Modeling and Simulating Social Systems with …2013-10-14 Modeling and Simulating Social Systems...

2013-10-14© ETH Zürich |

Modeling and Simulating Social Systems with MATLAB

Lecture 4 – Cellular Automata

© ETH Zürich |

Chair of Sociology, in particular of

Modeling and Simulation

Tobias Kuhn and Olivia Woolley

2013-10-14 Modeling and Simulating Social Systems with MATLAB 22

Schedule of the courseIntroduction to MATLAB

Introduction to social-science modeling and simulations

Working on projects (seminar thesis)

Handing in seminar thesis and giving a presentation

Create and Submit a Research Plan

DEADLINE: 21.10.

23.09.

30.09.

07.10.

14.10.

21.10.

28.10.

04.11.

11.11.

18.11.

25.11.

02.12.

09.12.

16.12.


Projects – Suggested Topics1 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


Repetition Dynamical systems described by a set of differential

equations (example: Lotka-Volterra)

Numerical solutions can be calculated iteratively using the Euler Method (example: Kermack-McKendrick)

The values and ranges of parameters critically matter for the system dynamics (Example 2, epidemiological threshold)

Time resolution in Euler 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


Simulation Models

What do we mean when we speak of

(simulation) models? Formalized (computational) representation of social,

economic etc. dynamics Reduction of complexity, i.e. highly simplified

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? 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? 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? 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.

Cellular-automata simulations are discrete both

in time and space.


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

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:1. Deactivate active cell if Ni <2 or Ni >3.2. Activate inactive cell if Ni =3

i


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


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

life


Complex Patterns

17

“Gosper Glider Gun”:


Life in Life

18

The Game of Life can be used to simulate the

Game of Life:

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

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


Life in Life

19

The Game of Life can be used to calculate

everything (if a Turing machine can):

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

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


Cellular Automata: A New Kind of Science?

A thorough study of

all 256 “behavioral”

rules of a 1-D cellular

automata.


Cellular Automata: A New Kind of Science?


Cellular Automaton: Principles

1. Self Organization: Patterns appear without a

designer

2. Emergence: High-level phenomena can

appear such as gliders, glider guns, etc.

3. Complexity: Simple rules produce complex

phenomena


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


Highway Simulation

We have prepared some files for the highway

simulations: draw_car.m : Draws a car, with the functiondraw_car(x0, y0, w, h)

simulate_cars.m: Runs the simulation, with the functionsimulate_cars(moveProb, inFlow, withGraphics)


Highway Simulation

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


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.



The Kermack-McKendrick model is specified as:

S: Susceptible persons

I: Infected persons

R: Removed (immune)

persons

β: Infection rate

γ: Immunity rate

S

IR

β transmission

γrecovery





I: Infected persons

R: Removed (immune)

persons

β: Infection rate

γ: Immunity rate





I: Infected persons

R: Removed (immune)

persons

β: Infection rate

γ: Immunity rate )(

)( )()(

)()(

tIdt

dR

tItStIdt

dI

tStIdt

dS

γ

γβ

β

=

−=

−=





I: Infected persons

R: Removed (immune)

persons

β: Infection rate

γ: Immunity rate )(

)( )()(

)()(

tIdt

dR

tItStIdt

dI

tStIdt

dS

γ

γβ

β

=

−=

−=





I: Infected persons

R: Removed (immune)

persons

β: Infection rate

γ: Immunity rate




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

0 0 0 0 0 0 0

1 1 1 0 0 0 0

1 1 1 1 0 0 0

2 1 1 1 0 0 0

2 1 1 1 0 0 0

1 1 1 0 0 0 0

1 0 0 0 0 0 0




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



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

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, tIterate over all cells, i=1..N, j=1..N

Iterate over all neighbors, k=1..M

End k-iterationEnd i-iteration

End t-iteration

…


Cellular-Automaton Implementation

Sequential update:


Cellular-automaton implementation

Sequential update:



Sequential update:



Sequential update:



Sequential update:



Sequential update:



Random update:



Random update:



Random update:



Random update:



Random update:



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!


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

Set parameter values



Define grid, x



Define neighborhood



Main loop. Iterate the time variable, t



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



For each cell i, j: Iterate over the neighbors



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


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.

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.

http://www.conwaylife.com/wiki/


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.

http://www.soms.ethz.ch/matlab

https://github.com/msssm/lecture_files


Exercise 1b

Generate a video of an interesting case in your

traffic simulation.

We have uploaded an example file,

simulate_cars_video.m, and

simulate_cars_video_w7.m


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 individualsto change state back to Susceptible.

What changes?

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