Benevento, Spring 2011
Costas Siettos
School of Applied Mathematics and Physical Sciences, NTUA, Athens, Greece
Cellular Automata Models : Intro and Paradigms
Cellular Automata Models : Intro and Paradigms
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Fokker Planck εξίσωση
Fokker Planck εξίσωση
Micro-scale
Meso-scale
Macroscale
Fokker Planck
Moments
Spherical Harmonics Wavelets
ODE’s PDE’s
The Analysis and Control is usually sought at this level
Different time and space scales Macro scales much much bigger than the bigger Microscopic scale
Microscopic/
Stochastic models
Brownian Dynamics Monte Carlo Molecular Dynamics
Cellular Automata
A Big! number of available Microscopic/ Stochastic/ Models Simulating the Time Evolution of Real World- Complex Systems
(Fluid Mechanics, Material Science, Bio, Ecology, Process Engineering,…)
Motivation
Real Complex Problems: Fire-Spreading
Greece: Summer of 2007: -2x10 5 hectares of Forest Burned -74 Died
Island of Spetses, 1990: Burned the 1/3 of the Island (around 8 km
2)
Real Complex Problems: Fire-Spreading Modeling
Atomistic/Stochastic Models Like Cellular Automata
Can! Predict Large & Multiscale Complex Problems Evolution!
Agents:
Tree =f (Type, Density, Height,
Ground Slope)
A cell can take each time one of the three states:
• 1:Black,empty/burned
• 2:Green,trees.
• 3:Red: Fire
The update rule has as follows :
• The fire on a site will spread to the trees at its nearest-neighbor sites at the next time step with probability p.
• All trees on fire will burn down and return to empty sites at the next time step.
At time t+1 fire
With probability p
fire
At time t
fire
At time t
At time t+1 Empty
sites
Cellular Automata: A simplistic model
Probability of Spreading :
42% Probability of Spreading :
46%
Cellular Automata: Phase transitions
Coarse-Grained Computations and Control for Cellular Automata Models of Randomly Connected Individuals
Two CA Models
A. Network of Neurons
B. Infection Spreading among Individuals
• Each neuron is described by 2 states
• Every neuron has 4 links (5 with itself) which influence the fate of its state.
• The topology depends on the number of remote & local connections
Neurons Connections-Topology of the network
a(t)=1: activated a(t)=0: inactivated
Local connections
Remote connections
Determined by two functions:
• Arousal function s(x): the probability that an inactivated neuron becomes activated .
, i,j
if or 1 if not
i j 2
s x a x
The CA model: The Evolution of the network
• the depression function r(x): the probability that an activated Neuron becomes inactivated
, i,j
1 if or if not
i j 2
r x a x
where |Λ(x)| =the number of connections of each neuron (including itself) (here 5)
,
i,j i j
2
a x
: Majority rule
, i,j
3 5
2 2
i j
a x
, i,j
2 5
2 2
i j
a x
s(x)=ε
(if ε<<1: small probability to become activated)
s(x)=1-ε
The rules: an example
(bigger probability to become activated)
, i,j
3 5
2 2
i j
a x
r(x)=ε
(small probability to become inactivated)
, i,j
2 5
2 2
i j
a x
r(x)=1-ε
(bigger probability to become inactivated)
Temporal Simulations
Why two states? Bifurcation Analysis Transition rates? Rare Events Analysis
ε=0.215, 2 remote neighbors
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105 0.35
0.4 0.45 0.5 0.55 0.6 0.65 0.7
time
p
Coarse-Grained Computations for Infection Spreading
Among Individuals
• N number of individuals
• Each individual can be in one of the 3 following states
1: S usceptible : not yet infected;
probabilistic potential to be infected
2: I nfected
3: R ecovered; recovers from the infection;
immunized from infection
• He/ She interacts with 4 other
The CA model
Rules of Evolution:
• A susceptible gets infected with probability p S->I if one of his links is infected
• An infected recovers with probability p
I->R• A recovered becomes susceptible again with probability p R->S
otherwise has immunity
Temporal Simulations
0 10 20 30 40 50
0 0.2 0.4 0.6 0.8 1
time (days)
% o f th e po pu la tio n Susceptible Infected
0 5 10 15 20 25 30 35 40 45 50
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
time (days)
% o f th e po pu la tio n
Susceptible Infected