaton model in MATLAB is done like this: The iteration over the cells can be done either sequentially, or randomly. Iterate the time variable, t Iterate over all cells, i=1..N, j=1..N Iterate over all neighbors, k=1..M … End k-iteration End i-iteration End t-iteration

Cellular-Automaton Implementation
Cellular-Automaton Implementation Sequential update:

Cellular-automaton implementation
Cellular-automaton implementation Sequential update:

Cellular-automaton implementation Random update:

Cellular-automaton implementation Attention: Simulation results may be sensitive to the type of update used! Random update is usually preferable but also not always the best solution Just be aware of this potential complication and check your results for dependency on the updating scheme!

Boundary Conditions The boundary conditions can be any of the following: Periodic: The grid is wrapped, so that what exits on one side reappears at the other side of the grid. Fixed: Agents are not influenced by what happens at the other side of a border.

Boundary Conditions The boundary conditions can be any of the following: Fixed boundaries Periodic boundaries

MATLAB Implementation of the Kermack-McKendrick Model
MATLAB Implementation of the Kermack-McKendrick Model

MATLAB implementation
Set parameter values MATLAB implementation

Define grid, x MATLAB implementation

Define neighborhood MATLAB implementation

Main loop. Iterate the time variable, t MATLAB implementation

Iterate over all cells, i=1..N, j=1..N MATLAB implementation

For each cell i, j: Iterate over the neighbors MATLAB implementation

The model, i.e. updating rule goes here. MATLAB implementation

Breaking Execution When running large computations or animations, the execution can be stopped by pressing Ctrl+C in the main window:

References Wolfram, Stephen, A New Kind of Science. Wolfram Media, Inc., May 14, 2002. potsdam.de/HyFISCH/Produzieren/lis_projekt/pr oj_gamelife/ConwayScientificAmerican.htm Schelling, Thomas C. (1971). Dynamic Models of Segregation. Journal of Mathematical Sociology 1:

Exercise 1 Download the files draw_car.m and simulate_cars.m from the course web page, Investigate how the flow (moving vehicles per time step) depends on the density (occupancy 0%..100%) in the simulator. This relation is called the fundamental diagram in transportation engineering.

Exercise 1b Generate a video of an interesting case in your traffic simulation. We have uploaded an example file, simulate_cars_video.m, to the website.

Exercise 2 Download the file disease.m which is an implementation of the Kermack-McKendrick model as a Cellular Automaton. Plot the relative fractions of the states S, I, R, as a function of time, and see if the curves look the same as for the implementation last class

Exercise 3 Modify the model Kermack-McKendrick model in the following ways: Change from the 1st order Moore neighborhood to a 2nd and 3rd order Moore neighborhood. Make it possible for Removed individuals to change state back to Susceptible. What changes?

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