Percolation/Site percolation

Given an ${\displaystyle M\times N}$ rectangular array of cells numbered ${\displaystyle \mathrm {cell} [0..M-1,0..N-1]}$ assume ${\displaystyle M}$ is horizontal and ${\displaystyle N}$ is downwards.

Assume that the probability of any cell being filled is a constant ${\displaystyle p}$ where

{\displaystyle 0.0\leq p\leq 1.0}

The task

Simulate creating the array of cells with probability ${\displaystyle p}$ and then testing if there is a route through adjacent filled cells from any on row ${\displaystyle 0}$ to any on row ${\displaystyle N}$ , i.e. testing for site percolation.

Given ${\displaystyle p}$ repeat the percolation ${\displaystyle t}$ times to estimate the proportion of times that the fluid can percolate to the bottom for any given ${\displaystyle p}$ .

Show how the probability of percolating through the random grid changes with ${\displaystyle p}$ going from ${\displaystyle 0.0}$ to ${\displaystyle 1.0}$ in ${\displaystyle 0.1}$ increments and with the number of repetitions to estimate the fraction at any given ${\displaystyle p}$ as ${\displaystyle t=100}$ .

Use an ${\displaystyle M=15,N=15}$ grid of cells for all cases.

Optionally depict a successfull percolation path through a cell grid graphically.

Show all output on this page.

C

Translation of: D

<lang c>#include <stdio.h>

include <stdlib.h>
include <time.h>
include <string.h>
include <stdbool.h>

define N_COLS 15
define N_ROWS 15

// Probability granularity 0.0, 0.1, ... 1.0

define N_STEPS 11

// Simulation tries

define N_TRIES 100

typedef unsigned char Cell; enum { EMPTY_CELL = ' ',

      FILLED_CELL  = '#',
      VISITED_CELL = '.' };

typedef Cell Grid[N_ROWS][N_COLS];

void initialize(Grid grid, const double probability) {

   for (size_t r = 0; r < N_ROWS; r++)
       for (size_t c = 0; c < N_COLS; c++) {
           const double rnd = rand() / (double)RAND_MAX;
           grid[r][c] = (rnd < probability) ? EMPTY_CELL : FILLED_CELL;
       }

}

void show(Grid grid) {

   char line[N_COLS + 3];
   memset(&line[0], '-', N_COLS + 2);
   line[0] = '+';
   line[N_COLS + 1] = '+';
   line[N_COLS + 2] = '\0';

   printf("%s\n", line);
   for (size_t r = 0; r < N_ROWS; r++) {
       putchar('|');
       for (size_t c = 0; c < N_COLS; c++)
           putchar(grid[r][c]);
       puts("|");
   }
   printf("%s\n", line);

}

bool walk(Grid grid, const size_t r, const size_t c) {

   const size_t bottom = N_ROWS - 1;
   grid[r][c] = VISITED_CELL;

   if (r < bottom && grid[r + 1][c] == EMPTY_CELL) { // Down.
       if (walk(grid, r + 1, c))
           return true;
   } else if (r == bottom)
       return true;

   if (c && grid[r][c - 1] == EMPTY_CELL) // Left.
       if (walk(grid, r, c - 1))
           return true;

   if (c < N_COLS - 1 && grid[r][c + 1] == EMPTY_CELL) // Right.
       if (walk(grid, r, c + 1))
           return true;

   if (r && grid[r - 1][c] == EMPTY_CELL) // Up.
       if (walk(grid, r - 1, c))
           return true;

   return false;

}

bool percolate(Grid grid) {

   const size_t startR = 0;
   for (size_t c = 0; c < N_COLS; c++)
       if (grid[startR][c] == EMPTY_CELL)
           if (walk(grid, startR, c))
               return true;
   return false;

}

typedef struct {

   double prob;
   size_t count;

