Modeling and Simulating Social Systems with MATLAB

Transcription

1 Modeling and Simulating Social Systems with MATLAB Lecture 4 Cellular Automata Olivia Woolley, Tobias Kuhn, Dario Biasini, Dirk Helbing Chair of Sociology, in particular of Modeling and Simulation ETH Zürich

2 Schedule of the course Introduction to MATLAB Working on projects (seminar thesis) Introduction to social-science modeling and simulations Handing in seminar thesis and giving a presentation canceled 2

3 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

4 Seminar thesis Studying a scientific paper and/or a data set Reproducing and extending results / running a simulation in MATLAB Writing a report and giving a talk 4

5 Strategies for Research Projects Data project : Choose a dataset for which you have an interesting question Come up with a research question that can be tested with that data Run simulations that either use the dataset as input or that reproduce certain aspects of the data Look for scientific publications on the topic to place your project in the context of existing work 5

6 Strategies for Research Projects Paper project : Choose a scientific paper (or several papers) on a modeling and simulation approach that looks interesting Come up with a research question that builds upon the paper but is different from their research questions Extend, simplify, modify, and/or combine existing models for testing your research question Run simulations for your model and compare the results to the ones for the original model 6

7 Research Plan Structure 1-2 (not more!) pages stating: Brief, general introduction to the problem How you represent the problem with a model Research questions you want to try to answer Existing literature and previous projects you will base your model on and possible extensions. Research methods you are planning to use 7

8 Research Plan Upon submission, the Research Plan can: be accepted be accepted under revision be modified later on only if change is justified (create new version) Talk to us if you are not sure Final deadline for signing-up for a project is: 17 March 2014 (midnight) 8

9 Repetition Dynamical systems described by a set of differential equations; numerical solutions can be calculated iteratively using Euler's method (examples: Lotka-Volterra and Kermack-McKendrick) The values and ranges of parameters critically matter for the system dynamics (Example 2, epidemiological threshold) Time resolution in Euler's method must be sufficiently high to capture fast system dynamics (Example 3) MATLAB provides the commands ode23 and ode45 for solving differential equations (Example 3) 9

10 Simulation Models What do we mean when we speak of (simulation) models? Formalized (computational) representation of social (or other kinds of) dynamics Reduction of complexity, i.e. highly simplifying assumptions The goal is not to reproduce reality in general (only very specific aspects of it) Formal framework to test and evaluate causal hypotheses against empirical data (or stylized facts) 10

11 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 analytical models fail (or where dynamics are too complex for analytical modeling) 11

12 Simulation Models Weakness of (simulation) models? The choice of model parameters and implementation details can have a strong influence on the simulation outcome We can only model aspects of a system, i.e. the models are necessarily incomplete & reductionist More complex models are generally not better (only when simplification is not possible) It can be hard to relate simulation results to a realistic and relevant empirical question 12

13 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 specificationdependence of simulations Validate the model mechanism both with observations and causal theory Use empirical data to evaluate the predictive power of the simulation model 13

14 Cellular Automaton (plural: Automata) 14

15 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 Cellular-automata simulations are discrete in time and space 15

16 Cellular Automaton The grid can have an arbitrary number of dimensions: 1-dimensional cellular automaton 2-dimensional cellular automaton 16

17 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 17

18 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 18

19 Game of Life Ni = Number of 1st order Moore neighbors to cell i that are activated. For each cell i: Deactivate active cell iff Ni <2 or Ni >3. Activate inactive cell iff Ni =3 i 19

20 Game of Life Ni = Number of 1st order Moore neighbors to cell i that are activated. For each cell i: Deactivate active cell if Ni <2 or Ni >3. Activate inactive cell if Ni =3 20

