馬可夫鏈 (Markov chain) 或稱馬可夫過程 (Markov process)是一種隨機過程,在數學上只要這傳遞矩陣有一些好的性質,就可以證明經由這馬可夫鏈抽樣的分佈會收斂到想要的機率分佈。
這邊我簡單模擬有一群人,分為AB兩種狀態,並且一開始兩種狀態人數相同,而每回合 中的每個人會有 aRate 機率變 B中的每個人會有 bRate 的機率變 A。如此進行多回合。(這是 Agent-Based 而不是直接矩陣運算)
程式模擬:
A to B: ( 0 ~ 1 )

B to A: ( 0 ~ 1 )

HTML Code:
<!DOCTYPE html>
<html>
<head>
    <title>Simulation</title>
 <meta charset="utf-8">
 <style>
  body {
   font: 14px sans-serif;
  }
  .axis path,
  .axis line {
   fill: none;
   stroke: #111;
   shape-rendering: crispEdges;
  }
  .x.axis path {
   display: none;
  }
 </style>
 <script src="https://d3js.org/d3.v3.min.js"></script>
 <script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.0/jquery.min.js"></script>
</head>
<body>
<p>Exchange System</p>
<p>A to B: ( 0 ~ 1 )</p>
<input type="number" id="aRate" value='0.2' min='0.0' max='1.0'>
<p>B to A: ( 0 ~ 1 )</p>
<input type="number" id="bRate" value='0.1' min='0.0' max='1.0'>
<button>Result</button>
<div id="Res"></div>

<script>
var aRate, bRate;
$(document).ready(function(){
$("button").click(function(){
aRate = $("#aRate").val();
bRate = $("#bRate").val();
f();
});
});
var Pss = { // person state
A: 1,
B: 2
};

function f() {
var res = System.runRounds(), data = [];
for(var i=0; i<= System.NumRounds; i++){
data.push({ Terms: i, A: res[i].A, B: res[i].B});
}
showResult(data);
}

var System = (function (){
var People = [];

var runRounds = function(){
var result = [];
initialize();
result.push({ A: System.NumPeople, B: System.NumPeople}); // initial
for(var i= 0; i< System.NumRounds;i++){
result.push( aRound());
}
return result;
};

var initialize = function(){
People = [];
for(var i= 0; i< System.NumPeople; i++){
People.push(new Person(Pss.A));
People.push(new Person(Pss.B));
}
};

var aRound = function(){
People.forEach(function(ps){
ps.change();
});
return count();
};

var count = function(){
var Pp = People;
var data = { A: 0, B: 0};
for(var i in Pp){
if(Pp[i].State == Pss.A) data.A++;
else data.B++;
}
return data;
};

return{
NumPeople: 10000,
NumRounds: 30,
runRounds: runRounds
};
})();

function Person(state){
this.State = state;
}

Person.prototype.change = function(){
this.Rate = (this.State == Pss.A)? aRate: bRate;
this.State = (Math.random()< this.Rate)? 3^this.State: this.State;
}

function showResult(data) {
var margin = {top: 20, right: 20, bottom: 30, left: 40},
width = 900 - margin.left - margin.right,
height = 500 - margin.top - margin.bottom;

var x0 = d3.scale.ordinal()
.rangeRoundBands([0, width-20], .1);

var x1 = d3.scale.ordinal();

var y = d3.scale.linear()
.range([height, 0]);

var color = d3.scale.ordinal()
.range([ "#aa4748","#ff8c00"]);

var xAxis = d3.svg.axis()
.scale(x0)
.orient("bottom");

var yAxis = d3.svg.axis()
.scale(y)
.orient("left")
.tickFormat(d3.format(".2s"));
$("#Res").empty();

var svg = d3.select("#Res").append("svg")
.attr("width", width + margin.left + margin.right)
.attr("height", height + margin.top + margin.bottom)
.append("g")
.attr("transform", "translate(" + margin.left + "," + margin.top + ")");

var X_Names = ['A','B',];

data.forEach(function(d) {
d.xs = X_Names.map(function(name) { return {name: name, value: +d[name]}; });
});

x0.domain(data.map(function(d) { return d.Terms; }));
x1.domain(X_Names).rangeRoundBands([0, x0.rangeBand()]);
y.domain([0, d3.max(data, function(d) { return d3.max(d.xs, function(d) { return d.value; }); })]);

svg.append("g")
.attr("class", "x axis")
.call(xAxis)
.attr("transform", "translate(0," + height + ")")
.append("text")
.attr("x", width+10)
.attr("y", 18)
.style("text-anchor", "end")
.text("Terms");;

svg.append("g")
.attr("class", "y axis")
.call(yAxis)
.append("text")
.attr("transform", "rotate(-90)")
.attr("y",6)
.attr("dy", ".71em")
.style("text-anchor", "end")
.text("People");

var terms = svg.selectAll(".terms")
.data(data)
.enter().append("g")
.attr("class", "terms")
.attr("transform", function(d) { return "translate(" + x0(d.Terms) + ",0)"; });

terms.selectAll("rect")
.data(function(d) { return d.xs; })
.enter().append("rect")
.attr("width", x1.rangeBand())
.attr("x", function(d) { return x1(d.name); })
.attr("y", function(d) { return y(d.value); })
.attr("height", function(d) { return height - y(d.value); })
.style("fill", function(d) { return color(d.name); });

var legend = svg.selectAll(".legend")
.data(X_Names.slice().reverse())
.enter().append("g")
.attr("class", "legend")
.attr("transform", function(d, i) { return "translate(0," + i * 20 + ")"; });

legend.append("rect")
.attr("x", width - 16)
.attr("width", 18)
.attr("height", 18)
.style("fill", color);

legend.append("text")
.attr("x", width+12 )
.attr("y", 9)
.attr("dy", ".35em")
.style("text-anchor", "end")
.text(function(d) { return d; });

};

</script>
</body>
</html>