Modeling and Simulating Social Systems with MATLAB Lecture 5 - - PowerPoint PPT Presentation

2015-10-26 ETH Zürich

Modeling and Simulating Social Systems with MATLAB

Lecture 5 – Cellular Automata

Olivia Woolley, Stefano Balietti, Lloyd Sanders Computational Social Science (D GESS)

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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

• Implement an existing model form the scientific literature
• Run simulations that either use the dataset as input or

that reproduce certain aspects of the data

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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

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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)

4

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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

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Simulation Models

• Strength of (simulation) models?
• Computational laboratory: test how micro dynamics

lead to macro patters

• Experiments with computer models are particularly

useful in the social sciences, where experiments in the real world are typically not possible

• Suitable where analytical models fail or where

dynamics are too complex for analytical models

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Simulation Models

• Weakness of (simulation) models?
• The choice of model parameters and implementation

details can have a strong influence on the simulation

• utcome
• We can only model aspects of a system, i.e. the

models are necessarily incomplete and 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

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Simulation Models

• How to we deal with the known limitations?
• Simple models with only few parameters
• Test whether simulation results depend on details of

the implementation or model

• Validate the model mechanism with observations and

causal plausibility

• Use empirical data to evaluate the predictive power of

the simulation model

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Cellular Automaton (plural: Automata)

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Cellular Automaton (plural: Automata)

• A cellular automaton is a set of rules, defining

how the state of a cell in a grid is updated, depending on the states of its neighboring cells

• Cellular-automata simulations are discrete in

time and space

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Cellular Automaton

• The grid can have an arbitrary number of

dimensions:

1-dimensional cellular automaton 2-dimensional cellular automaton

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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

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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

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Game of Life

• Ni = Number of 1st order Moore neighbors to cell i

that are activated.

• For each cell i:
• 1. Deactivate active cell iff Ni <2 or Ni >3.
• 2. Activate inactive cell iff Ni =3

i

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Game of Life

• Ni = Number of 1st order Moore neighbors to cell i

that are activated.

• For each cell i:
• 1. Deactivate active cell if Ni <2 or Ni >3.
• 2. Activate inactive cell if Ni =3

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Game of Life

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

life

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Complex Patterns

17

• “Gosper Glider Gun”:

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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

• nly 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

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Surprising discoveries continue

• “The loafer” is a very small spaceship
• It is the slowest known orthogonal spaceship

(speed of c/7)

• It was discovered only in 2013

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Life in Life

20

• The Game of Life can be used to simulate the

Game of Life:

http://www.youtube.com/watch?v=xP5-iIeKXE8

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Life in Life

21

• The Game of Life can be used to calculate

everything (if a Turing machine can):

http://www.youtube.com/watch?v=My8AsV7bA94

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1-dimensional Cellular Automata

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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

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Highway Simulation

• Simple example of a 1-dimensional cellular

automaton

• Rules for each car at cell i:

1.

Stay: If the cell directly to the right is occupied.

2.

Move: Otherwise, move one step to the right, with probability p

Move to the next cell, with the probability p

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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)

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Highway Simulation

• Running the simulation is done like this:

simulate_cars(0.9, 0.2, true)

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Kermack-McKendrick Model

• In lecture 4, 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

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Kermack-McKendrick model

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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

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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

) ( ) ( ) ( ) ( ) ( ) ( t I dt dR t I t S t I dt dI t S t I dt dS         

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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

) ( ) ( ) ( ) ( ) ( ) ( t I dt dR t I t S t I dt dI t S t I dt dS         

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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

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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 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1

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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

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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.

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Cellular-Automaton Implementation

• Implementation of a 2-dimensional cellular

automaton in MATLAB can be done like this:

• The iteration over the cells can be done either

sequentially, in parallel, 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 …

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Cellular-automaton implementation

• Sequential update:

...

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Cellular-automaton implementation

• Parallel update (use a copy of the grid):

... ...

data from last iteration: working copy:

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Cellular-automaton implementation

• Random update:

...

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Cellular-automaton implementation

• Attention:
• Simulation results can be very sensitive to the type of

update used

• Random update is usually preferable for social

simulations (but also not always the best solution)

• Just be aware of this potential complication and

check your results for dependency on the updating scheme!

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Boundary Conditions

• The boundary conditions can be:
• Periodic: The grid is wrapped, so that what exits on
• ne side reappears at the other side of the grid.
• Fixed: Agents are not influenced by what happens at

the other side of a border.

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Boundary Conditions

• Periodic boundaries are usually preferable:

Fixed boundaries Periodic boundaries

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MATLAB Implementation of the Kermack- McKendrick Model

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MATLAB implementation Set parameter values

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MATLAB implementation Define grid, x

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MATLAB implementation Define neighborhood

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MATLAB implementation Main loop. Iterate the time variable, t

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

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MATLAB implementation For each cell i, j: Iterate

• ver the neighbors

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

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Aborting Execution

• When running large computations or animations, the

execution can be aborted by pressing Ctrl+C in the main window:

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References

• Wolfram, Stephen, A New Kind of Science. Wolfram

Media, Inc., May 14, 2002.

• http://www.conwaylife.com/wiki/
• Martin Gardner. The fantastic combinations of John

Conway's new solitaire game "life". Scientific American 223 (October 1970): 120-123.

• Schelling, Thomas C. (1971). Dynamic Models of
• Segregation. Journal of Mathematical Sociology 1:143-

186.

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Exercise 1

• Get draw_car.m and simulate_cars.m from

www.soms.ethz.ch/matlab or from https://github.com/msssm/lecture_files 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.

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Exercise 1b

• Generate a video of an interesting case in your

traffic simulation.

• We have uploaded an example file:

simulate_cars_video.m

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Exercise 2

• Download the file disease.m which is an

implementation of the Kermack-McKendrick model as a cellular automaton

• Plot the relative fractions
• f the states S, I, R, as

a function of time, and see if the curves look the same as for the implementation last class

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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?