doc/3-4-diskrete_mathematik.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Diskrete Mathematik\n",
    "\n",
    "Python bietet diverse Funktionen, um grundlegende Konzepte aus der Kombinatorik umzusetzen.\n",
    "Zum Beispiel lassen sich alle Kombinationen von zwei oder mehr Mengen mittels der Funktion `product` bilden, oder geordnete Auswahlen aus einer Liste mit oder ohne zurücklegen treffen.\n",
    "\n",
    "Dokumentation: [itertools](https://docs.python.org/2/library/itertools.html)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Für die folgenden Beispiele definieren wir uns die folgenden zwei diskrete Mengen (eine Liste ist OK)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "animals = [\"cat\", \"dog\", \"fish\"]\n",
    "owners = [\"susi\", \"tim\", \"franziska\", \"henry\"]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Produktbildung\n",
    "\n",
    "Alle Kombinationen durchprobieren"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cat   belongs to susi      \n",
      "cat   belongs to tim       \n",
      "cat   belongs to franziska \n",
      "cat   belongs to henry     \n",
      "dog   belongs to susi      \n",
      "dog   belongs to tim       \n",
      "dog   belongs to franziska \n",
      "dog   belongs to henry     \n",
      "fish  belongs to susi      \n",
      "fish  belongs to tim       \n",
      "fish  belongs to franziska \n",
      "fish  belongs to henry     \n"
     ]
    }
   ],
   "source": [
    "from itertools import product\n",
    "for animal, owner in product(animals, owners):\n",
    "    print(\"%-5s belongs to %-10s\" % (animal, owner))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kombinationen aus einer Menge ziehen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "susi      \n",
      "tim       \n",
      "franziska \n",
      "henry     \n"
     ]
    }
   ],
   "source": [
    "# Ziehe 1 ...\n",
    "from itertools import combinations\n",
    "for owner1 in combinations(owners, 1):\n",
    "    print(\"%-10s\" % (owner1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "susi       tim       \n",
      "susi       franziska \n",
      "susi       henry     \n",
      "tim        franziska \n",
      "tim        henry     \n",
      "franziska  henry     \n"
     ]
    }
   ],
   "source": [
    "# ... ziehe 2 ...\n",
    "from itertools import combinations\n",
    "for owner1, owner2 in combinations(owners, 2):\n",
    "    print(\"%-10s %-10s\" % (owner1, owner2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "susi       tim        franziska \n",
      "susi       tim        henry     \n",
      "susi       franziska  henry     \n",
      "tim        franziska  henry     \n"
     ]
    }
   ],
   "source": [
    "# ... oder 3 ....\n",
    "from itertools import combinations\n",
    "for owner1, owner2, owner3 in combinations(owners, 3):\n",
    "    print(\"%-10s %-10s %-10s\" % (owner1, owner2, owner3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "susi       tim        franziska  henry     \n"
     ]
    }
   ],
   "source": [
    "# ... oder gar 4!\n",
    "from itertools import combinations\n",
    "for owner1, owner2, owner3, owner4 in combinations(owners, 4):\n",
    "    print(\"%-10s %-10s %-10s %-10s\" % (owner1, owner2, owner3, owner4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kombinationen mit Zurücklegen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "susi       susi      \n",
      "susi       tim       \n",
      "susi       franziska \n",
      "susi       henry     \n",
      "tim        tim       \n",
      "tim        franziska \n",
      "tim        henry     \n",
      "franziska  franziska \n",
      "franziska  henry     \n",
      "henry      henry     \n"
     ]
    }
   ],
   "source": [
    "from itertools import combinations_with_replacement\n",
    "for owner1, owner2 in combinations_with_replacement(owners, 2):\n",
    "    print(\"%-10s %-10s\" % (owner1, owner2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Permutationen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "susi       tim       \n",
      "susi       franziska \n",
      "susi       henry     \n",
      "tim        susi      \n",
      "tim        franziska \n",
      "tim        henry     \n",
      "franziska  susi      \n",
      "franziska  tim       \n",
      "franziska  henry     \n",
      "henry      susi      \n",
      "henry      tim       \n",
      "henry      franziska \n"
     ]
    }
   ],
   "source": [
    "from itertools import permutations\n",
    "for owner1, owner2 in permutations(owners, 2):\n",
    "    print(\"%-10s %-10s\" % (owner1, owner2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "susi       tim        franziska \n",
      "susi       tim        henry     \n",
      "susi       franziska  tim       \n",
      "susi       franziska  henry     \n",
      "susi       henry      tim       \n",
      "susi       henry      franziska \n",
      "tim        susi       franziska \n",
      "tim        susi       henry     \n",
      "tim        franziska  susi      \n",
      "tim        franziska  henry     \n",
      "tim        henry      susi      \n",
      "tim        henry      franziska \n",
      "franziska  susi       tim       \n",
      "franziska  susi       henry     \n",
      "franziska  tim        susi      \n",
      "franziska  tim        henry     \n",
      "franziska  henry      susi      \n",
      "franziska  henry      tim       \n",
      "henry      susi       tim       \n",
      "henry      susi       franziska \n",
      "henry      tim        susi      \n",
      "henry      tim        franziska \n",
      "henry      franziska  susi      \n",
      "henry      franziska  tim       \n"
     ]
    }
   ],
   "source": [
    "from itertools import permutations\n",
    "for owner1, owner2, owner3 in permutations(owners, 3):\n",
    "    print(\"%-10s %-10s %-10s\" % (owner1, owner2, owner3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graphen\n",
    "\n",
    "[NetworkX](https://networkx.github.io/) ist eine der am weitesten verbreiteten Python-Bibliotheken für Graphentheorie\n",
    "mit einer [Menge von Algorithmen](http://networkx.github.io/documentation/latest/reference/algorithms.html).\n",
    "\n",
    "\n",
    "[NetworkX Dokumentation](http://networkx.github.io/documentation/networkx-1.9.1/)\n",
    "\n",
    "(Eine andere Pythonbibliothek wäre [SageMath / Graph Theory](http://doc.sagemath.org/html/en/reference/graphs/index.html))\n",
    "\n",
    "Ein Graph ist eine Menge von Knoten und Kanten (Vertices and Edges),\n",
    "wobei die Kanten für Verbindungen zwischen den Knoten stehen.\n",
    "Es gibt ungerichtete und gerichtete Verbindungen -- letzteres ist dann ein \"gerichteter Graph\".\n",
    "Eine Implementierung im Computer besteht nun darin,\n",
    "in einer Datenstruktur diese Kanten und Knoten effizient zu verwalten\n",
    "und dann darauf aufbauend Algorithmen aus der Graphentheorie anzuwenden.\n",
    "\n",
    "Bemerkung: Falls die Bibliothek `networkx` nicht installiert sein sollte, so kann man dies entweder in der Kommandozeile oder in Canopy unter Tools &rarr; Canopy Terminal mittels\n",
    "\n",
    "    pip install networkx\n",
    "    \n",
    "nachholen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'1.9.1'"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import networkx as nx\n",
    "%matplotlib inline\n",
    "nx.__version__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hier wird ein einfacher Graph instanziert. Die Knoten sind einige Hauptstädte, die Kanten stellen (Zug-) Verbindungen zwischen ihnen dar. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "g1 = nx.Graph()\n",
    "\n",
    "g1.add_node(\"Wien\")\n",
    "g1.add_node(\"Graz\")\n",
    "g1.add_node(\"Linz\")\n",
    "g1.add_node(\"Salzburg\")\n",
    "g1.add_node(\"Innsbruck\")\n",
    "g1.add_node(\"Villach\")\n",
    "g1.add_node(\"St. Pölten\")\n",
    "\n",
    "g1.add_edge(\"Graz\", \"Salzburg\")\n",
    "g1.add_edge(\"Wien\", \"Salzburg\")\n",
    "g1.add_edge(\"Salzburg\", \"St. Pölten\")\n",
    "g1.add_path([\"Wien\", \"Graz\", \"Linz\", \"Salzburg\", \"Innsbruck\", \"Villach\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot des Graphen, wobei mit `spring_layout` eine gleichmäßig verteilte Positionierung der Knoten berechnet wird.\n",
    "Anschließend werden nicht nur die Knoten und Kanten,\n",
    "sondern auch die Namen der Knoten geplottet."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Created with matplotlib (http://matplotlib.org/) -->\n",
       "<svg height=\"256pt\" version=\"1.1\" viewBox=\"0 0 382 256\" width=\"382pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"\n",
       "M0 256.117\n",
       "L382.492 256.117\n",
       "L382.492 0\n",
       "L0 0\n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"LineCollection_1\">\n",
       "    <path clip-path=\"url(#pe5892d72bd)\" d=\"\n",
       "M198.174 54.7233\n",
       "L199.325 143.24\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pe5892d72bd)\" d=\"\n",
       "M198.174 54.7233\n",
       "L296.131 43.9248\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pe5892d72bd)\" d=\"\n",
       "M199.325 143.24\n",
       "L218.566 199.625\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pe5892d72bd)\" d=\"\n",
       "M199.325 143.24\n",
       "L122.324 203.353\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pe5892d72bd)\" d=\"\n",
       "M199.325 143.24\n",
       "L322.341 183.384\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pe5892d72bd)\" d=\"\n",
       "M199.325 143.24\n",
       "L89.2219 157.6\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pe5892d72bd)\" d=\"\n",
       "M322.341 183.384\n",
       "L218.566 199.625\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pe5892d72bd)\" d=\"\n",
       "M218.566 199.625\n",
       "L89.2219 157.6\" style=\"fill:none;stroke:#000000;\"/>\n",
       "   </g>\n",
       "   <g id=\"PathCollection_1\">\n",
       "    <defs>\n",
       "     <path d=\"\n",
       "M0 8.66025\n",
       "C2.29673 8.66025 4.49969 7.74776 6.12372 6.12372\n",
       "C7.74776 4.49969 8.66025 2.29673 8.66025 0\n",
       "C8.66025 -2.29673 7.74776 -4.49969 6.12372 -6.12372\n",
       "C4.49969 -7.74776 2.29673 -8.66025 0 -8.66025\n",
       "C-2.29673 -8.66025 -4.49969 -7.74776 -6.12372 -6.12372\n",
       "C-7.74776 -4.49969 -8.66025 -2.29673 -8.66025 0\n",
       "C-8.66025 2.29673 -7.74776 4.49969 -6.12372 6.12372\n",
       "C-4.49969 7.74776 -2.29673 8.66025 0 8.66025\n",
       "z\n",
       "\" id=\"mf9612d42da\" style=\"stroke:#000000;\"/>\n",
       "    </defs>\n",
       "    <g clip-path=\"url(#pe5892d72bd)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"198.174040833\" xlink:href=\"#mf9612d42da\" y=\"54.7232693626\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"199.325196056\" xlink:href=\"#mf9612d42da\" y=\"143.240194679\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"322.34129115\" xlink:href=\"#mf9612d42da\" y=\"183.383941931\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"296.131365812\" xlink:href=\"#mf9612d42da\" y=\"43.9247767857\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"218.565501093\" xlink:href=\"#mf9612d42da\" y=\"199.625358974\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"122.324375048\" xlink:href=\"#mf9612d42da\" y=\"203.353348214\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"89.221875\" xlink:href=\"#mf9612d42da\" y=\"157.600397628\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_1\">\n",
       "    <g clip-path=\"url(#pe5892d72bd)\">\n",
       "     <!-- Graz -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M41.1094 46.2969\n",
       "Q39.5938 47.1719 37.8125 47.5781\n",
       "Q36.0312 48 33.8906 48\n",
