2.DeepLearning/13. Graph Convolutional Networks.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Graph convolutional Network\n",
    "\n",
    "This tutorial based on \n",
    "1. [Theoretical Foundations of Graph Neural Networks](https://www.youtube.com/watch?v=uF53xsT7mjc) and \n",
    "2. [Graph Neural Networks](https://www.youtube.com/playlist?list=PLSgGvve8UweGx4_6hhrF3n4wpHf_RV76_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Convolutional Graph Neural Network in Detail\n",
    "\n",
    "\n",
    "Let \n",
    "+ $X \\in \\mathbb{R}^{N \\times d}$ be a matrix of node feature vectors.\n",
    "+ $A \\in \\mathbb{R}^{N \\times N}$ be an adjacency matrix\n",
    "+ $D$ be a degree matrix ,i.e., $D_{ii}= \\sum_j A_{ij}$\n",
    "\n",
    "$$ H^{l+1} = \\sigma \\,( \\tilde{D}^{-\\frac{1}{2}} \\, \\tilde{A} \\, \\tilde{D}^{-\\frac{1}{2}} \\, H^l \\, W^l)$$\n",
    "\n",
    "\n",
    "+ $\\tilde{A} = A + I$ is the adjacency matrix of the undirected graph G with added self-connections,i.e., $I$ is the indendity matrix.\n",
    "\n",
    "\n",
    "+ $\\tilde{D}_{ii}=\\sum_j \\tilde{A}_{ij}$\n",
    "\n",
    "\n",
    "+ $H^l \\in \\mathbb{R}^{N \\times D}$ is the matrix in the l.th layer. Hence $H^0$= X is a matrix of node feature vectors.\n",
    "\n",
    "$$ \n",
    "A=\\begin{bmatrix} \n",
    "0 \\quad 1 \\quad 0 \\quad 0 \\quad 0 \\\\\n",
    "1 \\quad 0 \\quad 1 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 1 \\quad 0 \\quad 1 \\quad 1 \\\\\n",
    "0 \\quad 0 \\quad 1 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad 1 \\quad 0 \\quad 0 \\\\\n",
    "\\end{bmatrix} \n",
    "$$\n",
    "$$\n",
    "D=\\begin{bmatrix} \n",
    "1 \\quad 0 \\quad 0 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 2 \\quad 0 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad 3 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad 0 \\quad 1 \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad 0 \\quad 0 \\quad 1 \\\\\n",
    "\\end{bmatrix} \n",
    "$$ and $D^{-1}$ denotes the inverse of $D$.\n",
    "$$\n",
    "D^{-1}=\\begin{bmatrix} \n",
    "1.0 \\quad 0 \\quad 0 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad .5 \\quad 0 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad .3 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad 0 \\quad 1. \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad 0 \\quad 0 \\quad 1. \\\\\n",
    "\\end{bmatrix} \n",
    "$$\n",
    "\n",
    "$$ D^{-1} \\cdot A \\cdot x=y,$$\n",
    "\n",
    "$$ \n",
    "\\begin{bmatrix} \n",
    "1.0 \\quad 0 \\quad 0 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad .5 \\quad 0 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad .3 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad 0 \\quad 1. \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad 0 \\quad 0 \\quad 1. \\\\\n",
    "\\end{bmatrix} \n",
    "\\cdot\n",
    "\\begin{bmatrix} \n",
    "0 \\quad 1 \\quad 0 \\quad 0 \\quad 0 \\\\\n",
    "1 \\quad 0 \\quad 1 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 1 \\quad 0 \\quad 1 \\quad 1 \\\\\n",
    "0 \\quad 0 \\quad 1 \\quad 0 \\quad 0 \\\\\n",
    "0 \\quad 0 \\quad 1 \\quad 0 \\quad 0 \\\\\n",
    "\\end{bmatrix} \n",
    "\\cdot\n",
    "\\begin{bmatrix} 1 \\\\\n",
    "2 \\\\\n",
    "3 \\\\\n",
    "4 \\\\\n",
    "5 \\\\\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix} \n",
    "2. \\\\\n",
    "2. \\\\\n",
    "3.3 \\\\\n",
    "3.0 \\\\\n",
    "3.0 \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "+ The vector representation of the $i-$th node is represented with $x_i$.\n",
    "+ $A_{i:} \\cdot x=y_i$ update the vector representation of the $i-$th node.\n",
    "+ This means that $y_i$ is a sum of the features of the neighbouring nodes.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.linalg import sqrtm \n",
    "\n",
    "import networkx as nx\n",
    "from scipy.linalg import sqrtm \n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import animation\n",
    "%matplotlib inline\n",
    "from IPython.display import HTML\n",
    "\n",
    "# (1) A is given\n",
    "A = np.array(\n",
    "    [[0, 1, 0, 0, 0], \n",
    "     [1, 0, 1, 0, 0], \n",
    "     [0, 1, 0, 1, 1], \n",
    "     [0, 0, 1, 0, 0], \n",
    "     [0, 0, 1, 0, 0]])\n",
    "\n",
    "# (2) Compute A tilde.\n",
    "A_tilda = A + np.identity(len(A))\n",
    "\n",
    "# (3) Compute D and D tilde.\n",
    "D = np.zeros(A.shape)\n",
    "np.fill_diagonal(D, A.sum(axis=0))\n",
    "D_tilda = D + np.identity(len(D))\n",
    "D_tilda_inv_sqrt=sqrtm(np.linalg.inv(D_tilda))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\tilde A = A + I$$\n",
    "$$ \\tilde D = D + I$$\n",
    "\n",
    "\n",
    "$$ \\hat A = \\tilde D ^{-\\frac{1}{2}} \\tilde A \\tilde D ^{-\\frac{1}{2}}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.5       , 0.40824829, 0.        , 0.        , 0.        ],\n",
       "       [0.40824829, 0.33333333, 0.28867513, 0.        , 0.        ],\n",
       "       [0.        , 0.28867513, 0.25      , 0.35355339, 0.35355339],\n",
       "       [0.        , 0.        , 0.35355339, 0.5       , 0.        ],\n",
       "       [0.        , 0.        , 0.35355339, 0.        , 0.5       ]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D_tilda_inv_sqrt @ A_tilda @ D_tilda_inv_sqrt "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\sigma \\,( \\tilde{D}^{-\\frac{1}{2}} \\, \\tilde{A} \\, \\tilde{D}^{-\\frac{1}{2}} \\, X \\, W)$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.04669446],\n",
       "       [ 1.08883771],\n",
       "       [-0.19833822],\n",
       "       [ 1.12762852],\n",
       "       [ 0.41792461]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X=np.random.randn(5,5)\n",
    "W=np.random.randn(5,1)\n",
    "\n",
    "# New Representation of input.\n",
    "D_tilda_inv_sqrt @ A_tilda @ D_tilda_inv_sqrt @ X @ W"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Visualising feature updates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# (1) Create nx Network\n",
    "g = nx.from_numpy_array(A)\n",
    "# (2) Compute tilda\n",
    "A_mod = A + np.eye(g.number_of_nodes())\n",
    "\n",
    "# D for A_mod:\n",
    "D_mod = np.zeros_like(A_mod)\n",
    "np.fill_diagonal(D_mod, A_mod.sum(axis=1).flatten())\n",
    "\n",
    "# Inverse square root of D:\n",
    "D_mod_invroot = np.linalg.inv(sqrtm(D_mod))\n",
    "node_labels = {i: i+1 for i in range(g.number_of_nodes())}\n",
    "pos = nx.planar_layout(g)\n",
    "fig, ax = plt.subplots(figsize=(10,10))\n",
    "nx.draw(g, pos, with_labels=True, labels=node_labels, node_color='#83C167', edge_color='gray', node_size=1500, font_size=30, font_family='serif')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial signal input: [1. 0. 0. 0. 0.]\n",
      "Final signal output after running 20 steps of message-passing:  [0.1559171  0.18976912 0.21665888 0.15263111 0.15263111]\n"
     ]
    }
   ],
   "source": [
    "\n",
    "A_hat = D_mod_invroot @ A_mod @ D_mod_invroot\n",
    "A_hat\n",
    "\n",
    "\n",
    "H = np.zeros((g.number_of_nodes(), 1))\n",
    "H\n",
    "\n",
    "H[0,0] = 1 # the \"water drop\"\n",
    "iters = 20\n",
    "results = [H.flatten()]\n",
    "for i in range(iters):\n",
    "    H = A_hat @ H\n",
    "    results.append(H.flatten())\n",
    "print(f\"Initial signal input: {results[0]}\")\n",
    "print(f\"Final signal output after running {iters} steps of message-passing:  {results[-1]}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<video width=\"720\" height=\"720\" controls autoplay loop>\n",
       "  <source type=\"video/mp4\" src=\"data:video/mp4;base64,AAAAIGZ0eXBNNFYgAAACAE00ViBpc29taXNvMmF2YzEAAAAIZnJlZQAAZRdtZGF0AAAAnmW4AAQA\n",
       "ABHkxQABC0k5OTk5OTk5OTk5OTk5OTk5OTk5OTk5OTk5OTk5OTk5OTk5OTk5OTk5OTk5Ouuuuuuu\n",
       "uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu\n",
       "uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuvAAACamUBauAAEAAAR5MUAAQt\n",
       "JOTk5OTk5OTk5OTk5OTk5OTk5OT4h4fGtwAUxpzMK9RDT1et4lMZxXBaJg9XvBlPHHIjfu9aTxh8\n",
       "KozTz1eD1JAKk9o5lwf38DfIwFNx3u93uRMcF8ADkHFg2pNqDrQS+cQmcq37+dpQnvgXEnDvhEYI\n",
       "4TwEd4jRK/3//rwZny2U99/AQHeXZUAi1COH/4C5YTh9RRToG+/A/GiBopTIYBVbR5qANCxRozAO\n",
       "rGU0II1zUROgQkTJ5QqeGU2QyHP3Q1/DDMT+/oc67a7AGm2sHFENyI9eelIYFawFPoHlRJAzxDJr\n",
       "mOdCgLGs9QcZ7nmN4AEQW4imPonACJfJlmP1OFyhWRhg/56i8IeYliZ3/mN2WAFwj1klP/+Y/9jU\n",
       "vdgABAMP9IcRQT7IjeQMAoA5HCcSQfkrt3sAhTM1IZTqWhmJ4qI4QlewZqr2dANphPMXUwim3ct4\n",
       "+pAxBiPbP/EwAABB2k5OTk5OTk5OTk5OTk5OTk5OTk66666666666666666666677w33hrrrrrrr\n",
       "rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr\n",
       "rrrjsABDZnd2Zmd3Zn//s4APwtY8FSoMPve8XErOvl4AEm2hJFOpCOkE2WC3QOvoAC/KLgY4zTO7\n",
       "DUB5NjmeAf6bbtTFnfzgw/G186n/oJdR8AJho0QWfc9Z8O0yMmf+e2tcB//PBZT6s6Ep393HD6T+\n",
       "wQQAFIaSYukiFOHc5Rhfw5HPXXXXXXXXXXXXXXXXXXXXgAAAAaFlALS4AAQAABHkxQABC0k5OTk5\n",
       "OTk5OTk5OTk5OTk5OTk5P/h/OFMAWCpsgAQAlJrFpiAsKDvveZHoIbE8xYlCPcEZ4ABvx8n8PtD4\n",
       "XwBEhi9hIAMj2+pkIx82PgAOozumBGFHtX/iwmLxGrycnJycnJycnJycnJycnJycnJyddddddddd\n",
