Benevento, Spring 2011

Costas Siettos

School of Applied Mathematics and Physical Sciences, NTUA, Athens, Greece

Cellular Automata Models : Intro and ParadigmsCellular Automata Models : Intro and Paradigms

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Fokker Planckεξίσωση

Fokker Planckεξίσωση

Micro-scale

Meso-scale

Macroscale

Fokker PlanckMoments

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/ ModelsSimulating 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: -2x105 hectares of Forest Burned-74 Died

Island of Spetses, 1990: Burned the 1/3 of the Island (around 8 km2)

Real Complex Problems: Fire-Spreading Modeling

Atomistic/Stochastic ModelsLike Cellular AutomataCan! 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.

fireAt time t+1

With probability pfire

At time t

fireAt 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 not2i j

xs x a

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 not2i j

xr x a

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

,

i,j

2i j

xa

: Majority rule

,

i,j

532 2i j

xa

,

i,j

522 2i j

xa

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

532 2i j

xa

r(x)=ε

(small probability to become inactivated)

,

i,j

522 2i j

xa

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

Local connections

Remote connectionsLocal

connections

Remote connections

Coarse-Grained Computations for Infection Spreading

Among Individuals

• N number of individuals• Each individual can be in one of the

3 following states

1:Susceptible : not yet infected; probabilistic potential to be infected

2:Infected

3:Recovered; 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 pS->I if one of his links is

infected• An infected recovers with probability pI->R

• A recovered becomes susceptible again with probability pR->Sotherwise has immunity

Temporal Simulations

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

time (days)

% o

f the

pop

ulat

ion

SusceptibleInfected

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time (days)

% o

f the

pop

ulat

ion

SusceptibleInfected

pS->I = 0.9S=95%, I=5% S=90%, I=10%