       "Q26.2656 48 22.1875 43.0469\n",
       "Q18.1094 38.0938 18.1094 28.8125\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "L9.07812 54.6875\n",
       "L18.1094 54.6875\n",
       "L18.1094 46.1875\n",
       "Q20.9531 51.1719 25.4844 53.5781\n",
       "Q30.0312 56 36.5312 56\n",
       "Q37.4531 56 38.5781 55.875\n",
       "Q39.7031 55.7656 41.0625 55.5156\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-72\"/>\n",
       "      <path d=\"\n",
       "M34.2812 27.4844\n",
       "Q23.3906 27.4844 19.1875 25\n",
       "Q14.9844 22.5156 14.9844 16.5\n",
       "Q14.9844 11.7188 18.1406 8.90625\n",
       "Q21.2969 6.10938 26.7031 6.10938\n",
       "Q34.1875 6.10938 38.7031 11.4062\n",
       "Q43.2188 16.7031 43.2188 25.4844\n",
       "L43.2188 27.4844\n",
       "z\n",
       "\n",
       "M52.2031 31.2031\n",
       "L52.2031 0\n",
       "L43.2188 0\n",
       "L43.2188 8.29688\n",
       "Q40.1406 3.32812 35.5469 0.953125\n",
       "Q30.9531 -1.42188 24.3125 -1.42188\n",
       "Q15.9219 -1.42188 10.9531 3.29688\n",
       "Q6 8.01562 6 15.9219\n",
       "Q6 25.1406 12.1719 29.8281\n",
       "Q18.3594 34.5156 30.6094 34.5156\n",
       "L43.2188 34.5156\n",
       "L43.2188 35.4062\n",
       "Q43.2188 41.6094 39.1406 45\n",
       "Q35.0625 48.3906 27.6875 48.3906\n",
       "Q23 48.3906 18.5469 47.2656\n",
       "Q14.1094 46.1406 10.0156 43.8906\n",
       "L10.0156 52.2031\n",
       "Q14.9375 54.1094 19.5781 55.0469\n",
       "Q24.2188 56 28.6094 56\n",
       "Q40.4844 56 46.3438 49.8438\n",
       "Q52.2031 43.7031 52.2031 31.2031\" id=\"BitstreamVeraSans-Roman-61\"/>\n",
       "      <path d=\"\n",
       "M59.5156 10.4062\n",
       "L59.5156 29.9844\n",
       "L43.4062 29.9844\n",
       "L43.4062 38.0938\n",
       "L69.2812 38.0938\n",
       "L69.2812 6.78125\n",
       "Q63.5781 2.73438 56.6875 0.65625\n",
       "Q49.8125 -1.42188 42 -1.42188\n",
       "Q24.9062 -1.42188 15.25 8.5625\n",
       "Q5.60938 18.5625 5.60938 36.375\n",
       "Q5.60938 54.25 15.25 64.2344\n",
       "Q24.9062 74.2188 42 74.2188\n",
       "Q49.125 74.2188 55.5469 72.4531\n",
       "Q61.9688 70.7031 67.3906 67.2812\n",
       "L67.3906 56.7812\n",
       "Q61.9219 61.4219 55.7656 63.7656\n",
       "Q49.6094 66.1094 42.8281 66.1094\n",
       "Q29.4375 66.1094 22.7188 58.6406\n",
       "Q16.0156 51.1719 16.0156 36.375\n",
       "Q16.0156 21.625 22.7188 14.1562\n",
       "Q29.4375 6.6875 42.8281 6.6875\n",
       "Q48.0469 6.6875 52.1406 7.59375\n",
       "Q56.25 8.5 59.5156 10.4062\" id=\"BitstreamVeraSans-Roman-47\"/>\n",
       "      <path d=\"\n",
       "M5.51562 54.6875\n",
       "L48.1875 54.6875\n",
       "L48.1875 46.4844\n",
       "L14.4062 7.17188\n",
       "L48.1875 7.17188\n",
       "L48.1875 0\n",
       "L4.29688 0\n",
       "L4.29688 8.20312\n",
       "L38.0938 47.5156\n",
       "L5.51562 47.5156\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-7a\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(205.218313593 202.936608974)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-47\"/>\n",
       "      <use x=\"77.490234375\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"118.603515625\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"179.8828125\" xlink:href=\"#BitstreamVeraSans-Roman-7a\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_2\">\n",
       "    <g clip-path=\"url(#pe5892d72bd)\">\n",
       "     <!-- Salzburg -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M45.4062 27.9844\n",
       "Q45.4062 37.75 41.375 43.1094\n",
       "Q37.3594 48.4844 30.0781 48.4844\n",
       "Q22.8594 48.4844 18.8281 43.1094\n",
       "Q14.7969 37.75 14.7969 27.9844\n",
       "Q14.7969 18.2656 18.8281 12.8906\n",
       "Q22.8594 7.51562 30.0781 7.51562\n",
       "Q37.3594 7.51562 41.375 12.8906\n",
       "Q45.4062 18.2656 45.4062 27.9844\n",
       "M54.3906 6.78125\n",
       "Q54.3906 -7.17188 48.1875 -13.9844\n",
       "Q42 -20.7969 29.2031 -20.7969\n",
       "Q24.4688 -20.7969 20.2656 -20.0938\n",
       "Q16.0625 -19.3906 12.1094 -17.9219\n",
       "L12.1094 -9.1875\n",
       "Q16.0625 -11.3281 19.9219 -12.3438\n",
       "Q23.7812 -13.375 27.7812 -13.375\n",
       "Q36.625 -13.375 41.0156 -8.76562\n",
       "Q45.4062 -4.15625 45.4062 5.17188\n",
       "L45.4062 9.625\n",
       "Q42.625 4.78125 38.2812 2.39062\n",
       "Q33.9375 0 27.875 0\n",
       "Q17.8281 0 11.6719 7.65625\n",
       "Q5.51562 15.3281 5.51562 27.9844\n",
       "Q5.51562 40.6719 11.6719 48.3281\n",
       "Q17.8281 56 27.875 56\n",
       "Q33.9375 56 38.2812 53.6094\n",
       "Q42.625 51.2188 45.4062 46.3906\n",
       "L45.4062 54.6875\n",
       "L54.3906 54.6875\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-67\"/>\n",
       "      <path d=\"\n",
       "M8.5 21.5781\n",
       "L8.5 54.6875\n",
       "L17.4844 54.6875\n",
       "L17.4844 21.9219\n",
       "Q17.4844 14.1562 20.5 10.2656\n",
       "Q23.5312 6.39062 29.5938 6.39062\n",
       "Q36.8594 6.39062 41.0781 11.0312\n",
       "Q45.3125 15.6719 45.3125 23.6875\n",
       "L45.3125 54.6875\n",
       "L54.2969 54.6875\n",
       "L54.2969 0\n",
       "L45.3125 0\n",
       "L45.3125 8.40625\n",
       "Q42.0469 3.42188 37.7188 1\n",
       "Q33.4062 -1.42188 27.6875 -1.42188\n",
       "Q18.2656 -1.42188 13.375 4.4375\n",
       "Q8.5 10.2969 8.5 21.5781\" id=\"BitstreamVeraSans-Roman-75\"/>\n",
       "      <path d=\"\n",
       "M53.5156 70.5156\n",
       "L53.5156 60.8906\n",
       "Q47.9062 63.5781 42.9219 64.8906\n",
       "Q37.9375 66.2188 33.2969 66.2188\n",
       "Q25.25 66.2188 20.875 63.0938\n",
       "Q16.5 59.9688 16.5 54.2031\n",
       "Q16.5 49.3594 19.4062 46.8906\n",
       "Q22.3125 44.4375 30.4219 42.9219\n",
       "L36.375 41.7031\n",
       "Q47.4062 39.5938 52.6562 34.2969\n",
       "Q57.9062 29 57.9062 20.125\n",
       "Q57.9062 9.51562 50.7969 4.04688\n",
       "Q43.7031 -1.42188 29.9844 -1.42188\n",
       "Q24.8125 -1.42188 18.9688 -0.25\n",
       "Q13.1406 0.921875 6.89062 3.21875\n",
       "L6.89062 13.375\n",
       "Q12.8906 10.0156 18.6562 8.29688\n",
       "Q24.4219 6.59375 29.9844 6.59375\n",
       "Q38.4219 6.59375 43.0156 9.90625\n",
       "Q47.6094 13.2344 47.6094 19.3906\n",
       "Q47.6094 24.75 44.3125 27.7812\n",
       "Q41.0156 30.8125 33.5 32.3281\n",
       "L27.4844 33.5\n",
       "Q16.4531 35.6875 11.5156 40.375\n",
       "Q6.59375 45.0625 6.59375 53.4219\n",
       "Q6.59375 63.0938 13.4062 68.6562\n",
       "Q20.2188 74.2188 32.1719 74.2188\n",
       "Q37.3125 74.2188 42.625 73.2812\n",
       "Q47.9531 72.3594 53.5156 70.5156\" id=\"BitstreamVeraSans-Roman-53\"/>\n",
       "      <path d=\"\n",
       "M48.6875 27.2969\n",
       "Q48.6875 37.2031 44.6094 42.8438\n",
       "Q40.5312 48.4844 33.4062 48.4844\n",
       "Q26.2656 48.4844 22.1875 42.8438\n",
       "Q18.1094 37.2031 18.1094 27.2969\n",
       "Q18.1094 17.3906 22.1875 11.75\n",
       "Q26.2656 6.10938 33.4062 6.10938\n",
       "Q40.5312 6.10938 44.6094 11.75\n",
       "Q48.6875 17.3906 48.6875 27.2969\n",
       "M18.1094 46.3906\n",
       "Q20.9531 51.2656 25.2656 53.625\n",
       "Q29.5938 56 35.5938 56\n",
       "Q45.5625 56 51.7812 48.0938\n",
       "Q58.0156 40.1875 58.0156 27.2969\n",
       "Q58.0156 14.4062 51.7812 6.48438\n",
       "Q45.5625 -1.42188 35.5938 -1.42188\n",
       "Q29.5938 -1.42188 25.2656 0.953125\n",
       "Q20.9531 3.32812 18.1094 8.20312\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "L9.07812 75.9844\n",
       "L18.1094 75.9844\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-62\"/>\n",
       "      <path d=\"\n",
       "M9.42188 75.9844\n",
       "L18.4062 75.9844\n",
       "L18.4062 0\n",
       "L9.42188 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-6c\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(174.182383556 146.551444679)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-53\"/>\n",
       "      <use x=\"63.4765625\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"124.755859375\" xlink:href=\"#BitstreamVeraSans-Roman-6c\"/>\n",
       "      <use x=\"152.5390625\" xlink:href=\"#BitstreamVeraSans-Roman-7a\"/>\n",
       "      <use x=\"205.029296875\" xlink:href=\"#BitstreamVeraSans-Roman-62\"/>\n",
       "      <use x=\"268.505859375\" xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"331.884765625\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"371.248046875\" xlink:href=\"#BitstreamVeraSans-Roman-67\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_3\">\n",
       "    <g clip-path=\"url(#pe5892d72bd)\">\n",
       "     <!-- Wien -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M56.2031 29.5938\n",
       "L56.2031 25.2031\n",
       "L14.8906 25.2031\n",
       "Q15.4844 15.9219 20.4844 11.0625\n",
       "Q25.4844 6.20312 34.4219 6.20312\n",
       "Q39.5938 6.20312 44.4531 7.46875\n",
       "Q49.3125 8.73438 54.1094 11.2812\n",
       "L54.1094 2.78125\n",
       "Q49.2656 0.734375 44.1875 -0.34375\n",
       "Q39.1094 -1.42188 33.8906 -1.42188\n",
       "Q20.7969 -1.42188 13.1562 6.1875\n",
       "Q5.51562 13.8125 5.51562 26.8125\n",
       "Q5.51562 40.2344 12.7656 48.1094\n",
       "Q20.0156 56 32.3281 56\n",
       "Q43.3594 56 49.7812 48.8906\n",
       "Q56.2031 41.7969 56.2031 29.5938\n",
       "M47.2188 32.2344\n",
       "Q47.125 39.5938 43.0938 43.9844\n",
       "Q39.0625 48.3906 32.4219 48.3906\n",
       "Q24.9062 48.3906 20.3906 44.1406\n",
       "Q15.875 39.8906 15.1875 32.1719\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-65\"/>\n",
       "      <path d=\"\n",
       "M3.32812 72.9062\n",
       "L13.2812 72.9062\n",
       "L28.6094 11.2812\n",
       "L43.8906 72.9062\n",
       "L54.9844 72.9062\n",
       "L70.3125 11.2812\n",
       "L85.5938 72.9062\n",
       "L95.6094 72.9062\n",
       "L77.2969 0\n",
       "L64.8906 0\n",
       "L49.5156 63.2812\n",
       "L33.9844 0\n",
       "L21.5781 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-57\"/>\n",
       "      <path d=\"\n",
       "M54.8906 33.0156\n",
       "L54.8906 0\n",
       "L45.9062 0\n",
       "L45.9062 32.7188\n",
       "Q45.9062 40.4844 42.875 44.3281\n",
       "Q39.8438 48.1875 33.7969 48.1875\n",
       "Q26.5156 48.1875 22.3125 43.5469\n",
       "Q18.1094 38.9219 18.1094 30.9062\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "L9.07812 54.6875\n",
       "L18.1094 54.6875\n",
       "L18.1094 46.1875\n",
       "Q21.3438 51.125 25.7031 53.5625\n",
       "Q30.0781 56 35.7969 56\n",
       "Q45.2188 56 50.0469 50.1719\n",
       "Q54.8906 44.3438 54.8906 33.0156\" id=\"BitstreamVeraSans-Roman-6e\"/>\n",
       "      <path d=\"\n",
       "M9.42188 54.6875\n",
       "L18.4062 54.6875\n",
       "L18.4062 0\n",
       "L9.42188 0\n",
       "z\n",
       "\n",
       "M9.42188 75.9844\n",
       "L18.4062 75.9844\n",
       "L18.4062 64.5938\n",
       "L9.42188 64.5938\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-69\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(308.09129115 186.695191931)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-57\"/>\n",
       "      <use x=\"96.626953125\" xlink:href=\"#BitstreamVeraSans-Roman-69\"/>\n",
       "      <use x=\"124.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"185.93359375\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_4\">\n",
       "    <g clip-path=\"url(#pe5892d72bd)\">\n",
       "     <!-- Villach -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M54.8906 33.0156\n",
       "L54.8906 0\n",
       "L45.9062 0\n",
       "L45.9062 32.7188\n",
       "Q45.9062 40.4844 42.875 44.3281\n",
       "Q39.8438 48.1875 33.7969 48.1875\n",
       "Q26.5156 48.1875 22.3125 43.5469\n",
       "Q18.1094 38.9219 18.1094 30.9062\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "L9.07812 75.9844\n",
