Percolation Simulating percolation models Guillermo Amaral Caesar Systems - Argentina

ESUG 2009 Guillermo Amaral 2

ESUG 2009 Guillermo Amaral 3

ESUG 2009 Guillermo Amaral 4

ESUG 2009 A virtual lab Guillermo Amaral 5

ESUG 2009 Percolation deals with…

ESUG 2009 Propagation of diseases Guillermo Amaral 7

ESUG 2009 Propagation of fire Guillermo Amaral 8

ESUG 2009 Oil & gas in reservoirs Guillermo Amaral 9

ESUG 2009 Gelation & Polymerization Guillermo Amaral 10

ESUG 2009 The problem

ESUG 2009 Original problem (Broadbent - Hammersley, 1957) Guillermo Amaral 12 What is the probability that the water reaches the center of the rock?

ESUG 2009 The simulation

ESUG 2009 The mathematical model

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

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

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 θ(p)θ(p) p pc?pc? pc?pc?

ESUG 2009 Known critical probabilities Guillermo Amaral 18 BondSite Square ½ … Bow-tie 1 − p − 6p 2 - 6p 3 − p 5 = 0 (0.4045…) … Hexagonal 1- 2 sin(π/18) (0.6527…) … Triangular 2 sin(π/18) (0.3472…) ½ Kagomé0.5244…0.6527…

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

ESUG 2009 Areas of interest  Large-graph representation  Pseudo-random numbers  Graph exploration  Analysis of connected components Guillermo Amaral 20

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

ESUG 2009 Simulation process Guillermo Amaral Build the model 2. Generate a “random” configuration 3. Search for percolating clusters 4. Collect results of output variables

ESUG 2009 The simulator

ESUG 2009 My experience…

ESUG 2009 Guillermo Amaral 25 Programming with a solution in mind leads to answers, but modeling the problem also raises new questions

ESUG 2009 Questions

ESUG 2009 A case of study

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’ ) ?

ESUG 2009 Scope analysis visualization Guillermo Amaral 29 > = Mirror coloringScale coloring

ESUG 2009 Object design

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

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

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