       "ddddddddddddd+lTzx1HioAHnYtUyyPjPzv4QAWD8W1juZ5tw/Vnt7FkagQuw8Vx//tf3XQzAAzb\n",
       "b0SM0AA+T/aAmDtItZEpH4eUQ/7aynzAEITO4fE1S/ozLWlgRV7DAb8APB3eM6jn1yUv4j7BF4kl\n",
       "V7OZvgw71111111111111111111111111111111111111111110seoAEL3d338HrXNTBft//ph5o\n",
       "I/CfRDADxJnCflPrnhbpziM++WiHwA6IvyPyh3ui666666666666666666666666666666666666\n",
       "6666666+ekfmnFYr7omfbv6IYAdRekLSh3unhXzrJ0sR34NA8hRwL1i66666666666666666668A\n",
       "AAE/ZQBDrgABAAAEeTFAAELSTk5OTk5OTk5OTk5OTk5OTk5OTk5OTk4QwAIXu7vv4CWZGKj3/++H\n",
       "mn8J75thQa8/gZMB5CuBv1ntQ2nHpDQ6XeArF4QsUO90JuTk5OTk5OTk5OTk5OTk5OTk5Ouuuuuu\n",
       "uuuuuuuuuuuuuuuuuuvnp/NOK2obTj36DB0vAVIviPnDvdPahWuPD/Qvz4B5JlidHPri66666666\n",
       "666666666666666666666666666666666664//GIZlsjP/9MPNP4Tifdj+twDwztWdHPrntTdx4f\n",
       "di/B8BYaSo7OguLrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr56fzTiuH7Sb5y7z/5c\n",
       "AovwrlI7vfPzHYFGfn/wYOJ+FcpHd74uuuuuuuuuuuuuuuuuvAAAAo5lAFouAAEAAAR5MUAAQtJO\n",
       "Tk5OTk5OTk5OTk5OTk5OTk5OTk5OTk5OEMABGRGZkREZmRH8Aie+9v/wA6KEoeTBB/HpxByB4ayA\n",
       "yFlkO/uBwpPCs5nEa8/AE2NkmPB9zlgJj5RGQDKGjfsBvvT2xRVvXjVbbP/2g1+/j+mXD4LnU4vg\n",
       "JGTTx58nVnAEP8rKn/0/AbNfSZN/B7v606c+BG1/plo/39QjF0qJycnJycnJycnJycnJycnXXXXX\n",
       "XXXXXXXXXXXXXXXXXXXXX/Jg8EjDEAGNq31np3kkS68GvkDWI/VBvkoALRFSzw4KEsnQ5+9NEYP3\n",
       "8//Qf+CWasnoNn4/Cnw/sJ/gApUhlseN+PGcAC0kU9jw8ozZkUe9dddddddddddddddddddddddd\n",
       "dddddddddddddddcdgARH6r+v//gBx1GxI+GfMBTu7uyGQekND5OGfS8Y8bEij7wCLMsfm/l//+g\n",
       "A6oPxM0FemCwfSQiK6e3CuStLn4/529U1ifKZXI7w/+dV1BhXJLdemPjn/4AVBnt72n5h9mJhdY/\n",
       "Nc/+DlpGeAB9meQVYgcLLPX7AECRLGpH4CsKWT1SHH/AA6p+k7feP+LeZmh9QDBTGiwmQ64O0v/7\n",
       "D0F9GZuY1M8OuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuOwAEg/SaeyLSIX/4AWkm54STY\n",
       "AM+t9D8HAr3pERz29m+QWmhb+sH+5E/+Jwh21gA55WeEhX9pLGCyZ7x3QBUPLWzCh9e6sZn7T/2H\n",
       "jwcZW/QcfMH8mG1/iafzQUhYM7xncc+ueFrTXEZ98v8APBTrGdRz65C3TuJn3yHYfUz/+Key0ef6\n",
       "666666666666668AAAKoZQBwrgABAAAEeTFAAELSTk5OTk5OTk5OTk5OTk5OTk5OTk5OEMACI7Sr\n",
       "bSf//ACssPyEyvb/Td3LjMAMxt6WPgAdRZuwse+2WkElTwqlXABmNmyTHwmZgmKBOpDpU/D7u0OP\n",
       "j/Se6g/3d//8PYKMwkL4H3DtfvL7oH5OTk5OEMAHWO/HnOP/7oALEX5H6908JvXMjC/64p2RDz3/\n",
       "x1F+QtO90hlPzIwvz75TmRDz3k1JycnJycnJycnJycnJydddddddddddddddddddddcdgAs+zP9u\n",
       "bH2+/+AFTwfTNHfQAY9GxI/WB4P7QiKj28rKjw1ftA/c3N3PlxOAG2zZEYOAA/VIPzQkFfrQVT9y\n",
       "dA4mZBNzh945/v0b3IPz8XC/kw11111/yNP5wpG4H0hXA38s8ZbrJ0sR34NA8hXAvWQI7tPfDsAC\n",
       "D677v//DNrMiD/Pv9ddddddddddddddddddddddddddddddddcdgAJpNtJJtpP//ADPrfQ/Bwmzw\n",
       "kzntx2API3G54E6nASP02gBx1mxI+EB4fS0iIj28Y5savhgAOeS88JP3SBOCyYtd0fImxqc8DD/3\n",
       "y//D0Og0kbO4fXuFcDrrrrrrrv4Gn89MXDsXhCxw73T2obTj1Bg6X4CpF8R84d7pHGa1x74/cLL1\n",
       "1111111111111111111111111111111x2AE8EfyGgr33P4AzG3pA9gDMbNkgeFzMEwzqAG5LH4TO\n",
       "/+IcoWhev/ifj5tsHg/eW/+JwAfkm54SFtgujPBNDh946G7/1/9NP5QWmhb++LCh4L+T9ddddddd\n",
       "dcfgALGrS1JZKY/8BZJlifXrnjEMy2RgPtW1IH+3+Hh3as6XrkYhmW0YOwAIPrvu///wfNJukD/b\n",
       "/XXXXXXXXXXXXXgAAAMKZQAhy4AAQAABHkxQABC0k5OTk5OTk5OTk5OTk5OThDAAiOZFWZkR//4A\n",
       "x6NiR+sDwf2hEVHt/97uRHxPgBts2RGDoAOxYfkJhWvYOCZoEncPvHN99G45ABepB+aEgp9aTKn7\n",
       "ntCqML7uX/+HsFKn8nXBhybk5OTk5OTk5OTk5OEMAHsHVVCorj/+6ACw9qjsgueMQ2towLmkfIHe\n",
       "1/8KH8K5Wb3yMTa+M+FAdnlkP+95NScnJycnJycnJycnJ111111111111x2AAmk20km2k//+AKpB\n",
       "CrIUn5/9x6wrwUvhz8isf+wVgbDeOvzDyZY8rPwIYA9paJC94O//gUhprZNM/x2ABeoKC7ERCIbr\n",
       "+AFHVmwkftICR+mwAKeS3PCT9aROC0SFrpg8PpaRER7ecfNjV+sD/3e9+XhKyACoaWtmg45YHQaS\n",
       "NncPr3Jillat7If8mDDl8nXXXXXXXXXXXXH4ACw+SW6XbmP/ChfhXKRvfPMT33sFAdHlkP+7/Dl4\n",
       "VnIjXmX72HYACS1pabWm///AUBkLLDp3X+uuuuuuuuuuuuuuuuuuuuuuuv+5FaC+XwAJshWfxxyh\n",
       "jv/4AD2oWOXQoDCDb8/Tre89H9uz9/+krj/v0AQkhqhHjxjm0xNCahgguBfAuBf/2pz0f0pis685\n",
       "w/9gRtfy7z/U8Zlac2u7CPrfWPwUJs8JMKOe3XaiFZlUUz+PGqBYno1mRz3hyxmThWHPfefw/sPa\n",
       "yv36c9wYdd9ddddddddddddd+KU/0oEXCR4VnphXZ9+kQCq54G7xBw5SWymFb3f+AYtvA3Y11111\n",
       "11111111111111111114Q/Sedi93l9hC3crAxKdHNI94bHIuWI9Og5c5rWvffDCXOs35vT/h/aVj\n",
       "PMAOgsImvW4NPfwAEK+rawlxkHhJwF0Xp9tf+/4dPXXXXXXXXXXXXXXXXH4ACaDpJbN9OQvgHQdW\n",
       "tlN+4x+79vCgvg5lK3vnwqDu1spv3vjsAietNrTf/+Mf1zM8ODwcRXRPn+uuuuuuuuuuvAAABW1l\n",
       "ACdrgABAAAEeTFAAELSTk5OTk5OTk5OTk/4//D8ADbTEs5nKR1lMTFkcBtcCXff8f3/4Swwy0UyC\n",
       "5wMPsiMaK/BA0GPCZkAG0Y/DEO5L4LZxkLoPAO7//8JZE6ymYQ4vw+CiU/P8F777777777777777\n",
       "7774/AB7B5VRSFUf/3QALTTLpgv2nZ9aZnh4M7xHe9cD/g9aadMF+3xjOy0zPDwd3jOt65777777\n",
       "77777666666666666666666666666666666666/zQR+fCkLWncRn3y0Q+AIEmYn5V7pC3TdEz7d8\n",
       "dgAQT1ptab/+KeyyPPHUX5C073T/XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXmnhn\n",
       "p4uFbnWR0tEPgCcD6QtcO90jLdZOl+112DQPIVwL1muuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu\n",
       "uuuuuuuuuv80/nwpARWdZNRZiV+AmA+ELHDvdLUK0498dgAhD1kZlkZ//hmNbIg/z7x2L4hYd7p/\n",
       "rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrvzTDz08XahWuPUGDpfgLJMgnzr3SOM1rj\n",
       "3x/oX5wDw7pWdHPrnrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrj8AK5kZGJ7z3jEM\n",
       "y2RgPtTdIH+3h4Z2qO1658Yhmtoz47DyRmRGce//wfNI+kD/bw8NpKjtT5/rrrrrrrrrrrrrrrrr\n",
       "rrrrrrrrrrrrrrrrrrrrrrrrrrvxT0+mkXDE2vWYJu/H+DwCi/CuUgrvfIYm19mfON+P8jgHLwrF\n",
       "IKI156666666666666664rBa+CEhO2cJCpb/H8PkG93S78AgR8GmMHHeutv+f+cwgqlFBU4/BPRt\n",
       "N6euuuuuuuuuuuuuuuuuuuuuuuuv+l+BFxEyt97C2fl5EBiLQknOpGdZTPWSkAZ6foAEQXxEjH0R\n",
       "eyXu4qzIYQG+/oFtKUw/vsDDtxfO/+gtAWohqyK/BJwYkLoXY+h8BI6m19Gn3vzwKSjWhoy5e9T+\n",
       "cApi/xAZCWc+ef+AIY7mwLnt/8ALocnkXv+9enrrrrrrrrrrrrhZxD+v1/43Cu52s0/MsyQsoAEX\n",
       "d6daf3gC1ENW08/3iCrKzuRvNk/j/H7D35vvvvvvvvvvvvvvvvvvvvvvvvv7QH9PYX4GpB/oBWcR\n",
       "7q42pu480+7b/ge0+4XlL99zV/p/2wpCvNiZTkbPeJnwTh2A45Ew2j3g43yIph/z3p6666666666\n",
       "664XUNPeu3/hWm4LJbH+H9VwDgqglbwlGwHR7/2Iid8vBbrgWIIsOuJVb8bvz/by999999999999\n",
       "999999999999/PSZ/XQrBs2L9Ipin/T/wAsy+lwi7qALN667r21+fn//6hg/CsF8YpEV1JrAlgfl\n",
       "MmLef4eJFGzJ+AjsJGmo8N3n/tc5XX8hcAFab4p84ADaY/hue1hnsIiSg/zmUI/9tb/ez2254d/v\n",
       "6P/nof3p66666666666666Wlrrrrrrrrrrrrrrrrrrrrrrrrrrrpb66666666666666666666666\n",
       "6666666666666666666666666666666666666666666666666666666666666666666666666666\n",
       "6666666666666666666666666666666666666666666666666666666666666666666666666666\n",
       "6666666666666666666666666666666666666666666666666666666666666666666666666666\n",
       "66666666666666666666666668AAAAAKYeAAQACcgjwFrAAAAJBhAWuAAQACcgjwtxMIsHvHg4Vq\n",
       "JmDhWomYOFaiZri/Ahfhu1jwVySwOCuSWsIsFyGYAdJGA+I5uwDBKcCsWnGavUVjsLUk4YUf/Qc8\n",
       "AAVDqoiDfTvJ/oA+psB5UCPIfp4wpWQfGQxnkgslLMNUwJFPRiErGpevXB9KTPqwpii2TrRRJWxX\n",
       "j7EcAyQsb8TwE1AAAAAMYQC04ABAAJyCPAWsAAAAEGEAQ7gAEAAnII8DpyYrwIMAAAASYQBaOAAQ\n",
       "ACcgjwO3L4T4ZoD+AAAADGEAcLgAEAAnII8BawAAABxhACHOAAQACcgjweBP/+ACENNbJpn+CwYv\n",
       "oE2AAAADYWEAJ24ABAAJyCPAKHwv4AUeg+YhR8MACXlMMhiv/TVjSqAAEBk7GaQmMADmPAAACAzC\n",
       "VYQAEcl3wELXvpOWYyrWm3cfX8g3AFaYyZ34M7VRj4uoDrfLANVI2n5+D8DMDA4aMC4ZcDxaWsta\n",
       "aD4XgVEHczwssLmCp/5zSDD8BZOOeAEK07Erv7mAD/avQcEfS7veAcD5gK8XmPwEjpD9sBExoDQv\n",
       "p/gEqpjETtg3sEGPg9wN0D5heuAmMwzLfABtjcEAhHgAHp4AB6KAAehCIhwgACAFEvQXww998BGi\n",
       "Qro2ADjkJXz0/ff52mi+EPjoNQG02Sb8EnGSPQf3/hgrfSICGu8eeGIaNJofDRQgERD7Rgz5pD9v\n",
       "gFkP6+C7Lyy+w/+HtuDQnx54AB6AYoZBBeBQAD08AA94Ca2JEDvCInKEAARBRDSJHreQKIB3w86D\n",
       "vGjltZhkGRzRg41oF+HsfgEZRvJ9ZZ6YhI3qPQxDNbTM+EX07od44waUNlcs2Y2wDUrGHtmBKXPP\n",
       "bMRj3zfSnw2HOAIGRpsVHTMEmZiL+qbcAGBBWQGBGqqYSnwk4FJSBq5BZra7Fl3poG9elrobNYQ/\n",
       "p7De4oAIhk91bqJzfbU5gB7fFzP60NGBATE25pS2NACWqeF24A7YluWz2W5PrN6EH9rDkcWiY2j3\n",
       "rS0zERzSTeDn9JTP+e8sCkFluNjBxXCRfizyq4If5Fnh3DUP+FChhaf4eAAIDPwAX/j0sCqHnXgC\n",
       "QgEhSY0g/gCaf4R4+/Nh9QGAcOxkNEBw+Lx35+PEUeWAWko2yfn4fNhDBTgkE7YVQ5fwB/ZBfYMC\n",
       "NFE3yI3fLxvgAun1iw63bcADTfSEcKQb7YFX75/9/km2ZuDHj34FgjAaEChI/JIs1rV354BJ3pAd\n",
       "H2QmZ8JiSpU8DiXxHiPEeI8R4jxHiPEeI8R4jxHiPEeI8R4jxHiPEeI8R4jxH6CuK866N9uawAC2\n",
       "x4AB6ZZxGnAJ+mABFARGzhAAEwQs7Brdxl86zG1FyOin7o5sPqGD6DsAJsqTbFrCgDWkd+P8Qkpt\n",
       "PzBuUVBx1/t/jBQgCMo0wanLC6fhwJBAziGm4O0ISrjoMxQAD0BEmiAAgNrAFBxICeHxlQ4ITMwg\n",
       "ACQYooFlmX/A3jpjDR/wChQAAAAKYeAAgAEchvAWsAAAAINhAWuAAgAEchvDLUIxmABrkpu2U/oC\n",
       "XWt/tVmUa3v/W2ihuFpE/xEpteNkiBDfexq57HI0kykCSwARVgGRnM5/0I9L+vgZqGCvp7//qgKx\n",
       "UTraTsQrBD/4EGgTDRNdUeYPl3sULGlFFzdccf/bSIkFBl+S80CSv1eAAXClAqvFBYYBOQAAAAxh\n",
       "ALTgAIABHIbwFrAAAAAMYQBDuAAgAEchvAWsAAABBmEAWjgAIABHIbw5w/DCCUf41kLosYsDLAM8\n",
       "AwFAMsAMUAxCaAwABHhvwjBcn4YigGB2EooBlgB8QAeGSHhAN2EIGxvwhbg1MjIFfC9AIPgH4IVe\n",
       "DwDKKQFgGVFDBLAMS4EgAiVkhCoCglRHcjeRx2YfP/0CzAGsaYlqCqPy+v6Wi/hb8ITHmUqmPXoD\n",
       "fdSDJXMFXBfVULANSOwdF6UkQCQhIy+mAEkP42KAYoBlgGKAYoBigGKAYoBiQDywDJgoUmhQDFAM\n",
       "eLCsXhoAoKTAAEAb+B4uYJcvxsUAxQDFAMUAxQDFADFAMFoqBQDFAMmD4w1AcA0f8DOGEIJgG+Ee\n",
       "NgqlYD8kMH0AAAAMYQBwuAAgAEchvAWsAAAAH2EAIc4ACAARyG8HgTwAIPrvu//wAMJaRlsSsofg\n",
       "eYAAAAKKYQAnbgAIABHIbwCh8L+AhHU+ADXRMEwaoBu/wAZ/VUgN/54AAAjCvgrZfCPO75BdjkvG\n",
       "4A7t0ivjSgM2HgAEZAEsagtnG5ZYABGeAAwAACMUAAjFAAIxQACMUABigAMUAAjFAAIxEBIDjgCF\n",
