percolation

PercolationSimulating percolation models

Guillermo AmaralCaesar Systems - Argentina

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A virtual lab

5

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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)

Guillermo Amaral

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 typesDirection

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

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