       "L18.1094 75.9844\n",
       "L18.1094 46.1875\n",
       "Q21.3438 51.125 25.7031 53.5625\n",
       "Q30.0781 56 35.7969 56\n",
       "Q45.2188 56 50.0469 50.1719\n",
       "Q54.8906 44.3438 54.8906 33.0156\" id=\"BitstreamVeraSans-Roman-68\"/>\n",
       "      <path d=\"\n",
       "M48.7812 52.5938\n",
       "L48.7812 44.1875\n",
       "Q44.9688 46.2969 41.1406 47.3438\n",
       "Q37.3125 48.3906 33.4062 48.3906\n",
       "Q24.6562 48.3906 19.8125 42.8438\n",
       "Q14.9844 37.3125 14.9844 27.2969\n",
       "Q14.9844 17.2812 19.8125 11.7344\n",
       "Q24.6562 6.20312 33.4062 6.20312\n",
       "Q37.3125 6.20312 41.1406 7.25\n",
       "Q44.9688 8.29688 48.7812 10.4062\n",
       "L48.7812 2.09375\n",
       "Q45.0156 0.34375 40.9844 -0.53125\n",
       "Q36.9688 -1.42188 32.4219 -1.42188\n",
       "Q20.0625 -1.42188 12.7812 6.34375\n",
       "Q5.51562 14.1094 5.51562 27.2969\n",
       "Q5.51562 40.6719 12.8594 48.3281\n",
       "Q20.2188 56 33.0156 56\n",
       "Q37.1562 56 41.1094 55.1406\n",
       "Q45.0625 54.2969 48.7812 52.5938\" id=\"BitstreamVeraSans-Roman-63\"/>\n",
       "      <path d=\"\n",
       "M28.6094 0\n",
       "L0.78125 72.9062\n",
       "L11.0781 72.9062\n",
       "L34.1875 11.5312\n",
       "L57.3281 72.9062\n",
       "L67.5781 72.9062\n",
       "L39.7969 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-56\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(276.938865812 47.2360267857)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-56\"/>\n",
       "      <use x=\"66.158203125\" xlink:href=\"#BitstreamVeraSans-Roman-69\"/>\n",
       "      <use x=\"93.94140625\" xlink:href=\"#BitstreamVeraSans-Roman-6c\"/>\n",
       "      <use x=\"121.724609375\" xlink:href=\"#BitstreamVeraSans-Roman-6c\"/>\n",
       "      <use x=\"149.5078125\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"210.787109375\" xlink:href=\"#BitstreamVeraSans-Roman-63\"/>\n",
       "      <use x=\"265.767578125\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_5\">\n",
       "    <g clip-path=\"url(#pe5892d72bd)\">\n",
       "     <!-- Innsbruck -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M44.2812 53.0781\n",
       "L44.2812 44.5781\n",
       "Q40.4844 46.5312 36.375 47.5\n",
       "Q32.2812 48.4844 27.875 48.4844\n",
       "Q21.1875 48.4844 17.8438 46.4375\n",
       "Q14.5 44.3906 14.5 40.2812\n",
       "Q14.5 37.1562 16.8906 35.375\n",
       "Q19.2812 33.5938 26.5156 31.9844\n",
       "L29.5938 31.2969\n",
       "Q39.1562 29.25 43.1875 25.5156\n",
       "Q47.2188 21.7812 47.2188 15.0938\n",
       "Q47.2188 7.46875 41.1875 3.01562\n",
       "Q35.1562 -1.42188 24.6094 -1.42188\n",
       "Q20.2188 -1.42188 15.4531 -0.5625\n",
       "Q10.6875 0.296875 5.42188 2\n",
       "L5.42188 11.2812\n",
       "Q10.4062 8.6875 15.2344 7.39062\n",
       "Q20.0625 6.10938 24.8125 6.10938\n",
       "Q31.1562 6.10938 34.5625 8.28125\n",
       "Q37.9844 10.4531 37.9844 14.4062\n",
       "Q37.9844 18.0625 35.5156 20.0156\n",
       "Q33.0625 21.9688 24.7031 23.7812\n",
       "L21.5781 24.5156\n",
       "Q13.2344 26.2656 9.51562 29.9062\n",
       "Q5.8125 33.5469 5.8125 39.8906\n",
       "Q5.8125 47.6094 11.2812 51.7969\n",
       "Q16.75 56 26.8125 56\n",
       "Q31.7812 56 36.1719 55.2656\n",
       "Q40.5781 54.5469 44.2812 53.0781\" id=\"BitstreamVeraSans-Roman-73\"/>\n",
       "      <path d=\"\n",
       "M9.07812 75.9844\n",
       "L18.1094 75.9844\n",
       "L18.1094 31.1094\n",
       "L44.9219 54.6875\n",
       "L56.3906 54.6875\n",
       "L27.3906 29.1094\n",
       "L57.625 0\n",
       "L45.9062 0\n",
       "L18.1094 26.7031\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-6b\"/>\n",
       "      <path d=\"\n",
       "M9.8125 72.9062\n",
       "L19.6719 72.9062\n",
       "L19.6719 0\n",
       "L9.8125 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-49\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(169.427478333 58.0345193626)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-49\"/>\n",
       "      <use x=\"29.4921875\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"92.87109375\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"156.25\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "      <use x=\"208.349609375\" xlink:href=\"#BitstreamVeraSans-Roman-62\"/>\n",
       "      <use x=\"271.826171875\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"312.939453125\" xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"376.318359375\" xlink:href=\"#BitstreamVeraSans-Roman-63\"/>\n",
       "      <use x=\"431.298828125\" xlink:href=\"#BitstreamVeraSans-Roman-6b\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_6\">\n",
       "    <g clip-path=\"url(#pe5892d72bd)\">\n",
       "     <!-- St. Pölten -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M18.3125 70.2188\n",
       "L18.3125 54.6875\n",
       "L36.8125 54.6875\n",
       "L36.8125 47.7031\n",
       "L18.3125 47.7031\n",
       "L18.3125 18.0156\n",
       "Q18.3125 11.3281 20.1406 9.42188\n",
       "Q21.9688 7.51562 27.5938 7.51562\n",
       "L36.8125 7.51562\n",
       "L36.8125 0\n",
       "L27.5938 0\n",
       "Q17.1875 0 13.2344 3.875\n",
       "Q9.28125 7.76562 9.28125 18.0156\n",
       "L9.28125 47.7031\n",
       "L2.6875 47.7031\n",
       "L2.6875 54.6875\n",
       "L9.28125 54.6875\n",
       "L9.28125 70.2188\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-74\"/>\n",
       "      <path d=\"\n",
       "M30.6094 48.3906\n",
       "Q23.3906 48.3906 19.1875 42.75\n",
       "Q14.9844 37.1094 14.9844 27.2969\n",
       "Q14.9844 17.4844 19.1562 11.8438\n",
       "Q23.3438 6.20312 30.6094 6.20312\n",
       "Q37.7969 6.20312 41.9844 11.8594\n",
       "Q46.1875 17.5312 46.1875 27.2969\n",
       "Q46.1875 37.0156 41.9844 42.7031\n",
       "Q37.7969 48.3906 30.6094 48.3906\n",
       "M30.6094 56\n",
       "Q42.3281 56 49.0156 48.375\n",
       "Q55.7188 40.7656 55.7188 27.2969\n",
       "Q55.7188 13.875 49.0156 6.21875\n",
       "Q42.3281 -1.42188 30.6094 -1.42188\n",
       "Q18.8438 -1.42188 12.1719 6.21875\n",
       "Q5.51562 13.875 5.51562 27.2969\n",
       "Q5.51562 40.7656 12.1719 48.375\n",
       "Q18.8438 56 30.6094 56\n",
       "M35.2188 75.7812\n",
       "L45.125 75.7812\n",
       "L45.125 65.9219\n",
       "L35.2188 65.9219\n",
       "z\n",
       "\n",
       "M16.125 75.7812\n",
       "L26.0312 75.7812\n",
       "L26.0312 65.9219\n",
       "L16.125 65.9219\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-f6\"/>\n",
       "      <path d=\"\n",
       "M19.6719 64.7969\n",
       "L19.6719 37.4062\n",
       "L32.0781 37.4062\n",
       "Q38.9688 37.4062 42.7188 40.9688\n",
       "Q46.4844 44.5312 46.4844 51.125\n",
       "Q46.4844 57.6719 42.7188 61.2344\n",
       "Q38.9688 64.7969 32.0781 64.7969\n",
       "z\n",
       "\n",
       "M9.8125 72.9062\n",
       "L32.0781 72.9062\n",
       "Q44.3438 72.9062 50.6094 67.3594\n",
       "Q56.8906 61.8125 56.8906 51.125\n",
       "Q56.8906 40.3281 50.6094 34.8125\n",
       "Q44.3438 29.2969 32.0781 29.2969\n",
       "L19.6719 29.2969\n",
       "L19.6719 0\n",
       "L9.8125 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-50\"/>\n",
       "      <path id=\"BitstreamVeraSans-Roman-20\"/>\n",
       "      <path d=\"\n",
       "M10.6875 12.4062\n",
       "L21 12.4062\n",
       "L21 0\n",
       "L10.6875 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-2e\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(94.2734375478 206.664598214)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-53\"/>\n",
       "      <use x=\"63.4765625\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"102.685546875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "      <use x=\"134.47265625\" xlink:href=\"#BitstreamVeraSans-Roman-20\"/>\n",
       "      <use x=\"166.259765625\" xlink:href=\"#BitstreamVeraSans-Roman-50\"/>\n",
       "      <use x=\"222.9375\" xlink:href=\"#BitstreamVeraSans-Roman-f6\"/>\n",
       "      <use x=\"284.119140625\" xlink:href=\"#BitstreamVeraSans-Roman-6c\"/>\n",
       "      <use x=\"311.90234375\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"351.111328125\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"412.634765625\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_7\">\n",
       "    <g clip-path=\"url(#pe5892d72bd)\">\n",
       "     <!-- Linz -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M9.8125 72.9062\n",
       "L19.6719 72.9062\n",
       "L19.6719 8.29688\n",
       "L55.1719 8.29688\n",
       "L55.1719 0\n",
       "L9.8125 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-4c\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(78.106875 160.911647628)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-4c\"/>\n",
       "      <use x=\"55.712890625\" xlink:href=\"#BitstreamVeraSans-Roman-69\"/>\n",
       "      <use x=\"83.49609375\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"146.875\" xlink:href=\"#BitstreamVeraSans-Roman-7a\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pe5892d72bd\">\n",
       "   <rect height=\"223.2\" width=\"334.8\" x=\"33.421875\" y=\"12.0390625\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2b68565b56a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "pos = nx.spring_layout(g1)\n",
    "nx.draw_networkx_labels(g1, pos, font_size = 12)\n",
    "nx.draw(g1, pos)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ausgehend von diesem Graph lassen sich einige Kennzahlen und Parameter ausrechnen:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ist er zusammenhängend?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.is_connected(g1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Grad von Graz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.degree(g1, \"Graz\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Durchmesser des ganzen Graphs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.diameter(g1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exzentrizität"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'Graz': 3,\n",
       " 'Innsbruck': 2,\n",
       " 'Linz': 3,\n",
       " 'Salzburg': 2,\n",
       " 'St. Pölten': 3,\n",
       " 'Villach': 3,\n",
       " 'Wien': 3}"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.eccentricity(g1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Kürzester Weg von Linz nach Innsbruck"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['Linz', 'Salzburg', 'Innsbruck']"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.shortest_path(g1, \"Linz\", \"Innsbruck\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Es gibt außerdem eine große Bibliothek für Graphen.\n",
    "Hier wird ein Dodekaedergraph geladen und der 10-te Knoten entfernt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Created with matplotlib (http://matplotlib.org/) -->\n",
       "<svg height=\"320pt\" version=\"1.1\" viewBox=\"0 0 479 320\" width=\"479pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"\n",
       "M0 320.917\n",
       "L479.523 320.917\n",
       "L479.523 0\n",
       "L0 0\n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"LineCollection_1\">\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M105.818 200.845\n",
       "L140.947 230.854\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M105.818 200.845\n",
       "L95.1362 169.201\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M140.947 230.854\n",
       "L215.494 164.244\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M140.947 230.854\n",
       "L227.665 258.896\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M227.665 258.896\n",
       "L175.563 237.235\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M227.665 258.896\n",