       "WJdzQHsD3f+H/ALtYDcTCzNI8AAjPAAIxQACMUAAjARZGZ2S8xtWA3xiixP8DqEEjYjeagRCqPAj\n",
       "+W9g3wWY8LimbXO3DAgAoJE7+DwZ4MUZoSp+NpE/2GHOnpPGWDEe8D/ZPngw0DshRijFBigxRlgx\n",
       "QYoMaUJAa8Wxb/BXyEzOyWeDD4g4BFBlg+G0AjoECxxrQL8NLH4AS4mLI/J8RYKRFOQFqJzYo9BS\n",
       "0aMVVlc+Mv6bXzvHUCCCDZXHwS0VgFRC1FzzxYK2RoeeesRfYzeKQ/2HPgDpyxCpHUhVBqyE6Fcx\n",
       "hx9G0V4BQQlyCIvsBlOcIcv8PQKASQONXILNbXbcAE8q3nQ6Zifpbzwb43QAJOEsGWDFBigxQaEF\n",
       "xMpKDOFgxQZYMUGNMSA/i8Thnf42KDFBigxQZsLGozgcigxQYoMUGlBTMlFBjDEBF/PCeA7hqU8x\n",
       "Oe8HgAEfgBmXWaB32QlNf8CPoxwsf8bFAAIxQACM8AAjFAAIxQACMUAAjFAAIxQACMBDm82zoscK\n",
       "ZD/XCDlhwS5xAiIGAIy/4TeQms/DsUAAjFAAIxQACMAnqPB/wAGXdMwfucGv3wHv8Sp7SgfwpIgo\n",
       "kT46VdSiCQNBeG+GogRglgxsUSAEvg6AQ/KGBhB4zcbFBigywYoMUGKDFBigwkExnIlH3vwEv+Tj\n",
       "KEENfv0++FooMUGHwQdCoH9vnoS2/DiUIh8AoUAAAAAKYeAAwAGcgS8BawAAAT5hAWuAAwAGcgS8\n",
       "Kif4yAM7Xf97jQA40yRWx3qrBDcfclfS8+1aZ9q/g8IuDeCwEB8IcgtDsEmLakaYDc9VZnWzTDeo\n",
       "wTCbjerGBd/hDh1eoXgr4zTh0mImoCf4+5I5KX31DEZgDpM4sz3QJHsDXfAYluejQlS/xV1WA/d2\n",
       "rtfgJpHAoRRm/SAHlyg1DKLt982n/ntgYsr0ABhmKFKfTPZMWLYwQIgOLI27cZ7aJnzx40A/IgJu\n",
       "SFaqLT9hKDrU39GjYY6VnCjRBpnoPv8CoR4j4jgsPwWaNooC22OnnxvImHSESBJ5+0qCY/ws04Qq\n",
       "HhnhihlhimKGKGVAdYLxvj3f42JF6Lya33IK48zzBS+nXR/vs93lCF6O2oKYpihihlndA8KGWGMU\n",
       "BmAvEug1u/huoQYD93If/y9Ab/C/sKwAAADFYQC04ADAAZyBLwtwvjgARd4ZCDsNFP6IivxDCWHh\n",
       "SXimWYpimKYpimWYplmKYpimKYpjjHAAKYn+FophiGBFPEh5RcETIu8eeIb4pxTgqEL8PQSdHZB+\n",
       "ZMXvUWmFxcJ96rdH+owuH86A4a3gzmP+Ni4uLi4uGguMASPyAuLxYlSNWEPQ5PwFS2v0vNnsPHQY\n",
       "GN+DdxNwW8mEKxtr+JhK/683P1uPZOJda/NYxS+mQyQLePyK2B+wujkt2h6UFBqSK8Kz8KQAAABV\n",
       "YQBDuAAwAGcgS8M8mEjA+d/iepTHCtyL4D/LoBx0rAW8ftljgGA5k4y90MOoCTQw98F3Ey5Dctjm\n",
       "ru9EB828NGgnwLeP1QOvaMGhwi5egAjXF6XwnAAAAKlhAFo4ADAAZyBLw3wvhxDIPwEStwEsl/gR\n",
       "722BcNUXGAKG16jsfBVzeGZXJcbVVVVVWArAwR+XwC7UGRVNBwGViVL/xtVVVJBepiw/DM8DZoI8\n",
       "AMvcI/aNs9YydJpoqn3CA3z4KOL3h1FeEYDwPqV+OwJ/698DJOjbwIWqnltx/5D+sfP+JXEL8VQW\n",
       "BE9p//vIZvIezjITwAIPrvu///ALOwlKakFM/whAAAAA3WEAcLgAMABnIEvC/NxcJODEvCOBI3gf\n",
       "9IH4OjkAfIpGxf4SrQezlx/+/IyBoYbHeQT/EwTtgeXvRwEhO27XhgbwhBLxfCHbWihRxuHaBLKX\n",
       "E/HY2ABzFSBs/d5ngdw9Avm7D+b+fio6TOrDATHBzrWkxSxcGCPm4IMzyKgqeEsEWgNPSWENBHTg\n",
       "sBdCF/jd2f73Fm3JDecTxMoelkOAYDRoZLggP/CPhE4erxcErZ3+EyK4gk84RX+IwW4b8rF7mCDa\n",
       "poJ5/i+KgJG7A/bP0DQZizVGhmFY/L4gRAY6AAABS2EAIc4ADAAZyBLwlxnjwIaEuBlswwLN9Ag3\n",
       "e8gN5uHxHaT+f0hOO4mEeLbHokH4pJd2ZMHLaWGWhrm8EToDbseGOEv9VmmPABtpS1Bgw8kiKX4h\n",
       "0Q6uXD8F3OCCG17+B8RhuewMzvol3Px/FcJMMkKQD8Jn7tUtPl9X0XzAfq7HYc+GD/wQTBgNgIPw\n",
       "CATv+nwR5xweCBLuHecH55imKYpjBgdMC+eoo+GF/i4uKL/+ReNhiDAhF8BMcF7ggACRPAftXDVB\n",
       "3fWBRcsxcXIHnpvnMm1ng4SfGu8RgERH8Thtw+/XzX2erTuwoRNtYYP/GxjZW0OFo87UE8SAQhgU\n",
       "xTLMU93hFweRlmKZIQIBLMYMCBAXiBYjvc7/DsXFxcXtEmzILR/o4uLi4cxh4MEn4jA1cJcVDSA4\n",
       "MHmhvvRJZOggX8s0wbSyO5j+TGQAAAMaYQAnbgAMABnIEvG8NQ2iFWIi/X4738K2xqI/b0AfQyCg\n",
       "phU0EO7tMgpDoQLLyyxS/wzMGD2zRD9fGO+E+E5wyrDxpZ7DceDNgt4rRhG5S6JtYH0v2UJL+Wjd\n",
       "GR+Z9gwW90sH+ViXrwaCC3io0GJDFmugoaup/lqgRtUGBwW8J4IwGgSP4H2winYahWMFvJsBXcP5\n",
       "5MwjyyiUI5gnDhA/n4c4TqwUjBxmJ3hgIIfAfa+Qmx08RzvH8L9QAx69Lf3AEwqz8T41Ad7saDUD\n",
       "aAABgSoJS468q8QsfB5rmXjcCX2bg0aoHoPABqJOWEIr0ULAGeADAAGKAMUAYoAxQBigAxQBigDE\n",
       "QkJAaLWJd59KDMv+H/CgIekH3uGNrLYoPAGeAMUAYoAwCB6Zhmt7sAFIAlbvBDUUDFjDeV4TNCiB\n",
       "VeJSDO8NcEEwxakwvRm3JmmlDjoNrAAns9isA4dH+OKCR8/5ddg+G8bZhu44kEHgJ42CwTvlFjJV\n",
       "iWxWKxWKy2KxWNoSGmw60cuwwyf4f9WuWwmdIVlvj9MWQN38xw+J0s/7DHcKYE/sfnliYgIAAxQA\n",
       "ZYAMUAHIzY9ZeiPABigAyQAEEELABiAtEAnxnSqYfSH+w5wPBzXEHRCUkLoHWNiC89bnEr6HsQF1\n",
       "ABwEJcgaUa3cHeSSKJH/hPAoBaHBiXIFhrWegEwABAWJtN4hUMUhfWX8DfG5hc0iKTxlsVisVkDi\n",
       "QSSRM1m3WKz2KxuJDvEMLgwHw+Vrjff42KxWKxWQewK43KYxWKxWKzOw6QzVwsZIiHi9cVxLg7hq\n",
       "RBR7w8AHICZzjA+9AFMwkJmv/jYoAMUAGeAMUAGKADFABigAxQAYBFvuUsEcBfh5h+NhGB8vDOKI\n",
       "CdcR0KGXh9EubXw7FABigAxQAYSiGyBI420HXdznCBtmb+bA+/wNRAcczqrxLBjzCW/p4b4e7ikc\n",
       "iOY+lOzHgEHioz5iqFcx1FZvTJWhaEjcT8bFYrLYrFYrFYrqDq0uUXkhSleMoQQ1jEtm9sOifPwr\n",
       "FYrCrhXWoHX+gO+ET1rAeuHgFCgAAAAKYeABAAIcge8BawAAASZhAWuABAAIcge8KieRjWFg2MuA\n",
       "C8UrqyRVzgAOxi4tERLXd8/zKD7oxso+qRfAIdqOvnUaB+RBsx6ywYc3leA4IEZKer3kGAF1MiIi\n",
       "Enx0UmauHynzYAg46B9I2DolA0CIUcEzSnvRoucN6BANh4n2BWHaaVQIieAqwQ2iUyseRdPfEP/3\n",
       "sJBox5lpFFCInDf4copJoqOdx6VbfhwlhqAIh/MSA8GIin2kOKJdZNE4uO/3G4VA9h1k/Hwjqx+B\n",
       "aIwwWH4LON4SsACED6AvVAAXtUl7bi/2WKACgDLMyVTjzLMUxTFMUxTA2GEngJMm4ZRmY9Bu/6f4\n",
       "2Iu4COaJ+O30eZ50mG4tEQcxKQCQpimKYplmkDeAplmAoIVwCJpflXlmo0wPC0AAAACzYQC04AEA\n",
       "AhyB7wtwvhMMJEnggMzGfKiQcJLAX8QMIeFV4UnmKZZimKYpimKZZimWYpimKYpimODHAAJYn+Fo\n",
       "IeXbAAoXsZC1MCf0B3bNpVrzwIJCuK4nDDxbGCvh6CF0xQeYETf4fNtQUGiKYpnkAAJZggeA0NrM\n",
       "2XgZsDh7eFlB4a0/jYpimKYoYpghxAVAWkKdpmDLAl0aYP0X0eLaxnroCDHcDAY3w1MWGf3qYVV9\n",
       "wN8AAAAYYQBDuABAAIcge8CQfgWwn//ATN22/wlAAAABTmEAWjgAQACHIHvDnG0Dglb32tO82Zqj\n",
       "JnDASPWi70Y1+ydk9yAu3AA3IEGguw+LDLDA4YoUDpMFMPhxpwQJ4yL/D/Nc65wz/vxO/kZt8IsP\n",
       "DPDguoc18/iA+sADDuahaClsITgtmClmXwj95QnPOMYCvheBNrA0KukKxOtVw8wIWpoODqPOQtB4\n",
       "8MCiYMCxE1EYuQJHzWmqPkchF/Pn/paWbxP/0CzgKjCh4SKInUHmfoaATp+EHO1tD3wV8EEI/zOG\n",
       "e8H1nA7ACjuukyfwBI/XsZHuw/43kx8Bl1+8vPW3d1W2lvgHffX4qYC6EJIeFFNzFEnd+NimKYBi\n",
       "mKYpimKcXZLJHnUw/zMwUxTM6IsdwYMDcAv8ROKjLFjU/jYpimKYpi4aCSwKYjZIAWFMUzKq9G1R\n",
       "UHCFl+cXBwDEJJAWXXh6wkVoYKmJkCsSuE4AAAATYQBwuABAAIcge8CEfsTwKh+HIAAAANdhACHO\n",
       "ABAAIcge8PieHOGOF/jNIuNpyEFLh4JUZZtTxDoTNgCs/8b4EzcN1Q0CBGanXDsjwTz2wlIEzhWr\n",
       "5MTzFM48swa4OCbJUXw6ojd+GiyWfGr41UfGwSYnlMswoRkCLDL64Fus4NG6D4plmKYpkVUfEfNY\n",
       "ilYicHCaX7hCNJ4QqTAqP/G0kuITmisFM2kBIOPLM8xT0pEBrREnB5eYKpFmBcwIEEL0ctwypSHO\n",
       "/xsUxQxTFDtBdmPXimKGKGEtPwCM0hyv/AoOnAoCaK+cYBC35WKIPwAAAz5hACduABAAIcge8dwr\n",
       "C0AYWbxIgaH5oUw2WZWwfjrsDLYjoQkhVirwrMyuMeFRPifE+J8T4nxPifE+J8TwJ5+BbEcCAfz8\n",
       "NCOj8fwv4TdQIWTjSMGAsSMMOPAYBwlEfyAmDK7xuADtNkuc/v7gJNQziBYDg8AAgPqJYAXX8F9H\n",
       "uf/+cWAzwGeAMsBigMUAYoDFAYoAxQBgsDDMAA4zOv/CUDPwiCIL1MwLw/u7rpQBEQJM2OJFgN4B\n",
       "oKAMsAYBB5Nhl6+dwwDwAT1d/vr/5A6iAwEoo8QfBA8a0REux+E5YQg3wQSOBE0faYJW1zo+mNSj\n",
       "QATF7cDw88ZYxQYuhCRDjoKoct/Y7uHowEz1ZsgbvGxN7/pN3rA/E4EAIfBwIaZJQ/j/fND3n7/t\n",
       "8HIrwwP/mCwZYMUGKDLBngxQYoxEkJANViVLyWVpISt/w//I7KFAzxhOtwhRlg7goucwGhUNRqa0\n",
       "y54Rr+88bcmWau3RWLIdxuAUesPUHWEYeAxQGWAxQHBjabtErWZqHgMUB3gECoLAYiKIAEPErF4Z\n",
       "DrPRVxYaYx8742KAxQGKAxQGeA2ViAigMotRQGKAxQGKAxQGVFsRQGEb3gJ+MAxBAAP+cMIVHC4i\n",
       "77vzY/x/D3fEeI8R4jxHiPEeI8R4jxHiPEeI8R4jxHiPEeI8R4jxH8bkg0D68R4ywYoxRijVg4TC\n",
       "MORpARCFgxQZYMUYiUCwjrUx5n286KUNnSNFPwcXnYr42KDFBigxQZgmEHITmyzFBigxQYo7YPYs\n",
       "8AxIoMCOf7XoxI5IY7boO4aiHoxwgeA4rfvvRJ8BVBEyP8gkwjDj+NigMUBlgMUBigMUBigMUBgg\n",
       "bNZOGMXH2QEMEBK0exX4hEgcAEFWEe2dm8FVDHT3w7FAYoDFAYIhzNEEljTp9d4DcFf0/4AQgSuU\n",
       "ItfBhghr08jx00BR2OCPd8N8PYTN2frfsAg4BLBmPZE4gAJiZ7VfFiCtMyd+uaFegiZjFWluFfGx\n",
       "QYoMsGKDFBigxQYoOZZiuTS3GZi/mO2USvmeJJA4CKv4NXLAlb/Mt4dijDoIQgUEfBtiQrOmf+MM\n",
       "3C4GqCWzClfhtEhQz6LE/vwChQAAAAth4AFAApyAq8BawAAAAS5hAWuABQAKcgKvDOhlIAHnAMJY\n",
       "iyCJYaGN3gA4bAzAqnRvIxiK7XlrC2OKgCwZ+DDHaPuBOCOHQwbERyZ8Qk4CgmidVmdXIa3EAgAZ\n",
       "KCgkDeqc7LQYaIn3NYwYwP8VwLJWQYFTA5NmsAAQAaAeG7vcBqAOCmk4jECeq0pG564MEDHbbbl/\n",
       "fn8B8bhanV37ARZDj1//2kExgOIpDDQaBHMJFuAGigDy0AvL4A1ioEb1WcEeKIqF2uiAABBwmB//\n",
       "7wBeo5WAgAKdBhE4lJrAELxvCZ7xkF0KQSsPJGwwcyPDLDEL6H5Q866SRYYoYoYoYpjEAODxh70f\n",
       "AjzxrHYy76+X/Gz+AqwAUWZZgDNr/6fue7PjcZ1IUMUMUxTLDJiIIKZZjUAMKAF5qDYGQxnhaAAA\n",
       "AKJhALTgAUACnICrwvwpFDFMUxTFMUxTFMUxTFMUxTFMUxTGjHAALYn+NihkZFMQet2SvKiTS5N6\n",
       "C9CE/VtTxCUj8rEErprPEbSS54K+HoIOgCR4cAYYNSFDFDOCEASzIiEGkfx3wSoJFWDvY+AMB/03\n",
       "b3+HHDxsUMUxTBNi3YKYVgwLo5JIeGpFVtQzwL9wgXVlECdBhxqYMfvBIqG9EDWzwOEAAAAdYQBD\n",
       "uABQAKcgKvA6cmGEXS4Lwn//ABCa712f4SgAAADkYQBaOABQAKcgKvDnGxkPBl4gWKNAIMzaDq0y\n",
       "eIEbpZzkh+V4H5+Lghhji4T6YTwiewAhd79f/G1XVVWZQas9YvZ3TkkEIlk2ngcUsyIIcJ11R4/H\n",
       "I8ABXwvYg+QdCG4Nk/xUYLMSbgPuxyWGmvw6khfCN/TPwpFy8XFy+RcXFxcXFxcXFxfjgGCzjYuL\n",
       "i4uLi4uLgSegNtY88IDpNAKFMXOkFLDRAWoEgEPCbjsvBA8fZlWheNi4uLi4uF5gXREjApi5nRDk\n",
       "EmQI2htpHo7cASOMA0BADKJw1ZHCXRpHNNPoXQPoAAAAE2EAcLgAUACnICrwJnLwUwNlgaYAAAEF\n",
       "YQAhzgAUACnICrwjzYa5YCFdUnDRusC/5K4yAD+pXv33r64quf99u7rW7u4LOI1Wqqs/BPxvh1SY\n",
       "BE9ck/OySLwEAYPObq+AAbT7RNG1/eFhoeYplmKYAAgdGbwELV+3wH2fbPogPw4vW3tt/+dRbb+p\n",
       "r8bAkeI9iDuK5jw6ahMtQwEEDQH2+wCz4xQyzFDFDWLiLSAuceDPDERI+RPPAIlS3hSMnuBVq175\n",
       "/4KT/xsBkFr02kZfFJnBsGnkxLMUyzFOESqjSLMUw+FOCzEuBoxjMR7GEszzIzcOETd/Zzv8bFMU\n",
       "xTF7RTM1k/RimIwFyLoDgrmj4HFYB4hPpi2BzoGr3cPwAAADQ2EAJ24AFAApyAq8dwrEBDASzGhT\n",
       "GBhFhBIJKTZQwu2gwjsQki7Xhcy04bn64i7u7u7xPifE+J8T4nxPifE+J8T4nxPifE+J8TwHsfgs\n",
       "4XqsBJ9aE6AjawMM2NAwahDZHgAwDIQLM8fH1Sr4YfX+okpK7eNwEhboDemkUOoeANRuWNOG+RYA\n",
       "zwAYHgDFAGKAMUAYgAcLAGWAMUAYOhIDAYApR2JKvn4ycNaYCVP/w/IPzv4OGVhzIKswJw8AHaAB\n",
