modeling and simulating social systems with matlab · 2016-10-17 modeling and simulating social...

2016-10-17ETH Zürich

Modeling and Simulating Social Systems with MATLAB

Lecture 4 – Cellular Automata

Chair of Computational Social Science

Olivia Woolley, Lloyd Sanders, Dirk Helbing

2016-10-17 Modeling and Simulating Social Systems with MATLAB 2


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 32 Network/systemic resilience


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


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


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


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. https://github.com/msssm/interesting_projects/wiki

Research methods you are planning to use


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

Deadline for submitting a proposal:

23 October 2016 (midnight)


Recap: Dynamical Systems

Dynamical systems are 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


Recap: 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)



Strengths 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)



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



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 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 iff Ni <2 or Ni >3.2. Activate inactive cell iff 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

22

“Gosper Glider Gun”:


Game of Life: Some Bits of History1970: 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 pattern 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


Life in Life

24

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

25

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


1-dimensional Cellular Automata


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


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

use the approach of cellular automata


Kermack-McKendrick model



The Kermack-McKendrick model is specified as:

S: Susceptible persons

I: Infected persons

R: Removed (immune)

persons

β: Infection rate

γ: Recovery rate





I: Infected persons

R: Removed (immune)

persons

β: Infection rate

γ: Recovery rate

dSdt

=−β I ( t )S ( t )

dIdt

= β I (t )S ( t )−γ I ( t )

dRdt

=γ I (t )





I: Infected persons

R: Removed (immune)

persons

β: Infection rate

γ: Immunity rate

dSdt

=−β I ( t )S ( t )

dIdt

= β I (t )S ( t )−γ I ( t )

dRdt

=γ I (t )





I: Infected persons

R: Removed (immune)

persons

β: Infection rate

γ: Recovery 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



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.




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


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:



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.


Aborting Execution

When running large computations or animations, the

execution can be aborted 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

http://www.coss.ethz.ch/education/matlab.html 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.

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


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 back to Susceptible

What changes?

modeling and simulating social systems with matlab · 2016-10-17 modeling and simulating social...

Documents