Simulating percolation models Guillermo Amaral Caesar Systems Argentina Guillermo Amaral 2 Guillermo Amaral 3 Guillermo Amaral 4 A virtual lab Guillermo Amaral 5 Percolation deals with ID: 598944
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Slide1
Percolation
Simulating percolation models
Guillermo
Amaral
Caesar
Systems
- ArgentinaSlide2
Guillermo Amaral
2Slide3
Guillermo Amaral
3Slide4
Guillermo Amaral
4Slide5
A virtual lab
Guillermo Amaral
5Slide6
Percolation deals with…Slide7
Propagation
of
diseases
Guillermo Amaral
7Slide8
Propagation
of
fire
Guillermo Amaral
8Slide9
Oil
& gas in
reservoirs
Guillermo Amaral
9Slide10
Gelation
&
Polymerization
Guillermo Amaral
10Slide11
The problemSlide12
Original problem (Broadbent -
Hammersley
, 1957)
Guillermo
Amaral
12
What is the
probability
that the water reaches the center of the rock?Slide13
The simulationSlide14
The mathematical modelSlide15
The simplest model
Guillermo Amaral
15
v
ϵ
ℤ
2
v
u
at
distance
1
from
v
u
v
P(
e
“open”)
=
p
P(
e
“close”)
= 1 -
p
e
Open
path
from
u
to
v
v
u
Percolating
cluster
Open
cluster
from
v
vSlide16
Dimensions
3-D
n-D…
2-D
Element
being
open/
close
Bond
Site
Both
…
Structure
Square
Bow-tie
Hexagonal
Kagomé
Other
…
Model types
Guillermo Amaral
16
Direction
Anisotropic
p
1
p
2
Isotropic
p
pSlide17
θ
(p) =
P
p
(
a
given
vertex
belongs
to
a
percolating
cluster
)
θ
(p) = 0
si
p = 0
θ
(p) = 1
si
p = 1
θ
(p)
is monotonically
non-decrescent
There
is pc Є
[0, 1]
such
that
:
θ
(
p
) = 0
if
p <
p
c
θ
(p) > 0
if
p >
p
c
When
is
p =
p
c
?
Phase transition: Critical probabilityGuillermo Amaral17pc110θ(p)ppc?Slide18
Known critical probabilities
Guillermo Amaral
18
Bond
Site
Square
½
0.5927…
Bow-tie
1 − p − 6p
2
- 6p
3
− p
5
= 0
(
0.4045…)
0.5472…
Hexagonal
1- 2 sin(
π
/18)
(
0.6527…)
0.6970…
Triangular
2 sin(
π
/18)
(
0.3472…)
½
Kagomé
0.5244…
0.6527…Slide19
Why simulation?
Problems very hard to prove analytically
Square bond model critical probability =
0.5
Clues for a formal proof
Application to practical cases
Guillermo Amaral
19Slide20
Areas of interest
Large-graph representation
Pseudo-random numbers
Graph exploration
Analysis of connected components
Guillermo Amaral
20Slide21
Simulation variables
Guillermo Amaral
21
SimulationSlide22
Simulation process
Guillermo Amaral
22
1. Build the model
2. Generate a “random” configuration
3. Search for
percolating clusters
4. Collect results of output variablesSlide23
The simulatorSlide24
My experience… Slide25
Guillermo Amaral
25
Programming with a solution in mind leads to answers, but modeling the problem also raises new questionsSlide26
QuestionsSlide27
A case of studySlide28
Scope analysis
Guillermo
Amaral
28
v
=
(
x, y
)
v
’
=
(
y, x
)
v
’
v
p
v
p
H
x
0
(
x
0
↔
v
)
(
x
0
↔
v
’
)
If
p
H
<
p
v
,
P
(
x
0
↔
v
)
<
P
(
x
0
↔
v
’
)
?Slide29
Scope analysis visualization
Guillermo Amaral
29
>
=
Mirror
coloring
Scale
coloringSlide30
Object designSlide31
Objects (1)
Guillermo Amaral
31
PercolationModel
BondPercolation
SitePercolation
Lattice
SquareLattice
GraphPattern
SubgraphPattern
NodeBasedPattern
LatticeGraph
Square1KVertical1Horizontal
Square1Vertical1KHorizontal
…
OpenPolicy
SiteOpenPolicy
BondOpenPolicy
IsotropicPolicy
AnisotropicPolicy
AdjacencySolver
PatternAdjacencySolver
MatrixAdjacencySolver
CubicLattice
SquareVerticalHorizontal
…
CaesarSlide32
Objects (2)
Guillermo Amaral
32
AdjacencyMatrix
PSBitMatix
PSFloatMatrix
PSSparseMatrix
PSSparseFloatMatrix
GraphAlgorithm
GraphSearchAlgorithm
QuickUnionFind
BreathFirstSearch
DepthFirstSearch
WeightedQuickUnionFind
WQUFPC
ModelSampler
CriticalRangeFinder
CompositeSampler
NodeScopeAnalizer
VariableWalker
ModelEvaluator
ModelHistory
UnionFindAnalizer
…
…
CaesarSlide33
Objects (3)
Guillermo Amaral
33
PSDrawer
CriticalRangeDrawer
ChartDrawer
SquareLatticeGraphDrawer
BondPercolationGraphDrawer
SitePercolationGraphDrawer
PieChartDrawer
XYChartDrawer
ChartObject
ChartAxis
Chart
ChartSerie
RangeMark
XYSerieMarker
PieChar
XYChart
DrawerTool
NodeLocator
XYChartPointLocator
EdgeLocator
ClusterPainter
Caesar