       "L317.374 241.942\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M175.563 237.235\n",
       "L95.1362 169.201\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M175.563 237.235\n",
       "L268.211 209.285\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M268.211 209.285\n",
       "L241.997 124.679\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M268.211 209.285\n",
       "L373.851 217.438\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M373.851 217.438\n",
       "L317.374 241.942\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M373.851 217.438\n",
       "L400.76 154.595\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M317.374 241.942\n",
       "L337.046 180.397\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M337.046 180.397\n",
       "L215.494 164.244\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M337.046 180.397\n",
       "L385.136 120.252\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M215.494 164.244\n",
       "L228.677 76.0123\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M228.677 76.0123\n",
       "L320.384 65.2339\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M170.547 66.273\n",
       "L269.115 53.1819\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M170.547 66.273\n",
       "L117.583 105.836\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M269.115 53.1819\n",
       "L352.315 89.0279\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M269.115 53.1819\n",
       "L320.384 65.2339\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M320.384 65.2339\n",
       "L385.136 120.252\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M385.136 120.252\n",
       "L400.76 154.595\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M400.76 154.595\n",
       "L352.315 89.0279\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M352.315 89.0279\n",
       "L241.997 124.679\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M241.997 124.679\n",
       "L117.583 105.836\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M117.583 105.836\n",
       "L95.1362 169.201\" style=\"fill:none;stroke:#000000;\"/>\n",
       "   </g>\n",
       "   <g id=\"PathCollection_1\">\n",
       "    <defs>\n",
       "     <path d=\"\n",
       "M0 8.66025\n",
       "C2.29673 8.66025 4.49969 7.74776 6.12372 6.12372\n",
       "C7.74776 4.49969 8.66025 2.29673 8.66025 0\n",
       "C8.66025 -2.29673 7.74776 -4.49969 6.12372 -6.12372\n",
       "C4.49969 -7.74776 2.29673 -8.66025 0 -8.66025\n",
       "C-2.29673 -8.66025 -4.49969 -7.74776 -6.12372 -6.12372\n",
       "C-7.74776 -4.49969 -8.66025 -2.29673 -8.66025 0\n",
       "C-8.66025 2.29673 -7.74776 4.49969 -6.12372 6.12372\n",
       "C-4.49969 7.74776 -2.29673 8.66025 0 8.66025\n",
       "z\n",
       "\" id=\"mf9612d42da\" style=\"stroke:#000000;\"/>\n",
       "    </defs>\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"105.818133206\" xlink:href=\"#mf9612d42da\" y=\"200.844872954\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"140.947280131\" xlink:href=\"#mf9612d42da\" y=\"230.854253882\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"227.665408583\" xlink:href=\"#mf9612d42da\" y=\"258.896205357\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"175.563108091\" xlink:href=\"#mf9612d42da\" y=\"237.234544698\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"268.210643332\" xlink:href=\"#mf9612d42da\" y=\"209.285446821\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"373.850767003\" xlink:href=\"#mf9612d42da\" y=\"217.438029022\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"317.373644478\" xlink:href=\"#mf9612d42da\" y=\"241.942045218\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"337.046480838\" xlink:href=\"#mf9612d42da\" y=\"180.396628962\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"215.494002682\" xlink:href=\"#mf9612d42da\" y=\"164.243937487\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"228.677336419\" xlink:href=\"#mf9612d42da\" y=\"76.0122916709\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"170.547011034\" xlink:href=\"#mf9612d42da\" y=\"66.2729509869\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"269.114879203\" xlink:href=\"#mf9612d42da\" y=\"53.1819196429\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"320.383897735\" xlink:href=\"#mf9612d42da\" y=\"65.2338904158\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"385.136042996\" xlink:href=\"#mf9612d42da\" y=\"120.251573216\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"400.760167915\" xlink:href=\"#mf9612d42da\" y=\"154.594653408\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"352.315424746\" xlink:href=\"#mf9612d42da\" y=\"89.0279000608\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"241.996934346\" xlink:href=\"#mf9612d42da\" y=\"124.678675156\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"117.582639695\" xlink:href=\"#mf9612d42da\" y=\"105.836027312\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"95.1361607143\" xlink:href=\"#mf9612d42da\" y=\"169.200548952\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pb36153c835\">\n",
       "   <rect height=\"288.0\" width=\"432.0\" x=\"33.421875\" y=\"12.0390625\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2b68565eca58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g2=nx.dodecahedral_graph()\n",
    "g2.remove_node(10)\n",
    "nx.draw(g2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Durchmesser"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.diameter(g2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vergleich der Dichte beider Graphen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.38095238095238093, 0.15789473684210525)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.density(g1), nx.density(g2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Histogramdaten für die Grade aller Knoten:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 0, 3, 16]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.degree_histogram(g2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gerichteter Graph\n",
    "\n",
    "Ein **gerichteter Graph** für die Wörter in einem Text."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "text = '''11:15 Restate my assumptions:\n",
    "1. Mathematics is the language of nature.\n",
    "2. Everything around us can be represented and understood through numbers. \n",
    "3. If you graph these numbers, patterns emerge.\n",
    "Therefore: There are patterns everywhere in nature.'''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import re\n",
    "words = re.compile(r'[a-zA-Z]+')\n",
    "tokens = words.findall(text)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "text_digraph = nx.MultiDiGraph()\n",
    "for t1, t2 in zip(tokens[:-1], tokens[1:]):\n",
    "    text_digraph.add_edge(t1, t2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Created with matplotlib (http://matplotlib.org/) -->\n",
       "<svg height=\"320pt\" version=\"1.1\" viewBox=\"0 0 479 320\" width=\"479pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"\n",
       "M0 320.917\n",
       "L479.523 320.917\n",
       "L479.523 0\n",
       "L0 0\n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"LineCollection_1\">\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M156.866 157.856\n",
       "L119.411 159.938\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M220.586 82.3312\n",
       "L384.366 130.533\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M182.409 246.896\n",
       "L95.1362 191.435\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M384.366 130.533\n",
       "L378.387 124.295\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M360.459 213.113\n",
       "L254.628 258.896\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M246.068 133.886\n",
       "L191.506 143.733\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M191.506 143.733\n",
       "L195.733 251.541\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M317.733 170.227\n",
       "L246.068 133.886\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M95.1362 191.435\n",
       "L103.043 123.696\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M342.827 108.744\n",
       "L303.457 199.134\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M204.699 53.1819\n",
       "L288.424 56.3969\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M159.465 63.4024\n",
       "L204.699 53.1819\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M127.345 219.795\n",
       "L254.628 258.896\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M244.985 213.195\n",
       "L156.866 157.856\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M110.768 211.145\n",
       "L127.345 219.795\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M103.043 123.696\n",
       "L148.128 93.3679\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M297.424 248.893\n",
       "L182.409 246.896\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M225.757 243.895\n",
       "L244.985 213.195\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M225.757 243.895\n",
       "L110.768 211.145\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M119.411 159.938\n",
       "L117.982 229.571\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M254.628 258.896\n",
       "L297.424 248.893\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M392.416 171.188\n",
       "L360.459 213.113\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M117.982 229.571\n",
       "L225.757 243.895\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M148.128 93.3679\n",
       "L220.586 82.3312\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M368.776 104.346\n",
       "L392.416 171.188\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M329.308 71.185\n",
       "L368.776 104.346\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M260.536 53.4764\n",
       "L329.308 71.185\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M288.424 56.3969\n",
       "L260.536 53.4764\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M303.457 199.134\n",
       "L317.733 170.227\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M303.457 199.134\n",
       "L225.757 243.895\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M195.733 251.541\n",
       "L303.457 199.134\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M378.387 124.295\n",
       "L342.827 108.744\" style=\"fill:none;stroke:#000000;\"/>\n",
       "   </g>\n",
       "   <g id=\"LineCollection_2\">\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M128.775 159.417\n",
       "L119.411 159.938\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M343.421 118.482\n",
       "L384.366 130.533\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M116.954 205.3\n",
       "L95.1362 191.435\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M379.882 125.854\n",
       "L378.387 124.295\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M281.086 247.45\n",
       "L254.628 258.896\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M205.146 141.271\n",
       "L191.506 143.733\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M194.676 224.589\n",
       "L195.733 251.541\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M263.984 142.971\n",
       "L246.068 133.886\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M101.066 140.631\n",
       "L103.043 123.696\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M313.299 176.537\n",
       "L303.457 199.134\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M267.493 55.5932\n",
       "L288.424 56.3969\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M193.39 55.737\n",
       "L204.699 53.1819\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M222.807 249.121\n",
       "L254.628 258.896\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M178.895 171.691\n",
