The document discusses simulating percolation models. Percolation deals with the propagation of things through structures like diseases through populations or fluids through materials. The simulation models this using graphs with nodes/edges that can be open or closed randomly. It explores finding the critical probability where a percolating cluster first forms by varying the open probability in simulations. Programming the simulation requires objects for the percolation model, lattice graphs, algorithms for graph searches and analysis of results.

1. Percolation Simulating percolation models Guillermo Amaral CaesarSystems - Argentina

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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…

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

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

38. Scope analysis visualization Guillermo Amaral 29 Mirrorcoloring Scalecoloring &gt; =

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