       "iQoAywBha2FjCZ6DrXdGB2QUIo6VOp9QJnsm+z/AReQzX6DfBBAlbzR0/U8xgymCDQwiQkfM6ChP\n",
       "ajwZ4MUYqAkhPKlhiwwAzbvl+kHvgR+NlBAgAsVdYJKFzGH++DxnjuIo8xaeCWMzm6oKDFGKMUYo\n",
       "ywYoxQY0oQgGg+zuA/1w1MeY1VeN7ilhVGPGbDvbkw6R3jX6RYM8exyzFx4lmQUmMpaOPyoCH3V+\n",
       "YEFIvveO43BvpFrYjwBigDLAGKAOQFcfkh/XmzwBigDPAGKAMQSg4ABAODCX63qQGOlzzZjZggMY\n",
       "/d8bFAGKAMUAYoAzwBgmtgJwKANYjsBQBigDFAGKAMUAYi7GevAsAJskuzswHhRBAAEAHC3pLEJN\n",
       "zRkNFptx4jb7u7uGuNvIxgcqATU+P2+HwaHjFBlgxQej04IHwyCz9FjFGeMUGNxCAf6YKGSkQHjO\n",
       "kXjYoMUGKDFBhXslmlBEJigxRijFBkfsBPOkmC4OPg4I/B6hAl9Z7SA+K9CDCg7hqAh2RgdaPAHY\n",
       "i+gEu3DC5wTcFaeFlB8Aj1Mt9/xsUAYoA7QANULAGKAMUAYoAxQBgTbdhWTExmaX7f0Y+noF4x/H\n",
       "/A3EBwAYD8vBnX4QC4Tkdx+/w7FAGKAMUBgSl420IuxjiVjU29/zl17S73+BkYfhVIIQ3Oe1AQIx\n",
       "Fgh+NBKLYgVd7M8N8PVwRvLBigwju4JgQ0DoWqcrCJkanDkYehFeN8IWYeii7fjYoMUGWDFGKDFB\n",
       "ijFGhXaI3e9u6i52ILsbpX4lIMGD9mglqZVyVMyURA/GxQYgYAE9XnHYSttokL90y4fGdH3e8Akz\n",
       "YYiqf+BS8uKRsfc1iLG8AoUAAAALYeABgAMcgNvAWsAAAAFWYQFrgAYADHIDbwyrLGYAFRZQOcsT\n",
       "RYCFh+I1zwARSg6I0pbzdzhDMAAIAkQeA1dxdQSjp+5FTgACAgDgvzV1LrL5R7zDOABDF+MNiKb+\n",
       "ZFsp1jYCqBZ+GQNzr6N/XAwP2hE/SX8WVDH7f/gBmACIB5h6zBs2sda9l8gqBLQd4Q8WAEgdfQDI\n",
       "CHrx5JS1sO0zgEADrDCr2hnjsHVk73rLghlEbpfiTI3e3EqX2jEKZSAi6uKYWh28CAEQNU7EEm9/\n",
       "pGTNXbHgBl0BaSaXEqNd3//AQvG7IyEzUyCJ75mKqR5l4k8/FMUxTFMUxTDKHCfBDoKZ8hUHvDqE\n",
       "CQsBkd9K/+NlVIRRXizLM1YCJZx4Azv3Q+mFMUxTFMs7IXqBTLMSDQbAoUyzJU77oL2YTe4nem+G\n",
       "4A5dRPMJD3rwitstkACP2vCrwhNLgRKRr7f3APUfiVRL7bwrAAAA0mEAtOABgAMcgNvC3C+BB2Eh\n",
       "0YQ5gluJuxhJZAZlfGXoCbAoSg6FCESSaAWVYeehqyxfL++8bPMUM8xTFMUxTFM8xTtBqCzFMUxT\n",
       "FMcBjgAEL4e8P/xP8LQuRIMUHHDHBBBHiXfnAGImlrOMB0CChXDK3OJMMErJ74wV8PQga0BpI8/C\n",
       "BTgFxTSm+Cw14yoIEXhs4Ya0eBAeMYNAE/WscNkfmIfGxTFMUMKkhChhAkMrEhCxqYQST2l9aO7Q\n",
       "D42CiGO4YYSU0PkFfL2wEv5Spm+BwgAAABhhAEO4AGAAxyA28BRhP/+ACE0160z/CUAAAAFMYQBa\n",
       "OABgAMcgNvDnGwwnc+tcUSHUjRwnoZyKkIMtUxiebDzOCAAE+QTuFmB8YGBrF2onzbBAuvzy3h/j\n",
       "oPOYxDdAJdoDwqJzg8O1MciORuEEsQoDfCTQsxCXmIMYk/Pic6MSieCrhewEFySAlYtAfka46kHm\n",
       "KqPiWYniAphbSdYgJPZn7DxTds3eif8bLMUxTFM8ycioRTPMBTFMUxTFMsxTFxwAGOAAWwJ9fXH/\n",
       "M+Pef2FoAIS6a9NNhyhZyB8tY//BmGTW8Rn5vgwxmAXzHgQsCwJLFtcOjLcwC2h5rHhnhlYISJhR\n",
       "zwV8bFDFMUxTFMUxTFMEnIijzgGEyhBCmKZsrhjVTDQFA0AkxwTPCuHdYQHJkQXV9fjYpimKYpih\n",
       "imLgpmEXFM7gm1OAwWHRVaQJZBYMDhw0Mvg6yYwQGBD+OuoEXyjWj5bIH0AAAAASYQBwuABgAMcg\n",
       "NvA8COCQTwzAAAABGWEAIc4AGAAxyA28HfDHAeHWoCMbYR902YH3CDPQcvxxjLQloDgxsMhnYoqn\n",
       "wDVEH7gCP5Bw34BZxvhKp6A1Sm6RPJ4pDzpw5AjwvRdIkCzFMsMUwacC0EHVu1HkxTPOGQjckwQJ\n",
       "nWkIcX+Pj+Pj/86He2u36+v+NheAgrMFmQmJ3QIcGFMaKo2sUzzFMU1lAEWZ0cw7ODhJE61DysAQ\n",
       "tOR7Wg74SKjgFfGwDwBhEESnVH3U7KQHPOWjpn4QBLDFM48s+kBG9tBb7mgZ5imgCaRZg8cCBBKE\n",
       "jwzOqkBfYzI5dhDE3zv8bFDFxQxTIzIRdypihiAsCmRsgBDfDszwNR+7/wcQ4OQRHXABfP83V2QA\n",
       "GLf0pUMXch+AAAADMWEAJ24AGAAxyA28dwrDpPoPNSMNDAAQqR8EzlEq282ULgjoDZeEp7BbpzwC\n",
       "a8EHgnrINA41GAQMtbVgBE+QNDQuR4MO2whIRYwIMNTl2UjMMb/CbSVDxuAIABGVi7Sh+BM0hgR7\n",
       "isE4Ai8BhmVxOrUbtVS3rAANKAItRPBlgxQGKDFAZYDFBigxoUDAQFTYna81hRFgwAUA3Dh1+H60\n",
       "GS66GB5glK81IsHYAFyAoMsGEDTM16+wAIbIkuODUUKEUUCFncX+FYwQJ/ZM1T/AGb6kf7BvggoB\n",
       "jYF4ftbHyZhrAoTFkwKR+PBngxQGBlwQiR5dDyLhUEV/h6MIajRg1Vfx43hlefJwZt0HgxErwHiz\n",
       "Jq+A0SADWjwYoMUGKDLBlgxQYoMaFCBB0GhTklRO/mcimYR6LP7joW5nw/uJFbAPmpK2AM4RSzAJ\n",
       "Fg6CG5YItxvhfoi36MDqQUjDNWEabamSTm+cjOVBp/odxuQdmoi5PQeAxQZYMUHAme5o7sydi2Hg\n",
       "xQdwASIWDEEUHAAIA9MwD4JsE7JzR6A6hNLfxnACA742KDFBigxQZYMCd/Nv54FB0ATEKFBigxQY\n",
       "oMUGEezYF0BEjdZLXuw4GIAgHslI1zfS7AdGP/eW3THoG+NzgePgoHAMCwYoMsGKDwuAhMmJGRCw\n",
       "GKDLAYoMRKIEFoBDUPILNe7NOEKI8Cb9EY/dKM6A3xsUGWDFBigMiLNjx3gFJYoMUBigMUGQFjOJ\n",
       "tm20s8CSQOAh7sx10Iwlfj/VkLbN4P42KDFBiQEYJYDFBigxQYoMBJ+YZWF5b81nQaCc+vEkgwAG\n",
       "AXhqjxoUAj1mjMlCO3Tw7FBigxQGAMLV4xzaE140TcAy+XHEa3n+2Bl8G0IQynQaD4cEF8Agevun\n",
       "RhftoiVP+mA8b0B0NcPYRtzAyb06KN3YSgnABg1fiM/ORBgfRIQHgUoKZDYGQzy5HbGheI3kfy0w\n",
       "Jn/l7vGywYoMsGKAxQYoMUBigM1FiFhVMGAvCrmJAos9XYkSEAEOlKJ4I1BnfTJ2BDfkPC0UGBI+\n",
       "MfiAqD5of2fDX3H74NTCFs0BG7vy/BJbg3VYA4DgFCgAAAALYeABwAOcgOvAWsAAAAEyYQFrgAcA\n",
       "DnIDrwywiwWEGYALABmAIAMD6dKJ9DQ7EM3aAATyGgIsLraRzFycLa+NZDmvvEHQPwoFWqDxPPGs\n",
       "Td+AdQBhzoHaDFCWOvgFSCEvxgARgHAbz/g907H/AskAkeEEH45SZYBJMsbXJQFYI/kgCYY8JA7J\n",
       "lctUCpveLBRAYiQilEIkHPAkDMUA8yBGqSo5mHnchaITSEGxhXOIUcABiADmUkMABHJ3V4Lofxxm\n",
       "YAAgMlOUcAJbA3aZu2P/aHwXAIKNvFgDD894J7//YEEYohSncpHOfAQnG6ZjFCQkhjQ2YdKPMsME\n",
       "LpkjwyzFMUxTFMUxggBweEIMIY03Ahrx9NLXixfRRn+NrhAlRCLMsNcEFk8pFgwRTFMUxTLMsxTF\n",
       "DAUQpwAvk7LETuQzfhaAAAAAlmEAtOABwAOcgOvC/GxTFMUxTFMUxTFMUxTCpHIWYpimKYpjjHAA\n",
       "MXl/4n+FopkZFNBTBJSKDAJ8RB3eI8GCnw+EjnpAi4HBXw9EraXR58AxgICFDFM8kEs4wR8NuBwP\n",
       "YeMRxH0IDhRQHhvdfGxTFDFMUxcSEAAITSSqadxbGZQX/7AXoGxnDILUBgMJSPccGYgqGmoDhAAA\n",
       "AA1hAEO4AHAA5yA68BawAAABBWEAWjgAcADnIDrw5wQcEutckPQWhVBAROVODjsmRZlhinEDDl+E\n",
       "DlQleR+GImGAmwsy8cABskOFwykIOpegWhB/iBXwvh84BCyDw1VCzVULDQcVhFi5mEoQvYMnwjfy\n",
       "oBfhSKYpimKZZkkgimcRARTFMUxTFMsxTFDRwDBXxvl0JC5oA2vgAFMqA9n8oYfUYJ9AIWj/vbiD\n",
       "UECR72jmGd+oO/ZnN5WtNJNvm7wSKCBZ2cOneLh2ADvcvcDXeTG1LP2XjZZim8FgsMUxTFMUN/yz\n",
       "xxqKYoZJySHKwHwX/C4J2H42KYpihimKYksChoCgKGKGs8oAAYnDnw6C/D9lGDhE1LIH0AAAAA1h\n",
       "AHC4AHAA5yA68BawAAAApmEAIc4AHAA5yA68DPxvhCyQ8yPJieZYZg0+YFZSAWUeFgsy8UwQgFgi\n",
       "eBNvDT1/Lo2Lg4RWdJtNvW3+H4UOAIm+g70izPMUxTFMUxxcm0zw0I3wSLkTQ7hkrgK+Nl0sykzz\n",
       "xcUyzFOGSw1ZtLMUMyUpZjBgQCE85YwRNnhkSHIT/xsUxQxTEHCY0zMoozFMsxTDRcBpmX8XwPCE\n",
       "+ROTvBK1m6AgCH4AAAKmYQAnbgAcADnIDrx3DsUxASCceQZYswqWyA++PkQkjpSRM+FzGownT8Am\n",
       "vBB4ED2BCdYKNIwI3mBh1CDHcjdpBgaA8AYBkIFmfhtOsgBkf/r/4mfSiN53jalJUIauwARcCCQV\n",
       "qpiA3Tx4AN4AaHgDLAGKADFAGKADLABigAxQBg6EgwwBYdToPSH7wEIBDAG58f4ftAAGI6w5kVtu\n",
       "w8Ab4AhgIoAywBgitgCu2ACsCdrwC4DqIDAXHLHCZhigjOW8v8CJkZoMG+CCQCowTaHzNifo2lzA\n",
       "sGA1JtUeM8YoML6CSIP4GKjICJ6PHWNv8bDs85EmZ4eWMydyjLGFRMFjFGKMUYoyxijFGCwMSANe\n",
       "Q5TEEB2f/h/kqqoUhwiAmy3AWM8ExexNHSwO0EUN26mVCT2VH3a6JWsdcO43BZ16yal6w8AGKADL\n",
       "AGKAOVulGgbBYAxQBk4A5FgDEEUQACH/BRk89ek8Z0B3xsUAYoAxQBigDLAGuXgKAMomICKAMUAY\n",
       "oAxQBigDFhMQQUAZu54BEIRAAEAHRksifUVVHXkWmcgN8boAEnApsnGEFjFGWMUcsaiEAAIoMUZY\n",
       "MUY0MCyAP7IRfgc5NNUo47iP42KMsYoxQZgmGszgQQBijFGKDFGjGwzVdRiRQcEPi8kN8TGE6csO\n",
       "4aglGQMKBYABW0DgTjY8Km9A5bCAJzHhKWPItV4AwAlcDN87/jYoAxQBuwBZFgAxQBigDFAGKAMw\n",
       "HcUUEZC8oj8LCeN4G4gcAEHXLAYZ/CJYWcdPfDsUAYoAwqADgjLDaQCLbGgm+e+BKvlu2lsB+Bsg\n",
       "hDc4QM7JWxMsI7fBHuPdV36G/i44g5xsUYoxRigxRijFBigwkXNIlH3lF+IL/XiESEAQqxxnGe0Y\n",
       "TrumYphLgFDgAAAAC2HgAgAEHIDrwFrAAAABHWEBa4AIABByA68Ml+EzjpA+QZgANCYKJfaFFVyU\n",
       "xZkhA7AATyHgYYTrSRw6MJWPgOAJJ44ii14ImvhcGRL0gAXA4IHVtEOgspx6ffcgABAHsABCx21b\n",
       "AYTHInTP+4DC77QBjcFRHIxBAhkfuCAAQIQihQOlFtV89mH/+BAfgAMIsDDmbC6nnM7RDgRBX+AE\n",
       "LAEvIIjKzncwVDEXh38EAQigQLKXUMeMorGXGLPEAAY5YANikMTjAVbGMdef94FWwAEeHjHttcMo\n",
       "DTGGPgANmA4PK+ufPjhJiaXwCF43ohPmwHVhZlmF6AsyzFMUxTFMUygB1hePmEaadF+NieBqMsy/\n",
       "cKmITeFMUxTFMsyWHgEUy/gEAWY4i5fEeGDwtAAAAG9hALTgAgAEHIDrwtwvghWJJRixVk2BEXJI\n",
       "L1uCxBL8JiZfokJ2j3hSeYplmKYpimKYplmKbk+LMUxTFMU0cAOJ4LOHruWcMmgUxTODhZpDA8H8\n",
       "HkvPDfw7FMXFMUxQxcKUJRGYx7nQkl89hjREBwgAAAANYQBDuACAAQcgOvAWsAAAAHJhAFo4AIAB\n",
       "ByA68CSF8LFAh0PZWSHQ9lZDAFAKAAU8KVVVUTxTRWwqqqyqjjHAAIQWcbVVVVWFNycYLCqrPF40\n",
       "QAEADnGE14K8QRVL8bVaqRhSeABFJQK3xWRiQhoXPcaKA+Ao29lCAVYm4ZpyT/2IwfQAAAANYQBw\n",
       "uACAAQcgOvAWsAAAAKhhACHOACAAQcgOvB3wxwUSgIqIvZVF4OyUqEElmqX4LznZiTIuCzggh+wk\n",
       "5uFmXliHyodUWZZimKGKcgWkOFaxI0M5CRzx/wivt/AoF+H4VKFUWYS1OCmWYpimMAgOVlTwDxRs\n",
       "QTziAkCAOMDICvjcU4gwFDFMsxTheY5Zi4XNCzQIEXiuNCPHP/DsUxTFMUM752kUxTFMMGQoCRYy\n",
       "NhE8QNkA7bRiMPwAAAJpYQAnbgAgAEHIDrxvDUAL12lxkWaMCKAgbsP9MAJCASFJAYWPc8AQtGBm\n",
       "CJRs+ATbhfwk4uASNZpJwArYAVh4MA4SkdA4/0ZPc2MWLxtwhe+gbhhATwGt8Z+A5YAywGeAxQGK\n",
       "AMUBigMUAYoDFAYLAwMYAP5sG3Bx3/h/ChNGEOGd8WDPARgigMsBgh+zHeAEBqBgrAxMMYZeCB5q\n",
       "gI+t6f4EDRu9Bvgg2Y+SFplGTE2vZE6BwBLGeMUHkJRHP0B0qBlVqA8HxsNIgSJlnhYLGYPSyxij\n",
       "LGKMUYoxRijFGKNEgaPpTPExhDBiv+H+FEECaBiwZO4FGWPhtBQOSFTP1hp0SQY0fTDuNww7EyLC\n",
       "IeAMUBlgxQHK+mEWDFAYYAWgsBiCKIBFhxXPfSEqgjnmOayFCD3+tfjYoDFAYoDFAZYDBdZEFAZR\n",
       "aAUBigMUBigMUBlFyCQQgCVeg4GNAAK7oRoZLhxTAtYG+NyRWehAEsYoyxijiB7mxMOQWDFGWDFG\n",
       "iEDiUYOPujE2MNy6HEfxsUYoxRigyVXNF+WKMUYoMUaUHRjv6IIP+ToyW4P42KAxQGKAxQGKAxQG\n",
       "KAxQGEfUWWKF46PCieXEIkDgI60xiXhK2BknePw7FAYoDCwDCBBmZzcQXPH9HEa/Y/4WkEj7/jsp\n",
       "R8gp75cfqDfD2QdtYvW8ACwcACQQgZpgTirQMtE6JC1fA/gglYvxPudWkJoU3w/GxRijLGKDFGKM\n",
       "UGKDXyiVIXoRKUne8iEBFX7Q8qdo8OxQBhsAB4NgDxgXvw3YE7/5vD5wGuLhaggSlcPR0fBGWY3r\n",
       "pdOAUKAAAAALYeACQAScgOvAWsAAAADtYQFrgAkAEnIDrwyX1bzRmAC8iM6Fndj/mAAPmSHGVr3/\n",
       "neQwOI1nywKoX2GaCVrQEFAufs6CUFP9dwAB4Bi/Ys5AnXE9+/fLDSgO4lwY6BwzbDzkAAICIhAu\n",
       "w5PZcHsjB8n7oHwKAoA+sxbzDED7mP+v1++AOMAIFtLmpvdAD+4EjuUuGDKAc4zyfgPAAEJ4BZdV\n",
