{
 "metadata": {
  "name": "",
  "signature": "sha256:aebfca84760b53334a8875cb96270087d1a7ea3790b5e830ea9cd0d23b42cb08"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This code completes the discussion starting on p.6 of \n",
      "[these notes](https://courses.cit.cornell.edu/info2950_2014fa/markovpr.pdf):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.display import Image\n",
      "Image('w4.png')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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4pDYBbtJxvpCS3O/5XeLP+uW0He7KKIBPlgsOP9FPv+3psH/3UxoQV/cJSKHm/p3cv4Nd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\neIBAChBAiZ4CRkYSQeD/Ac51pAGoWMp7AAAAAElFTkSuQmCC\n",
       "prompt_number": 1,
       "text": [
        "<IPython.core.display.Image at 0x10bdbda50>"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mweb={1:[2,3],2:[3],3:[1],4:[3]}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "T= array([[1./len(mweb[i]) if m in mweb[i] else 0. for m in mweb] for i in mweb])\n",
      "T"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "array([[ 0. ,  0.5,  0.5,  0. ],\n",
        "       [ 0. ,  0. ,  1. ,  0. ],\n",
        "       [ 1. ,  0. ,  0. ,  0. ],\n",
        "       [ 0. ,  0. ,  1. ,  0. ]])"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "around(reduce(dot,[T]*25),3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "array([[ 0.4,  0.2,  0.4,  0. ],\n",
        "       [ 0.4,  0.2,  0.4,  0. ],\n",
        "       [ 0.4,  0.2,  0.4,  0. ],\n",
        "       [ 0.4,  0.2,  0.4,  0. ]])"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "&nbsp;\n",
      "Now add the \"teleportation\" term, with p=.2:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p=.2\n",
      "n=len(T)\n",
      "Tp=p*ones([n,n])/n + (1-p)*T"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Tp"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "array([[ 0.05,  0.45,  0.45,  0.05],\n",
        "       [ 0.05,  0.05,  0.85,  0.05],\n",
        "       [ 0.85,  0.05,  0.05,  0.05],\n",
        "       [ 0.05,  0.05,  0.85,  0.05]])"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Converges reasonably well after just 8 iterations:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "linalg.matrix_power(Tp,8)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "array([[ 0.3698848 ,  0.19271104,  0.38740416,  0.05      ],\n",
        "       [ 0.35939904,  0.2031968 ,  0.38740416,  0.05      ],\n",
        "       [ 0.35939904,  0.19271104,  0.39788992,  0.05      ],\n",
        "       [ 0.35939904,  0.2031968 ,  0.38740416,  0.05      ]])"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The result starting from the equiprobability state $(1/4,1/4,1/4,1/4)$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(1/4.)*ones(4).dot(linalg.matrix_power(Tp,8))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "array([ 0.36202048,  0.19795392,  0.3900256 ,  0.05      ])"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}