} Counter;

int main() {

   const double probability_step = 1.0 / (N_STEPS - 1);
   Counter counters[N_STEPS];
   for (size_t i = 0; i < N_STEPS; i++)
       counters[i] = (Counter){ i * probability_step, 0 };

   bool sample_shown = false;
   static Grid grid;
   srand(time(NULL));

   for (size_t i = 0; i < N_STEPS; i++) {
       for (size_t t = 0; t < N_TRIES; t++) {
           initialize(grid, counters[i].prob);
           if (percolate(grid)) {
               counters[i].count++;
               if (!sample_shown) {
                   printf("Percolating sample (%dx%d,"
                          " probability =%5.2f):\n",
                          N_COLS, N_ROWS, counters[i].prob);
                   show(grid);
                   sample_shown = true;
               }
           }
       }
   }

   printf("\nFraction of %d tries that percolate through:\n", N_TRIES);
   for (size_t i = 0; i < N_STEPS; i++)
       printf("%1.1f %1.3f\n", counters[i].prob,
              counters[i].count / (double)N_TRIES);

   return 0;

} </lang>

Output:

Percolating sample (15x15, probability = 0.40):
+---------------+
|###.  #  # #  #|
|###.. #  ##### |
|   #. ######  #|
|###....  ######|
|######.  ### # |
| #####.######  |
|#......... ##  |
|...#...##.# ## |
|##.#...##.### #|
| ###..# #. #   |
|# #######. # ##|
|   # ##...#### |
| ##  # .#####  |
|#######.##  ###|
|# ##   .## # # |
+---------------+

Fraction of 100 tries that percolate through:
0.0 0.000
0.1 0.000
0.2 0.000
0.3 0.000
0.4 0.010
0.5 0.070
0.6 0.630
0.7 0.970
0.8 1.000
0.9 1.000
1.0 1.000

D

Translation of: Python

<lang d>import std.stdio, std.random, std.array, std.datetime;

enum size_t nCols = 15,

           nRows = 15,
           nSteps = 11,    // Probability granularity: 0.0, 0.1 ... 1.0
           nTries = 100;   // Simulation tries.

alias BaseType = char; enum Cell : BaseType { empty = ' ',

                      filled  = '#',
                      visited = '.' }

alias Grid = Cell[nCols][nRows];

void initialize(ref Grid grid, in double probability,

               ref Xorshift rng) {
   foreach (ref row; grid)
       foreach (ref cell; row) {
           immutable r = rng.front / cast(double)rng.max;
           rng.popFront;
           cell = (r < probability) ? Cell.empty : Cell.filled;
       }

}

void show(in ref Grid grid) {

   immutable static line = '+' ~ "-".replicate(nCols) ~ "+";
   line.writeln;
   foreach (const ref row; grid)
       writeln('|', cast(BaseType[nCols])row, '|');
   line.writeln;

}

bool percolate(ref Grid grid) pure nothrow {

   bool walk(in size_t r, in size_t c) nothrow {
       enum bottom = nRows - 1;
       grid[r][c] = Cell.visited;

       if (r < bottom && grid[r + 1][c] == Cell.empty) { // Down.
           if (walk(r + 1, c))
               return true;
       } else if (r == bottom)
           return true;

       if (c && grid[r][c - 1] == Cell.empty) // Left.
           if (walk(r, c - 1))
               return true;

       if (c < nCols - 1 && grid[r][c + 1] == Cell.empty) // Right.
           if (walk(r, c + 1))
               return true;

       if (r && grid[r - 1][c] == Cell.empty) // Up.
           if (walk(r - 1, c))
               return true;

       return false;
   }

   enum startR = 0;
   foreach (immutable c; 0 .. nCols)
       if (grid[startR][c] == Cell.empty)
           if (walk(startR, c))
               return true;
   return false;

}

void main() {

   static struct Counter {
       double prob;
       size_t count;
   }

   StopWatch sw;
   sw.start;

   enum probabilityStep = 1.0 / (nSteps - 1);
   Counter[nSteps] counters;
   foreach (immutable i, ref co; counters)
       co.prob = i * probabilityStep;