       "L156.866 157.856\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M123.201 217.633\n",
       "L127.345 219.795\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M136.857 100.95\n",
       "L148.128 93.3679\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M211.162 247.396\n",
       "L182.409 246.896\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M240.178 220.87\n",
       "L244.985 213.195\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M139.515 219.332\n",
       "L110.768 211.145\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M118.339 212.162\n",
       "L117.982 229.571\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M286.725 251.394\n",
       "L297.424 248.893\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M368.448 202.632\n",
       "L360.459 213.113\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M198.814 240.314\n",
       "L225.757 243.895\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M202.472 85.0904\n",
       "L220.586 82.3312\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M386.506 154.477\n",
       "L392.416 171.188\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M358.909 96.0554\n",
       "L368.776 104.346\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M312.115 66.7579\n",
       "L329.308 71.185\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M267.508 54.2065\n",
       "L260.536 53.4764\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M314.164 177.454\n",
       "L317.733 170.227\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M245.182 232.704\n",
       "L225.757 243.895\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M276.526 212.236\n",
       "L303.457 199.134\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M351.717 112.632\n",
       "L342.827 108.744\" style=\"fill:none;stroke:#000000;stroke-width:4.0;\"/>\n",
       "   </g>\n",
       "   <g id=\"PathCollection_1\">\n",
       "    <defs>\n",
       "     <path d=\"\n",
       "M0 8.66025\n",
       "C2.29673 8.66025 4.49969 7.74776 6.12372 6.12372\n",
       "C7.74776 4.49969 8.66025 2.29673 8.66025 0\n",
       "C8.66025 -2.29673 7.74776 -4.49969 6.12372 -6.12372\n",
       "C4.49969 -7.74776 2.29673 -8.66025 0 -8.66025\n",
       "C-2.29673 -8.66025 -4.49969 -7.74776 -6.12372 -6.12372\n",
       "C-7.74776 -4.49969 -8.66025 -2.29673 -8.66025 0\n",
       "C-8.66025 2.29673 -7.74776 4.49969 -6.12372 6.12372\n",
       "C-4.49969 7.74776 -2.29673 8.66025 0 8.66025\n",
       "z\n",
       "\" id=\"mf9612d42da\" style=\"stroke:#000000;\"/>\n",
       "    </defs>\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"156.86555529\" xlink:href=\"#mf9612d42da\" y=\"157.855849519\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"220.586380178\" xlink:href=\"#mf9612d42da\" y=\"82.3312139563\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"182.408529046\" xlink:href=\"#mf9612d42da\" y=\"246.896266992\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"384.366041986\" xlink:href=\"#mf9612d42da\" y=\"130.532575312\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"360.459195326\" xlink:href=\"#mf9612d42da\" y=\"213.113048358\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"246.068047571\" xlink:href=\"#mf9612d42da\" y=\"133.886229471\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"191.505656466\" xlink:href=\"#mf9612d42da\" y=\"143.732937584\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"317.732840078\" xlink:href=\"#mf9612d42da\" y=\"170.227220505\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"95.1361607143\" xlink:href=\"#mf9612d42da\" y=\"191.434924139\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"342.82709696\" xlink:href=\"#mf9612d42da\" y=\"108.744411131\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"204.698553427\" xlink:href=\"#mf9612d42da\" y=\"53.1819196429\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"159.465063873\" xlink:href=\"#mf9612d42da\" y=\"63.4024396196\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"127.344632149\" xlink:href=\"#mf9612d42da\" y=\"219.795246457\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"244.98491727\" xlink:href=\"#mf9612d42da\" y=\"213.195130134\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"110.768107182\" xlink:href=\"#mf9612d42da\" y=\"211.144606322\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"103.042653523\" xlink:href=\"#mf9612d42da\" y=\"123.695973457\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"297.423635667\" xlink:href=\"#mf9612d42da\" y=\"248.893266253\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"225.757491905\" xlink:href=\"#mf9612d42da\" y=\"243.894608512\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"119.411428585\" xlink:href=\"#mf9612d42da\" y=\"159.937836629\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"254.62770238\" xlink:href=\"#mf9612d42da\" y=\"258.896205357\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"392.415782951\" xlink:href=\"#mf9612d42da\" y=\"171.187918995\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"117.981544592\" xlink:href=\"#mf9612d42da\" y=\"229.57068123\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"148.128260575\" xlink:href=\"#mf9612d42da\" y=\"93.3678893655\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"368.775973053\" xlink:href=\"#mf9612d42da\" y=\"104.345582852\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"329.307749437\" xlink:href=\"#mf9612d42da\" y=\"71.1850382925\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"260.535652317\" xlink:href=\"#mf9612d42da\" y=\"53.4763742865\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"288.424003534\" xlink:href=\"#mf9612d42da\" y=\"56.3969147594\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"303.456614067\" xlink:href=\"#mf9612d42da\" y=\"199.133921264\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"195.733402319\" xlink:href=\"#mf9612d42da\" y=\"251.541029648\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"378.386968366\" xlink:href=\"#mf9612d42da\" y=\"124.294815364\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_1\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- represented -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M56.2031 29.5938\n",
       "L56.2031 25.2031\n",
       "L14.8906 25.2031\n",
       "Q15.4844 15.9219 20.4844 11.0625\n",
       "Q25.4844 6.20312 34.4219 6.20312\n",
       "Q39.5938 6.20312 44.4531 7.46875\n",
       "Q49.3125 8.73438 54.1094 11.2812\n",
       "L54.1094 2.78125\n",
       "Q49.2656 0.734375 44.1875 -0.34375\n",
       "Q39.1094 -1.42188 33.8906 -1.42188\n",
       "Q20.7969 -1.42188 13.1562 6.1875\n",
       "Q5.51562 13.8125 5.51562 26.8125\n",
       "Q5.51562 40.2344 12.7656 48.1094\n",
       "Q20.0156 56 32.3281 56\n",
       "Q43.3594 56 49.7812 48.8906\n",
       "Q56.2031 41.7969 56.2031 29.5938\n",
       "M47.2188 32.2344\n",
       "Q47.125 39.5938 43.0938 43.9844\n",
       "Q39.0625 48.3906 32.4219 48.3906\n",
       "Q24.9062 48.3906 20.3906 44.1406\n",
       "Q15.875 39.8906 15.1875 32.1719\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-65\"/>\n",
       "      <path d=\"\n",
       "M18.3125 70.2188\n",
       "L18.3125 54.6875\n",
       "L36.8125 54.6875\n",
       "L36.8125 47.7031\n",
       "L18.3125 47.7031\n",
       "L18.3125 18.0156\n",
       "Q18.3125 11.3281 20.1406 9.42188\n",
       "Q21.9688 7.51562 27.5938 7.51562\n",
       "L36.8125 7.51562\n",
       "L36.8125 0\n",
       "L27.5938 0\n",
       "Q17.1875 0 13.2344 3.875\n",
       "Q9.28125 7.76562 9.28125 18.0156\n",
       "L9.28125 47.7031\n",
       "L2.6875 47.7031\n",
       "L2.6875 54.6875\n",
       "L9.28125 54.6875\n",
       "L9.28125 70.2188\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-74\"/>\n",
       "      <path d=\"\n",
       "M44.2812 53.0781\n",
       "L44.2812 44.5781\n",
       "Q40.4844 46.5312 36.375 47.5\n",
       "Q32.2812 48.4844 27.875 48.4844\n",
       "Q21.1875 48.4844 17.8438 46.4375\n",
       "Q14.5 44.3906 14.5 40.2812\n",
       "Q14.5 37.1562 16.8906 35.375\n",
       "Q19.2812 33.5938 26.5156 31.9844\n",
       "L29.5938 31.2969\n",
       "Q39.1562 29.25 43.1875 25.5156\n",
       "Q47.2188 21.7812 47.2188 15.0938\n",
       "Q47.2188 7.46875 41.1875 3.01562\n",
       "Q35.1562 -1.42188 24.6094 -1.42188\n",
       "Q20.2188 -1.42188 15.4531 -0.5625\n",
       "Q10.6875 0.296875 5.42188 2\n",
       "L5.42188 11.2812\n",
       "Q10.4062 8.6875 15.2344 7.39062\n",
       "Q20.0625 6.10938 24.8125 6.10938\n",
       "Q31.1562 6.10938 34.5625 8.28125\n",
       "Q37.9844 10.4531 37.9844 14.4062\n",
       "Q37.9844 18.0625 35.5156 20.0156\n",
       "Q33.0625 21.9688 24.7031 23.7812\n",
       "L21.5781 24.5156\n",
       "Q13.2344 26.2656 9.51562 29.9062\n",
       "Q5.8125 33.5469 5.8125 39.8906\n",
       "Q5.8125 47.6094 11.2812 51.7969\n",
       "Q16.75 56 26.8125 56\n",
       "Q31.7812 56 36.1719 55.2656\n",
       "Q40.5781 54.5469 44.2812 53.0781\" id=\"BitstreamVeraSans-Roman-73\"/>\n",
       "      <path d=\"\n",
       "M45.4062 46.3906\n",
       "L45.4062 75.9844\n",
       "L54.3906 75.9844\n",
       "L54.3906 0\n",
       "L45.4062 0\n",
       "L45.4062 8.20312\n",
       "Q42.5781 3.32812 38.25 0.953125\n",
       "Q33.9375 -1.42188 27.875 -1.42188\n",
       "Q17.9688 -1.42188 11.7344 6.48438\n",
       "Q5.51562 14.4062 5.51562 27.2969\n",
       "Q5.51562 40.1875 11.7344 48.0938\n",
       "Q17.9688 56 27.875 56\n",
       "Q33.9375 56 38.25 53.625\n",
       "Q42.5781 51.2656 45.4062 46.3906\n",
       "M14.7969 27.2969\n",
       "Q14.7969 17.3906 18.875 11.75\n",
       "Q22.9531 6.10938 30.0781 6.10938\n",
       "Q37.2031 6.10938 41.2969 11.75\n",
       "Q45.4062 17.3906 45.4062 27.2969\n",
       "Q45.4062 37.2031 41.2969 42.8438\n",
       "Q37.2031 48.4844 30.0781 48.4844\n",
       "Q22.9531 48.4844 18.875 42.8438\n",
       "Q14.7969 37.2031 14.7969 27.2969\" id=\"BitstreamVeraSans-Roman-64\"/>\n",
       "      <path d=\"\n",
       "M41.1094 46.2969\n",
       "Q39.5938 47.1719 37.8125 47.5781\n",
       "Q36.0312 48 33.8906 48\n",
       "Q26.2656 48 22.1875 43.0469\n",
       "Q18.1094 38.0938 18.1094 28.8125\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "L9.07812 54.6875\n",
       "L18.1094 54.6875\n",
       "L18.1094 46.1875\n",
       "Q20.9531 51.1719 25.4844 53.5781\n",
       "Q30.0312 56 36.5312 56\n",
       "Q37.4531 56 38.5781 55.875\n",
       "Q39.7031 55.7656 41.0625 55.5156\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-72\"/>\n",
       "      <path d=\"\n",
       "M18.1094 8.20312\n",
       "L18.1094 -20.7969\n",
       "L9.07812 -20.7969\n",
       "L9.07812 54.6875\n",
       "L18.1094 54.6875\n",
       "L18.1094 46.3906\n",
       "Q20.9531 51.2656 25.2656 53.625\n",
       "Q29.5938 56 35.5938 56\n",
       "Q45.5625 56 51.7812 48.0938\n",
       "Q58.0156 40.1875 58.0156 27.2969\n",
       "Q58.0156 14.4062 51.7812 6.48438\n",
       "Q45.5625 -1.42188 35.5938 -1.42188\n",
       "Q29.5938 -1.42188 25.2656 0.953125\n",
       "Q20.9531 3.32812 18.1094 8.20312\n",
       "M48.6875 27.2969\n",
       "Q48.6875 37.2031 44.6094 42.8438\n",
       "Q40.5312 48.4844 33.4062 48.4844\n",
       "Q26.2656 48.4844 22.1875 42.8438\n",
       "Q18.1094 37.2031 18.1094 27.2969\n",
       "Q18.1094 17.3906 22.1875 11.75\n",
       "Q26.2656 6.10938 33.4062 6.10938\n",
       "Q40.5312 6.10938 44.6094 11.75\n",
       "Q48.6875 17.3906 48.6875 27.2969\" id=\"BitstreamVeraSans-Roman-70\"/>\n",
       "      <path d=\"\n",
       "M54.8906 33.0156\n",
       "L54.8906 0\n",
       "L45.9062 0\n",
       "L45.9062 32.7188\n",
       "Q45.9062 40.4844 42.875 44.3281\n",
       "Q39.8438 48.1875 33.7969 48.1875\n",
       "Q26.5156 48.1875 22.3125 43.5469\n",
