{ "metadata": { "name": "", "signature": "sha256:c90b43dea8879d8147fa8b3f4a9cd35a896a21b6418b2ae61ce712725e42bdc0" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Random walks and random stumbles" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Random walks are a gold mine for a wide variety of stochastic theory and practice. They are easy to explain, easy to code, and easy to misunderstand. In this section, we start out with the simplest imaginable random walk and then show how things can go wrong. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's examine the problem of the one-dimensional random walk. Consider a particle at the origin ($x=0$), with $p$ probability of moving to the right and $q=1-p$ probability of moving to the left. When the particle reaches $x=1$, the experiment terminates. On average, how many steps are required for this to terminate if $p=1/2$? This seems like a reasonable question, doesn't it?\n", "\n", "The first thing we have to do is establish the conditions under which termination occurs. Thus, we need the probability that the particle ultimately reaches $x=1$. We'll call this probability $P$. On average, how many steps does it make to terminate?\n", "\n", "This experiment is easy enough to code as shown below:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def walk():\n", " 'starting at x=0, step left/right with probability 1/2 until x=1'\n", " x=0\n", " while x!=1:\n", " x+=random.choice([-1,1]) # equi-probable left-right move\n", " yield x" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Some classical deft reasoning leads to:\n", "\n", "$$P = p + q P^2$$\n", "\n", "where $p$ is the probability it moves to $x=1$ from $x=0$ on the first try, and $q$ is the probability it does not, but then ultimately makes it back to $x=0$ (with probability $P$) and then again makes it from there to $x=1$, again with probability $P$ (thus, $P^2$). \n", "\n", "There are two solutions to $P$, $P=1$ and $P=p/(1-p)$. The crossover point is $p=1/2$. This is drawn below." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from __future__ import division # want floating point division\n", "\n", "fig,ax=subplots()\n", "p = linspace(0,0.5,20)\n", "ax.plot(p,p/(1-p),label=r'$p<1/2$',lw=3.)\n", "ax.plot([0.5,1],[1,1],label=r'$p > 1/2$',lw=3.)\n", "ax.axis(ymax=1.1)\n", "ax.legend(loc=0)\n", "ax.set_xlabel('$p$',fontsize=16)\n", "ax.set_ylabel('$P$',fontsize=16)\n", "ax.set_title(r'Probability of reaching $x=1$ from $x=0$',fontsize=16)\n", "ax.grid()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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U+fnA5MlS+7PPWCiIKorFwgqwP1ZSUi5WrRIvZgQATk7i9SpsAd8XEubCcCwW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\n3TogMFDZuIjI8nDMwkrt3i1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"text": [ "" ] } ], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Average number of steps to termination" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A straight-forward question to ask is what is the average number of steps it takes to ultimately terminate this random walk? The quick analysis above says that the particle will *ultimately* terminate, but, on average, how many steps are required for this to happen?\n", "\n", "Unfortunately, the analysis above is not very helpful here because the statement is about the probability of ultimate termination, not the probability of termination *given* a particular point somewhere on the left of the origin.\n", "\n", "Fortunately, we have all the Python-based tools to experimentally get at this. The following generator describes the $p=1/2$ random walker." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def walk():\n", " 'starting at x=0, step left/right with probability 1/2 until x=1'\n", " x=0\n", " while x!=1:\n", " x+=random.choice([-1,1]) # equi-probable left-right move\n", " yield x" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's try generating a realization of this walk." ] }, { "cell_type": "code", "collapsed": false, "input": [ "random.seed(123) # set seed for reproducibility\n", "\n", "s = list(walk()) # generate the random steps\n", "\n", "fig,ax=subplots()\n", "ax.plot(s)\n", "ax.set_ylabel(\"particle's x-position\",fontsize=16)\n", "ax.set_xlabel('step index k',fontsize=16)\n", "ax.set_title('Example of random walk',fontsize=16)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 4, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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nCGtj6u+yLCNJRObNm4cZM2aoto8cOQJ/f39ER0ejUaNGuHTpktUMtCRSPJCE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\nLCJERKTbnxSA+RU3HRY3AAAAAElFTkSuQmCC\n", "text": [ "" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, that we're set up, we can generate a whole list of these walks and then average their lengths to estimate the mean of these walks." ] }, { "cell_type": "code", "collapsed": false, "input": [ "s = [list(walk()) for i in range(50)] # generate 50 random walks\n", "len_walk=map(len,s) # lengths of each walk" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following plots out a few of these random walks so we can get a feel of what's going on with the average length." ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig,ax=subplots()\n", "for i in s:\n", " ax.plot(i)\n", "ax.set_ylabel(\"particle's x-position\",fontsize=16)\n", "ax.set_xlabel('step index k',fontsize=16)\n", "ax.set_title('average length=%3.2f'%(mean(len_walk)))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 6, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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EIqLX6xkZGZn03ujoKHr99O5CPiryBvNJzTIgiimkpI+gLdPSFN9EijeFtXeu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0l8P1EOlR+yY4RENNge9rtzQFciERGdf31u3rwZmzdv7rE/LCxM/bet863/9Kc/hcVigclkAgAs\nXLgQu3btQkhICBITExESEoIJEyZg165dvJ01RFwmuMDf0x9fXfsKsx+ZDaBzSK+7s7s6pPdXT/zK\nzlES0Uii/DcT3dfdt4n6PaGiDMporQelKEq/zzhQ/5KPJCM+IB5rjGsAAAn/lwDvid64cP0CTtec\nRuPmRjg5OPVzFiIaLR72u3NAVyIPkkRodOsaodWlvLYcB549gNj3YhGuDWcCISIrnGOdrBgf6X6H\n1o3bN9BwqwHfn/F9uDu78+l0IuqBSYSs3P3A4dnaswibFgYHjYP6qnciorsN6HYWjR++k31habeg\ntrnWakjvlu9vgWGqwc7REdFIwyRCVhRFQYQ2Amfrzlo9XGjSm+wcGRGNRLydRT2ob+mt41t6iej+\nmESohwhtBMquluFc3TmEa8PtHQ4RjWBMItRDhDYC+ZX58HL1whSXKfYOh4hGMCYR6iHUOxQNtxp4\nK4uI+sUkQj24ObkhwCuAQ3qJqF9MItSr2JmxiPqfKHuHQUQj3IDenTUa8N1ZREQP7mG/O3klQkRE\nNmMSISIimzGJEBGRzZhEiIjIZkwiRERkMyYRIiKyGZMIERHZjEmEiIhsxiRCREQ2YxIhIiKbMYkQ\nEZHNmESIiMhmTCJERGQzJhEiIrIZkwgREdmMSYSIiGw2YpLIxo0bMWvWLBiNRixfvhyNjY0AgKqq\nKri6umLOnDmYM2cO1q5da+dIiYioy4hJInFxcaioqEB5eTkCAwORnZ2t7gsICEBpaSlKS0uxa9cu\nO0Y5OpSUlNg7hBGDbdGNbdGNbTF4RkwSMZlM0Gg6w5k/fz4uXbpk54hGL/4P0o1t0Y1t0Y1tMXhG\nTBK52759+/D000+r62azGXPmzEF0dDQ+/fRTO0ZGRER3mzCcH2YymXD16tUe219//XUsXboUAJCV\nlQUnJyesXr0aAPDoo4+iuroanp6e+PLLL7Fs2TJUVFRg8uTJwxk6ERH1RkaQ/fv3S2RkpNy6davP\nMtHR0fKPf/yjx3a9Xi8AuHDhwoXLAyx6vf6hvreH9UrkfgoLC/HGG2/g+PHjcHFxUbdfu3YNnp6e\ncHBwwMWLF1FZWQl/f/8ex1+4cGE4wyUiIgCKiIi9gwAAg8EAi8UCLy8vAMDChQuxa9cuHDlyBFu3\nboWjoyM0Gg1effVVPPPMM3aOloiIgBGURIiIaPQZkaOzHlRhYSGCg4NhMBiQk5Nj73CGnZ+fHyIi\nIjBnzhzMmzcPANDQ0ACTyYTAwEDExcXhxo0bdo5yaKSlpUGr1SI8PFzddr+6Z2dnw2AwIDg4GEVF\nRfYIecj01hYZGRnQ6XTqw7oFBQXqvrHaFtXV1YiJiUFoaCjCwsLwm9/8BsD47Bd9tcWg9ouH+kVl\nBGhraxO9Xi9ms1ksFosYjUY5f/68vcMaVn5+fvLtt99abdu4caPk5OSIiMi2bdtk06ZN9ghtyJ04\ncUK+/PJLCQsLU7f1VfeKigoxGo1isVjEbDaLXq+X9vZ2u8Q9FHpri4yMDHnrrbd6lB3LbXHlyhUp\nLS0VEZGmpiYJDAyU8+fPj8t+0VdbDGa/GPVXIqdPn0ZAQAD8/Pzg6OiIpKQkHD161N5hDTu5567k\nn//8Z6SmpgIAUlNT8ac//ckeYQ25J554Ap6enlbb+qr70aNHkZycDEdHR/j5+SEgIACnT58e9piH\nSm9tAfTsG8DYbotHHnkEs2fPBgBMmjQJs2bNQk1NzbjsF321BTB4/WLUJ5GamhpMnz5dXdfpdGoj\njReKomDx4sWYO3cu9uzZAwCora2FVqsFAGi1WtTW1tozxGHVV90vX74MnU6nlhsvfWXHjh0wGo1I\nT09Xb+GMl7aoqqpCaWkp5s+fP+77RVdbLFiwAMDg9YtRn0QURbF3CHZ38uRJlJaWoqCgADt37sQn\nn3xitV9RlHHbTv3Vfay3y4svvgiz2YyysjL4+PjgF7/4RZ9lx1pbNDc3Y8WKFdi+fXuPh5PHW79o\nbm7GypUrsX37dkyaNGlQ+8WoTyK+vr6orq5W16urq60y6Xjg4+MDAPD29sZzzz2H06dPQ6vVqm8H\nuHLlCqZNm2bPEIdVX3W/t69cunQJvr6+dolxuEybNk39wnz++efVWxNjvS3u3LmDFStWICUlBcuW\nLQMwfvtFV1v88Ic/VNtiMPvFqE8ic+fORWVlJaqqqmCxWJCXl4eEhAR7hzVsWlpa0NTUBAC4efMm\nioqKEB4ejoSEBBw8eBAAcPDgQbXzjAd91T0hIQG5ubmwWCwwm82orKxUR7ONVVeuXFH//cc//lEd\nuTWW20JEkJ6ejpCQEKxfv17dPh77RV9tMaj9YrBHA9hDfn6+BAYGil6vl9dff93e4QyrixcvitFo\nFKPRKKGhoWr9v/32W4mNjRWDwSAmk0muX79u50iHRlJSkvj4+Iijo6PodDrZt2/ffeuelZUler1e\ngoKCpLCw0I6RD7572+Ldd9+VlJQUCQ8Pl4iICHn22Wfl6tWravmx2haffPKJKIoiRqNRZs+eLbNn\nz5aCgoJx2S96a4v8/PxB7Rd82JCIiGw26m9nERGR/TCJEBGRzZhEiIjIZkwiRERkMyYRIiKyGZMI\nERHZjEmExoSysjJkZGTg+vXrdvn86OhoxMTEDNr5SkpKoNFocOLEiUE7Z38yMjKg0WjQ0dExbJ9J\no9+ImR6X6GGUlZXh1VdfxZo1a3p9k+1Qe+edd4b9M4fCWHtnFA09JhEaU+z17GxwcLBdPnew8dlj\nelC8nUWjwr/+9S8899xz0Gq1cHV1xYwZM5CYmIj29nYcOHAAaWlpAACDwQCNRgONRoNvvvkGANDW\n1obs7GwEBwfDxcUFvr6++OUvf4nW1lb1/FVVVdBoNNi9ezc2bNgArVYLNzc3LF26FF9//XW/8d17\nO6vrdtSxY8fw8ssvw9vbG97e3khJSUFjY6PVsfX19Vi9ejU8PDzg6emJ1NTUPmei/PDDD7FgwQK4\nubnB09MTiYmJVi/M27JlC5ydnfHFF1+o227evImgoCB873vfQ3t7+wBau1thYSEmTZqEdevWMcFQ\n74bolS1EgyogIEDmz58vH374oZw4cUIOHz4sKSkpYrFYpL6+XrZs2SKKosiRI0fk888/l88//1xa\nW1tFRGTVqlXi5uYmr732mnz00UeyY8cOmTJliqxYsUI9v9lsFkVRZPr06ZKQkCD5+fmyf/9+8fHx\nkcDAQLlz585944uOjpaYmBh1/eOPPxZFUWTmzJmybt06KS4ulh07doirq6ukpqZaHRsVFSUeHh6y\nc+dOKSoqkrS0NNHpdKIoihw/flwtt3v3blEURdLT06WgoEDy8vJk1qxZMnPmTGlqahKRzpk+IyMj\nxWAwSHNzs4iIpKamypQpU6Sqquq+ddi6dasoiqLOZHfw4EFxcnKSrKysfv7r0HjGJEIjXn19vSiK\nIseOHeuzzP79+0VRFPn3v/9ttf3EiROiKIocOnTIavsf/vAHURRFysrKRKQ7iYSGhlqVO3nypCiK\nIu++++59Y1y0aFGvSeRHP/qRVbmXX35ZXFxc1PWioiJRFEXy8vKsysXHx1slkaamJnF3d5f09HSr\ncmazWZycnOTtt99Wt1VVVcmUKVMkNTVVDh8+LIqiSG5u7n3jF+lOIm1tbZKTkyOOjo791puIt7No\nxPvOd74Df39/bNq0CXv37kVlZeWAjy0sLISTkxOWL1+OtrY2dTGZTADQY/TTypUrrdYjIyOh0+nw\nt7/9zabYn3nmGav1sLAwtLa2oq6uDgBw6tQpODg4YMWKFVblkpKSrNZPnTqFpqYmrF692qoeOp0O\nQUFBVvWYMWMG3nnnHbz33ntIS0tDamoqVq1aNeCY169fj4yMDBw5ckS9TUjUFyYRGhWKi4sxd+5c\nvPLKKwgKCoJerx/QiKi6ujpYLBa4ubnByclJXbRaLRRFQUNDg1X5rulT7zZt2jSbp0v18vKyWnd2\ndgYA3L59G0DnvA6enp5wcHDo8Zn31gMAFi9ebFUPJycnnDt3rkc9nn76aXh5ecFiseDnP//5A8Wc\nm5uL8PBwxMbGPtBxND5xdBaNCjNnzlQnFCovL8dvf/tbrF27Fn5+fliyZEmfx02dOhUuLi749NNP\ne93fNStkl66Z7+5WW1uLxx577CGi75uPjw+uX7+O9vZ2q0TSNf93l6lTpwLonEwpNDS0x3nunf71\npZdeQkdHB/z9/fGTn/wEJ0+e7JGo+vLXv/4VJpMJ8fHxyM/Ph5ub24NWi8YRXonQqGM0GvHWW28B\nACoqKgB0/4Xf0tJiVTY+Ph63b9/GjRs38Nhjj/VY7k0iH3zwgdUopJMnT6KmpgYLFy4ckrpERkai\nvb0dH3zwgdX23NzcHuUmT56MysrKXuthMBjUsocPH8ahQ4ewZ88e5OXlobS0FFu2bBlwTKGhoSgp\nKUFlZSXi4+Nx8+bNh6skjWm8EqER78yZM/jZz36GpKQk6PV6dVivo6MjnnzySQBASEgIAGDnzp1Y\ns2YNHB0dYTQasWjRIiQnJ2PlypXYsGEDHn/8cWg0GlRVVaGgoAA5OTlWX8DNzc1YtmwZXnjhBdTV\n1eGVV15BYGAg1qxZ02+cYsMQ2MWLFyMqKgovvPACrl27hoCAAOTl5anJsYu7uzveeOMNvPTSS6iv\nr8eSJUvg4eGBmpoaHD9+HDExMUhOTobZbMaLL76I559/Xv2dJSsrC5s3b0ZcXByio6MHFFdwcDBK\nSkoQExODp556Sh3qS9SDvX/ZJ+pPXV2dpKamSmBgoEycOFG8vLwkOjpaioqKrMplZmaKr6+vODg4\niEajka+//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"text": [ "" ] } ], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, here's where things get interesting! Fifty samples of the random walk is really not that many, so we'd like to hopefully get a better average by generating a lot more. If you try to generate, say, 1000 realizations, you'd be in for a long wait! This is because some of the random walks are really, really long! Furthermore, this is a **persistent** phenomenon; it's not just a bad draw from the random deck. Even if there is only one really long walk, it seriously distorts the average. It's tempting to conclude that this is just some outlier and get on with it, but **not** doing so will lead us to a very powerful theorem in stochastic processes." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To get a better picture of what's going on here, let's re-define our random walker function so we can limit how far it can go and thereby how long we'd have to wait." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def walk(limit=50):\n", " 'limited version of random walker'\n", " x=0\n", " while x!=1 and abs(x)" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "What's interesting about the above plot is how steep the slope is. For a path length of one (terminating at the first step), we already have 50% of the probability accounted for. By a path-length of 100, we already have about 90% of the probability. The problem is squeezing out the remaining probability mass means computing random walks longer than 500 (our arbitrary limit). We can do all the above steps for higher limits far above 500, but this observation still holds. Thus, the problem with averaging this is that getting more probability further out competes with the lengths of those further paths. For the average to converge, we want to asymptotically get more improbable paths relatively faster than those paths grow!\n", "\n", "Let's examine the standard deviation of our averages." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def estimate_std(limit=10,ncount=50):\n", " 'quick estimate of the standard deviation of the averages'\n", " ws= array([[ nwalk(limit) for i in range(ncount)] for k in range(ncount)])\n", " return (limit,ws.mean(), ws.mean(1).std())\n", "\n", "for limit in [10,20,50,100,200,300,500,1000]:\n", " print 'limit=%d,\\t average = %3.2f,\\t std=%3.2f'% estimate_std(limit)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "limit=10,\t average = 4.15,\t std=0.55\n", "limit=20,\t average = 6.25,\t std=1.08" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n", "limit=50,\t average = 10.00,\t std=2.36" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n", "limit=100,\t average = 15.55,\t std=3.52" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n", "limit=200,\t average = 21.59,\t std=7.20" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n", "limit=300,\t average = 25.60,\t std=8.70" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n", "limit=500,\t average = 32.20,\t std=12.71" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n", "limit=1000,\t average = 47.38,\t std=21.40" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n" ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "If this were converging, then the standard deviation of the mean (and the mean itself) should start converging as the walk limit increased. This is obviously not happening here." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Graph-based combinatorial approach" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So far, we have been looking at this using samples and started suspecting that there is something going on with the convergence of the average. However, this is not uncovering what's going on under the sheets with the convergence of the average. Yes, we've tinkered with varying the sample size, but it could still be the case that there is some much larger sample size out there that would cure all the problems we have so far experienced.\n", "\n", "The next code-block assembles some drawing utilities on the excellent networkx package so we can analyze this problem using graph combinatoric algorithms.\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import networkx as nx\n", "import itertools as it\n", "\n", "class Graph(nx.DiGraph):\n", " '''\n", " operations assuming pos attribute in nodes to support drawing and\n", " manipulating path lattice.\n", " '''\n", " def draw(self, ax=None,**kwds):\n", " '''\n", " Draw based on pos attribute and pass kwds to nx.draw\n", " :param: axes(optional, default is None)\n", " '''\n", " pos = self.get_pos()\n", " node_size=kwds.pop('node_size',200)\n", " alpha=kwds.pop('alpha',0.3)\n", " if ax is None: nx.draw(self,pos=pos,\n", " node_size=node_size,\n", " alpha=alpha,\n", " **kwds)\n", " else: nx.draw(self,pos=pos,\n", " node_size=node_size,\n", " alpha=alpha,\n", " ax=ax,**kwds)\n", "\n", " def get_pos(self,n=None):\n", " '''\n", " n := str name of node\n", " get positions as returned dictionary\n", " '''\n", " pos=dict([( i,j['pos'] ) for i,j in self.nodes(data=True)])\n", " if n is None:\n", " return pos\n", " else:\n", " return pos[n]\n", "\n", " def getx(self,x):\n", " return sorted([(i[0],i[1]['val'] ) for i in self.nodes(True) if i[0][0]==x])\n", "\n", " def gety(self,y):\n", " return sorted([(i[0],i[1]['val'] ) for i in self.nodes(True) if i[0][1]==y])\n", "\n", "# functions to allow diagonal lattice walking\n", "def diagwalk(level,n):\n", " x = level\n", " y = -level\n", " while y<=1 and x" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The lattice diagram above shows the potential paths to termination for the random walk. For example, the all paths that lead to the node labeled (5,1) are those paths that take exactly five steps to terminate. Note that this is a directed graph so the graph can only be tranversed in the direction of the arrows (arrowheads denoted by thick ends as shown). Fortunately, networkx has powerful tools for computing these paths as shown in the following." ] }, { "cell_type": "code", "collapsed": false, "input": [ "l5=list(nx.all_simple_paths(g,(0,0),(5,1)))\n", "for i in l5:\n", " print i\n", "\n", "fig,ax=subplots()\n", "fig.set_size_inches((6,6))\n", "g.draw(ax,with_labels=0)\n", "g.subgraph(l5[0]).draw(ax=ax,with_labels=1,node_color='b',node_size=700)\n", "g.subgraph(l5[1]).draw(ax=ax,with_labels=1,node_color='b',node_size=800)\n", "ax.set_title('Pathways that terminate in five steps',fontsize=16)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "[(0, 0), (1, -1), (2, 0), (3, -1), (4, 0), (5, 1)]\n", "[(0, 0), (1, -1), (2, -2), (3, -1), (4, 0), (5, 1)]\n" ] }, { "metadata": {}, "output_type": "pyout", "prompt_number": 15, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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WFra99x4LExLITk4mKSmJ7OzsPsf59jY2YsnIYPbZZ8sl3QPA5/OxZcsW2tra\n2L51K67t2zk7P5/oqChGjxlDbExMr3O8Ph/FDQ0kL15MXpj08EYC69Zt4bLLriEiIpvvf/89ABob\nG9iyZQsul4vWxiZSgBSzGYPRwJqqn/DtmX9kbNq8Huk4PR6aAgHSxo0j4hvWw2AwQFvbVqqrH2XP\nnk1yJabkxCGEoKSkhJqaGiwWC3V1daSmptJWXs6suDhmjBuHUd9zj+R2u50vrFbco0ZRtHgxev2x\n7aEs+Qqn00lxcTE2m40DBw7g8/mINBgIVlSwJC+P3IyMHg/HYDBIg9XKfpeL1LlzGT958jDmPvyw\nWCw8/3wxBw+OIiFhIpWVlZSU7MLt9tDW1kpEhAGjTkuqWsO0CROI/sYKY38gQIfLiU2tISU3t8+5\nn46OJozGMr797VhmzQqf4S0p4GGG3+9n27ZttLS0UF9fT2trK3l5eURGRjJu3Dg8Dge1O3Zgdrkw\nKgpBwCYE/oQEcgoLyc7JkT3vAdDZ2UlxcTF2u52Kigp0Oh2jR4/+cqx2NM0HD2Ldt48EIdACfqAV\nMOfmMmrSpD6HVSRH5s0317Bxo5rSUhW1tc24XC7a2zswmUyYTFGkpaWRnz8OW0sL2O1EqBQUuuvf\nrVZjSkomNiGhzz1o/H4f7e27yMtr5Jpr5hAbGzvk5TtepICHEW63m82bN9PR0UFVVRVOpzMk3jNm\nzCDuywUlgUAAi8WC1+tFURSMRqPcvGoQaG5uZvv27TgcDvbt20dsbCyZmZnEx8czY8aMkDi4XN02\nNr/fj0ajITY2Vjp+BkhzczM//vHD7NkTi8eTht3uIDY2BoPBwNixYykomBKK63I58Xp9CCHQaDQY\njQZUqv47La2tFSQkVPGtbyVSVDSl33gjETmJGSY4nU42btwY6vmpVCrGjRuHyWRi5syZPfb6VqvV\nYWGBCidqa2vZtWsXXV1dVFRUkJaWRnJyMunp6UyZMqWHp9tgMITVlqThwFNPPYXLVYbLFUlXl5Ok\npIkYDAYmTZpMbm5uj7gGg5GQS/AItLc3YDQ2Mn68oLBw4gnI+YlF2gjDBL1ej0qlory8nIiICHJz\nc4mPj2fOnDnyQw1DQHR0NDabjYqKCkaNGkVycjJjx45l6tSpckHOEJCVlUVzcz2wi5SUZtRqK4WF\nhb3E+2gRQmC11qHTVZKX18U550wPyy1+ZQ88THA6nXg8HjIzMzGbzaSkpDB16lQ5nj1EtLe3o9Fo\nyMvLIyqkv1FDAAAgAElEQVQqismTJ5OdPfDPekmOjMvloqqqisTERMaMicbl6mTMmDiCwVa6ukyY\nTMe2nsHrdWGzHcRkamHsWAcXXzwzNPwYbkgBDwNaW1vZvn07p512GrGxsdTU1JCfnx9Wn34KZ8rK\nymhubuaCCy6gra0NnU5HcnLycGfrlMBisXD//feTlpbGBx98wPLly7n44otpamrjv/+tprbWRWtr\nAnp9MpGRsYd9G3K57DidLShKC6mpHUyZYmD+/Dlh/YUqOYk5wqmrq6OsrIxp06bJicghJhgMsmPH\nDjweD9OnT0d3FPutSwaPyspKHnroIWbPns3VV1/d67jFYuG//y2hstJNa2skdrsRlcqEEEZUqm6b\nrBABwIkQTjQaGwkJLpKTPcyZk8v48XlhP/wlBXwEs3//fmpra3tNUkpOPF6vly