       "bwEIAKJAWogpEnmpomlM+GfWc4YfmEJRza17zDkAAZXDFoxXmugaAQvG+FZCHWvKmyRcXVVUqBYU\n",
       "qacOJnLWvNfxs2MKRcoopeNGSqqqXqLjFAyEvPoWTPhaAAAAOmEAtOACQAScgOvC/ClVVVVVVRcX\n",
       "VVUcY4AcT/C0UMYoKBoKYKBxB4IhPQYLAaW489DAYV+yaOjgJyAAAAANYQBDuACQAScgOvAWsAAA\n",
       "AJthAFo4AJABJyA68OcEHaHWo3mecPNCzLMsxcUwOZgYGkhndI8DQ/YQNdaF4YimBrEMUyw6opjS\n",
       "ECWEmi+EHgCnTHUYS/ccEAr4XqFzgOwqHmS0BZiyrAvBrgwKEcQVQxIgVEfuRvlQDvhSqqqs6yqq\n",
       "qqqjjQWcKVVVVVFxz04ZjVZJgLVnxtVF1UXUyVVZkROglGzgtmCdmVvg+gAAAA1hAHC4AJABJyA6\n",
       "8BawAAAAeGEAIc4AJABJyA68DPxvjWBg2F8i8vQDVRdVGMHRY8HhXQINogQRWHrHxqzp8bMkIZUm\n",
       "mxKrHDqXqrPZ4rg8vhAwwvCdrAFfG1F4uqqs3oci6qJcGQqxxE0y2miUThmd/jaqtiiHOGouLhqE\n",
       "IRNCc/iLgUS/MYKQfgAAAYFhACduACQAScgOvHcKxAHCwNAwxlhQMNAaQzQ/EdCBZeJ6KGFgeATb\n",
       "jaw0JchYy4XC2WMsYoxRijFBisUYoxkBAh0KsS7heZi/E5/h/iRGCGlWLBljFGKMbXJxwqEcBoGJ\n",
       "h+KfJ4RcOEHON2YoxWa9uKxWKxWKxWK3G0MBXphbO8fxuEXaKkAASxijLBijolMpnAsGKMsYoxFQ\n",
       "cAink8eQPrGcAJ/jYoxRijFGWMUWgKM94oxRijFGKNCOwKDTvD4EoA/kzckTtpg3xuFG5LZbd3hV\n",
       "uKrSC2Ky243EhysXhekaE2jiP424rcVj6DLcVisVuFEmNC7GPiI8cuZnhDYsg7hqBNyNPw8AB9Bg\n",
       "DO9Ghjm8CbgiZHDwoWRwBmdUyddf8bFGKM8YoxRijFGKNAIS5pR+M11HjKBwEdcaix/CFuL/+HYo\n",
       "xRijC6QQTdm9nz31im/PFEITMyJThNiTDqKUwX8am74g5xtxWKxWK3FYow6hZJWW9sFZC9DHH0yb\n",
       "4BQ4AAAAC2HgAoAFHIDrwFrAAAADRmEBa4AKABRyA68K8VgDwABAUAAUAAQKQA4CA2l0o4AAieAA\n",
       "CJCxBwABE8AAESFiDgACJ4AAIkLFqPQyB1AOQaUR61P6uAaRTEVTY9XvCA5jwBsXMvKCExd/ml4A\n",
       "ANYAAICcAFU4SAlGCjHML7xRgHQKKWMqGgHOxyWSt5IwDXE4co4tfmK8AAgGGGQlHmmq2JTMpHR1\n",
       "Ch4Qg1hMLmLXhJCKvhmkVzAABAaAA4eOJhgABwMK0mAA6DDnDObfNnkQts5Fr1xBAHHJpiTl0+J0\n",
       "YGCMKdxMEGTCSQWEo7Kul/73sxqAAIGe+GHMgACAgo4AR+GMGuQ7j2o8Hk8pbpoAgSjWJoAERJBH\n",
       "SbX/2m1DAB7HO4f0AAIAwphvYBx/o0nIM7VvgGDHjSjqx04FBj9ybCY4/gQAAgHAQu9gGSfXtmLp\n",
       "iGjkjyAFXBhhAD0MyHkRR2Nt0ZAYHgKyvsBJGMWhv+ZcL0nLGOV+73AMEdAwIk/deDgUe6MzDbGg\n",
       "iTTDAEg2wgYFrH8+SgsSRvN5tkNL9PIyEVlMdCiJShODCMy4+Tfge+dPV2QAFCILEuaeHId6fLVf\n",
       "X/e+vWH3IAOULI2OI+X5FT/4wAAQFQA5wKllr4zv9c/sBBJwcWnXS+kkjdoPDDGQAEQAAgBwCgwL\n",
       "aL1uyBrFaTVeIBiAGtBspTIyLwRYHR1JPIXBCSBrk1j6yxVLcEwDeIrkPEn6KNOvs/gwAEACBxOk\n",
       "K+MeiuAq3GzQ3h+wB0Dg9KLUtAAHhdFjzQY/EAAaj+CDkDEeDIGKLDC57BqAMp/oKAAQA2gjXHpJ\n",
       "wEAwC32agIe8GQt0z7Wq4oAAQBwAK7feAwY1YgFk4LPB2bEbmAoKAA1uEYLf/vPewAjlcFApUTEl\n",
       "NWYVfwBwAnuMPACQBqcRf/ieUgy+J8wABAAwAo8CqC6OWvABxACrvEBCnC2FDppMqFk7hyx8QZBC\n",
       "WJMNUSMRVWR8NxYTXcAHjxc7qsASgoh2SGfFBgIABjYNQW//ecAxVqARI2L4yb7AOLAZ//3gBjnA\n",
       "UJW2sPrE5/0NxJFUBCcbzhAI9oNYdkybyLi6qqjFAsrz8IXsb+Nmw2SX3C1seGKqqyXqLjRjD9Gi\n",
       "do6U8LQAAABZYQC04AKABRyA68L8KVVVVVVVFxdVVRxjnE8FnD0Iktgs8RwUxTPLBZwZvcGTQYe3\n",
       "wlOTxBYey/h2KYoYpimLhQKhNIhaaUrxpkvAdDDGy1P4kgjNLM3wOEAAAAANYQBDuACgAUcgOvAW\n",
       "sAAAABFhAFo4AKABRyA68CT1RwDA2QAAAA1hAHC4AKABRyA68BawAAAATmEAIc4AKABRyA68DPwQ\n",
       "R5S+RebtzIuqjGC57x0cnDChjLNNbTX/nVBZw9KyXxdVVVFw5ABUgpkD1NN9M/8O1VRcXk9VFxhC\n",
       "GsZ/U1Iw/AAAARBhACduACgAUcgOvAKHwv4ceCbpWAE8wiYbSPAGF+CBZn+nAm0XxvG3SQCjWGmW\n",
       "aQS2Wz2K3FYrFYrFYikDQ15OFLP8OcEGjQgmt0TNhwQhBBo6HjPGKDNCUj+CVoWBzGAh+FMpBsto\n",
       "HcJuQ4VisVisVlsVvEd+H+QPl4zwYUZQoyx8PKb8hUzNTKyOruETjQ7gg2KxRvDCcGCLG4rcbiE/\n",
       "lf5p4zgBP8bcVu4rufu7u7+K8kcByMV8N8KbRAUItlt3dhlEBdEqFsVlt0S8R/ClxW4raxXoMVis\n",
       "VvXejWP42K3FYrd3cf8/OvZLyF5EDiNejj8s1HxB3jbisVisVuK3aDKEK78+IOtfh0ac90AocAAA\n",
       "AAth4ALABZyA68BawAAAAKhhAWuACwAWcgOvDPH5QGh14ACABsAgnDTbYvc6mmfQE/yAAEADAMA5\n",
       "O7frgAOi+XNlAobJDPjgQFgAb2CIFr/3kBQIA5CU8AAoAFYSduIx2vvVhogn4YAYkINqhmYj3PMU\n",
       "RJoZh0IABoQIaCBwyI9N95BVaXeEQwyfYYAoOKR/Sf/vFa9/5whw4AAgNpACbaAeqMwDB+HH6yCA\n",
       "DrWNNAGPwsIJNN2gE3AAAAARYQC04ALABZyA68MdUcAwE7AAAAANYQBDuACwAWcgOvAWsAAAAG5h\n",
       "AFo4ALABZyA68CSHFC4BZMJhbCYX/jAUIIEmEwkkm2/xtVVVEYKXqqqqqo4xwACH4I/Pu4LONqqq\n",
       "qqLmBAGaGqqFbQOMaAB/hCeEOj42qi6qLqFspVZi4uENlKaDHvBbDwW8/oMPRATbB4QPoAAAAA1h\n",
       "AHC4ALABZyA68BawAAAAM2EAIc4ALABZyA68DR1RwA/GxGA02LPK2ouqqDGqTGC0Bwkh6K2+Wgf3\n",
       "DxDhIwRwFh3gggAAAS5hACduACwAWcgOvAKJwpVmPnC0DLMVlstitxWKxWKxXRgK8P8Jk8TYlgyx\n",
       "ijFGAha2B+l7wpB/gVh2OPkX1BArKHghzMoOcbpOKw803bisVisVisVitxzGKFWDWxxyXHx/BVhp\n",
       "SLfYrt3cVujHGEyN8bcVu5bf3f3d3exWPl7o1/bCWAb43EKhMtit3eK1KkFYrLbmjB+TzCA4j+Nu\n",
       "K3FYVRAc5YisVitzmGuMxx/LSOhIH8bFbuK3d3nzs+aNFZel/h2KMUYowyD8EqyF+4QGv9/D1A4s\n",
       "fhxzDMcgkZ/dvUG+HsdV8sGKDCypOCbMjfAzU4sQRMjGCiUEXwSPj6ZEw3M30LwpcVlsVitxW/bW\n",
       "N+X+8PqO/DMUBhCjSCgDpfVAbIIWz8BFp9FuW+AUKAAAAAth4AMABhyA68BawAAAAPNhAWuADAAY\n",
       "cgOvDPH5wwFjgsABuhsAywb2mhKLRHv8TCM5+F/vP3378xMUVPEEc01pBDfgBUCBIbYoq3PygHHT\n",
       "P+CHY2MifDJIAAQO1AAEBh+u8oHpmrwAgEJseSaL2xBgilvOAAWAWgIgCEliGEFp4PUAAcB2AHVi\n",
       "tMABcBCvXiAAEAYBCwANzYfd9BsMm7EPKH4YrxbqPSpsg0N+r5IPGAxSQnM+5isABk3TAIyZ6YQA\n",
       "74MCQsII7ZKadZaAQnG86oN2GPemROEi4uqquQ1ij+3GlA+dGPy2F+Nuw2SXhvUpk2qqqpeoucQU\n",
       "Bk19Vx6MPC0AAAAvYQC04AMABhyA68L8KVVVVVVVFxdVVUcAOJ/haKcUwxTBTgw4ezwaR4nDCemz\n",
       "ATkAAAANYQBDuADAAYcgOvAWsAAAABBhAFo4AMABhyA68CT1aBsgAAAADWEAcLgAwAGHIDrwFrAA\n",
       "AABnYQAhzgAwAGHIDrwM/G+G+EXkXl6JVF1Xg4iJKGhjIxVZGrIPDihjLNNbTX/uU0/4LONsQvCk\n",
       "yVZWf8OfpFy9URFMxHdaK5IlFJ/42qqLjlNJFwOouLh7kQQAH+g0X4Q4tIGkmxagPwAAAKNhACdu\n",
       "ADAAYcgOvHcKxTFMaVBlhTCxygOFkQkhCQrwQHgt7h3nfAJtxt3DiIKsd9EWxWWxW4rFYrFYrGlC\n",
       "Q0NmK+EqQ99Lvgs43HoatpYrt6ZIntxCY2NxAl40n40kcrGcAIDvjbit3FY2xd/d3d6xWG0AIOYi\n",
       "A8r5BgwskilBEQHOqOAMIcbFbuK3d3CDuoTNHq/IWm8ZQOCVegOCB6IPdALBAAAAC2HgA0AGnIDr\n",
       "wFrAAAAAsmEBa4ANABpyA68M8VgAUQd4KhPKJHM8Q3QPAU+TArDM6HilN+gCAgOoKTJNrDdWZVm7\n",
       "2z6AACAEAKDHgXQKqMwtjIBBEJqcKITpzSdPTsYUa59lt/+XgIK+DFAROCazoMhVDW+KGAMRvimp\n",
       "ifeSPr//cAwAckeG1JPLRUEy6vnzveABBuASa5oINUE5UsFwhi75wDOAQOiH7kTOmNMTN/AAYdQC\n",
       "j3ejzDL8vVDEUxnATcAAAAANYQC04ANABpyA68BawAAAAA1hAEO4ANABpyA68BawAAAAl2EAWjgA\n",
       "0AGnIDrw5wxDmqGapYsyzPMBTLDFMS0DAJl8ughXvn+GIgwCvIsy9N7LMaCCg3QSaF9OHdwRl0eg\n",
       "V8L4VgAECL34eZRSAs0pEQWGWDiL8Ejs1xTinkb5U+FKqqq4AAKyqqqqqo5gs42qqqqouNGJxypV\n",
       "brMQRWQxGTP/G1UXVRdU1VZ5QpNgc8gqfshppWEUH0AAAAANYQBwuADQAacgOvAWsAAAABBhACHO\n",
       "ADQAacgOvA0dXIE2AAABg2EAJ24ANABpyA68AofC/gSNY4kHEQGwAOoQAKh4MVIJRHYcLvQE3+fX\n",
       "VaRBeFKjQQXu4nFYrLYrcVisVisVgsoYK8P3iREBHSEWDLGKMUYSOUGkBNwa5y4DqIKJzlBAOPGA\n",
       "I/b0/w4KIN8EGzDXHIWYDVSk4ReEHPPGeMUGBngPRzIxvqoejDNONE/G0n7FaVzmIrFYrFYrLYrc\n",
       "bQwBXl2D//w/wyNGWDBF2ugUZ4+GRECwlqCpmgKbogm/bITceiDtqh3BBsVu8dBDe3dxuMfw8uV9\n",
       "LGdAd8bcVu5bobn7u7urQCvHxx+PZ/DfG7WIrLbu8VuXisVltx8SHuK7mM3xH8bcVuK2sbisVitw\n",
       "6tG/I5gyM/ueA7hqAU7Qz0HgD8EYoc8ALCJkcEav2wBwELQ0vfn/CkVv2K3d3Pb87fGUIteHYoxR\n",
       "ijD4sIXV+hQjcQX8PQEmZjjFJMmVwwscgPmai9C4g5xtxWKxWK3Fbnst5wAAq5XIhBNeQzDqaZi8\n",
       "7gFDgAAAAAth4AOABxyA68BawAAAARphAWuADgAccgOvDLxIIxXgAIEUFLCI0KoFBGDyn7H4QAUC\n",
       "XXRssPDDIcckclcwfQAG7ACOVwUClRMSU1ZhV5QAAgPySAACBaAAIA8AMl2ECCYACAANwCEnAKNp\n",
       "n7GgFeQMn4XbDFEI9MNmxRQ3Vnq94KgBA4GGSh2oRAt20QRWYk+Y0KInXPBob1qxOyvgEcQSjMaj\n",
       "US0raLEf/3AADKDClhx7jM8EjuYK6q8NjAAhoIKjzudPchWHy/yACHLxYSVzMkAyRhrEugITjeGr\n",
       "lODAWjZMrlkvF1VVAAQWFeNrWvGD+NmweykvcMLrqnGKqqyXqLjFAoOGnDn0Owsf4bgLLITtjC3M\n",
       "gBKveIuEJkXCj3wAV1/J5gn28KwAAAB5YQC04AOABxyA68LcL4KiLRF0JMlUjETM7zIFCUK7kFW6\n",
       "JoH48QkbjC+8KeVlVVVlT8XVVUcY4AcTwWcPQ3Awlngi9sFMU3RjCzeNMzDbhIqQHbtSlwIum/4/\n",
       "qGoD4dimLimKYoYfJdAXWDJKPEWgwON+CdoSMM+BwgAAAA1hAEO4AOABxyA68BawAAAADWEAWjgA\n",
       "4AHHIDrwFrAAAAANYQBwuADgAccgOvAWsAAAAJphACHOADgAccgOvB3wxwMHIQEqhezURY4gzKaZ\n",
       "EEljiqiHgmI7ICZQ46AWcEFRMYKl8P0xhzIuqjFBuX2Efivhxd5p5r7mtzX/nSXcf//h+FqtaWZN\n",
       "UUyzFMUw0VMrJcVweV87wiaW8cEICvjbYvC8aCrKzfDkV4ckXlRjAhEtpsVypGU7/G1VRea1UXUj\n",
       "HgtgxLPAtHjWStw/AAAADmEAJ24AOABxyA68AX9AAAAAC2HgA8AHnIDrwFrAAAAA0mEBa4APAB5y\n",
       "A68M8VgAIBIdwQDgX5N7DYx0AiXBZ+AAYCAcIkdcCBJ+AMrk3TjG34gABAYAAEAgDBRQYi/jBX0X\n",
       "wYcgNHgCAZZ5KQcXcNBccAmZiP+0QT8wDBQp74A2ZfZtVCSkR5PABgHpWASX0tlcphoq/wAwFQJd\n",
       "NPPMAPBAxgnHUpXiE5BUcS4IENK8GSjwaOEsywBUVof0AAIDpWAFnhijSpXjAhAYavywQk958AwE\n",
       "JEJR4dKqKGFAABAoBb18GFEHmJeJUPRB1Efjd9ATcAAAAA1hALTgA8AHnIDrwFrAAAAADWEAQ7gA\n",
       "8AHnIDrwFrAAAAANYQBaOADwAecgOvAWsAAAAA1hAHC4APAB5yA68BawAAAADWEAIc4APAB5yA68\n",
       "BawAAAAOYQAnbgA8AHnIDrwBf0AAAAALYeAEAAgcgOvAWsAAAAC8YQFrgBAAIHIDrwy+LFYAFiBv\n",
       "BdCtAg39Vd8Yd9AEDhwABAhAJy4xokH1kRJFz4E4CQX8gACAOmAOdohYohT+AAtgYg1eRL5cKROi\n",
       "kj1xgAhqGB05BhbQupUEUxH/1nFGQnwQE2BpQ8Z5NQV2nn/7QAfBMBR43GthGiSLv1s1YVf+2917\n",
       "+nIHovtH/fviA44W/Wlou8/3/7QoQExCjAGoUsfk+9ADSiGpFQjWcpXroAB8MiLCeG3cxV1oBNwA\n",
       "AAANYQC04AQACByA68BawAAAAA1hAEO4AQACByA68BawAAAANGEAWjgBAAIHIDrwOvBBATNte8fA\n",
       "IYD4CuwnYfAu/ELWBCf8CeEJDaHyyeARDWg0Z7naAIEAAAANYQBwuAEAAgcgOvAWsAAAAA1hACHO\n",
       "AEAAgcgOvAWsAAAAamEAJ24AQACByA68AsPG6RYFvcQ4KxWKxWKy2K3IhI15Ongs43Cis5YLb28V\n",
       "qOsRWKy26JD+Lxw0cR/G3Fbisfx1xWKxW7z+iI9YjCN3L6As424rFYrFbitz4vld+NIdETVsYlwJ\n",
       "K2fgFDgAAAALYeAEQAicgOvAWsAAAAC3YQFrgBEAInIDrwy8IgSxWAAgCARIn5QAjmn8XMyQFnoV\n",