   Grid grid;
   bool sampleShown = false;
   auto rng = Xorshift(unpredictableSeed);

   foreach (ref co; counters) {
       foreach (immutable _; 0 .. nTries) {
           grid.initialize(co.prob, rng);
           if (grid.percolate) {
               co.count++;
               if (!sampleShown) {
                   writefln("Percolating sample (%dx%d," ~
                            " probability =%5.2f):",
                            nCols, nRows, co.prob);
                   grid.show;
                   sampleShown = true;
               }
           }
       }
   }
   sw.stop;

   writefln("\nFraction of %d tries that percolate through:", nTries);
   foreach (const co; counters)
       writefln("%1.1f %1.3f", co.prob, co.count / cast(double)nTries);

   writefln("\nSimulations and grid printing performed" ~
            " in %2.3f seconds.", sw.peek.msecs / 1000.0);

}</lang>

Output:

Percolating sample (15x15, probability = 0.50):
+---------------+
|.....#. ####   |
|...#.#.# # # ##|
|####.#.....#  #|
|##...#.#.#.   #|
|..##..#..#. #  |
|#..#.##.##..# #|
|#....# ## #. # |
|##.#####....   |
|....###..##.## |
|#.#### .# #### |
|#.##.##. ##### |
|#....# .##   ##|
| ######.## ## #|
| # ##  .# #    |
| ##  # .### ###|
+---------------+

Fraction of 100 tries that percolate through:
0.0 0.000
0.1 0.000
0.2 0.000
0.3 0.000
0.4 0.000
0.5 0.090
0.6 0.570
0.7 0.950
0.8 1.000
0.9 1.000
1.0 1.000

Simulations and grid printing performed in 0.046 seconds.

Python

<lang python>from random import random import string from pprint import pprint as pp

M, N, t = 15, 15, 100

cell2char = ' #' + string.ascii_letters NOT_VISITED = 1 # filled cell not walked

class PercolatedException(Exception): pass

def newgrid(p):

   return [[int(random() < p) for m in range(M)] for n in range(N)] # cell

def pgrid(cell, percolated=None):

   for n in range(N):
       print( '%i)  ' % (n % 10) 
              + ' '.join(cell2char[cell[n][m]] for m in range(M)))
   if percolated: 
       where = percolated.args[0][0]
       print('!)  ' + '  ' * where + cell2char[cell[n][where]])

def check_from_top(cell):

   n, walk_index = 0, 1
   try:
       for m in range(M):
           if cell[n][m] == NOT_VISITED:
               walk_index += 1
               walk_maze(m, n, cell, walk_index)
   except PercolatedException as ex:
       return ex
   return None

def walk_maze(m, n, cell, indx):

   # fill cell 
   cell[n][m] = indx
   # down
   if n < N - 1 and cell[n+1][m] == NOT_VISITED:
       walk_maze(m, n+1, cell, indx)
   # THE bottom
   elif n == N - 1:
       raise PercolatedException((m, indx))
   # left
   if m and cell[n][m - 1] == NOT_VISITED:
       walk_maze(m-1, n, cell, indx)
   # right
   if m < M - 1 and cell[n][m + 1] == NOT_VISITED:
       walk_maze(m+1, n, cell, indx)
   # up
   if n and cell[n-1][m] == NOT_VISITED:
       walk_maze(m, n-1, cell, indx)

if __name__ == '__main__':

   sample_printed = False
   pcount = {}
   for p10 in range(11):
       p = p10 / 10.0
       pcount[p] = 0
       for tries in range(t):
           cell = newgrid(p)
           percolated = check_from_top(cell)
           if percolated:
               pcount[p] += 1
               if not sample_printed:
                   print('\nSample percolating %i x %i, p = %5.2f grid\n' % (M, N, p))
                   pgrid(cell, percolated)
                   sample_printed = True
   print('\n p: Fraction of %i tries that percolate through\n' % t )
   
   pp({p:c/float(t) for p, c in pcount.items()})</lang>

Output:

The Ascii art grid of cells has blanks for cells that were not filled. Filled cells start off as the '#', hash character and are changed to a succession of printable characters by successive tries to navigate from the top, (top - left actually), filled cell to the bottom.

The '!)' row shows where the percolation finished and you can follow the letter backwards from that row, (letter 'c' in this case), to get the route. The program stops after finding its first route through.