1bthAREUFhYWHY\n/6GHG5s3b+aZZ57h4osv5oILLug3nhACi8XCnj1V7N3biNOpxuHQ4fV2Dy2qVILISA8Gg4+MjBgK\nCrLIyjp5PjAtBXwEcmihjs1mo6ioSC66GWIOLdRJSUkJmz0xTibef/993njjDa699lpOP/30oz4v\nGAzS1dVFZ2cnXq8XIQRqtRqTyYTZbD4p36CkgI8w/H4/W7duRaVSMW3aNDlJOcR0dHSwZcsWcnNz\nycnJGe7snHK8+OKLrF+/nptuuom8vLzhzs6IRwr4CMLtdlNcXExcXByTJk2Sk5RDTHNzM7t27aKg\noEBOUg4xgUCAxx57jJqaGm677Ta5juEokQI+Qujq6qK4uJicnBzGjh073Nk55aiqqqKiooIZM2YQ\n08dGVJITh8vl4oEHHiAYDHLbbbfJ+Z5jQAr4COCQTXDixIlyBd8wcMgmOHPmTLnkfYj5uk3w17/+\ntRwyPEakgA8z0iY4fEib4PByJJug5MhIAR9GpE1w+JA2weHlaG2CksMjBXwYkDbB4UXaBIeX47UJ\nSnojBXyIkTbB4UXaBIcXaRMcXKSADyHSJji8SJvg8CFtgicGKeBDhLQJDi/SJjh8SJvgiUMK+BAg\nbYLDi7QJDh/SJnhikQJ+gpE2weFD2gSHF2kTPPFIAf8aHR0dWCwWGho6aWrqxOXyARARoSU52URa\nWgwJCfFHLcTSJjgwhBDHPU8gbYLHhtvtpqmpiZaWTurrO+jq8uD3+9HrtcTFRZKWZiYxMZbk5OSj\n6kVLm+DQcMoLuBCC2tpadu6sYv/+Djo7jTidWtxuNX5/t3ioVGAw+DEa/URHOxk71kRhYTZZWVl9\nNmZpEzx2AoEAjY2N1Ne3Ul/fSVubHb9foChgMulJSzOTlhZDZmbGEYdBpE3w6LFarezZc4Bduxqx\nWvV0dWlxuzX4fCqEUAF+IiIEBoMfk8lDWppg6tRs8vJGY+jnw5PSJjh0nNICbrPZ2LBhFzt22LBY\nTHg8MahUCej1kej1RjSa7j2DAwE/Ho8Tj8dBINCGXt9OXJyDSZN0LFxYSGxsbChNaRM8Nnw+H+Xl\nFezcWYfdbkavTyEyMgaDwRTqNXs8LpzOTpzONhSlntxcM4WFeT3q/RDSJnh0+P1+du7cw7p1NTQ3\nR2GzRQJJ6PXRREREotV2dzqCwSA+nxu324HP145a3UpMjIusLBdLlkxi1KicHulKm+DQcsoKeFVV\nNe+9t5vq6ki6upIwGjMxGqOP6lyXy47dXktUVAtZWZ2cddZ4xo3LlTbBY6SlpYU1a0pob08iMXEs\nev2RJxgDgQBWawNe715mz05n4sT80ENS2gSPDpvNxvvvf055ObS0xKLVZmMyxaNSHbmz4fN56Opq\nQqWqIzW1ndNPT2DevOkoiiJtgsPAKSngX3xxgH/9q4zq6jiEyCEmJvW4xLajo4lgsJqMDAsLF6Zg\nt3dIm+BRsndvBWvX1hAdXUB09LF9lBbA5/PS1LSbnBwnZ545k4aGBmkTPAo6Ozt5883PKCsz4nCk\nERMzKvSmeSx0d2K+IDGxhWnTVGzZshZA2gSHmFNOwOvr63nttR0cPJiAVptPVFTv1/BjweXqwmrd\ngaJs4wc/OJ0zzpg1SDk9edm7t4JPPqknLW1W6FX9eGls3IvPt5m8vFTOOOMMaRM8DC6Xizfe+C+7\ndhlwOnOIj88eUHp+v4+Ghl2Ul79KQUGAp59+BI1Gfid9KDmlpubdbjcffVRCTY15UMQbwOcT2Gwx\nKEoBJSVt2O32QcjpyUtLSwtr19YMingHg0EcDg1VVWY0migp3kfg889LKC/X4nBkDli8AWy2Lnbt\nasBonMmoUWdTU1M7CLmUHAun1ONy8+bdVFRo8fkyiI/vW7y7RcGOrbUNn9sFAjR6PaaEBEymqB7j\nhG1tbdhsNsaMycfhiKO6ej8bNuzknHPmDFWRwgqfz8eaNSVER0/pV7x9Ph+W5mbaGxrwez2oVWoM\ncXEkpadjMpl6xKuoqECn0zF79vmUlW0gL69eLpTqh9raWjZvbqOtLY2EhP7F2+Vy0Wltw2PrIiiC\nqDVajLGxmGNienwIuL6+nm3btjF+/DgyMtJobCzh44/3cOWVyfJBOoScMkMoRUVFuFxq5sx5kri4\ngj4nbLq6bFgOVqHz+TBrNOi13c83r99Pp9eHW6slPjMTc2wsLS3NeDwe0tLS0Wg0BINBrNY97N9/\nP/v2baCurm6oiziiue222wgGFXJyLiUj47Rex4PBINWVlVgrK4kTggSjEa1aTVAIulwuWvx+lLg4\nRk2ejEajYd++fcTGxpKZmQmA3d5BMLiZqCgXK1eu5LXXXhvqIo5YhBDk508kGEzgoov+0+dkvdfr\noam6mkCnDbNaTaROh6IoBIJBurweugBjUjLJaalUVh6gvLycqVOnkpGRAYDVWoPZXEFLyz/46KMP\nqampGeJSnpqcEgJeXl7OpEmT+OUv38brnU5sbBqbNv2BtWt/SzDoRa+P4YLzXyPCEUea0Yhe2/ek\njtfvp95up8ZZTvHmX+D12lCrdSxYcB+zZt1EZ6cFg2EPf/rT2axc+SpLly4d4pKOTCwWC4WFhdx9\n9zOYzYs4eLCEV175HQcObEelUjNx4nwWn3EdsV0eRsXFo+5n0U1bVxf73W7aFTdr1z5KVdVOEhOz\nuO66v1BQsJi6umIuvDCDc845h1dffZXJkycPcUlHJhs2bGDevPlcc80m0tKKehx76aVFVFevZcqE\nG5k7+grMxsg+0wiKIJYuO+Xt7TS2bqO29sEe7X/GjF/R2bmFwkIrP//5d3nllVdk+x8CTokx8P/7\nv/8jKSkNtzsRkymRtrZ9rF59M0VFv+Dmm7uIiRnDv//9HTKiIvsVbwCVohDhdrNx4/XExIzm//6v\nk6KiX7J69S20te3DZIrDZotk9OgJ3HXXXUNXwBHOihUrWLBgAR5PInq9EYejg3PP/Ql/+1s1f/tb\nNQE//PO1WxmbkNiveAPoVSqMFgvvvPUb8vNn8uqrVq644n5+//vvYrO1EhmZTUlJFZdddhl//etf\nh7CEI5ubbrqV2NgMoqJG9wg/eHANdXWbABVR0K94H6K9oYFgYwPVVfcSH5/bo/13dh4Akmlu1lBU\nVMTdd999wsoj+YpTQsA3btzI6NGnoSixaDRa1q+/F53OxJln/gGdLoqFc5bjD7iosWzqNw2f30er\nxUKbs4RA0MW8mX8hIiKaxYsfRq+PZt26e1Cp1KhUsYwfP4fy8vIhLOHI5sMPPyQ/fyJ6fbc3eNq0\nc5g9eykGQxQqlYaC7PlUNx++vjo6OmhubibKrKGl7QCLF/8UrVbP7NkXk5NzGhs3vonZnERtbSdz\n587lvffeG4qihQW7d+8iNXUakZE97ZVvvXUJ06b9CkVR0B/GPeIP+CktLcXr85Ke7sYfcLJ06b97\ntX+dLgabTc8ZZ8yX7X+IOCUEvLOzk9TUSUD35EpLSylmc/dEjtfrQeUEtUpHVXPfAu7xeGhtbSPa\nHE1zVwlqlZYInx6XywmA2ZyNxVIKgKJEkpo6g0AgQFNT04kvXBiwe/duIiLiewkIgKW5GUv9brKT\nxvR7vsViob2jg6zMTFq6Gkg2p2Gra+TQ6N+oUQXU1OxBpVIhRBTp6elUVVVJRxDdbdfpdJCSMg21\n+iuR/uijG1GrteRlXXXY891uNyUlJWi1WiZNmkSddRsqRYvP89Uc0qH2r9dH4nJpyM6egN/vl+1/\nCDglBFwIgUoVh0bTLeB+vwudrnuxgcPhJBJQq7S4fV29znU6nVjb24mLjcVoMOL22VCptJhUCg67\nAwCdzoTP5/zy/wbU6iSge+Zf0t179njAYDD1Ola24798vP01rl7881CY0+mkoqKCYDBIQ2Mjbreb\nzIwMtFotbq+TKIMJlcOBy+UCwGiMxuXqvndCfHWNjo6OE1yykY/D4UAIgcHw1crIrq4Gtm79C9/+\n9j8IOhy9zmlsaqS5uZkuexe7d+8mJiaW8ePGo1JUuH021CotDqs1FP9Q+9fpIvD5tAQC3cOQsv2f\neE4JG6GiKDgcnShK9/NKqzXi9Xb3zgKBABogEPQRoTX3OM/n91FZWUn+uHx02u6tSCO00QSDftSK\nCo+ve7dCj8eGVts9fqhSqbHbuxv3IYfEqU5sbCwOh7PXjoANDV+wYuX/cvXiXzAhswDoFt3S0lLc\nbjeVByrJyspmXH5+6NwInRGnx4GG7v08oNuBcshZoSgabDYbgFyRSXf7VhQFl+urh9k//nEhWVlz\nSUmZQbN1T4/4VVVV1Dc04PuybY8ZPbpHO47QRhMUPoI+fyisZ/vXYrG0ALL9DwWnRA/cZIqmoaEM\n6H7lTkycTGdnNQAqlYLDYyUQ9JKTPLPXucFgkP379+MPdDfYnOSZBIJe7C4LKnV39XV2VpOUNAno\n7u3X1GxFrVbL/SC+5LTTTqO5uaetsqWlmjvuWMKiOdcwZ8JZX4a1UFJSgtPloqmpiXZrOwcPHKCh\noSF0XlbiaJo7GnD7XCFRr6raRWbmxC9jCCorK8nJyZFLuunuvERERNLcvCMU1tpaRlXVWh55JJKX\n3p+BEAFWl/yBv31wOfUNDbjdLuxdXQQCfgKBAD6/L3Rud/v34fJ91QP/evuHIPv370aj0cj2PwSc\nEgI+Y8Z0amt34PN5AJg79zd4vV2sXn0zQrh4b9uv0GkiGZMyF4D/bPkN9/9zAlpN97ifSlGxu2Q3\nXq+XMSlz0WqM/GfrrwjiYfXqm/F6u5g7906ge7Ofysp1jBo1ut/8nGqcd955VFWV4/F0D3m0tdXz\n298u4vzzf8b8RdfQ5XJRU1NDWVlZ91awX6zl/foXSUlNISUlhdraWioqKgBIj88iJ2ksH259FUUR\nbNz4FtXVpcyefciy5mTbtm2cd955w1TakYVer2f06Ilfuk26ufba3Vx33W5+9KOdnDP/dRRFxZi4\nCxgTuQyHw8G+9hXsdN6E0WhEUSm0trbh9XoButu/2sD64l/idnf0aP+BgB8hvOzevVlu4ztEnBIC\nfu+992K11uF2d0+qxMfns3jxw2ze/DhPP52JzVnLd2cvD8XvsNcRbUwFQK1WM2HCBEzRJnaV7MLu\ncPC9M/5Kp7OOZ5/NYfPmx1m8+A/Ex+cC4Pc7qK8v4aab/m/oCzpCWbZsGaWlW2lv767/jz76G83N\nB3nttbu47Y4Z/PTpC7nxlStwOBy0tLQgdAEyYkaTnp7BtGnTmDptKh2dnZSWlhIIBrnmzFuwdNZw\n1VXJvPzyb7j11jeJju7+yIYQnbz77rtcd911w1nkEUNUVBSXXnoNnZ1VuFztAMTFjSExcSLJyZNJ\nzJiGEIDfgNfVPakfUNswqOPJzc0lOyubuNhY2qxWnF9O2i+e/gRdjmr++MfYHu3f43FiNPrZv7+U\nO++8cxhLfepwSizk8Xq9jBt3GkKkcfnlq3vtPGhtbcV98CBp5u4x8Ef+NZOLiv7A2LR5PeLV1dVR\n31BPXFo6MRMmkNjHK+L771/HgQNvUVNzoMfS71Od66+/nrY2M1de+ftQWCAQYP369ewuLibOaiXo\ndJKYmMh7X7zMsvk3MGPyVxuD+f1+9pSV4fF4UGVmMm7Rol5DJC6Xnc2b/0Rd3R65EvNrvP/+Z/zw\nhz8hKmoUl176TijcarWyYcN6PM3NGB1OFAWio6P5rPGXnFd4P5PHnBOK6/P5aGtrQ9FqcJpjyBk/\nvvffkbWOkpJbqKxc22PYS3LiOCUEHOCDDz7jk0/8BIOn9drEKhgMUld1EL21nSST6bBby+6rqaak\nqYnC02f12jbW5bLj8+1g3jw/S5cuPCHlCFecTicvvbSB5OTFqNVqPB4Pa9eupbW1lerqalwNDcxI\nTCQ2Kopx48aRlJTUKw2vz8cnO3ZgiYzk4u9+t9cDsqFhD3PmqJk4cdxQFSss+OKLSl59tZLGxlwS\nEkYB0NjYwJYtW3C53FiamogVgixzNJFGI+PHj8cQ0ftrO10uJ+UWCyl5+WRmZvb4OxFC0Na2k7y8\nWn70o4Wy8zJEnBJDKACnnZZDfLwDt7u51zGVSkV6dg6++Hhqu2x0uZwERTB0XAiBw+2iztaJITuH\nOYvPZO/eckpLS3uk43A0Ex/vYsqUge/0drJhNBrJzTVjtdZjs9n44IMPaGlpoaqqCp/PR1ZBAY0G\nAzFZWURG99yrwx8I0NTRQbnNxuQLLmDq9Ol89NFHWCyWUJxAwA/UMXq0rPtvkpWVSVKSB0Vpxv+l\ns+rzzz/H4XBisbRgiIpElZSIXR9BxqjRaLU9P/7s