       "YeAA3MAI7uCgUmIiSmrL6vzuTR6ORP/+5RP3IxqAAIL/dXrAGAAEJgC3gESgHWBI/IkhDSUWiya/\n",
       "7nAAYABhW8XJrARdlACYWf/2hYDDr5wufAqYlIIyf3+8NqAOAkp1MdtAsM06fuBd51mGArKloYTw\n",
       "1/3jAAEAYGCBzBKV10JwKddQ6+OOAXAxYKXx9ILAfThYAJuAAAAADWEAtOAEQAicgOvAWsAAAAAN\n",
       "YQBDuAEQAicgOvAWsAAAAA1hAFo4ARACJyA68BawAAAADWEAcLgBEAInIDrwFrAAAAANYQAhzgBE\n",
       "AInIDrwFrAAAAHVhACduAEQAicgOvAKJxt0nc9DWWy2KxW4rFYrFYrHwYhrwZmJ3eCzjcdpW0sV2\n",
       "9EgBKCUbbryRY3ByMHUjs8DQjiJxH8bcVu4rG8O/u7u78Fa/MfHAfx25TLrAWcbFbuK3d3YMbIXq\n",
       "axuGMH0xbw605OywCwQAAAALYeAEgAkcgOvAWsAAAADiYQFrgBIAJHIDrwy9lFWQAFoAAQDQphgK\n",
       "JdOkZGyhb6GKcwACCuAJXSCREAuYY2BCBZ/5QmfAAUAI0AeusiAvYOC5i1dg4gBEYThK7yX1ZEI9\n",
       "rvAQqQQbLHQi4rBYtrizaD7ZZonwACnAcSF/LoJ8RtVjV3DAQ4a4Y6DlBvDwLYjNeIeESwAKgqq8\n",
       "cC4036eCFaOGda70qSNPlHb///GAAIBQKC3j4zSeIP5wnC6Lap4yAB0IMYDSUgLjR4FxJFfkAB8G\n",
       "ACjp0H+wWutMJxY/febAMAQVv4WLNplbYViiEsBNwAAAAA1hALTgBIAJHIDrwFrAAAAADWEAQ7gB\n",
       "IAJHIDrwFrAAAAANYQBaOAEgAkcgOvAWsAAAAA1hAHC4ASACRyA68BawAAAADWEAIc4ASACRyA68\n",
       "BawAAAAOYQAnbgBIAJHIDrwBf0AAAAALYeAEwAmcgOvAWsAAAADCYQFrgBMAJnIDrwzxWABbxfE7\n",
       "qkaP/1A0AAwHABjBEiSPgHJNwL4tyGAIADFC/AGJKROM0fOX7Al7tTAPGAAIBhRxivQoFqjuRxOo\n",
       "Gr/skTxWuPgo9f/nWCxyfK+dmP+rhkAAIAXAFAXzrYg0Mg/QS594wg4SyQcInxYafsLZvIPpAACA\n",
       "cABgBs0gGC3HY2ACg7b8/34AAgUuUAAQFxwsEi3xIusw6YAF6VQB3+wChJXB3/vAEEHZrzBkrmap\n",
       "Wog2ATcAAAANYQC04ATACZyA68BawAAAAA1hAEO4ATACZyA68BawAAAADWEAWjgBMAJnIDrwFrAA\n",
       "AAANYQBwuAEwAmcgOvAWsAAAAA1hACHOAEwAmcgOvAWsAAAADmEAJ24ATACZyA68AX9AAAAAC2Hg\n",
       "BQAKHIDrwFrAAAABRGEBa4AUAChyA68MtWxkAChKM8Vsyh3olZGn///ABhOCicA43Sugrwsjsx+8\n",
       "YM45P200GqI1uI5c4AAQDygCWdx4B8itwABEqIPvVppGKqToq8AFAdVpwiy/U/FfrAVBhUQVCuHU\n",
       "gm0mucYGzgAHgSTVPNWwD1n3GCMCiVVDhroZTutt//eCAUL6BG20gKC8WC5w3jpx4MAAQAyQA2bg\n",
       "LCD72L1hltjz3M6AUkvgWI5Cb+rFo8f772bZmzDVV+gK4/AAEEIAAQGSgHtisIhn+vrsXpGKekX9\n",
       "/+ABgUKgs5EBcCKvuxMA7oT/L7SspInRZrr577tAYbeLYH/+Z1vYcsX/5sIkAWIAqh+mbc+7gIFk\n",
       "W+8FkC6GglBSSwwf/7QDMD/rwYsKLr0S5wcAaiTweP5TxbMo07f/aAZcnX0F5Za5//YCbgAAAA1h\n",
       "ALTgBQAKHIDrwFrAAAAADWEAQ7gBQAKHIDrwFrAAAAANYQBaOAFAAocgOvAWsAAAAA1hAHC4AUAC\n",
       "hyA68BawAAAADWEAIc4AUAChyA68BawAAAAOYQAnbgBQAKHIDrwBf0AAAANnbW9vdgAAAGxtdmhk\n",
       "AAAAAAAAAAAAAAAAAAAD6AAAUggAAQAAAQAAAAAAAAAAAAAAAAEAAAAAAAAAAAAAAAAAAAABAAAA\n",
       "AAAAAAAAAAAAAABAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAgAAApF0cmFrAAAAXHRr\n",
       "aGQAAAADAAAAAAAAAAAAAAABAAAAAAAAUggAAAAAAAAAAAAAAAAAAAAAAAEAAAAAAAAAAAAAAAAA\n",
       "AAABAAAAAAAAAAAAAAAAAABAAAAAAtAAAALQAAAAAAAkZWR0cwAAABxlbHN0AAAAAAAAAAEAAFII\n",
       "AAAAAAABAAAAAAIJbWRpYQAAACBtZGhkAAAAAAAAAAAAAAAAAABAAAAFQABVxAAAAAAALWhkbHIA\n",
       "AAAAAAAAAHZpZGUAAAAAAAAAAAAAAABWaWRlb0hhbmRsZXIAAAABtG1pbmYAAAAUdm1oZAAAAAEA\n",
       "AAAAAAAAAAAAACRkaW5mAAAAHGRyZWYAAAAAAAAAAQAAAAx1cmwgAAAAAQAAAXRzdGJsAAAAqHN0\n",
       "c2QAAAAAAAAAAQAAAJhhdmMxAAAAAAAAAAEAAAAAAAAAAAAAAAAAAAAAAtAC0ABIAAAASAAAAAAA\n",
       "AAABAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAGP//AAAAJmF2Y0MBQsAf/+EAD2dC\n",
       "wB+MjUBaC2QDwiEagAEABGjOPIAAAAAcdXVpZGtoQPJfJE/FujmlG88DI/MAAAAAAAAAGHN0dHMA\n",
       "AAAAAAAAAQAAABUAAEAAAAAAFHN0c3MAAAAAAAAAAQAAAAEAAAAcc3RzYwAAAAAAAAABAAAAAQAA\n",
       "ABUAAAABAAAAaHN0c3oAAAAAAAAAAAAAABUAABO1AAAEcQAABIAAAAhtAAAHkQAAB1cAAAgTAAAG\n",
       "XgAABVQAAAQAAAAFUwAAAs0AAAKBAAADLgAAAo0AAAFMAAABuQAAAZgAAAFcAAABPAAAAb4AAAAU\n",
       "c3RjbwAAAAAAAAABAAAAMAAAAGJ1ZHRhAAAAWm1ldGEAAAAAAAAAIWhkbHIAAAAAAAAAAG1kaXJh\n",
       "cHBsAAAAAAAAAAAAAAAALWlsc3QAAAAlqXRvbwAAAB1kYXRhAAAAAQAAAABMYXZmNTguNDUuMTAw\n",
       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "fig, ax = plt.subplots(figsize=(10, 10))\n",
    "kwargs = {'cmap': 'hot', 'node_size': 1500, 'edge_color': 'gray', \n",
    "          'vmin': np.array(results).min(), 'vmax': np.array(results).max()*1.1}\n",
    "def update(idx):\n",
    "    ax.clear()\n",
    "    colors = results[idx]\n",
    "    nx.draw(g, pos, node_color=colors, ax=ax, **kwargs)\n",
    "    ax.set_title(f\"Iter={idx}\", fontsize=20)\n",
    "HTML(animation.FuncAnimation(fig, update, frames=len(results), interval=1000, repeat=True).to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# More theoretical preliminaries"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "##  1. Permutation Invariance Property\n",
    "\n",
    "We require that our algorithms to satisfiy the permutation invariance property. This requirement stems from the fact that graph structured data does not involve ordering. Hence, the order of nodes should not matter for us. \n",
    "\n",
    "\n",
    "\n",
    "Let $X \\in \\mathbb{R}^{n \\times d}$ represent node features and $P$ be a permutation. A function $f$ permutation invariant if\n",
    "\n",
    "## $$ f(\\textbf{P} \\textbf{X}) = f(\\textbf{X})$$\n",
    "\n",
    "Let's take a look at some permutation invariant functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Deep Sets by Zaheer et al. NeurIPS 2017\n",
    "\n",
    "A generic form:\n",
    "## $$ f(\\textbf{X}) = f(\\textbf{P}\\textbf{X})$$\n",
    "## $$ f(\\textbf{X}) = \\phi \\Bigg(\\Sigma_i \\psi(x_i) \\Bigg),$$\n",
    "\n",
    "where $\\phi$ and $ \\psi $ are learnable functions. $\\phi$ is a __permutation invariant aggregation function__ (e.g. summation, averagining, maximization).\n",
    "\n",
    "##  2. Permutation Equivariance Property\n",
    "\n",
    "\n",
    "## $$ f(\\textbf{P}\\textbf{X}) = \\textbf{P}f(\\textbf{X})$$\n",
    "\n",
    "## $$ f(\\textbf{X}) = \\phi \\Bigg( \\Phi_i \\psi(x_i) \\Bigg),$$\n",
    "\n",
    "where $\\Phi$ is a permutation-invariant aggregator (such as sum, avg or max)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X:\n",
      "[[ 0.23130674 -0.17238292]\n",
      " [ 1.7595618  -0.71230477]\n",
      " [-0.30582224  0.31453775]\n",
      " [-1.16573429 -0.58862348]]\n",
      "\n",
      "PX:\n",
      "[[-0.30582224  0.31453775]\n",
      " [ 1.7595618  -0.71230477]\n",
      " [-1.16573429 -0.58862348]\n",
      " [ 0.23130674 -0.17238292]]\n",
      "\t\t\t\t Only order is changed.\n",
      "Σ_i Σ_j X_ij=> True, -0.639\n",
      "Σ_i X_ij=> True, [ 0.51931201 -1.15877342]\n",
      "Σ_j X_ij=> True, [ 0.05892382  1.04725703  0.00871551 -1.75435777]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "# Number of nodes and size of features\n",
    "n,d=4,2\n",
    "# (1) Node Representation\n",
    "X=np.random.randn(n,d)\n",
    "# (2) Permutation Matrix\n",
    "P=np.identity(n)[np.random.permutation(n)]\n",
    "print(f'X:\\n{X}\\n')\n",
    "print(f'PX:\\n{P@X}')\n",
    "print('\\t\\t\\t\\t Only order is changed.')\n",
    "# Sum\n",
    "sumX=np.einsum(\"ij->\",X)\n",
    "sumPX=np.einsum(\"ij->\",P@X)\n",
    "print(f'Σ_i Σ_j X_ij=> {np.allclose(sumX,sumPX,atol=1e4)}, {sumX:.3f}')\n",
    "\n",
    "# Column Sum\n",
    "colsumX=np.einsum(\"ij-> j \",X)\n",
    "colsumPX=np.einsum(\"ij-> j \",P@X)\n",
    "print(f'Σ_i X_ij=> {np.allclose(colsumX,colsumPX,atol=1e4)}, {colsumX}')\n",
    "\n",
    "# Row Sum\n",
    "rowsumX=np.einsum(\"ij-> i \",X)\n",
    "rowsumPX=np.einsum(\"ij-> i \",P@X)\n",
    "print(f'Σ_j X_ij=> {np.allclose(rowsumX,rowsumPX,atol=1e4)}, {rowsumX}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Learning on Graphs\n",
    "\n",
    "We can represent edges with a __A__ be a binary adjacency matrix.\n",
    "\n",
    "## Invariance: $$ f(\\textbf{P}\\textbf{X},\\textbf{P} \\textbf{A} \\textbf{P}^T) = f(\\textbf{X},\\textbf{A})$$\n",
    "\n",
    "\n",
    "## Equivariance: $$ f(\\textbf{P}\\textbf{X},\\textbf{P} \\textbf{A} \\textbf{P}^T) = \\textbf{P}f(\\textbf{X},\\textbf{A})$$\n",
    "\n",
    "\n",
    "\n",
    "## Neighbourhood:\n",
    "$$ \\mathcal{N}_i = \\{ j, (i,j) \\in \\mathcal{E} \\vee (j,i) \\in \\mathcal{E} \\} $$\n",
    "\n",
    "\n",
    "$$ \n",
    "f(\\textbf{X},\\textbf{A})=\\begin{bmatrix} \n",
    "- \\quad g( \\textbf{x}_1, \\textbf{X}_{\\mathcal{N}_1} ) \\quad-\\\\\n",
    "- \\quad g( \\textbf{x}_2, \\textbf{X}_{\\mathcal{N}_2} ) \\quad-\\\\\n",
    "\\cdots \\\\\n",
    "- \\quad g( \\textbf{x}_n, \\textbf{X}_{\\mathcal{N}_n} ) \\quad-\\\\\n",
    "\\end{bmatrix} \n",
    "$$\n",
    "\n",
    "g should be also permutation invariant.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Graph Convolutional Neural Network\n",
    "\n",
    "\n",
    "### $$ h_i = \\phi \\, \\Big( x_i , \\Phi_{j \\in \\mathcal{N}_i} c_{ij} \\, \\psi(x_j) \\Big)$$\n",
    "where __features of neighbours aggregated with fixed weight $c_{ij}$__. Useful for homophilous graphs and scaling up: when edges encode label similarity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Attention Graph Neural Network\n",
    "\n",
    "### $$ h_i = \\phi \\, \\big( x_i , \\Phi_{j \\in \\mathcal{N}_i} a(x_i,x_j) \\, \\psi(x_j) \\big)$$\n",
    "where __features of neighbours aggregated with implicit weights (via attention)__. Usefull: Edges need not encode homophily."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mesage-passing Graph Neural Network\n",
    "\n",
