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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

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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

)

(4)

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)

(5)

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

(6)

Probability of Spreading :

42% Probability of Spreading :

46%

Cellular Automata: Phase transitions

(7)

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

(8)

• 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

(9)

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 xa 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 xa x



  

 

    

 

  

where |Λ(x)| =the number of connections of each neuron (including itself) (here 5)

 

,

i,j i j

2

a x



 

 : Majority rule

(10)

 

, 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)

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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

(12)

Coarse-Grained Computations for Infection Spreading

Among Individuals

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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

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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

p

S->I

= 0.9

S=95%, I=5% S=90%, I=10%

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