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Percolation Simulating percolation models Guillermo Amaral Caesar Systems - Argentina
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ESUG 2009 Guillermo Amaral 2
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ESUG 2009 Guillermo Amaral 3
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ESUG 2009 Guillermo Amaral 4
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ESUG 2009 A virtual lab Guillermo Amaral 5
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ESUG 2009 Percolation deals with…
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ESUG 2009 Propagation of diseases Guillermo Amaral 7
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ESUG 2009 Propagation of fire Guillermo Amaral 8
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ESUG 2009 Oil & gas in reservoirs Guillermo Amaral 9
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ESUG 2009 Gelation & Polymerization Guillermo Amaral 10
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ESUG 2009 The problem
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ESUG 2009 Original problem (Broadbent - Hammersley, 1957) Guillermo Amaral 12 What is the probability that the water reaches the center of the rock?
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ESUG 2009 The simulation
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ESUG 2009 The mathematical model
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ESUG 2009 The simplest model Guillermo Amaral 15 v ℤ2v ℤ2 v ℤ2v ℤ2 v u at distance 1 from v u v P( e “open”) = p P( e “close”) = 1 - p P( e “open”) = p P( e “close”) = 1 - p e Open path from u to v Open path from u to v v u Percolating cluster Open cluster from v v
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ESUG 2009 Dimensions 3-D n-D… 2-D Element being open/close Bond Site Both… Structure SquareBow-tie HexagonalKagomé Other… Model types Guillermo Amaral 16 Direction Anisotropic p1p1 p2p2 Isotropic p p
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ESUG 2009 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 p c 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 probability Guillermo Amaral 17 pcpc 1 1 0 θ(p)θ(p) p pc?pc? pc?pc?
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ESUG 2009 Known critical probabilities Guillermo Amaral 18 BondSite 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 Kagomé0.5244…0.6527…
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ESUG 2009 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
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ESUG 2009 Areas of interest Large-graph representation Pseudo-random numbers Graph exploration Analysis of connected components Guillermo Amaral 20
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ESUG 2009 Simulation variables Guillermo Amaral 21 Simulation height, width Lattice parameters k Pattern parameters p p V, p H Open policy parameters Estimator θ(p) Percolating cluster size Simulation running time
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ESUG 2009 Simulation process Guillermo Amaral 22 1. Build the model 2. Generate a “random” configuration 3. Search for percolating clusters 4. Collect results of output variables
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ESUG 2009 The simulator
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ESUG 2009 My experience…
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ESUG 2009 Guillermo Amaral 25 Programming with a solution in mind leads to answers, but modeling the problem also raises new questions
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ESUG 2009 Questions
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ESUG 2009 A case of study
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ESUG 2009 Scope analysis Guillermo Amaral 28 v = ( x, y ) v ’ = ( y, x ) v = ( x, y ) v ’ = ( y, x ) v’v’ v pvpv pHpH x0x0 ( x 0 ↔ v ) ( x 0 ↔ v ’ ) ( x 0 ↔ v ) ( x 0 ↔ v ’ ) If p H < p v, P ( x 0 ↔ v ) <P ( x 0 ↔ v’ ) ? If p H < p v, P ( x 0 ↔ v ) <P ( x 0 ↔ v’ ) ?
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ESUG 2009 Scope analysis visualization Guillermo Amaral 29 > = Mirror coloringScale coloring
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ESUG 2009 Object design
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ESUG 2009 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 … … Caesar
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ESUG 2009 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 … … … … Caesar
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ESUG 2009 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
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