    "### $$ h_i = \\phi \\, \\big( x_i , \\Phi_{j \\in \\mathcal{N}_i} \\psi(x_i,x_j) \\big)$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Implementation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. What does matrix product of A and X imply ?\n",
    "\n",
    "Matrix product of an adjecency matrix $A$ with node representations $X$ \"updates\" each node representations $X_i$ with its neighbours $A_i$. \n",
    "\n",
    "Let's understand this computation in detail. Assume that $A$ is a idendity matrix, hence, the impact of neighbors of $i$th node is __zero__."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A:\n",
      "[[1. 0. 0. 0.]\n",
      " [0. 1. 0. 0.]\n",
      " [0. 0. 1. 0.]\n",
      " [0. 0. 0. 1.]]\n",
      "\n",
      "X:\n",
      "[[ 0.28889537  1.09321337]\n",
      " [ 1.83892694  0.18605693]\n",
      " [ 1.14810484 -1.63348288]\n",
      " [-1.01341867 -0.04361163]]\n",
      "\n",
      "AX:\n",
      "[[ 0.28889537  1.09321337]\n",
      " [ 1.83892694  0.18605693]\n",
      " [ 1.14810484 -1.63348288]\n",
      " [-1.01341867 -0.04361163]]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "n,d=4,2\n",
    "A=np.eye(n)\n",
    "X=np.random.randn(n,d)\n",
    "AX=np.einsum('ik,kj->ij', A, X)\n",
    "print(f'A:\\n{A}\\n')\n",
    "print(f'X:\\n{X}\\n')\n",
    "print(f'AX:\\n{AX}\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Why don't we extend the neighbourhood?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 1., 0., 0.],\n",
       "       [1., 1., 1., 0.],\n",
       "       [0., 1., 1., 0.],\n",
       "       [0., 0., 0., 1.]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Let's connect \n",
    "A[0][1]=1 # n_0 => n_1\n",
    "A[1][0]=1 # n_1 => n_0\n",
    "\n",
    "A[2][1]=1 # n_2 => n_1\n",
    "A[1][2]=1 # n_1 => n_2\n",
    "\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Node 0 and Node 2 are not connected\n",
    "assert A[0][2]==A[2][0]==0.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A:\n",
      "[[1.  1.  0.5 0. ]\n",
      " [1.  1.  1.  0. ]\n",
      " [0.5 1.  1.  0. ]\n",
      " [0.  0.  0.  1. ]]\n",
      "\n",
      "0.5\n",
      "0.5\n"
     ]
    }
   ],
   "source": [
    "# Neigbours of neighbours\n",
    "neighbours_of_neighbours_scalar=.5\n",
    "A+= ((A+A@A ==1) *1.0) * neighbours_of_neighbours_scalar \n",
    "print(f'A:\\n{A}\\n')\n",
    "print(A[0][2])\n",
    "print(A[2][0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A:\n",
      "[[1.  1.  0.5 0. ]\n",
      " [1.  1.  1.  0. ]\n",
      " [0.5 1.  1.  0. ]\n",
      " [0.  0.  0.  1. ]]\n",
      "\n",
      "AX:\n",
      "[[ 2.70187473  0.46252885]\n",
      " [ 3.27592715 -0.35421259]\n",
      " [ 3.13147947 -0.90081927]\n",
      " [-1.01341867 -0.04361163]]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "AX=np.einsum('ik,kj->ij', A, X)\n",
    "print(f'A:\\n{A}\\n')\n",
    "print(f'AX:\\n{AX}\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Node Classification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "import tensorflow as tf\n",
    "import spektral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Loading cora dataset\n",
      "Pre-processing node features\n"
     ]
    }
   ],
   "source": [
    "A, X, labels, train_mask, val_mask, test_mask = spektral.datasets.citation.load_data(dataset_name='cora')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X: (2708, 1433)\n",
      "A: (2708, 2708)\n",
      "Y: (2708, 7)\n"
     ]
    }
   ],
   "source": [
    "X,A = X.todense(),A.todense()\n",
    "\n",
    "A = A + np.eye(A.shape[0])\n",
    "X = X.astype('float32')\n",
    "A = A.astype('float32')\n",
    "\n",
    "print('X:',X.shape)\n",
    "print('A:',A.shape)\n",
    "# Labels\n",
    "print('Y:',labels.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0. 0. 0. ... 0. 0. 0.]]\n",
      "[[1. 0. 0. ... 0. 0. 0.]]\n",
      "[0 0 0 1 0 0 0]\n"
     ]
    }
   ],
   "source": [
    "print(X[0])\n",
    "print(A[0])\n",
    "print(labels[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "(2708,)\n",
      "140\n"
     ]
    }
   ],
   "source": [
    "print(train_mask[0])\n",
    "print(train_mask.shape)\n",
    "print(np.sum(train_mask)) # True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "False\n",
      "(2708,)\n",
      "500\n"
     ]
    }
   ],
   "source": [
    "print(val_mask[0])\n",
    "print(val_mask.shape)\n",
    "print(np.sum(val_mask)) # True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "False\n",
      "(2708,)\n",
      "1000\n"
     ]
    }
   ],
   "source": [
    "print(test_mask[0])\n",
    "print(test_mask.shape)\n",
    "print(np.sum(test_mask)) # True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "def masked_softmax_cross_entropy(logits, labels, masks):\n",
    "    loss = tf.nn.softmax_cross_entropy_with_logits(logits=logits, labels=labels)\n",
    "    masks = tf.cast(masks, dtype=tf.float32)\n",
    "    masks /= tf.reduce_mean(masks)\n",
    "    loss *= masks\n",
    "    return tf.reduce_mean(loss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def masked_accuracy(logits, labels, masks):\n",
    "    correct_prediction = tf.equal(tf.argmax(logits, 1), tf.argmax(labels, 1))\n",
    "    accuracy_all = tf.cast(correct_prediction, tf.float32)\n",
    "    masks = tf.cast(masks, dtype=tf.float32)\n",
    "    masks /= tf.reduce_mean(masks)\n",
    "    accuracy_all *= masks\n",
    "    return tf.reduce_mean(accuracy_all)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gnn(fts, adj, transform, activation):\n",
    "    seq_fts = transform(fts)\n",
    "    ret_fts = tf.matmul(adj, seq_fts)\n",
    "    return activation(ret_fts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "def train_cora(fts, adj, gnn_fn, units=32, epochs=200, lr=.01):\n",
    "    lyr_1 = tf.keras.layers.Dense(units)\n",
    "    lyr_2 = tf.keras.layers.Dense(7)  # for 7 classes\n",
    "\n",
    "    def cora_gnn(fts, adj):\n",
    "        hidden = gnn_fn(fts, adj, lyr_1, tf.nn.relu)\n",
    "        logits = gnn_fn(hidden, adj, lyr_2, tf.identity)\n",
    "        return logits\n",
    "\n",
    "    optimizer = tf.keras.optimizers.Adam(learning_rate=lr)\n",
    "    best_accuracy = 0.0\n",
    "\n",
    "    for ep in range(epochs + 1):\n",
    "        with tf.GradientTape() as t:\n",
    "            logits = cora_gnn(fts, adj)\n",
    "            loss = masked_softmax_cross_entropy(logits, labels, train_mask)\n",
    "        variables = t.watched_variables()\n",
    "        grad = t.gradient(loss, variables)\n",
    "        optimizer.apply_gradients(zip(grad, variables))\n",
    "\n",
    "        logits = cora_gnn(fts, adj)\n",
    "        val_accuracy = masked_accuracy(logits, labels, val_mask)\n",
    "        test_accuracy = masked_accuracy(logits, labels, test_mask)\n",
    "\n",
    "        if val_accuracy > best_accuracy:\n",
    "            best_accuracy = val_accuracy\n",
    "            print(f'Epoch {ep} | Traninng Loss: {loss.numpy():.3f} | Val acc: {val_accuracy.numpy():.3f} Test Acc: {test_accuracy.numpy():.3f}| ')\n",
    "    print('Completed')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sum Pooling of neighbours\n",
      "Epoch 0 | Traninng Loss: 2.013 | Val acc: 0.074 Test Acc: 0.097| \n",
      "Epoch 1 | Traninng Loss: 2.036 | Val acc: 0.424 Test Acc: 0.443| \n",
      "Epoch 2 | Traninng Loss: 1.687 | Val acc: 0.432 Test Acc: 0.436| \n",
      "Epoch 3 | Traninng Loss: 1.500 | Val acc: 0.540 Test Acc: 0.566| \n",
      "Epoch 4 | Traninng Loss: 1.375 | Val acc: 0.598 Test Acc: 0.618| \n",
      "Epoch 5 | Traninng Loss: 1.242 | Val acc: 0.680 Test Acc: 0.668| \n",
      "Epoch 6 | Traninng Loss: 1.146 | Val acc: 0.718 Test Acc: 0.716| \n",
      "Epoch 7 | Traninng Loss: 1.050 | Val acc: 0.746 Test Acc: 0.742| \n",
      "Epoch 8 | Traninng Loss: 0.954 | Val acc: 0.758 Test Acc: 0.760| \n",
      "Epoch 9 | Traninng Loss: 0.865 | Val acc: 0.770 Test Acc: 0.771| \n",
      "Completed\n"
     ]
    }
   ],
   "source": [
    "print('Sum Pooling of neighbours')\n",
    "train_cora(X, A, gnn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "# neighbours of neighbours\n",
    "neighbours_of_neighbours_scalar=.1\n",
    "A_neg= A+((A+A@A ==1) *1.0) * neighbours_of_neighbours_scalar \n",
    "A_neg = A_neg.astype('float32')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sum Pooling of neighbours\n",
      "Epoch 0 | Traninng Loss: 1.872 | Val acc: 0.112 Test Acc: 0.144| \n",
      "Epoch 1 | Traninng Loss: 1.867 | Val acc: 0.382 Test Acc: 0.387| \n",
      "Epoch 2 | Traninng Loss: 1.913 | Val acc: 0.422 Test Acc: 0.435| \n",
      "Epoch 3 | Traninng Loss: 1.869 | Val acc: 0.646 Test Acc: 0.636| \n",
      "Epoch 4 | Traninng Loss: 1.491 | Val acc: 0.700 Test Acc: 0.706| \n",
      "Epoch 5 | Traninng Loss: 1.250 | Val acc: 0.702 Test Acc: 0.696| \n",
      "Epoch 14 | Traninng Loss: 0.770 | Val acc: 0.708 Test Acc: 0.722| \n",
      "Epoch 17 | Traninng Loss: 0.627 | Val acc: 0.732 Test Acc: 0.753| \n",
      "Epoch 19 | Traninng Loss: 0.580 | Val acc: 0.742 Test Acc: 0.752| \n",
      "Epoch 20 | Traninng Loss: 0.567 | Val acc: 0.748 Test Acc: 0.756| \n",