       "Q18.1094 38.9219 18.1094 30.9062\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "L9.07812 54.6875\n",
       "L18.1094 54.6875\n",
       "L18.1094 46.1875\n",
       "Q21.3438 51.125 25.7031 53.5625\n",
       "Q30.0781 56 35.7969 56\n",
       "Q45.2188 56 50.0469 50.1719\n",
       "Q54.8906 44.3438 54.8906 33.0156\" id=\"BitstreamVeraSans-Roman-6e\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(185.347630178 85.6424639563)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"38.86328125\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"100.38671875\" xlink:href=\"#BitstreamVeraSans-Roman-70\"/>\n",
       "      <use x=\"163.86328125\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"202.7265625\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"264.25\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "      <use x=\"316.349609375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"377.873046875\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"441.251953125\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"480.4609375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"541.984375\" xlink:href=\"#BitstreamVeraSans-Roman-64\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_2\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- around -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M34.2812 27.4844\n",
       "Q23.3906 27.4844 19.1875 25\n",
       "Q14.9844 22.5156 14.9844 16.5\n",
       "Q14.9844 11.7188 18.1406 8.90625\n",
       "Q21.2969 6.10938 26.7031 6.10938\n",
       "Q34.1875 6.10938 38.7031 11.4062\n",
       "Q43.2188 16.7031 43.2188 25.4844\n",
       "L43.2188 27.4844\n",
       "z\n",
       "\n",
       "M52.2031 31.2031\n",
       "L52.2031 0\n",
       "L43.2188 0\n",
       "L43.2188 8.29688\n",
       "Q40.1406 3.32812 35.5469 0.953125\n",
       "Q30.9531 -1.42188 24.3125 -1.42188\n",
       "Q15.9219 -1.42188 10.9531 3.29688\n",
       "Q6 8.01562 6 15.9219\n",
       "Q6 25.1406 12.1719 29.8281\n",
       "Q18.3594 34.5156 30.6094 34.5156\n",
       "L43.2188 34.5156\n",
       "L43.2188 35.4062\n",
       "Q43.2188 41.6094 39.1406 45\n",
       "Q35.0625 48.3906 27.6875 48.3906\n",
       "Q23 48.3906 18.5469 47.2656\n",
       "Q14.1094 46.1406 10.0156 43.8906\n",
       "L10.0156 52.2031\n",
       "Q14.9375 54.1094 19.5781 55.0469\n",
       "Q24.2188 56 28.6094 56\n",
       "Q40.4844 56 46.3438 49.8438\n",
       "Q52.2031 43.7031 52.2031 31.2031\" id=\"BitstreamVeraSans-Roman-61\"/>\n",
       "      <path d=\"\n",
       "M30.6094 48.3906\n",
       "Q23.3906 48.3906 19.1875 42.75\n",
       "Q14.9844 37.1094 14.9844 27.2969\n",
       "Q14.9844 17.4844 19.1562 11.8438\n",
       "Q23.3438 6.20312 30.6094 6.20312\n",
       "Q37.7969 6.20312 41.9844 11.8594\n",
       "Q46.1875 17.5312 46.1875 27.2969\n",
       "Q46.1875 37.0156 41.9844 42.7031\n",
       "Q37.7969 48.3906 30.6094 48.3906\n",
       "M30.6094 56\n",
       "Q42.3281 56 49.0156 48.375\n",
       "Q55.7188 40.7656 55.7188 27.2969\n",
       "Q55.7188 13.875 49.0156 6.21875\n",
       "Q42.3281 -1.42188 30.6094 -1.42188\n",
       "Q18.8438 -1.42188 12.1719 6.21875\n",
       "Q5.51562 13.875 5.51562 27.2969\n",
       "Q5.51562 40.7656 12.1719 48.375\n",
       "Q18.8438 56 30.6094 56\" id=\"BitstreamVeraSans-Roman-6f\"/>\n",
       "      <path d=\"\n",
       "M8.5 21.5781\n",
       "L8.5 54.6875\n",
       "L17.4844 54.6875\n",
       "L17.4844 21.9219\n",
       "Q17.4844 14.1562 20.5 10.2656\n",
       "Q23.5312 6.39062 29.5938 6.39062\n",
       "Q36.8594 6.39062 41.0781 11.0312\n",
       "Q45.3125 15.6719 45.3125 23.6875\n",
       "L45.3125 54.6875\n",
       "L54.2969 54.6875\n",
       "L54.2969 0\n",
       "L45.3125 0\n",
       "L45.3125 8.40625\n",
       "Q42.0469 3.42188 37.7188 1\n",
       "Q33.4062 -1.42188 27.6875 -1.42188\n",
       "Q18.2656 -1.42188 13.375 4.4375\n",
       "Q8.5 10.2969 8.5 21.5781\" id=\"BitstreamVeraSans-Roman-75\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(162.220404046 250.207516992)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"61.279296875\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"100.142578125\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"161.32421875\" xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"224.703125\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"288.08203125\" xlink:href=\"#BitstreamVeraSans-Roman-64\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_3\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- and -->\n",
       "     <g transform=\"translate(373.983229486 133.843825312)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"61.279296875\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"124.658203125\" xlink:href=\"#BitstreamVeraSans-Roman-64\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_4\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- of -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M37.1094 75.9844\n",
       "L37.1094 68.5\n",
       "L28.5156 68.5\n",
       "Q23.6875 68.5 21.7969 66.5469\n",
       "Q19.9219 64.5938 19.9219 59.5156\n",
       "L19.9219 54.6875\n",
       "L34.7188 54.6875\n",
       "L34.7188 47.7031\n",
       "L19.9219 47.7031\n",
       "L19.9219 0\n",
       "L10.8906 0\n",
       "L10.8906 47.7031\n",
       "L2.29688 47.7031\n",
       "L2.29688 54.6875\n",
       "L10.8906 54.6875\n",
       "L10.8906 58.5\n",
       "Q10.8906 67.625 15.1406 71.7969\n",
       "Q19.3906 75.9844 28.6094 75.9844\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-66\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(354.892320326 216.424298358)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"61.181640625\" xlink:href=\"#BitstreamVeraSans-Roman-66\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_5\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- you -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M32.1719 -5.07812\n",
       "Q28.375 -14.8438 24.75 -17.8125\n",
       "Q21.1406 -20.7969 15.0938 -20.7969\n",
       "L7.90625 -20.7969\n",
       "L7.90625 -13.2812\n",
       "L13.1875 -13.2812\n",
       "Q16.8906 -13.2812 18.9375 -11.5156\n",
       "Q21 -9.76562 23.4844 -3.21875\n",
       "L25.0938 0.875\n",
       "L2.98438 54.6875\n",
       "L12.5 54.6875\n",
       "L29.5938 11.9219\n",
       "L46.6875 54.6875\n",
       "L56.2031 54.6875\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-79\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(235.766797571 137.197479471)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-79\"/>\n",
       "      <use x=\"59.1796875\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"120.361328125\" xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_6\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- graph -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M45.4062 27.9844\n",
       "Q45.4062 37.75 41.375 43.1094\n",
       "Q37.3594 48.4844 30.0781 48.4844\n",
       "Q22.8594 48.4844 18.8281 43.1094\n",
       "Q14.7969 37.75 14.7969 27.9844\n",
       "Q14.7969 18.2656 18.8281 12.8906\n",
       "Q22.8594 7.51562 30.0781 7.51562\n",
       "Q37.3594 7.51562 41.375 12.8906\n",
       "Q45.4062 18.2656 45.4062 27.9844\n",
       "M54.3906 6.78125\n",
       "Q54.3906 -7.17188 48.1875 -13.9844\n",
       "Q42 -20.7969 29.2031 -20.7969\n",
       "Q24.4688 -20.7969 20.2656 -20.0938\n",
       "Q16.0625 -19.3906 12.1094 -17.9219\n",
       "L12.1094 -9.1875\n",
       "Q16.0625 -11.3281 19.9219 -12.3438\n",
       "Q23.7812 -13.375 27.7812 -13.375\n",
       "Q36.625 -13.375 41.0156 -8.76562\n",
       "Q45.4062 -4.15625 45.4062 5.17188\n",
       "L45.4062 9.625\n",
       "Q42.625 4.78125 38.2812 2.39062\n",
       "Q33.9375 0 27.875 0\n",
       "Q17.8281 0 11.6719 7.65625\n",
       "Q5.51562 15.3281 5.51562 27.9844\n",
       "Q5.51562 40.6719 11.6719 48.3281\n",
       "Q17.8281 56 27.875 56\n",
       "Q33.9375 56 38.2812 53.6094\n",
       "Q42.625 51.2188 45.4062 46.3906\n",
       "L45.4062 54.6875\n",
       "L54.3906 54.6875\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-67\"/>\n",
       "      <path d=\"\n",
       "M54.8906 33.0156\n",
       "L54.8906 0\n",
       "L45.9062 0\n",
       "L45.9062 32.7188\n",
       "Q45.9062 40.4844 42.875 44.3281\n",
       "Q39.8438 48.1875 33.7969 48.1875\n",
       "Q26.5156 48.1875 22.3125 43.5469\n",
       "Q18.1094 38.9219 18.1094 30.9062\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "L9.07812 75.9844\n",
       "L18.1094 75.9844\n",
       "L18.1094 46.1875\n",
       "Q21.3438 51.125 25.7031 53.5625\n",
       "Q30.0781 56 35.7969 56\n",
       "Q45.2188 56 50.0469 50.1719\n",
       "Q54.8906 44.3438 54.8906 33.0156\" id=\"BitstreamVeraSans-Roman-68\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(174.781593966 147.044187584)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-67\"/>\n",
       "      <use x=\"63.4765625\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"104.58984375\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"165.869140625\" xlink:href=\"#BitstreamVeraSans-Roman-70\"/>\n",
       "      <use x=\"229.345703125\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_7\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- If -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M9.8125 72.9062\n",
       "L19.6719 72.9062\n",
       "L19.6719 0\n",
       "L9.8125 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-49\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(314.325027578 173.538470505)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-49\"/>\n",
       "      <use x=\"29.4921875\" xlink:href=\"#BitstreamVeraSans-Roman-66\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_8\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- nature -->\n",
       "     <g transform=\"translate(235.83457738 262.207455357)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"63.37890625\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"124.658203125\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"163.8671875\" xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"227.24609375\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"266.109375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_9\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- through -->\n",
       "     <g transform=\"translate(319.92584696 112.055661131)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"39.208984375\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "      <use x=\"102.587890625\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"141.451171875\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"202.6328125\" xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"266.01171875\" xlink:href=\"#BitstreamVeraSans-Roman-67\"/>\n",
       "      <use x=\"329.48828125\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_10\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- my -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M52 44.1875\n",
       "Q55.375 50.25 60.0625 53.125\n",
       "Q64.75 56 71.0938 56\n",
       "Q79.6406 56 84.2812 50.0156\n",
       "Q88.9219 44.0469 88.9219 33.0156\n",
       "L88.9219 0\n",
       "L79.8906 0\n",
       "L79.8906 32.7188\n",
       "Q79.8906 40.5781 77.0938 44.375\n",
       "Q74.3125 48.1875 68.6094 48.1875\n",
       "Q61.625 48.1875 57.5625 43.5469\n",
       "Q53.5156 38.9219 53.5156 30.9062\n",
       "L53.5156 0\n",
       "L44.4844 0\n",
       "L44.4844 32.7188\n",
       "Q44.4844 40.625 41.7031 44.4062\n",
       "Q38.9219 48.1875 33.1094 48.1875\n",
       "Q26.2188 48.1875 22.1562 43.5312\n",
       "Q18.1094 38.875 18.1094 30.9062\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "L9.07812 54.6875\n",
       "L18.1094 54.6875\n",
       "L18.1094 46.1875\n",
       "Q21.1875 51.2188 25.4844 53.6094\n",
