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
-
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%
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