percolation simulating percolation models guillermo amaral caesar systems - argentina
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PercolationSimulating percolation models
Guillermo AmaralCaesar Systems - Argentina
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A virtual lab
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Percolation deals with…
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Propagation of diseases
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Propagation of fire
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Oil & gas in reservoirs
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Gelation & Polymerization
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The problem
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Original problem (Broadbent - Hammersley, 1957)
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What is the
probability that the
water reaches the center of
the rock?
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The simulation
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The mathematical model
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The simplest model
v ϵ ℤ2
vu at distance 1
from v
u v
P(e “open”) = pP(e “close”) = 1 - p
e
Open path fromu to v
v
u
Percolating cluster
Open cluster from v
v
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Dimensions
3-D
n-D…
2-D
Element being open/close
Bond
Site
Both…
Structure
Square Bow-tie
Hexagonal Kagomé
Other…
Model types
Direction
Anisotropicp1
p 2
Isotropicp
p
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θ(p) = Pp(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 < pc
θ(p) > 0 if p > pc
When is p = pc?
Phase transition: Critical probability
pc
1
10
θ(p)
p
pc?
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Known critical probabilities
Bond Site
Square ½ 0.5927…
Bow-tie 1 − p − 6p2 - 6p3 − p5 = 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…
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Why simulation?
Problems very hard to prove analytically Square bond model critical probability = 0.5
Clues for a formal proof
Application to practical cases
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Areas of interest
Large-graph representation
Pseudo-random numbers
Graph exploration
Analysis of connected components
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Simulation variables
SimulationLattice parameters
• height, • width
Pattern parameters • k
Open policy parameters
• p• pV, pH
Estimator θ(p)
Percolating cluster size
Simulation running time
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Simulation process
1. Build the model
2. Generate a “random”
configuration
3. Search for percolating
clusters
4. Collect results of output variables
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The simulator
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My experience…
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Programming with a solution in mind leads to answers, but
modeling the problem also raises new questions
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Questions
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A case of study
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Scope analysis
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v = (x, y) v’ = (y, x)
v’
v
p vpH
x0 (x0↔v) (x0↔v’ )
If pH < pv,P(x0↔v) <P(x0↔v’)?
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Scope analysis visualization
>
=
Mirror coloring Scale coloring
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Object design
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Objects (1)
PercolationModel
BondPercolation SitePercolation
Lattice
SquareLatticeGraphPattern
SubgraphPattern NodeBasedPattern
LatticeGraph
Square1KVertical1Horizontal Square1Vertical1KHorizontal …
OpenPolicy
SiteOpenPolicyBondOpenPolicy
IsotropicPolicy AnisotropicPolicy
AdjacencySolver
PatternAdjacencySolver MatrixAdjacencySolver
CubicLatticeSquareVerticalHorizontal …
Caesar
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Objects (2)
AdjacencyMatrix
PSBitMatix
PSFloatMatrix PSSparseMatrix
PSSparseFloatMatrix
GraphAlgorithm
GraphSearchAlgorithmQuickUnionFind
BreathFirstSearch DepthFirstSearchWeightedQuickUnionFind
WQUFPC
ModelSampler
CriticalRangeFinder
CompositeSampler
NodeScopeAnalizer
VariableWalker
ModelEvaluator
ModelHistory
UnionFindAnalizer …
…
Caesar
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Objects (3)
PSDrawer
CriticalRangeDrawerChartDrawer SquareLatticeGraphDrawer
BondPercolationGraphDrawer
SitePercolationGraphDrawerPieChartDrawer XYChartDrawer
ChartObject
ChartAxis
Chart ChartSerieRangeMark
XYSerieMarker
PieChar XYChart
DrawerTool
NodeLocator XYChartPointLocator
EdgeLocator
ClusterPainter
Caesar