       "Q29.7812 56 35.6875 56\n",
       "Q41.6562 56 45.8281 52.9688\n",
       "Q50 49.9531 52 44.1875\" id=\"BitstreamVeraSans-Roman-6d\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(196.026678427 56.4931696429)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-6d\"/>\n",
       "      <use x=\"97.412109375\" xlink:href=\"#BitstreamVeraSans-Roman-79\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_11\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- Restate -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M44.3906 34.1875\n",
       "Q47.5625 33.1094 50.5625 29.5938\n",
       "Q53.5625 26.0781 56.5938 19.9219\n",
       "L66.6094 0\n",
       "L56 0\n",
       "L46.6875 18.7031\n",
       "Q43.0625 26.0312 39.6719 28.4219\n",
       "Q36.2812 30.8125 30.4219 30.8125\n",
       "L19.6719 30.8125\n",
       "L19.6719 0\n",
       "L9.8125 0\n",
       "L9.8125 72.9062\n",
       "L32.0781 72.9062\n",
       "Q44.5781 72.9062 50.7344 67.6719\n",
       "Q56.8906 62.4531 56.8906 51.9062\n",
       "Q56.8906 45.0156 53.6875 40.4688\n",
       "Q50.4844 35.9375 44.3906 34.1875\n",
       "M19.6719 64.7969\n",
       "L19.6719 38.9219\n",
       "L32.0781 38.9219\n",
       "Q39.2031 38.9219 42.8438 42.2188\n",
       "Q46.4844 45.5156 46.4844 51.9062\n",
       "Q46.4844 58.2969 42.8438 61.5469\n",
       "Q39.2031 64.7969 32.0781 64.7969\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-52\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(137.583813873 66.7136896196)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-52\"/>\n",
       "      <use x=\"64.982421875\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"126.505859375\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "      <use x=\"178.60546875\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"217.814453125\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"279.09375\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"318.302734375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_12\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- numbers -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M48.6875 27.2969\n",
       "Q48.6875 37.2031 44.6094 42.8438\n",
       "Q40.5312 48.4844 33.4062 48.4844\n",
       "Q26.2656 48.4844 22.1875 42.8438\n",
       "Q18.1094 37.2031 18.1094 27.2969\n",
       "Q18.1094 17.3906 22.1875 11.75\n",
       "Q26.2656 6.10938 33.4062 6.10938\n",
       "Q40.5312 6.10938 44.6094 11.75\n",
       "Q48.6875 17.3906 48.6875 27.2969\n",
       "M18.1094 46.3906\n",
       "Q20.9531 51.2656 25.2656 53.625\n",
       "Q29.5938 56 35.5938 56\n",
       "Q45.5625 56 51.7812 48.0938\n",
       "Q58.0156 40.1875 58.0156 27.2969\n",
       "Q58.0156 14.4062 51.7812 6.48438\n",
       "Q45.5625 -1.42188 35.5938 -1.42188\n",
       "Q29.5938 -1.42188 25.2656 0.953125\n",
       "Q20.9531 3.32812 18.1094 8.20312\n",
       "L18.1094 0\n",
       "L9.07812 0\n",
       "L9.07812 75.9844\n",
       "L18.1094 75.9844\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-62\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(277.751301567 202.445171264)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"63.37890625\" xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"126.7578125\" xlink:href=\"#BitstreamVeraSans-Roman-6d\"/>\n",
       "      <use x=\"224.169921875\" xlink:href=\"#BitstreamVeraSans-Roman-62\"/>\n",
       "      <use x=\"287.646484375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"349.169921875\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"390.283203125\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_13\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- in -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M9.42188 54.6875\n",
       "L18.4062 54.6875\n",
       "L18.4062 0\n",
       "L9.42188 0\n",
       "z\n",
       "\n",
       "M9.42188 75.9844\n",
       "L18.4062 75.9844\n",
       "L18.4062 64.5938\n",
       "L9.42188 64.5938\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-69\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(122.949632149 223.106496457)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-69\"/>\n",
       "      <use x=\"27.783203125\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_14\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- emerge -->\n",
       "     <g transform=\"translate(222.54491727 216.506380134)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"61.5234375\" xlink:href=\"#BitstreamVeraSans-Roman-6d\"/>\n",
       "      <use x=\"158.935546875\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"220.458984375\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"259.822265625\" xlink:href=\"#BitstreamVeraSans-Roman-67\"/>\n",
       "      <use x=\"323.298828125\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_15\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- can -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M48.7812 52.5938\n",
       "L48.7812 44.1875\n",
       "Q44.9688 46.2969 41.1406 47.3438\n",
       "Q37.3125 48.3906 33.4062 48.3906\n",
       "Q24.6562 48.3906 19.8125 42.8438\n",
       "Q14.9844 37.3125 14.9844 27.2969\n",
       "Q14.9844 17.2812 19.8125 11.7344\n",
       "Q24.6562 6.20312 33.4062 6.20312\n",
       "Q37.3125 6.20312 41.1406 7.25\n",
       "Q44.9688 8.29688 48.7812 10.4062\n",
       "L48.7812 2.09375\n",
       "Q45.0156 0.34375 40.9844 -0.53125\n",
       "Q36.9688 -1.42188 32.4219 -1.42188\n",
       "Q20.0625 -1.42188 12.7812 6.34375\n",
       "Q5.51562 14.1094 5.51562 27.2969\n",
       "Q5.51562 40.6719 12.8594 48.3281\n",
       "Q20.2188 56 33.0156 56\n",
       "Q37.1562 56 41.1094 55.1406\n",
       "Q45.0625 54.2969 48.7812 52.5938\" id=\"BitstreamVeraSans-Roman-63\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(93.1042160233 127.007223457)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-63\"/>\n",
       "      <use x=\"54.98046875\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"116.259765625\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_16\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- Everything -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M2.98438 54.6875\n",
       "L12.5 54.6875\n",
       "L29.5938 8.79688\n",
       "L46.6875 54.6875\n",
       "L56.2031 54.6875\n",
       "L35.6875 0\n",
       "L23.4844 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-76\"/>\n",
       "      <path d=\"\n",
       "M9.8125 72.9062\n",
       "L55.9062 72.9062\n",
       "L55.9062 64.5938\n",
       "L19.6719 64.5938\n",
       "L19.6719 43.0156\n",
       "L54.3906 43.0156\n",
       "L54.3906 34.7188\n",
       "L19.6719 34.7188\n",
       "L19.6719 8.29688\n",
       "L56.7812 8.29688\n",
       "L56.7812 0\n",
       "L9.8125 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-45\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(266.072698167 252.204516253)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-45\"/>\n",
       "      <use x=\"63.18359375\" xlink:href=\"#BitstreamVeraSans-Roman-76\"/>\n",
       "      <use x=\"122.36328125\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"183.88671875\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"225.0\" xlink:href=\"#BitstreamVeraSans-Roman-79\"/>\n",
       "      <use x=\"284.1796875\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"323.388671875\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "      <use x=\"386.767578125\" xlink:href=\"#BitstreamVeraSans-Roman-69\"/>\n",
       "      <use x=\"414.55078125\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"477.9296875\" xlink:href=\"#BitstreamVeraSans-Roman-67\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_17\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- patterns -->\n",
       "     <g transform=\"translate(201.422804405 247.205858512)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-70\"/>\n",
       "      <use x=\"63.4765625\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"124.755859375\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"163.96484375\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"203.173828125\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"264.697265625\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"304.060546875\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"367.439453125\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_18\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- are -->\n",
       "     <g transform=\"translate(108.960919592 232.88193123)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"61.279296875\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"100.142578125\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_19\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- Therefore -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M-0.296875 72.9062\n",
       "L61.375 72.9062\n",
       "L61.375 64.5938\n",
       "L35.5 64.5938\n",
       "L35.5 0\n",
       "L25.5938 0\n",
       "L25.5938 64.5938\n",
       "L-0.296875 64.5938\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-54\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(128.17805529 161.167099519)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-54\"/>\n",
       "      <use x=\"61.083984375\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "      <use x=\"124.462890625\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"185.986328125\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"224.849609375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"286.373046875\" xlink:href=\"#BitstreamVeraSans-Roman-66\"/>\n",
       "      <use x=\"321.578125\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"382.759765625\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"421.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_20\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- language -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M9.42188 75.9844\n",
       "L18.4062 75.9844\n",
       "L18.4062 0\n",
       "L9.42188 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-6c\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(365.365157951 174.499168995)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-6c\"/>\n",
       "      <use x=\"27.783203125\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"89.0625\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"152.44140625\" xlink:href=\"#BitstreamVeraSans-Roman-67\"/>\n",
       "      <use x=\"215.91796875\" xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"279.296875\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"340.576171875\" xlink:href=\"#BitstreamVeraSans-Roman-67\"/>\n",
       "      <use x=\"404.052734375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_21\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- be -->\n",
       "     <g transform=\"translate(141.491698075 96.6791393655)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-62\"/>\n",
       "      <use x=\"63.4765625\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_22\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- There -->\n",
       "     <g transform=\"translate(102.530803585 163.249086629)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-54\"/>\n",
       "      <use x=\"61.083984375\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "      <use x=\"124.462890625\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"185.986328125\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"224.849609375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_23\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- us -->\n",
