Percolation

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Simulation of percolation models by Guillermo Amaral, ESUG09, Brest, France

Tecnologia Negócios

Percolation Simulating percolation models Guillermo Amaral CaesarSystems - Argentina

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Gelation & Polymerization Guillermo Amaral 10

Original problem (Broadbent - Hammersley, 1957) Guillermo Amaral 12 What is the probability that the water reaches the center of the rock?

vϵℤ2 u v u at distance 1 fromv v Open pathfrom u tov u Open clusterfromv e Percolatingcluster v P(e“open”) = p P(e“close”) = 1 - p v The simplest model Guillermo Amaral 15

Dimensions Elementbeing open/close Structure Direction 3-D Square Bow-tie p1 p Bond p2 p Hexagonal Kagomé Site Model types Guillermo Amaral 16 Isotropic Anisotropic 2-D n-D… Other… Both…

Whenis p = pc?Phase transition: Critical probability Guillermo Amaral 17 θ(p) 1 pc? p pc 0 1

Known critical probabilities Guillermo Amaral 18

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 19

Areas of interest Large-graph representation Pseudo-random numbers Graph exploration Analysis of connected components Guillermo Amaral 20

Simulation variables Guillermo Amaral 21 Simulation

Simulation process Guillermo Amaral 22 2. Generate a “random” configuration 1. Build the model 3. Search for percolating clusters 4. Collect results of output variables

Guillermo Amaral 25 Programming with a solution in mind leads to answers, but modeling the problem also raises new questions

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Percolation

1. Percolation Simulating percolation models Guillermo Amaral CaesarSystems - Argentina

2. Guillermo Amaral 2

3. Guillermo Amaral 3

4. Guillermo Amaral 4

5. A virtual lab Guillermo Amaral 5

6. Percolation deals with…

7. Propagation of diseases Guillermo Amaral 7

8. Propagation of fire Guillermo Amaral 8

9. Oil & gas in reservoirs Guillermo Amaral 9

10. Gelation & Polymerization Guillermo Amaral 10

11. The problem

12. Original problem (Broadbent - Hammersley, 1957) Guillermo Amaral 12 What is the probability that the water reaches the center of the rock?

13. The simulation

14. The mathematical model

15. vϵℤ2 u v u at distance 1 fromv v Open pathfrom u tov u Open clusterfromv e Percolatingcluster v P(e“open”) = p P(e“close”) = 1 - p v The simplest model Guillermo Amaral 15

16. Dimensions Elementbeing open/close Structure Direction 3-D Square Bow-tie p1 p Bond p2 p Hexagonal Kagomé Site Model types Guillermo Amaral 16 Isotropic Anisotropic 2-D n-D… Other… Both…

17.

18. θ(p) = 0 si p = 0

19. θ(p) = 1 si p = 1

20. θ(p) ismonotonically non-decrescent

21. ThereispcЄ[0, 1] suchthat:

22. θ(p) = 0 if p < pc

23. θ(p) > 0 if p > pc

24. Whenis p = pc?Phase transition: Critical probability Guillermo Amaral 17 θ(p) 1 pc? p pc 0 1

25. Known critical probabilities Guillermo Amaral 18

26. 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 19

27. Areas of interest Large-graph representation Pseudo-random numbers Graph exploration Analysis of connected components Guillermo Amaral 20

28. Simulation variables Guillermo Amaral 21 Simulation

29. Simulation process Guillermo Amaral 22 2. Generate a “random” configuration 1. Build the model 3. Search for percolating clusters 4. Collect results of output variables

30. The simulator

31. My experience…

32. Guillermo Amaral 25 Programming with a solution in mind leads to answers, but modeling the problem also raises new questions

33. Questions

34. A case of study

35.

36.

37. v’ = (y, x)IfpH < pv, P(x0↔v) <P(x0↔v’)? v’ Scope analysis Guillermo Amaral 28

38. Scope analysis visualization Guillermo Amaral 29 Mirrorcoloring Scalecoloring > =

39. Object design

40. Objects (1) Guillermo Amaral 31 PercolationModel OpenPolicy BondPercolation SitePercolation SiteOpenPolicy BondOpenPolicy LatticeGraph IsotropicPolicy AnisotropicPolicy Lattice SquareVerticalHorizontal … GraphPattern SquareLattice CubicLattice SubgraphPattern NodeBasedPattern AdjacencySolver Square1KVertical1Horizontal Square1Vertical1KHorizontal … PatternAdjacencySolver MatrixAdjacencySolver Caesar

41. Objects (2) Guillermo Amaral 32 GraphAlgorithm AdjacencyMatrix PSBitMatix PSSparseFloatMatrix GraphSearchAlgorithm QuickUnionFind PSFloatMatrix PSSparseMatrix BreathFirstSearch DepthFirstSearch WeightedQuickUnionFind WQUFPC ModelSampler CriticalRangeFinder NodeScopeAnalizer … ModelEvaluator CompositeSampler ModelHistory VariableWalker UnionFindAnalizer … Caesar

42. Objects (3) Guillermo Amaral 33 ChartObject ChartAxis XYSerieMarker Chart ChartSerie RangeMark DrawerTool PieChar XYChart NodeLocator XYChartPointLocator ClusterPainter EdgeLocator PSDrawer CriticalRangeDrawer ChartDrawer SquareLatticeGraphDrawer BondPercolationGraphDrawer PieChartDrawer XYChartDrawer SitePercolationGraphDrawer Caesar

Percolation

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