9flo7urCoihMOH0WKpWKhoYGgsFAKI7d3o7R\naGPChAQp3kPIKWEjBEhNTSUnZw8tLS10dSVhMsX1OK5Wq0nPycGekECnpQVLeweaLzsYfgGa6Ghi\nckYRFRWFSqVi/vz5bNjwGU6nk+nTp+F2O9DpmsjI8JOVJe1TfVFYmMemTasoKxO4XG6qq6vRaDSM\nHj0as9nMnKVL8Xg8fFFZidLWhlZRCAJulYqYrCzGZGQQFRVFJhAZGcmnn37KzJlFZGVl09RUzrRp\nKf1+p/FURqfTMX16NnV1zWza9AmNjd0e+vb2DkwmEyZTFGlp6UyZUoCtvZ2q5hZ0IogCBICgTo85\nO5usmBg0Gg2RkZG0tDRTV1dHWlo6KpWC211NTk4Xp502bbiLe0pxygyhQPcnt158cTOVlanExhYc\n9kskPp8P/5f2KbVag06n6xXH7Xbz2WcbUKvV5OWZyMtr4bLLJpGVlXXCyhDOlJaW8tRTy9m3L4q2\nNg0mk4m0tDQSEhJYuHBh6OPPQghcLhd+vx+VSoVer++xlekhmpubWLduPaNHZzBunIOlS+f3GU/S\nvU3sD394K1u2dH/MweHwERsbg8FgYOzYsRQUTAnFDQQCeL3eLxfAddd/X8OKVmsbnZ02DAYfSUm1\nnH22iXnzinrFk5w4TpkhFIDk5GTmzcsgKamN9vb9X752941Wq8VgMGIwGPsUb4CIiAjmzp2Hx9PI\n7t2vM26cRop3P+zZs4fXX38dRQlisfwXg8FLWloaGRkZLFmyJCTe0O1dNhqNREdHExUV1a8oJyen\nMGfOTEpL30QIixTvw/DEE09gtx/E7d6Gzbab2FgTkZGRnHbaaT3EG7rfRg0GA0ajkYiIiH7nhOLi\n4tFqfbS1FZOaaqWoqPc2wZITyykl4ADTp5/G7NmRJCQ0YLXuDXnDjwe/30dnZyVnnpnMtGkq/vOf\nf3Lw4MFBzO3JQ25uLn6/n7KyMoqKJpCVZSU+XsW8efNCHyU+Vmy2VmAvv//9T2hvb2flypWh1ZmS\nnixevPjLpe0VZGS0oNPVM2nSOHJzj+/L8UIIrNY6YmKaOf10LcnJ3R84kQwt6rtOsX1PVSoV2dmp\nCNFCe3srVmsnPp+WiIhjW7XX1dWGw1FOUlITM2fquOGGKwF49tlnyc7OlqvQvsGmTZuorq4mNTWV\nmJgYLrnk2xQUpFJRUYFaHYNOF3HUaQUCfhob92A0VnL++YVkZGRQUFBASUkJW7ZsYfz48bI3/jVc\nLhfPPvssarWa1NRkEhNVzJo1lYgIA3Z7AL3e1Gubg8Ph8Tjp6KjAZKolL8/OZZfNZdy4cezcuZNg\nMEhcXNyRE5EMCqfUGPjXCQQC7Nmzj7VrK6mpMeFwxKDVJhMVldDv2Hgg4Kerqw2/vwWj0Up6uo35\n87MpKJgQ+pjrunXrWLFiBZdddhlLliwZyiKNWN555x2++OILli1bht/vp7W1lUmTupde19fX8+mn\ne7DZzERGZmM2J/UrJi6Xnfb2aqCOqVNTmDJlQg+hDgaDvPvuuxw8eJBly5ZJIaF7J8f777+ftLQ0\nfv3rX7Nq1SqmTp1KMBjk4493UVGhpa0tGkVJIioqCb2+78+hBYNBXC4bLlcLarWFlBQbU6dGMXfu\nlJDrxO12U1xcTFxcHJMmTTqsHVcyOJyyAn6I9vZ2Nm3aQ3m5ldbWSGw2PUKYUBQjQmi+bIR+hHAg\nhJ3oaA8JCQ7y8qKZPXsCiYmJvdIsLS3l8ccfZ+HChXz/+98f+kKNEPx+P6+99hoOh4PLL7+8371J\ngsEgjY2NlJRUUVvbiRBRX94DDd371zgRohOzWc1pp2UwenT2Yd0mn376KZs3b+bSSy89peckKisr\neeihh5g9ezZXX311r+Mej4cdO8rYtq0OiyUCqzUCny8ClcoE6AEFCAIuAoEujEYPCQlO0tNh1qxc\ncnPH9BJpv9/Ptm3bUBSFadOmHffwmOToOOUF/BCdnZ3s319FZWUrLS1duFxaAgE1QoBaHcRg8JGQ\nYGD06ETy87OP2Lurr6/nwQcfJD8/n5/+9KenXEO22+289NJLmEwmLr300tAbypEIBAJ0dXXR1dUV\n2kkvIiKCmJiYHhOdR2Lnzp188MEHXHjhhUycOPHIJ5xkbN68mWeeeYaLL76YCy644LBxPR4PVVU1\nlJU10NjYgcOhxufTAAqKEkSn82E2q8jMjGfixEzS0tIOO+QihKCkpASbzUZRUdEx3TfJsSEFvA+8\nXi82my20paZGoyE6OvqYG2JnZycPPPAARqORm2+++ZTxKLe2tvLiiy+Sl5d3RPE4kRw8eJDXXnuN\nBQsWMGvWrCOfcJLw/vvv88Ybb3Dttddy+umnH9O5hx6gbrebYDCIWq0mMjLyuHZ2rKiooKamhpkz\nZ8qdIU8QUsBPMF6vl0ceeQSLxcJvf/vbk35ctrq6mpUrVzJnzhzmzJkz3NmhpaWFl156ifHjx58S\nOxS++OKLrF+/nptuuom8vONzmAwmdXV1lJWVMW3aNOLj44c7OycdUsCHiOXLl7Njxw5uueUWRo0a\nNdzZOSGUlpby7rvv8q1vfSs0STkSsNvtvPjii8TGxvI///M/Rz2cE04EAgEee+wxampquO2220aU\nC6q1tZXt27czceJE0tPThzs7JxVSwIeQt99+m3fffZdf/OIXTJky5cgnhBEbNmxgw4YNXHbZZWRn\nj7z9SLxeL6+99hput5srrrgCo7Fvt0U44nK5eOCBBwgGg9x2220jcriiq6uL4uJicnJyem0CJzl+\npIAPMSejzfDrNsGEhIThzk6/nIw2w2/aBEfyZLm0GQ4+UsCHgZPFZni0NsGRxsliMzySTXAkIm2G\ng4sU8GHikM1w7Nix/PznPw+7hny8NsGRQrjbDI/FJjjSkDbDwUMK+DASrjbDkWITHCjhajMciE1w\nJCFthgNHCvgwE242w5FmExwo4WYzHGk2wYEibYYDQwr4CCEcbIYj1SY4UMLBZjiSbYIDRdoMjx8p\n4COIkWwzHOk2wYEykm2G4WATHCjSZnh8SAEfYYxEm2G42AQHyki0GYaTTXCgSJvhsSMFfAQyGDbD\nQCBAIND90VmNRnNM+z0fIlxtggNlMGyGPp+PYDCISqU67r3Jw9EmOFCkzfDYkAI+QjlWm6HP56Ou\nro66OiuNjZ00N9sIBhUUBTQaSEuLJS3NTEZGIqmpqUfs3YS7TXCgHKvNsKuri5qaeurrO2hoaKez\n0w2oUZQgUVFa0tPjSE+PITMzDbPZfMT0wtkmOFCkzfDokQI+gjkam6HT6WTPngq2b6+hqUlHV1cE\nTqcKISJRqbp7fsGgG7XajdHoJybGSVaWhmnTRpGXN6bPB8PJYhMcKEdjM2xpaWHHji8oK2ulrS0S\nu12Dx6NFpYqk+4uFQQIBJxERPoxGH/HxDsaPj6OwcAypqal9pnmy2AQHirQZHhkp4COcb9oMzWYz\nlZWV5OXlcfBgFR9/XEpVlYGOjkggiYiIOPT6yF5fFfJ63Xg8DjweC2p1KwkJDiZOVLNoUWEP+9bJ\nZhMcKN+0GVqtViIiItBoNGzfvof16+toaorG6YxGrU4iIsKMXm9ApfrqwRgMBvF4nLjdti+/5tRJ\ncnIXZ5yRwowZk3v0ME82m+BA+abN0GazER0dPdzZGjFIAQ8TDtkMExMT+eKLL5g5cz5OZyaNjTEE\ngxlER6f1+ym4b+L1urDZajAam8jM7OKcc/IZNy73pLUJDpRDNkODwUBrayuBQID4+DxqakxYLLFE\nRIwiKiruqCbdhBA4HB24XAdJSLCSl+fjwgtnYTKZTlqb4EA5ZDNMSUmhpqaG7OxsOcn5JVLAw4gb\nb7yRt956i/j4DNzucaSmns2UKediMBzf62Vnp4VA4ABZWa2kpXVSX19z0toEB4rT6eSnP/0pDQ0N\naDRp2O3jyM4+m4yMSUf94Pw6gYCf9vZqIiPrGTPGSmXlBvR6/UlrExwo9fX1vPDCC8THx5OWlkZy\ncrKc5KR7kE4SBlRWVtLQ0EBCQiqVlZG0tWVRXy/YvbuMYDBwXGmazYkYDBP4739tvP76Ts4991wp\n3v2wfv16UlJSsNm0lJUZ6ezMpKami46OjuNKT63WkJAwBqs1hb/9bSc1NW5uv/12Kd79UFFRQX5+\nPlarlYMHD9LU1MTGjRvxeDzDnbVhRX3XXXfdNdyZkByZuLg4bDYbGze2EgwW4nSaEULB43HT1tZG\nWu0yey8AACAASURBVFr6MfdGgsEAe/fux+3WMWrUGMDK+PEZx217O5nJyspiw4Yt1NTE43bn43Rq\n0Ot1WK3toU/uHStWq5UtW3ZiNmcwZcokDIZOcnLkSsS+iI+Pp7W1FbPZjMViCc1FNDc3k5SUhE6n\nG+4sDgtSwMMEv99PebmFtrZRtLfHYTSa6OzsxOfzEgwKGhsbSUtLRYggVksrzQcO0Fpfh7Wxka6O\nToJqFVqtLuQH93o97Ny5C7VazdSp03C7VbhcNgyGDikifdDa2kp5eYD6+kTU6jQ0Gg2trW1oNBrs\n9i78/gCxsbE4nQ4s9fW0VFfRWlfP/7d3n8FxnXee77+dE9ABjUYDjQwiEGAOIERBtCRK8lqSx57r\nHV/b5TBez3g9d+rWVs3Y+8L2BO3EvdeeN767WzV3Le/Ycvasq3bG46uxJUuyAkWKEhMSkUOjERro\nBjrH0/cFCVggIJEiAXTA//MS6D58zinw16ef8zvnCSwsEI/HUet06HS69Xnb2dlZzp8/z/79HRw5\n0s3ycohodIHmZotcpNuCwWDA4/EQCASwWCyEw2Hm5uYoKytjfn6eiooKtFot01NTXHnhBfpfeYXh\nixeZHBwkls1iLisryTqizIEXuNOnT1NTU8Pf/d3f8cwzw8zNtQB2zp17nXg8ztLSEmq1CqfTiToH\nHTU1VFvM2EwmtGoNOXIk0xlWk0kiajXOxgYMRhNXr17F6aygre1G00FRsgQCl2lvn+X3f/9hcrkc\nLpeLq1ev0tHRkd+DkEdrx//zn/8izz+fIZM5SCAQYWJigng8jt/vx263YbFY0GUVGux2HHodZUYT\nGrUaJacQTSQIptNkLRZqmluYmZlhcHCQ48ePU1dXB0A0ukIud5WHHsrx4Q8/SCgUkuO/hbUbfRYX\nF/H5fPj9ftrb24lGIhiDQfZbLDRardgtFlQqFYlUiplgkOlsFvvBgxw7fbqk5s0lwAvY4OAgBw8e\nJBgM8txzb/HccymeeeazRKN+crkMdXVfwGJ5hOXlJbLJJI1GE5VmMx3t7Tgcjk3bS2czjPqXmE0k\naN+/H5/vH3n55b8kmVxBpdLyh384hdU6yEc/6ubYsYM8+eSTLCwscPHixd3f+QKwdvynpqb48Y/f\n4he/GOf5579KJhMHcmg0Zurq/h25XDeGTJYmixmn1UpTUxNazeYbn8KJOG9OT7OaVXjggQcYGfn2\nhuP/hS+8RUvLJL/3e71UVFTs+eP/TnK5HNeuXWNqaoqlpSWuXr6Me3WV02437fu27tcrisI1n4/I\nvn3c98gjJRPichGzgH3pS1+ivb2dXC7H+HiYVMpObe1pPvCBb6BSqWlra6eiwoGt3EoVQDRKKpVk\n6PoQc/Pzm7YXjUTILi9RpdPhdDoxmZwcOPAJDh78NADl5S5WVkxcujQFwFNPPcVbb71FLBbbxb0u\nHGvHf2UlzPKyBbe7l0984v/jK1/J8JWvZGlsfISpqf+GzWjCkUqRjsaIRWOMjY2RSqU2bEvJKcxO\nTmFYWaWtvg6Hw7Hp+Gu1VSwvW5iY8AJy/N+JSqXi8OHD7N+/H6PRiDMcxhOPk04mmZmZYXJyklvP\nS9VqNUfq6jCNjjJw5UqeRr79JMAL2Llz5zh79iwrKyvEYkYMhmo++tH/yYkT/wegQqfT8b73PYjd\nYMBttWI0Ggmthkgmk0yMjzM9Pb2+Lb/fz5xvjuamZhoddlb8fo4c+V2eeOK/UVV1o/Ot1epQqays\nrGSJRqN0d3ej1Wp5+umn83QE8mvt+Hu9QSIRHVVVXTQ2vg+1Wo2ipDAYjGg0BmwqFW6nk1wux8rK\nCol4grGxsfXgzWQz9PX1kUqnOHn8GDYlRyQS2XT8zWYb0agOr/dGs2WvH//baWtrw24ycbCsjOam\nJpZuXtxcXFxkZGRk/VlAb3eguprZy5dJp9N5GPH2kwAvYCsrK5w+fRq/f5VIRItOZ9n0mlgsxuGm\nJmqrazCZzVgsFiLhCPFEnBmvl+HhYbyzswQCAVpaWrBYLJSbTCQCy1v+EedyFuJx/Xo9zul08tJL\nL+34vhaitePv8wVJJnXo9TceZfBXf6XiP/9nE0NDP+XRh77J4eZmzBYzNrsNjVZLcCVIIplkYmIc\nv9/P1atX0el0HDx4EJ1Wh12vY2VhYdO/p9cbyWQMzM4GURQF2NvH/3aSySQqv58zJ05gNploaGgg\nHA6zsLBAMBhkaGho09+4QaejKpXC6/XmadTbSwK8gK1dSFxejpJMatYD5O1i4TBlWg379u2jsaEB\ng8FAWXk58XicWDTK/Pw8sWiUfa371q/Cq1VqzDm2/GquVpuIx3XrvzMajQSDwZ3d0QKVy+Ww2+1E\nIhk0Gst6g+dP/iTHH/+xn5qaEzz3wu9htZhpaW6hvNxKeXkZapWa+bk5IpEog0OD2O0OOvd3olbd\neL/ZYCQZCm3Z31erLSQSKhKJBLC3j//tBINBKhQFZ0UFnV1dmM1mamtrWVhYWP8GlM5kNr2vxmRi\naXJy9we8AyTAC5hKpcLv95PJKORyqi1vHc5lMuvBUFdXR1tbG0aDAavVSjqTxmw2s2/fPn708r/n\nr37cwV/9uIO/f/ZJNCrI5ZQt/k012SzrZ4CJRKIgnoudDyqVisXFRW4cio3/VczmSj7zmVfJKkmG\nfc+h0WhobGxAo1aTTqfJkcO/5KehvoGW5ma+9+Ln1o////uvH0SjAkXZqj+gJpdTy/G/A5lMhrU7\nFswmE+0dHQRXVjAYDMzOzmI2mzFv8QA4nVZL5pZrFMVqbz0jtMg4HA7OnTvHhz60D5Uqu+nCDIBK\no9nwc5fLhd6gZ2xsjKNHjrLoX2R6ZoaPn/nvG668z0fCqFSbP79zOQW1mvWzzeXlZR588MEd2LvC\n53A4eOONN+jo+ACwVdjeOLsz6MpvXKT0zpJVFHQ6HclkEpfLRSqVIhqL8smHvrXhnSPhMGr1Vs/y\nyKFW59Y/rPfy8b8djUbD2vl1MplkbHQUh93OnM+H2+0mHo+TTCY39b8z2SzaErnxR87AC9iZM2f4\n1a9+hc1mQq9XSKUSJBIhEokb89PpdBSVLkP05tfEn73xVf76J13YrDaOHzuO1WqlpaUFtUrF+MQ4\n6cyN+UAlpxDLgcGgJ5FYIZNJALmb217CZMpgMpm4ePEimUyGz33uc/k6BHl15swZXnzxRSwWDYoS\n47XX/i/6+39EJpMiFPLyrW/1oNEYqCw/yuTkJL8a+Bv+efBjZBWF6ppqLGYLDoeDSDhCKBRa324s\nmURfZgFUWx5/g0HBaDTu+eN/O3a7nYBKRTgSYWBggEgkwvz8PJWVlbS3t7/jHcWLsRiOEll7U87A\nC9jXvvY1Ojs7MZlUWCxp/P4Y3/hGE7ncjbnTF174Mi+88GUe6f0fuCw9rES8WM0bO7BqlZrGxkZ8\nPh9jo2M0NjWRySnoHXYuX/57nnvuj9Zf+/Wv29Dry/mbv/kuNpuNL3zhCxw/fryg1ofcTWvH32rV\nYjCkCIV8vPTSn5HNplCp1JSX1/GhD/0vrg2NYQXCcR96tQOH3Y5Op6O+vh6r1Uo2myUQCBAIBrDb\n7awkk9jq67lw4RtbHP8ynnzyn9FoNPz5n//5nj7+t2MymVC53bz04os4bk6bWCwWKisrKS8vp62t\nbVOIpzMZfBoNZ5ua8jPobSY38hS43t5eHA4HDz74B0xMtOB0dm16zeLcHNlZL9//1fv58Kmv0ep5\n35bbWlpeYmF+AbXTSf3x45senJTNZlhZucjx40t89KNnqKqq2vN3Avb29mI0mujp+QrhcBd2+28e\n8xqJRLhy5QrLXi/m0CoXF/+YE57/QJ2zh8bGRsym3wSvklNYCa4QSyaJ223sO3RowzPD4cbTIU2m\nPj7yEQednS1yJ+ZtTE9P8/LLLzPz/PPUxWJUuVzY7XYcDgctLS1b3qzTNztL+uhRjpXIQhlyBl7g\nXn31VXK5HD/+8fPMzCyTSEQxGjfWCSvdVXgjET79yHO43uVpdna7g8VEkplYDEcksinAw+ElbLYY\nhw/XYbPZ9vyT3uDG8Q8Ggzz99MsEg4vAjQAPBAIMDPSTSCQJpVPEUfFI6zexWcpoamra9HAltUqN\n0WxmOpXEZDSSzSqbAjydXqS2NkpLy0GsVqsc/3dx/fp1hoeHURQFn9mMOp2mqawMd3U1DfX1W17w\nH5qbY6mmht4TJ/Iw4p0hAV4EVCoVx483MTQ0zsLCAkZjy4bfq9UaaltamJueZmp5GbtGg9VsWm+n\npLMZVuNxQjmo6uqi1mikv7+fRCJB082vkoqikE7PU1kZob39+G7vYkFzOBy0t9vxeleIRlcIheIM\nDw+TTCZZXFzAYrFQ5q4mnExSXVVFWlF4e3zHkklWUyniOh3tJ7tJpZLMzEzj8dRiNBoBiMfD6PUB\nmptNuFyu/OxoEVAUhStXruD1evH7/Xi9Xk739hJeWWEyEKBCqyWWTGK5eVyz2SyzgQCTySSalhbu\nf/jhknrapgR4kWhsbKChYYTlZR+xWCVm88Yn1mk0Guqam4m53awuLeFfWkKTy5FTARot1tpa6hwV\n62eGx44d5erVqyQSiZu3i/uw2wMcPlyJ3W7Pwx4WthMn9jEwcJlXXnmFYNBMIpFgaWkJu92O1VqO\n2+2mvb2DSCTC0vw8c5HwzRUxQWu2YK+vw221olZrMJvNaDRafL5Z3G43JpOZSGSS+voQ3d23X0B5\nr8pkMrzxxhssLS3h9XpZXl6ms7MTi8XCww8/jNVqZWp8nFcvXyYXCKDK5choNFTt30/n/v0l+cEo\nc+BFxOv18r3vXWFiwkNFxSE0WzwwaY2iKOtdYo1Gs+VXynQ6zbVrV1GUNE1NaY4cWeVTn3pIFhXY\ngqIo/OVffp1f/CJCINBELKalsrISs/nGHYAtLftueX0WRVFQq9WbpkrWxONx5uZ8qNUJamsXOHvW\nwGOP3S9LhW0hHo9z4cIFVlZWmJiYIJlM0tbWRllZGd3d3Rse3pbL5Uin0yiKgl7/m0col6LS3bMS\nVFdXx333uXC5/AQCw++6Eo9arUar1aLVat8xEHQ6HZ2dHaRSE8zO/oLTpxslvN/BL3/5SxKJINns\nFSKRS9jteiwWC+3t7ZvCG25Ma2m1uncMb7jRorBa9cTjlzGbJ+jtPSrh/Q7Onz9PIBBYn/fu6OjA\nbrevX+R/O5VKhV6vx2g0lnR4gwR40entPc7p0yacTi/Ly4OkUvG73lY0ukI0OsjZszYeeaSG5577\nBYuLi9s42tJx5swZ5ufnMZvD7N8foaxsmtpa610vPqwoCoHANEbjJGfOWDh2rGo9nMRmra2tDA0N\nYTabaW1txeVy8cADD2CxbH4+0F4iUyhFKJ1O8/zz57lwIcLCgh21uhGr1X3HZxuZTJrV1Rl0ullq\na1d59NFGjh8/xLlz53jxxRf5+Mc/TnNz8w7vRfFIpVJ873vfIxQKEQ6HSafT2GwdLC25CQQqsVia\nN12TeDfxeIRIZAK73U9LS4zf/u1TOJ1OLl26RDqd5uTJkyV1oe1era6ucuHCBcrKylhaWqKmpoZj\nx46VzDO974UEeJFSFIWhoRGef36YmZkyQqFyNBo3ZnMFBoN501dxRcmSSESJx/3AIg5HlPb2NI89\ndgSPx7P+uv7+fv7pn/6Jxx9/nKNHj+7yXhWeUCjEd77zHaqqqvid3/mdm2fhZgwGA6++epkLFwIs\nLJSTSjnR6aowm63odJuX7spkUsRiIVKpRXS6ZdzuECdP2untPbI+bZXL5RgYGMDv99PT04Npi+d4\n7DWLi4tcvnyZQ4cOUVNTw9LSEk6nU6aabpIAL3Krq6u8+eYQfX0L+P1mQiE9yaQBjaYMWDtDSaMo\nIUymDA5HArc7zYkTDRw82LHlOoHT09P88Ic/5NSpUzz00EO7uTsFZX5+nmeeeYajR4/y2GOPbfma\nqalpLl4cY2wsyvKymWhUi6JYUKlM3JihzKEocdTqKBZLBqczRlOTgVOnWmlqatwyiMbHxxkfH6e7\nuxubzbazO1nApqenGRoa2nSRUvyGBHiJiMViTExMMz0dYHY2QCSSI5u9ESAaTRan04jHY6elpYq6\nutuvPB8IBPjOd75DY2MjH/7wh0v+YtCtRkZG+Md//EceffRRuru7b/v6paUlxsa8zM6u4POtkkqp\nyeVUQA6DQaG62kptrZ19+2pxuVy3PYOcm5vj2rVrHD16lKqqqm3aq+IxNDSEz+ejp6dnz89zvxsJ\n8BKUy+WIx+NkMhlUqhsr96zdMPJexGIxvvvd72I0Gvn4xz++6e7CUnXx4kV++ctf8pGPfOSubmPP\nZrPE4/H1GqHRaESrfe+3XASDQd544w32799PQ0PDe35/MVq7UScajXLq1Kk98zd3tyTAxbvKZDL8\n5Cc/IRgM8pnPfKbka4bPP/88b775Jp/61Kc2XBvIl2g0yvnz5/F4POzfvz/fw9lR6XSaixcvotPp\n5CLlHZIAF3fk5z//OYODg3z6058uya/0iqLw05/+lLm5OT796U8X1N2oqVSKCxcuYLFYOHLkSElO\nZ8Xjcc6fP4/L5aKrq0suUt4hCXBxx0q1ZrhWE1QUhU9+8pN3Nd2007LZbMnWDNdqgq2trSX1d7Ub\nJMDFe1JqNcNba4KFfHZbijXDW2uC4r2RABfvWanUDO+kJliISqVmKDXBeycBLu7KWs2wubmZ3/qt\n3yroM9etvNeaYKEp9pqh1AS3hwS4uGvFWjO815pgoSjGmqHUBLeXBLi4J8VWMyy0muC9KqaaodQE\nt58EuNgWhV4zLOSa4L0qhpqh1AR3hgS42DaFWjMshprgvSrkmqHUBHeOBLjYVoVWMyymmuC9KsSa\nodQEd5YEuNh291ozzGQyRKNRstksKpUKg8GA2Wx+z9sp1prgvSqUmqHUBHeeBLjYEe+1Zri8vMzo\n6AzT00ECgQRg4cbjcHPkcnFMJgWPx0ZbWzV1dXW3fThUsdcE71W+a4ZSE9wdEuBix9xJzdDn83H+\n/DDz86DTNVJeXonJVLbpIlc6nSQSWSEW82IwLHHsWB0HD3ZsGeSlUhO8V/moGUpNcHdJgIsddWvN\nMJPJEAwG8Xg8vPHGNa5ciWC3H8RqrbzjbaZSCRYXh3E4FnnkkSO4XK7135VaTfBe3VozXFpawm63\n39XjbW9HaoK7TwJc7Iqf//znXLlyhWw2SywWw27fh93ei9vdftcXFkOhJUKhK5w920hbW0vJ1gTv\n1VrNMJ1OE4lEsFqtnDp1alsvckpNMD8kwMWuyGQy/Omf/imvvPIKRmMDJlMPp0+f5eDBg/e03VQq\ngdf7CtHoRdxuV8nWBO/V6uoq3/72t0kmk7S2tlJWVkZPTw9W650vxvxu25aaYH6UbqdKFJSBgQHU\najUmk4eZGTuJhIrLly/z+uuvoyjKXW83k1G4di3M2Jiaxx9/XML7HYyNjdHU1ITZbGZwcJDV1VVe\nffVV/H7/PW13cXGR8+fPc/DgQQnvPNA89dRTT+V7EKL0ud1uvN55Bge1VFa24vP5UBSFZDLJ8vIy\ndXV1G+ZM0+k0qVRqfVmyrb6SB4NBnnvuOZqamjl58iwTE1fYv79e5l63UFVVRTQaRaPRkMvlmJyc\nxGKx4Pf7MRqNG+qGuVyORCJBMpkkl8u943z59PQ0fX19dHd3b7gOIXaPTKGIXREMBvnRj94knW7h\n9dcvEI1GmZqawmKx4PF4cDorePDBh4jFYixOTpBcDqAFFCBnMFDZ3ExVTQ0GgwG40V559dVXOXr0\nCG1t7Td/1s/x4xm6u4/kb0cL2NqNPuPj4wQCAaampmhubsZut9Pe3k5DQwNT4+NMX7qEOhpFA6Ry\nOUweD01Hj1JbW7v+4Sg1wcIgAS521Je//GWqq6vp7j7DwIADt7uZpaUlXnzxRSKRCFNTU2g0Gmpq\nasiurtLd0ECz04nNbF4/606m0yyEwywBnsOHCUejXL58mdOnT1NXV7f+b2UyaZaWnud3f/csuVyO\no0eP8vLLL1NZeecNl71gYmKC/v5+wuEwIyMjN45hLkdmcpLTHg/NTidlb7vA6V9dZSoSYcXh4MT7\n38/k5KTUBAuEBLjYMX6/n2PHjjEwMMCPfvQagUAZP/jBf2J8/C1UKjVOZzsHDnyMhYUVkot+Tjid\nuKxWDhw8iGOLFkkyneZX/f0s6LT81oc+zEsvfYsXXvgOi4tTWK2VPPHEH9Ld/RiPPmqhra2Vr33t\naywsLPD1r389D3tf2Obn53nrrbdu1Axffx3L7CxnGxtxV1bS2tq65bTJjN/PP01McPDRR3nggQdk\nqqoAyEVMsWP+4R/+gSeffJJgMEg26yYej/D443/AN785xdNPT9PU1MbVq89gN5po1euIBALEYjGu\nXr3KwsLChm0pisLY6Cjl8Rgdjor1Ctwf/dEz/OAHKzz11LP8y7/8FwYGLjIw4APgE5/4BN/+9rdJ\np9O7vu+Frrq6mtOnT2MwGHDGYrSp1awsL7OyssLg4CDJZHLD65PJJCvz85wwGFCCQQnvAiEBLnbM\ns88+y4MPPojfv4pW6+DEiQ9w//3/FpOpDIPBxIc+9B9YXLxOtcFAY5Ubq9XK/Pw8sWiUwcFBpqam\ngBsVxKtXr5JMJOjpPkWjycSCz8dHPvIfaWk5ilqtpra2nZ6eDzMxcZnFxQiKolBXV4fD4eDcuXN5\nPhKFyeFwUFtdTbvBQGtzM7lcDp/PRyQSYWBggFgsBty4GWhgYABXVRX3HTmCyudjeXk5z6MXIAEu\ndtC1a9fo6OhgdnYFi2XzlEhf36+pdrfS29FBfX09Vms5TqcTv99POBJhfGKcvr4+Ll26hE6v58iR\nI+i0Wlzl5QQmJshms+vbyuVy9PX9msbGQyiKhVAoBEBnZydXrlzZtX0uJoqisDw0xKM3++A1Ny8S\nz8zMEL35ITo9Pc3w8DCNjY1Uu90ANBkMTA4N5Xn0AiTAxQ5aWVmhvLycUCiBwbDxrr+Jiav8+Md/\nyb953xewmUy0trbS1tqGxWKhyl3FysoKS/4lJiYnqXA6OdDVtX7Hpk6rxZjNEo/H17f3gx88BcCj\nj/47wEwikQCgvLyclZWVXdnfYhOLxdDF49jKyujo6KCiogKn00kqlaKvr4+FhQVmZ2dpa2ujoqJi\n/X1VNhsrMzN5HLlYs/0PRBDiJofDQTgcJptVUKl+c67g843yF3/xBJ///DdwaBvWzyLq6uowGA0M\nDgzidruJRaM0NzfR0tzMj1/5B/7na98G4KFDj/NQz++v3wD0s5/9F1588bv87d++jFarA9TrvwuH\nw/Io03eQzWZZm8nWaDQ0NTUxOzsL3Lg1fnZ2luMnTmxaJk+jVpOV6woFQQJc7JjDhw9z/fp1tNpK\nFCUL6FhcnOLP/uwxPvaxP+Ohhz5J//nzZG+eLQO4Kl0Yjt74Gt/R0cH8/Dxzc3P82/s/w//+wGfX\nX9cXCKDRaPjlL7/FT3/6f/O3f/trnM61h1dl1y+yDQ4O8qUvfWn3drqIaLVaUjdLaMlkkuHhYYxG\nIyqVioqKChoaGhi+fh2NWr3hRp10JoP2Zh9f5JdMoYgd88QTT/DSSy9RUWEhkYiwvDzLn/zJWZ58\n8v/kAx/49wDYaqoJ3JwKef7Kz/j9/+e3sVqtHDhwAK1Wi8fjQaVSMev1kslkAIinkqSNRi5c+Cnf\n/e5X+Yu/+AVud9P6v5vLRTCbzczOzhIIBLjvvvt2fd+LgcViQVVRgc/vZ2BggEgkgs/nw2KxrE+p\nHD58GJ/Ph9frXX+fb3UVV2trHkcu1kiAix3zmc98hp///OdUVpqIxVb5xS++ycLCBD/84VN87GPl\nfOxj5XzxPx5iCcgqCv7QAl31G++iVKvV1NTUYDabmZmZIZVKsRiO4Gpp5vvf/3PC4QBf/GL3+vb+\n63/9A/T6JGVlZXz/+9/ns5/9bEGtD1loypua+NVbbxGLxZiZmcFgMFBTU4PVaqWrqwu73U5XVxeh\nUIixsTGy2SxT2SxNbW35HrpAbuQRO+yrX/0qer0et/sJamu3XhlndGAA7dQU/+Nf/xOf/zdfpM7Z\nuOXrVldXmfH5CFdWcvzRR9dvq3+7YHCBqqoxHn74hNyJeRvT09O8+eabXH3uOWrm52nyeLDb7Tgc\nDlpaWjZ0vbPZLOPj4wwvLGB/9FF699ASdYVM5sDFjvrrv/5rstkszzzzHKlUAr1+89MCmzs6GIxG\n+NwHnqL2XS44qnQ6Vq1WcjYr0Wh0ywCPxabp6qrDYDAwODi4rftSSq5fv87w8DCRSIR4eTmz8TgN\nOh3u6moa6us3PTxMrVaTsVhYqq1Fz42LnIWwaPJeJ2fgYldcutTHhQtaPJ79W/4+k8kwPjREdGaG\nKrUap8WCTqtFURTC8TiLySRxi4V9N1d6GRkZobq6murq6vVtJJNxotFf86lPPSp3Cr6DtSXPvF4v\nfr8fr9dLW1sbKpUKZWmJRo2GJpOJSqsVjVpNKpPBFwwylc1ibm3lxPveh9frLYhFk4UEuNglsViM\n73//Zez2MxgM77zCfDweY8E3x4rXSzaVQq3RYHI4cDU04HA41rvga60Jm81KfX0DKpWKmZk3ePhh\nG52d7bu1W0VlbcmzpaUlvF4vy8vLdHR0YLFYOH78OG63m4WFBaYGBlidnSWbTqMzGnG1tdHU3r4h\nrPO9aLK4QQJc7JqxsXGefXaehob7t2V7mUyG0dFRNBoNNpsel2uSD37wzF0v0VbK1pY8W11dZWJi\ngmQySVtbG2VlZXR3d99VVz4fiyaLjeQvXeyalpZmOjrA5xvYlu1ptVra29tJJqOMjPyM06e7JLzf\nwcDAAMFgkOHhYRRFoaOjA7vdTm9v713f6ORwOOjt7WV0dJQhubU+L+QMXOyqdDrNs8++xtxcFTU1\nnfe8vUhkhXD4Ap2dJtLptCww8A5CoRBPP/00er2e+vp6nE4n3d3d2/I877VFky0WC0eOHJEPnSv9\nfgAAFC5JREFU0V0kAS52XSqV4oUX3mBsTEtV1eFNz0m5E7lcDr9/HI1mjCeeuDEPOz09zdDQ0F1P\nCZSqtUWHa2tr8Xq9VFRUcOzmxeDtks1muXTpEul0mpMnT0r3fpdIgIu8yOVyjIyM8etfj5PLteB0\nNqDT3dnZ4MrKIuHwMO3tGnp7j26osy0uLnL58mUOHTpETU3NTg2/aNx6PGKxGCaTacs1Ru/V2pJt\nfr+fnp4eqRnuAglwkVfhcJj+/lH6+hZIp90YDJVYLDZMpvL1kEmnk0QiK8TjKyiKl/p6PUePNm9Y\nTu3tVldXeeONN2hpaaGlpWU3d6eg5OsbycTEBGNjY1Iz3AUS4KIgpFIpZmdnmZtbwedbYXk5Si6n\nBhRMJh21tXY8HhseTzX2LZZbu9Va68LlctHV1bUjZ5yFLN+LDkvNcHdIgIuClc1mUavVdx2+a71n\nnU637XO+hWrtRp1CWHRYaoY7TwJclLRCCrSdVogfWNFolPPnz+PxeNi/f+u7cMXdkwAXe0K+pxR2\nWiFPGUnNcOdIgIs9o1Rrhms1wdbWVpqbm/M9nC1JzXBnSICLPaXUaobFtD9SM9x+EuBizymVmmGx\nfqOQmuH2kQAXe1IhzxnfiWKf05ea4faQABd7ViG2Nm6nlFo1UjO8dxLgYk8rpkAsxg+c25Ga4b2R\nABeCwp+SKPYpn3cjNcO7JwEuxE2FelGwGGqC90pqhndHAlyItym0Wl6hjWcnSc3wvZMAF+IWhVIz\nLNRvBDtNaoZ3TgJciC3ke8650Ofkd5rUDO+MBLgQ7yAfrY9iasXsNKkZ3p4EuBDv4tZADQaD6PX6\nHZnSKMWa4L26tWY4NjZGQ0ODXOS8SQJciDswNDTE9evXyWazGI1Gjh8/TnV19bZtP99TNoVsrWa4\nuLiIRqPBZrPJRc6bpHApxB2or68nHA7T19fH6uoqFy9eZGJiYlu2vbq6yquvvkpjYyMHDhyQ8L6F\nXq+ntraWyclJhoeHCQaDvPLKK4RCoXwPLe8kwIW4A4uLi9hsNpqbmxkZGWF5eZm+vj76+/u5ly+x\ni4uLnD9/ngMHDpRsx3s7zM3N0draitlsZnBwcP1Dz+/353toeaV56qmnnsr3IIQodA6HA5PJRCgU\nory8fP3sO51OEw6Hcbvd63cQZjIZgsEgoVCIRCKBTqfbcj57enqa/v5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"text": [ "" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The figure shows the two pathways that terminate in five steps. The other fact we need is how many pathways do *not* terminate in five steps, then we'll be on our way to computing a probability. Fortunately, we already have this built into our initial graph construction." ] }, { "cell_type": "code", "collapsed": false, "input": [ "for i in g.getx(5):\n", " print 'node (%d,%d)\\t'%(i[0]),\n", " print 'number of paths = %d'%(i[1])" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "node (5,-5)\tnumber of paths = 1\n", "node (5,-3)\tnumber of paths = 4\n", "node (5,-1)\tnumber of paths = 5\n", "node (5,1)\tnumber of paths = 2\n" ] } ], "prompt_number": 16 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we can compute the conditional probability of terminating in five steps as simply:\n", "\n", "$$\\mathbb{P}(5|\\hat{3},\\hat{1}) = \\frac{\\text{no. paths to (5,1)}}{\\text{total no. of paths of length 5}}$$\n", "\n", "This is a conditional probability because the only way to terminate in five steps is to not have terminated at step 1 or 3, which is emphasized by the hat on top of those digits in the above equation. To compute the unconditional probability, we need to account for \n", "\n", "$$\\mathbb{P}(\\hat{3},\\hat{1}) = \\mathbb{P}(\\hat{3}|\\hat{1}) \\mathbb{P}(\\hat{1})$$\n", "\n", "by computing this recursively,\n", "\n", "$$\\mathbb{P}(\\hat{3}|\\hat{1}) = 1- \\mathbb{P}({3}|\\hat{1})$$\n", "\n", "where $\\mathbb{P}(\\hat{1})=1/2$. Plugging and chugging with the above result gives\n", "\n", "$$\\mathbb{P}(5) = \\frac{1}{16}$$\n", "\n", "Because this is tedious to do by hand, we can automate this entire process as shown below." ] }, { "cell_type": "code", "collapsed": false, "input": [ "import operator as op\n", "\n", "def get_cond_prob(g):\n", " 'compute conditional probability for later' \n", " cp = OrderedDict()\n", " for i,j in g.gety(1):\n", " if (i[0],-i[0] ) not in g: continue\n", " cp[i[0]] = g.node[i]['val']/sum(map(op.itemgetter(1),g.getx(i[0])))\n", " return cp\n", "\n", "def get_prob(g,cp=None):\n", " 'get unconditional probability of termination '\n", " prob = OrderedDict()\n", " cq = OrderedDict()\n", " if cp is None: cp = get_cond_prob(g)\n", " for i,j in cp.iteritems(): cq[i]=1-j\n", " prob[1]= 0.5 # initial condition\n", " for i in cp.keys():\n", " prob[i]= cp[i]*prod(cq.values()[:(i-1)//2][::-1])\n", " return prob\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 17 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following quick check matches the result we computed earlier for termination in exactly five steps." ] }, { "cell_type": "code", "collapsed": false, "input": [ "g=construct_graph(15)\n", "p=get_prob(g)\n", "print 'prob of termination in 5 steps = ',p[5]" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "prob of termination in 5 steps = 0.0625\n" ] } ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following plot shows the probability of termination for each number of steps." ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig,ax=subplots()\n", "ax.axis(ymax=1)\n", "ax.plot(p.keys(),p.values(),'-o')\n", "ax.set_xlabel('exact no. of steps to terminate',fontsize=18)\n", "ax.set_ylabel('probability',fontsize=18)\n", "ax.axis(ymax=.6)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 19, "text": [ "(0.0, 16.0, 0.0, 0.6)" ] }, { "metadata": {}, "output_type": "display_data", "png": 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ERkgS0KqV9vL009ptGg3w6KP22LvXsP3WrXa6Q33d3bWX+6+7uXGehGoXJgyq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\n1ltvQQiBxMREREdHIyEhAeP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"text": [ "" ] } ], "prompt_number": 19 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thus, we have a computational method for computing the probability of terminating in a given number of steps. What we need now is a separate analytical result that confirms our work so far. This is where the book by Feller comes in (Feller, William. *An Introduction to Probability Theory and Its Applications: Volume One*. John Wiley & Sons, 1950.)." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Using Generating Functions (Feller, Vol I, p. 271)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In random walk terminology, the probability that first visit to $x=1$ takes place at the nth step is denoted as $\\phi_n$. Furthermore, the generating function is defined as:\n", "\n", "$$\\Phi(s) = \\sum_{n=0}^\\infty \\phi_n s^n$$\n", "\n", "we will need two other random variables. $N$ is the first time that the particle reaches $x=1$. This is a random variable with probability $\\phi_n$. Likewise, $N_1$ is the number of trials required to move the particle from anywhere to the left of $x=-1$ to $x=0$. Also, $N_2$ is the number of trials required to move the particle from $x=0$ to $x=1$. Now, here comes the key step: these three random variables are independent with the same probability distribution. With these definitions, we have\n", "\n", "$$N = 1 + N_1 + N_2$$\n", "\n", "In other words, you need $N_1$ trials to make it from wherever the particle was on the left of the origin back to the origin; and then you need $N_2$ trials to make it from there to $x=1$. The first $1$ accounts for the first step to the left from the origin. Thus we have,\n", "\n", "$$\\mathbb{E}(s^N|X_1=-1) = \\mathbb{E}(s^{1 + N_1 + N_2}|X_1=-1) = s\\Phi(s)^2$$\n", "\n", "where we can multiply through because of the mutual independence of the random variables. Additionally,\n", "\n", "$$\\mathbb{E}(s^N|X_1=1) = s$$\n", "\n", "because $N_1 = N_2 = 0$ in this case. Unwinding the conditional expectation yields,\n", "\n", "$$\\mathbb{E}(s^N) = \\mathbb{E}(s^N|X_1=1) p + \\mathbb{E}(s^N|X_1=-1) q$$\n", "\n", "Writing this out yields,\n", "\n", "$$\\Phi(s) = p s + s \\Phi(s)^2 (1-p)$$\n", "\n", "This is a quadratic equation in $\\Phi(s)$, so we have two solutions:\n", "\n", "$$\\Phi_1(s) = \\frac{-1-\\sqrt{1+4 (p-1) p s^2}}{2 s(p -1 )}$$\n", "\n", "and\n", "\n", "$$\\Phi_2(s) = \\frac{-1+\\sqrt{1+4 (p-1) p s^2}}{2 s(p -1 )}$$\n", "\n", "How do we pick between them? The limit of $\\Phi_2(s)$ is unbounded as $s \\rightarrow 0$. Thus, $\\Phi(s=1) = \\Phi_1(s=1)$.\n", "\n", "One of the properties of the $\\Phi$ function is that $\\Phi(s=1)$ should sum to one. In this case, we have\n", "\n", "$$\\Phi(s=1) = \\Phi_1(s=1)= \\frac{1-|p-q|}{2 q}$$\n", "\n", "Thus, if $p < q$, we have \n", "\n", "$$\\sum \\phi_n = p/q$$\n", "\n", "which doesn't equal one. The problem here is that the definition of the random variable $N$ only considered the conclusion that the particle would ultimately reach $x=1$. The other possibility, which accounts for the missing probability here, is that the particle *never* terminates. Similarly, when $p \\ge q$, we have\n", "\n", "$$\\sum \\phi_n = 1$$\n", "\n", "This should be no surprise because we saw this exact same result when we first started investigating this! Namely, $p \\ge q \\implies p \\ge 1/2$, which is what we concluded from our first plot. This is the situation where the particle *always* terminates.\n", "\n", "The first derivative of $\\Phi(s)$ function evaluated at $s=1$ is the mean. Thus,\n", "\n", "$$\\bar{N} = \\mathbb{E}(N) = \\Phi'(s=1)= -\\left( \\frac{-1 + 4\\,p\\,q + {\\sqrt{1 - 4\\,p\\,q}}} {-2\\,q + 8\\,p\\,q^2} \\right)$$\n", "\n", "Note that this is unbounded when $p=1/2$ which is exactly what we have been observing experimentally. \n", "\n", "To get at the individual probabilities for $p=1/2$, we can use a power series expansion on\n", "\n", "$$\\Phi(s)\\bigg|_{p=1/2}=-\\left( \\frac{-1 + {\\sqrt{1 - s^2}}}{s} \\right)$$\n", "\n", "Using the binomial expansion theorem, this gives\n", "\n", "$$\\phi_{2k-1} = (-1)^{k-1} \\binom{1/2}{k}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we can try this formula out against our graph construction and see if it matches. The downside is that scipy.misc.comb cannot handle fractional terms, so we need to use sympy instead, as shown below." ] }, { "cell_type": "code", "collapsed": false, "input": [ "import sympy\n", "\n", "# analytical result from Feller\n", "pfeller = dict([(2*k-1,sympy.binomial(1/2,k)*(-1)**(k-1)) for k in range(1,10)])\n", "\n", "for k,v in p.iteritems():\n", " assert v==pfeller[k] # matches every item!" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 20 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now that we know exactly why the mean does not converge for $p=1/2$, let's check the calculation for when we know it does converge, say, for $p=2/3$. In this case the average number of steps to terminate is\n", "\n", "$$\\mathbb{E}(N)\\bigg|_{p=2/3} = 3$$\n", "\n", "The code below redefines our earlier code for this case." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def nwalk(limit=500):\n", " 'p=2/3 version of random walker. Only returns length of path'\n", " n=x=0\n", " while x!=1 and n