       "     <g transform=\"translate(89.0105357143 194.746174139)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"63.37890625\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_24\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- everywhere -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M4.20312 54.6875\n",
       "L13.1875 54.6875\n",
       "L24.4219 12.0156\n",
       "L35.5938 54.6875\n",
       "L46.1875 54.6875\n",
       "L57.4219 12.0156\n",
       "L68.6094 54.6875\n",
       "L77.5938 54.6875\n",
       "L63.2812 0\n",
       "L52.6875 0\n",
       "L40.9219 44.8281\n",
       "L29.1094 0\n",
       "L18.5 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-77\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(76.0412321819 214.455856322)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"61.5234375\" xlink:href=\"#BitstreamVeraSans-Roman-76\"/>\n",
       "      <use x=\"120.703125\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"182.2265625\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"223.33984375\" xlink:href=\"#BitstreamVeraSans-Roman-79\"/>\n",
       "      <use x=\"282.51953125\" xlink:href=\"#BitstreamVeraSans-Roman-77\"/>\n",
       "      <use x=\"364.306640625\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "      <use x=\"427.685546875\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"489.208984375\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"528.072265625\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_25\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- Mathematics -->\n",
       "     <defs>\n",
       "      <path d=\"\n",
       "M9.8125 72.9062\n",
       "L24.5156 72.9062\n",
       "L43.1094 23.2969\n",
       "L61.8125 72.9062\n",
       "L76.5156 72.9062\n",
       "L76.5156 0\n",
       "L66.8906 0\n",
       "L66.8906 64.0156\n",
       "L48.0938 14.0156\n",
       "L38.1875 14.0156\n",
       "L19.3906 64.0156\n",
       "L19.3906 0\n",
       "L9.8125 0\n",
       "z\n",
       "\" id=\"BitstreamVeraSans-Roman-4d\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(222.751589817 56.7876242865)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-4d\"/>\n",
       "      <use x=\"86.279296875\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"147.55859375\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"186.767578125\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "      <use x=\"250.146484375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"311.669921875\" xlink:href=\"#BitstreamVeraSans-Roman-6d\"/>\n",
       "      <use x=\"409.08203125\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"470.361328125\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"509.5703125\" xlink:href=\"#BitstreamVeraSans-Roman-69\"/>\n",
       "      <use x=\"537.353515625\" xlink:href=\"#BitstreamVeraSans-Roman-63\"/>\n",
       "      <use x=\"592.333984375\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_26\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- assumptions -->\n",
       "     <g transform=\"translate(251.074003534 59.7081647594)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"61.279296875\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "      <use x=\"113.37890625\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "      <use x=\"165.478515625\" xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"228.857421875\" xlink:href=\"#BitstreamVeraSans-Roman-6d\"/>\n",
       "      <use x=\"326.26953125\" xlink:href=\"#BitstreamVeraSans-Roman-70\"/>\n",
       "      <use x=\"389.74609375\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"428.955078125\" xlink:href=\"#BitstreamVeraSans-Roman-69\"/>\n",
       "      <use x=\"456.73828125\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"517.919921875\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"581.298828125\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_27\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- is -->\n",
       "     <g transform=\"translate(325.373061937 74.4962882925)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-69\"/>\n",
       "      <use x=\"27.783203125\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_28\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- these -->\n",
       "     <g transform=\"translate(179.550277319 254.852279648)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"39.208984375\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "      <use x=\"102.587890625\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"164.111328125\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "      <use x=\"216.2109375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_29\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- the -->\n",
       "     <g transform=\"translate(359.410348053 107.656832852)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"39.208984375\" xlink:href=\"#BitstreamVeraSans-Roman-68\"/>\n",
       "      <use x=\"102.587890625\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_30\">\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <!-- understood -->\n",
       "     <g transform=\"translate(345.240718366 127.606065364)scale(0.12 -0.12)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-75\"/>\n",
       "      <use x=\"63.37890625\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"126.7578125\" xlink:href=\"#BitstreamVeraSans-Roman-64\"/>\n",
       "      <use x=\"190.234375\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"251.7578125\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"292.87109375\" xlink:href=\"#BitstreamVeraSans-Roman-73\"/>\n",
       "      <use x=\"344.970703125\" xlink:href=\"#BitstreamVeraSans-Roman-74\"/>\n",
       "      <use x=\"384.1796875\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"445.361328125\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"506.54296875\" xlink:href=\"#BitstreamVeraSans-Roman-64\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pb36153c835\">\n",
       "   <rect height=\"288.0\" width=\"432.0\" x=\"33.421875\" y=\"12.0390625\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2b68565cbac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "nx.draw(text_digraph, with_labels = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['Everything',\n",
       " 'around',\n",
       " 'us',\n",
       " 'can',\n",
       " 'be',\n",
       " 'represented',\n",
       " 'and',\n",
       " 'understood',\n",
       " 'through',\n",
       " 'numbers']"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.shortest_path(text_digraph, \"Everything\", \"numbers\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Baum (Tree)\n",
    "\n",
    "Ein Baum startet bei einem Wurzelknoten und verästelt sich ohne Schleifen zu bilden.\n",
    "Knoten ohne Kinder werden Blätter genannt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Created with matplotlib (http://matplotlib.org/) -->\n",
       "<svg height=\"320pt\" version=\"1.1\" viewBox=\"0 0 479 320\" width=\"479pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"\n",
       "M0 320.917\n",
       "L479.523 320.917\n",
       "L479.523 0\n",
       "L0 0\n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"LineCollection_1\">\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M185.323 205.23\n",
       "L176.198 128.899\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M185.323 205.23\n",
       "L272.827 258.478\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M176.198 128.899\n",
       "L238.336 59.3781\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M176.198 128.899\n",
       "L172.664 87.8281\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M272.827 258.478\n",
       "L226.746 258.896\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M272.827 258.478\n",
       "L258.863 210.646\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M238.336 59.3781\n",
       "L309.272 65.4999\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M238.336 59.3781\n",
       "L182.31 53.1819\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M172.664 87.8281\n",
       "L95.1362 96.4559\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M172.664 87.8281\n",
       "L299.251 97.4831\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M226.746 258.896\n",
       "L135.492 244.291\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M226.746 258.896\n",
       "L333.482 232.49\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M258.863 210.646\n",
       "L377.315 178.559\" style=\"fill:none;stroke:#000000;\"/>\n",
       "    <path clip-path=\"url(#pb36153c835)\" d=\"\n",
       "M258.863 210.646\n",
       "L99.4658 210.008\" style=\"fill:none;stroke:#000000;\"/>\n",
       "   </g>\n",
       "   <g id=\"PathCollection_1\">\n",
       "    <defs>\n",
       "     <path d=\"\n",
       "M0 8.66025\n",
       "C2.29673 8.66025 4.49969 7.74776 6.12372 6.12372\n",
       "C7.74776 4.49969 8.66025 2.29673 8.66025 0\n",
       "C8.66025 -2.29673 7.74776 -4.49969 6.12372 -6.12372\n",
       "C4.49969 -7.74776 2.29673 -8.66025 0 -8.66025\n",
       "C-2.29673 -8.66025 -4.49969 -7.74776 -6.12372 -6.12372\n",
       "C-7.74776 -4.49969 -8.66025 -2.29673 -8.66025 0\n",
       "C-8.66025 2.29673 -7.74776 4.49969 -6.12372 6.12372\n",
       "C-4.49969 7.74776 -2.29673 8.66025 0 8.66025\n",
       "z\n",
       "\" id=\"mf9612d42da\" style=\"stroke:#000000;\"/>\n",
       "    </defs>\n",
       "    <g clip-path=\"url(#pb36153c835)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"185.32302973\" xlink:href=\"#mf9612d42da\" y=\"205.230437462\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"176.197513778\" xlink:href=\"#mf9612d42da\" y=\"128.899220821\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"272.827235854\" xlink:href=\"#mf9612d42da\" y=\"258.478198598\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"238.335508106\" xlink:href=\"#mf9612d42da\" y=\"59.3781112151\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"172.663701325\" xlink:href=\"#mf9612d42da\" y=\"87.8280767834\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"226.746188305\" xlink:href=\"#mf9612d42da\" y=\"258.896205357\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"258.863198637\" xlink:href=\"#mf9612d42da\" y=\"210.646304785\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"182.309654477\" xlink:href=\"#mf9612d42da\" y=\"53.1819196429\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"309.271900527\" xlink:href=\"#mf9612d42da\" y=\"65.4998829394\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"299.250962421\" xlink:href=\"#mf9612d42da\" y=\"97.4831107882\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"95.1361607143\" xlink:href=\"#mf9612d42da\" y=\"96.4559484708\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"135.491880386\" xlink:href=\"#mf9612d42da\" y=\"244.291234769\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"333.481755575\" xlink:href=\"#mf9612d42da\" y=\"232.489824003\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"377.31526789\" xlink:href=\"#mf9612d42da\" y=\"178.559367951\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;\" x=\"99.4657992339\" xlink:href=\"#mf9612d42da\" y=\"210.007834143\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pb36153c835\">\n",
       "   <rect height=\"288.0\" width=\"432.0\" x=\"33.421875\" y=\"12.0390625\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2b6856a2af60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rtree = nx.balanced_tree(2, 3)\n",
    "%matplotlib inline\n",
    "nx.draw(rtree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.is_tree(rtree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Liste der Blätter-Knoten (Grad ist 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[7, 8, 9, 10, 11, 12, 13, 14]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ k for k, v in nx.degree(rtree).items() if v == 1]"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}