21 Game of Life Want to see the Game of Life in action? Type: life 21

22 Complex Patterns Gosper Glider Gun : 22

23 Game of Life: Some Bits of History 1970: Conway presents the Game of Life; discovery of the glider (speed c/4) and the standard spaceships (speed c/2); discovery of the Gosper glider gun (first pattern found with indefinite growth) 1971: First garden of eden (pattern that has no parent pattern and can only occur at generation 0) was discovered; first patter with quadratic growth (breeder) 1989: Discovery of the first spaceships with velocities c/3 and c/4 1996: Game of Life simulated in Game of Life 2000: A Turing machine was built in Game of Life 2004: Construction of the largest interesting pattern so far: caterpillar spaceship with speed 17c/45 consisting of 11,880,063 alive cells 2013: The first complete replicator (a pattern that produces an exact copy of itself) was built 2014: Game of Life wiki contains more than 3000 interesting patterns 23

24 Life in Life The Game of Life can be used to simulate the Game of Life: 24

25 Life in Life The Game of Life can be used to calculate everything (if a Turing machine can): 25

26 1-dimensional Cellular Automata 26

27 Cellular Automaton: Principles 1. Self Organization: Patterns appear from random start patterns 2. Emergence: High-level phenomena can appear such as gliders, glider guns, etc. 3. Complexity: Simple rules produce complex phenomena 27

28 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 28

29 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) 29

30 Highway Simulation Running the simulation is done like this: simulate_cars(0.9, 0.2, true) 30

31 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 use the approach of cellular automata 31

32 Kermack-McKendrick model 32

33 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 33

34 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 ds = β I (t ) S (t ) dt di = β I (t ) S (t ) γ I (t ) dt dr = γ I (t ) dt 34

35 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 ds = β I (t ) S (t ) dt di = β I (t ) S (t ) γ I (t ) dt dr = γ I (t ) dt 35

36 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 36

37 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 S S S S S S I I I S S S S I I I I S S S R I I I S S S R I I I S S S I I I S S S S I S S S S S S

38 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 38

39 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. 39

40 Cellular-Automaton Implementation Implementation of a 2-dimensional cellular automaton model in MATLAB is done like this: 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 The iteration over the cells can be done either sequentially, or randomly. 40

41 Cellular-Automaton Implementation Sequential update: 41

42 Cellular-automaton implementation Sequential update: 42

43 Cellular-automaton implementation Sequential update: 43

44 Cellular-automaton implementation Sequential update: 44

45 Cellular-automaton implementation Sequential update: 45

46 Cellular-automaton implementation Sequential update: 46

47 Cellular-automaton implementation Random update: 47

48 Cellular-automaton implementation Random update: 48

49 Cellular-automaton implementation Random update: 49

50 Cellular-automaton implementation Random update: 50

51 Cellular-automaton implementation Random update: 51

52 Cellular-automaton implementation Attention: Simulation results can be very 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! 52

53 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. 53

54 Boundary Conditions The boundary conditions can be any of the following: Fixed boundaries Periodic boundaries 54

55 MATLAB Implementation of the Kermack-McKendrick Model 55

56 Set parameter values MATLAB implementation 56

57 Define grid, x MATLAB implementation 57

58 Define neighborhood MATLAB implementation 58

59 Main loop. Iterate the time variable, t MATLAB implementation 59

60 Iterate over all cells, i=1..n, j=1..n MATLAB implementation 60

61 For each cell i, j: Iterate over the neighbors MATLAB implementation 61

62 The model, i.e. updating rule goes here. MATLAB implementation 62

63 Aborting Execution When running large computations or animations, the execution can be aborted by pressing Ctrl+C in the main window: 63

64 References Wolfram, Stephen, A New Kind of Science. Wolfram Media, Inc., May 14, Martin Gardner. The fantastic combinations of John Conway's new solitaire game "life". Scientific American 223 (October 1970): Schelling, Thomas C. (1971). Dynamic Models of Segregation. Journal of Mathematical Sociology 1:

65 Exercise 1 Get draw_car.m and simulate_cars.m from or from 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. 65

66 Exercise 1b Generate a video of an interesting case in your traffic simulation. We have uploaded an example file: simulate_cars_video.m 66

67 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 67

68 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 back to Susceptible What changes? 68