      "Epoch 73 | Traninng Loss: 0.106 | Val acc: 0.752 Test Acc: 0.738| \n",
      "Completed\n"
     ]
    }
   ],
   "source": [
    "print('Sum Pooling of neighbours')\n",
    "train_cora(X, A_neg, gnn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Only itself\n",
      "Epoch 0 | Traninng Loss: 1.947 | Val acc: 0.166 Test Acc: 0.167| \n",
      "Epoch 1 | Traninng Loss: 1.933 | Val acc: 0.304 Test Acc: 0.282| \n",
      "Epoch 2 | Traninng Loss: 1.914 | Val acc: 0.452 Test Acc: 0.416| \n",
      "Epoch 3 | Traninng Loss: 1.888 | Val acc: 0.498 Test Acc: 0.451| \n",
      "Epoch 8 | Traninng Loss: 1.713 | Val acc: 0.516 Test Acc: 0.468| \n",
      "Epoch 9 | Traninng Loss: 1.668 | Val acc: 0.530 Test Acc: 0.484| \n",
      "Epoch 10 | Traninng Loss: 1.621 | Val acc: 0.542 Test Acc: 0.490| \n",
      "Epoch 11 | Traninng Loss: 1.571 | Val acc: 0.550 Test Acc: 0.491| \n",
      "Epoch 16 | Traninng Loss: 1.290 | Val acc: 0.550 Test Acc: 0.502| \n",
      "Epoch 19 | Traninng Loss: 1.103 | Val acc: 0.554 Test Acc: 0.509| \n",
      "Epoch 20 | Traninng Loss: 1.039 | Val acc: 0.556 Test Acc: 0.513| \n",
      "Epoch 22 | Traninng Loss: 0.912 | Val acc: 0.558 Test Acc: 0.515| \n",
      "Epoch 31 | Traninng Loss: 0.420 | Val acc: 0.560 Test Acc: 0.526| \n",
      "Completed\n"
     ]
    }
   ],
   "source": [
    "# (2) Rely on yourself; Do not make use of your first neighbours: Node feature + Identity  matrix + GNN\n",
    "print('Only itself')\n",
    "train_cora(X, tf.eye(A.shape[0]), gnn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average Pooling\n",
      "Epoch 0 | Traninng Loss: 1.945 | Val acc: 0.180 Test Acc: 0.199| \n",
      "Epoch 1 | Traninng Loss: 1.931 | Val acc: 0.200 Test Acc: 0.231| \n",
      "Epoch 2 | Traninng Loss: 1.914 | Val acc: 0.204 Test Acc: 0.229| \n",
      "Epoch 5 | Traninng Loss: 1.846 | Val acc: 0.210 Test Acc: 0.246| \n",
      "Epoch 6 | Traninng Loss: 1.821 | Val acc: 0.238 Test Acc: 0.269| \n",
      "Epoch 7 | Traninng Loss: 1.794 | Val acc: 0.270 Test Acc: 0.297| \n",
      "Epoch 8 | Traninng Loss: 1.765 | Val acc: 0.288 Test Acc: 0.324| \n",
      "Epoch 9 | Traninng Loss: 1.735 | Val acc: 0.330 Test Acc: 0.364| \n",
      "Epoch 10 | Traninng Loss: 1.702 | Val acc: 0.394 Test Acc: 0.410| \n",
      "Epoch 11 | Traninng Loss: 1.667 | Val acc: 0.444 Test Acc: 0.458| \n",
      "Epoch 12 | Traninng Loss: 1.630 | Val acc: 0.520 Test Acc: 0.532| \n",
      "Epoch 13 | Traninng Loss: 1.592 | Val acc: 0.586 Test Acc: 0.590| \n",
      "Epoch 14 | Traninng Loss: 1.552 | Val acc: 0.642 Test Acc: 0.630| \n",
      "Epoch 15 | Traninng Loss: 1.511 | Val acc: 0.672 Test Acc: 0.668| \n",
      "Epoch 16 | Traninng Loss: 1.469 | Val acc: 0.686 Test Acc: 0.700| \n",
      "Epoch 17 | Traninng Loss: 1.425 | Val acc: 0.710 Test Acc: 0.717| \n",
      "Epoch 18 | Traninng Loss: 1.380 | Val acc: 0.722 Test Acc: 0.734| \n",
      "Epoch 19 | Traninng Loss: 1.333 | Val acc: 0.734 Test Acc: 0.748| \n",
      "Epoch 20 | Traninng Loss: 1.287 | Val acc: 0.738 Test Acc: 0.763| \n",
      "Epoch 21 | Traninng Loss: 1.239 | Val acc: 0.746 Test Acc: 0.768| \n",
      "Epoch 22 | Traninng Loss: 1.191 | Val acc: 0.752 Test Acc: 0.780| \n",
      "Epoch 23 | Traninng Loss: 1.143 | Val acc: 0.762 Test Acc: 0.792| \n",
      "Epoch 24 | Traninng Loss: 1.096 | Val acc: 0.768 Test Acc: 0.801| \n",
      "Epoch 25 | Traninng Loss: 1.048 | Val acc: 0.778 Test Acc: 0.802| \n",
      "Epoch 26 | Traninng Loss: 1.000 | Val acc: 0.780 Test Acc: 0.806| \n",
      "Epoch 28 | Traninng Loss: 0.908 | Val acc: 0.786 Test Acc: 0.812| \n",
      "Epoch 29 | Traninng Loss: 0.863 | Val acc: 0.788 Test Acc: 0.815| \n",
      "Epoch 31 | Traninng Loss: 0.777 | Val acc: 0.792 Test Acc: 0.815| \n",
      "Epoch 34 | Traninng Loss: 0.658 | Val acc: 0.796 Test Acc: 0.822| \n",
      "Epoch 38 | Traninng Loss: 0.522 | Val acc: 0.798 Test Acc: 0.824| \n",
      "Epoch 40 | Traninng Loss: 0.464 | Val acc: 0.800 Test Acc: 0.824| \n",
      "Epoch 41 | Traninng Loss: 0.437 | Val acc: 0.802 Test Acc: 0.826| \n",
      "Completed\n"
     ]
    }
   ],
   "source": [
    "# (3) Make use of average of  your first neighbours (1) Node feature + Degree normalized ADJ matrix + GNN\n",
    "print('Average Pooling')\n",
    "degree_matrix = tf.reduce_sum(A, axis=-1)\n",
    "train_cora(X, A / degree_matrix, gnn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Epoch 0 | Traninng Loss: 1.946 | Val acc: 0.130 Test Acc: 0.139| \n",
      "Epoch 6 | Traninng Loss: 1.885 | Val acc: 0.134 Test Acc: 0.146| \n",
      "Epoch 7 | Traninng Loss: 1.870 | Val acc: 0.202 Test Acc: 0.213| \n",
      "Epoch 8 | Traninng Loss: 1.854 | Val acc: 0.262 Test Acc: 0.300| \n",
      "Epoch 9 | Traninng Loss: 1.837 | Val acc: 0.356 Test Acc: 0.398| \n",
      "Epoch 10 | Traninng Loss: 1.820 | Val acc: 0.502 Test Acc: 0.516| \n",
      "Epoch 11 | Traninng Loss: 1.801 | Val acc: 0.574 Test Acc: 0.582| \n",
      "Epoch 12 | Traninng Loss: 1.780 | Val acc: 0.626 Test Acc: 0.625| \n",
      "Epoch 13 | Traninng Loss: 1.758 | Val acc: 0.654 Test Acc: 0.652| \n",
      "Epoch 14 | Traninng Loss: 1.735 | Val acc: 0.678 Test Acc: 0.681| \n",
      "Epoch 15 | Traninng Loss: 1.710 | Val acc: 0.692 Test Acc: 0.698| \n",
      "Epoch 16 | Traninng Loss: 1.683 | Val acc: 0.712 Test Acc: 0.718| \n",
      "Epoch 17 | Traninng Loss: 1.654 | Val acc: 0.718 Test Acc: 0.735| \n",
      "Epoch 18 | Traninng Loss: 1.624 | Val acc: 0.734 Test Acc: 0.751| \n",
      "Epoch 19 | Traninng Loss: 1.594 | Val acc: 0.750 Test Acc: 0.760| \n",
      "Epoch 20 | Traninng Loss: 1.562 | Val acc: 0.752 Test Acc: 0.772| \n",
      "Epoch 21 | Traninng Loss: 1.528 | Val acc: 0.756 Test Acc: 0.777| \n",
      "Epoch 22 | Traninng Loss: 1.494 | Val acc: 0.764 Test Acc: 0.778| \n",
      "Epoch 23 | Traninng Loss: 1.458 | Val acc: 0.770 Test Acc: 0.778| \n",
      "Epoch 27 | Traninng Loss: 1.308 | Val acc: 0.772 Test Acc: 0.780| \n",
      "Epoch 28 | Traninng Loss: 1.269 | Val acc: 0.774 Test Acc: 0.785| \n",
      "Epoch 41 | Traninng Loss: 0.806 | Val acc: 0.776 Test Acc: 0.787| \n",
      "Epoch 48 | Traninng Loss: 0.625 | Val acc: 0.778 Test Acc: 0.784| \n",
      "Epoch 57 | Traninng Loss: 0.462 | Val acc: 0.782 Test Acc: 0.789| \n",
      "Completed\n"
     ]
    }
   ],
   "source": [
    "# neighbours of neighbours\n",
    "neighbours_of_neighbours_scalar=1.0\n",
    "A_neg= A+((A+A@A ==1) *1.0) * neighbours_of_neighbours_scalar \n",
    "A_neg = A_neg.astype('float32')\n",
    "\n",
    "degree_matrix_A_neg = tf.reduce_sum(A_neg, axis=-1)\n",
    "train_cora(X, A_neg / degree_matrix_A_neg, gnn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Normalized Pooling\n",
      "Epoch 0 | Traninng Loss: 1.946 | Val acc: 0.358 Test Acc: 0.395| \n",
      "Epoch 1 | Traninng Loss: 1.935 | Val acc: 0.378 Test Acc: 0.429| \n",
      "Epoch 2 | Traninng Loss: 1.921 | Val acc: 0.414 Test Acc: 0.443| \n",
      "Epoch 3 | Traninng Loss: 1.901 | Val acc: 0.446 Test Acc: 0.475| \n",
      "Epoch 4 | Traninng Loss: 1.881 | Val acc: 0.482 Test Acc: 0.524| \n",
      "Epoch 5 | Traninng Loss: 1.860 | Val acc: 0.522 Test Acc: 0.555| \n",
      "Epoch 6 | Traninng Loss: 1.838 | Val acc: 0.554 Test Acc: 0.578| \n",
      "Epoch 7 | Traninng Loss: 1.814 | Val acc: 0.580 Test Acc: 0.622| \n",
      "Epoch 8 | Traninng Loss: 1.788 | Val acc: 0.620 Test Acc: 0.644| \n",
      "Epoch 9 | Traninng Loss: 1.760 | Val acc: 0.648 Test Acc: 0.677| \n",
      "Epoch 10 | Traninng Loss: 1.730 | Val acc: 0.664 Test Acc: 0.710| \n",
      "Epoch 11 | Traninng Loss: 1.698 | Val acc: 0.694 Test Acc: 0.720| \n",
      "Epoch 12 | Traninng Loss: 1.665 | Val acc: 0.712 Test Acc: 0.741| \n",
      "Epoch 13 | Traninng Loss: 1.630 | Val acc: 0.720 Test Acc: 0.756| \n",
      "Epoch 14 | Traninng Loss: 1.594 | Val acc: 0.740 Test Acc: 0.765| \n",
      "Epoch 15 | Traninng Loss: 1.556 | Val acc: 0.746 Test Acc: 0.774| \n",
      "Epoch 16 | Traninng Loss: 1.516 | Val acc: 0.750 Test Acc: 0.777| \n",
      "Epoch 17 | Traninng Loss: 1.474 | Val acc: 0.760 Test Acc: 0.780| \n",
      "Epoch 18 | Traninng Loss: 1.431 | Val acc: 0.764 Test Acc: 0.783| \n",
      "Epoch 19 | Traninng Loss: 1.387 | Val acc: 0.768 Test Acc: 0.784| \n",
      "Epoch 20 | Traninng Loss: 1.343 | Val acc: 0.776 Test Acc: 0.786| \n",
      "Epoch 21 | Traninng Loss: 1.297 | Val acc: 0.780 Test Acc: 0.788| \n",
      "Epoch 22 | Traninng Loss: 1.250 | Val acc: 0.788 Test Acc: 0.793| \n",
      "Epoch 23 | Traninng Loss: 1.203 | Val acc: 0.792 Test Acc: 0.800| \n",
      "Epoch 26 | Traninng Loss: 1.060 | Val acc: 0.800 Test Acc: 0.800| \n",
      "Completed\n"
     ]
    }
   ],
   "source": [
    "# (4) It is a good idea to normalize adjesonsy matrix\n",
    "norm_degr = tf.linalg.diag(1.0 / tf.sqrt(degree_matrix))\n",
    "norm_A = tf.linalg.matmul(norm_degr, tf.matmul(A, norm_degr))\n",
    "print('Normalized Pooling')\n",
    "train_cora(X, norm_A, gnn)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "pDL",
   "language": "python",
   "name": "pdl"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}