1. 851-0585-04L – Modeling and Simulating Social Systems with MATLAB
L – Modeling and Simulating Social Systems with MATLAB
Lecture 7 – Continuous Simulations
Giovanni Luca Ciampaglia, Stefano Balietti and Karsten Donnay
Chair of Sociology, in particular of Modeling and Simulation
© ETH Zürich |
© ETH Zürich |

2. Lesson 7 - Contents Micro-macro link Some basis of motion dynamics
Lesson 7 - Contents
Micro-macro link
Some basis of motion dynamics
Pedestrian simulation

3. Drunkard’s Walk
The drunkard takes a series of steps away from the lamp post, each with a random angle.
(Figure Resource: Sethna JP, 2006)

4. Random Walk and Diffusion Equation
Consider a general, uncorrelated random walk where at each time step the particle’s position changes by a step :
The probability distribution for each step is , which has zero mean and standard deviation .

5. For the particle to go from at time to at time , the step
For the particle to go from at time to at time , the step Such event occurs with probability
times the probability density Therefore, we have

6. The Micro-Macro Link Using Taylor expansion, we have
The following is the diffusion equation

7. The Continuous Assumption
Resources: Google

8. From discrete to continuous space
Cellular automaton (CA)
Continuous simulation

9. Modelling motion in continuous space
Agents are characterized by
position
X (m)
Y (m)
velocity
Vx (m/s)
Vy (m/s) Vy y x Vx

10. Modelling motion in continuous space
Next position after 1 second is given by:
x(t+1) = x(t) + Vx(t)
y(t+1) = y(t) + Vy(t) Vy
Time t+1 y
Time t x Vx

11. Modelling motion in continuous space
Time step dt is often different from 1:
x(t+dt) = x(t) + dt Vx(t)
y(t+dt) = y(t) + dt Vy(t)
Time t+dt dt>1 Vy y
Time t+dt dt<1
Time t x Vx

12. Modelling motion in continuous space
The velocity is also updated in time. y x

13. Modelling motion in continuous space
The velocity changes as a result of inter-individual interactions
Repulsion
Attraction
Examples: atoms, planets, pedestrians, animals swarms…

14. Modelling motion in continuous space
The changes of velocity vector is defined by an acceleration.

15. Modelling motion in continuous space
The velocity changes as a result of interactions with the environment
Repulsion
Attraction
Examples: bacteria, ants, pedestrian trails …

16. Modelling motion in continuous space
The velocity changes as a result of a random process
With or without bias
Example: exploration behaviour in many species of animals

17. Modelling motion in continuous space
How to define a random move?
Choose a random angle in a normal distribution
Rotate the velocity by this angle
Variance=1
In Matlab:
newAngle = randn()
Mean=0

18. Modelling motion in continuous space
How to define a random move?
Choose a random angle in a normal distribution
Rotate the velocity by this angle
In Matlab:
meanVal=2;
v=0.1;
newAngle = meanVal + v*randn()

19. Modelling motion in continuous space
How to define a BIASED random move?
Unbalance the distribution toward positive or negative values
In Matlab:
meanVal=2;
v=0.1;
newAngle = meanVal + v*randn()

20. Programming the simulator
Initialization: Four elements are required to define the state of individuals
X = [1 1 3 ];
Y = [2 1 1];
Vx = [ ];
Vy = [ ];
For t=1:dt:T
For p=1:N
[Vx(p) Vy(p)] = update( …) ;
X(p) = X(p) + dt*Vx(p) ;
Y(p) = Y(p) + dt*Vy(p) ;
end
The velocity is updated first, and then the position as a function of the velocity and the time step

21. Brownian Motion %Brownian Motion t = 100;n = 1000;dt=t/n;
x = zeros(1,n);y = zeros(1,n);
dx = zeros(1,n);dy = zeros(1,n);
x(1) = 0;y(1) = 0;
hold on;
k = 1;
for j=2:n
x(j) = x(j-1) + sqrt(dt)*randn;
y(j) = y(j-1) + sqrt(dt)*randn;
plot([x(j-1),x(j)],[y(j-1),y(j)]);
M(k) = getframe;
k = k + 1;
end 21

22. Pedestrian Simulation
CA model
Many-particle model

23. CA Model (A. Kirchner, A. Schadschneider, Phys. A, 2002)

24. Many-particle Model (Helbing, Nature, 2000)
Desired velocity
Repulsive force