Sample percolating 15 x 15, p =  0.40 grid

0)    a a a       b   c #        
1)    a a   #         c c   #   #
2)        # #   # #     c # # #  
3)  #   #       # # #   c        
4)    #     #         c c c c c c
5)  # # # # # #         c   c   c
6)        # # #         c   c   c
7)  #   #     # #     #   #   # c
8)  #   # #     #   #       c c c
9)    #       #         #   c    
0)  #       #   # # # #   c c # #
1)      #     #   #     # c      
2)  #     # # # # #   c c c   c  
3)  #   # # #         c   c c c  
4)      #           # c         #
!)                    c

 p: Fraction of 100 tries that percolate through

{0.0: 0.0,
 0.1: 0.0,
 0.2: 0.0,
 0.3: 0.0,
 0.4: 0.01,
 0.5: 0.11,
 0.6: 0.59,
 0.7: 0.94,
 0.8: 1.0,
 0.9: 1.0,
 1.0: 1.0}

Note the abrupt change in percolation at around p = 0.6. These abrupt changes are expected.

Tcl

Works with: Tcl version 8.6

<lang tcl>package require Tcl 8.6

oo::class create SitePercolation {

   variable cells w h
   constructor {width height probability} {

set w $width set h $height for {set cells {}} {[llength $cells] < $h} {lappend cells $row} { for {set row {}} {[llength $row] < $w} {lappend row $cell} { set cell [expr {rand() < $probability}] } }

   }
   method print {out} {

array set map {0 "#" 1 " " -1 .} puts "+[string repeat . $w]+" foreach row $cells { set s "|" foreach cell $row { append s $map($cell) } puts [append s "|"] } set outline [lrepeat $w "-"] foreach index $out { lset outline $index "." } puts "+[join $outline {}]+"

   }
   method percolate {} {

for {set work {}; set i 0} {$i < $w} {incr i} { if {[lindex $cells 0 $i]} {lappend work 0 $i} } try { my Fill $work return {} } trap PERCOLATED x { return [list $x] }

   }
   method Fill {queue} {

while {[llength $queue]} { set queue [lassign $queue y x] if {$y >= $h} {throw PERCOLATED $x} if {$y < 0 || $x < 0 || $x >= $w} continue if {[lindex $cells $y $x]<1} continue lset cells $y $x -1 lappend queue [expr {$y+1}] $x [expr {$y-1}] $x lappend queue $y [expr {$x-1}] $y [expr {$x+1}] }

}

Demonstrate one run

puts "Sample percolation, 15x15 p=0.6" SitePercolation create bp 15 15 0.6 bp print [bp percolate] bp destroy puts ""

Collect statistics

apply {{} {

   puts "Percentage of tries that percolate, varying p"
   set tries 100
   for {set pint 0} {$pint <= 10} {incr pint} {

set p [expr {$pint * 0.1}] set tot 0 for {set i 0} {$i < $tries} {incr i} { set bp [SitePercolation new 15 15 $p] if {[$bp percolate] ne ""} { incr tot } $bp destroy } puts [format "p=%.2f: %2.1f%%" $p [expr {$tot*100./$tries}]]

}}</lang>

Output:

Sample percolation, 15x15 p=0.6
+...............+
|..####...#.#.#.|
|.###..........#|
|..###....#..#..|
|#..#..##.#..#.#|
|......#.....##.|
|.#...#..#..##..|
|.#......#.....#|
|######..#...#..|
| #  #...#.#.###|
|# ##..#....## #|
| ##...#.###  ##|
|  #.###.# # # #|
|# ### #####   #|
|  ####  ## # # |
|    # ##  ##   |
+---------------+

Percentage of tries that percolate, varying p
p=0.00: 0.0%
p=0.10: 0.0%
p=0.20: 0.0%
p=0.30: 0.0%
p=0.40: 0.0%
p=0.50: 7.0%
p=0.60: 56.0%
p=0.70: 93.0%
p=0.80: 100.0%
p=0.90: 100.0%
p=1.00: 100.0%