3. Hubs and Paths (Instructor).ipynb

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   "source": [
    "import networkx as nx\n",
    "import matplotlib.pyplot as plt\n",
    "from collections import Counter\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "# Load the pickled data without the new individuals added in the previous notebook.\n",
    "\n",
    "G = nx.read_gpickle('Synthetic Social Network.pkl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Hubs: How do we evaluate the importance of some individuals in a network?\n",
    "\n",
    "Within a social network, there will be certain individuals which perform certain important functions. For example, there may be hyper-connected individuals who are connected to many, many more people. They would be of use in the spreading of information. Alternatively, if this were a disease contact network, identifying them would be useful in stopping the spread of diseases. How would one identify these people?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "## Approach 1: Neighbors\n",
    "\n",
    "One way we could compute this is to find out the number of people an individual is conencted to. NetworkX let's us do this by giving us a `G.neighbors(node)` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4, 2, 3, 12, 29]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Let's find out the number of neighbors that individual #7 has.\n",
    "G.neighbors(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Can you create a ranked list of the importance of each individual, based on the number of neighbors they have? \n",
    "\n",
    "Hint: One suggested output would be a list of tuples, where the first element in each tuple is the node ID (an integer number), and the second element is a list of its neighbors.\n",
    "\n",
    "Hint: Python's `sorted(iterable, key=lambda x:...., reverse=True)` function may be of help here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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    "scrolled": true,
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(19, [0, 16, 2, 4, 22, 17, 27, 12]),\n",
       " (1, [4, 2, 3, 12, 29]),\n",
       " (2, [16, 1, 3, 6, 19]),\n",
       " (3, [8, 1, 2, 6, 23]),\n",
       " (17, [8, 9, 19, 14, 16]),\n",
       " (12, [0, 1, 19, 29]),\n",
       " (14, [17, 11, 13, 25]),\n",
       " (16, [17, 2, 19, 13]),\n",
       " (24, [9, 7, 13, 15]),\n",
       " (0, [10, 19, 12]),\n",
       " (4, [1, 19, 28]),\n",
       " (6, [2, 3, 23]),\n",
       " (8, [17, 3, 22]),\n",
       " (9, [24, 17, 11]),\n",
       " (10, [0, 11, 21]),\n",
       " (11, [9, 10, 14]),\n",
       " (13, [16, 24, 14]),\n",
       " (21, [10, 27, 26]),\n",
       " (23, [3, 20, 6]),\n",
       " (25, [28, 14, 7]),\n",
       " (27, [19, 20, 21]),\n",
       " (28, [25, 4, 15]),\n",
       " (29, [1, 26, 12]),\n",
       " (7, [24, 25]),\n",
       " (15, [24, 28]),\n",
       " (20, [27, 23]),\n",
       " (22, [8, 19]),\n",
       " (26, [21, 29]),\n",
       " (5, []),\n",
       " (18, [])]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Possible Answers:\n",
    "sorted([n for n in G.nodes()], key=lambda x:len(G.neighbors(x)), reverse=True)\n",
    "sorted([(n, G.neighbors(n)) for n in G.nodes()], key=lambda x: len(x[1]), reverse=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Approach 2: Degree Centrality\n",
    "\n",
    "The number of other nodes that one node is connected to is a measure of its centrality. NetworkX implements a **degree centrality**, which is defined as the number of neighbors that a node has normalized to the number of individuals it could be connected to in the entire graph. This is accessed by using `nx.degree_centrality(G)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 0.10344827586206896,\n",
       " 1: 0.1724137931034483,\n",
       " 2: 0.1724137931034483,\n",
       " 3: 0.1724137931034483,\n",
       " 4: 0.10344827586206896,\n",
       " 5: 0.0,\n",
       " 6: 0.10344827586206896,\n",
       " 7: 0.06896551724137931,\n",
       " 8: 0.10344827586206896,\n",
       " 9: 0.10344827586206896,\n",
       " 10: 0.10344827586206896,\n",
       " 11: 0.10344827586206896,\n",
       " 12: 0.13793103448275862,\n",
       " 13: 0.10344827586206896,\n",
       " 14: 0.13793103448275862,\n",
       " 15: 0.06896551724137931,\n",
       " 16: 0.13793103448275862,\n",
       " 17: 0.1724137931034483,\n",
       " 18: 0.0,\n",
       " 19: 0.27586206896551724,\n",
       " 20: 0.06896551724137931,\n",
       " 21: 0.10344827586206896,\n",
       " 22: 0.06896551724137931,\n",
       " 23: 0.10344827586206896,\n",
       " 24: 0.13793103448275862,\n",
       " 25: 0.10344827586206896,\n",
       " 26: 0.06896551724137931,\n",
       " 27: 0.10344827586206896,\n",
       " 28: 0.10344827586206896,\n",
       " 29: 0.10344827586206896}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.degree_centrality(G) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "If you inspect the dictionary closely, you will find that node 19 is the one that has the highest degree centrality, just as we had measured by counting the number of neighbors.\n",
    "\n",
    "There are other measures of centrality, namely **betweenness centrality**, **flow centrality** and **load centrality**. You can take a look at their definitions on the NetworkX API docs and their cited references. You can also define your own measures if those don't fit your needs, but that is an advanced topic that won't be dealt with here.\n",
    "\n",
    "The NetworkX API docs that document the centrality measures are here: http://networkx.github.io/documentation/networkx-1.9.1/reference/algorithms.centrality.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercises\n",
    "\n",
    "1. Can you create a histogram of the distribution of degree centralities?\n",
    "2. Can you create a histogram of the distribution of number of neighbors?\n",
    "3. Can you create a scatterplot of the degree centralities against number of neighbors?\n",
    "4. If I have `n` nodes, then how many possible edges are there in total, assuming self-edges are allowed? What if self-edges are not allowed?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Hint: You may want to use:\n",
    "\n",
    "    plt.hist(list_of_values)\n",
    "\n",
    "and \n",
    "\n",
    "    plt.scatter(x_values, y_values)\n",
    "\n",
    "If you know the Matplotlib API, feel free to get fancy :)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "slideshow": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10bae7d68>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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UGINdkhpjsEtSY3oHe5Ljk7wvyceTXJ3kJbNsmCSpn1RVvx2To4Gjq+ojSQ4C/hF4RlXt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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109e78b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Possible Answers:\n",
    "fig = plt.figure(0)\n",
    "degree_centralities = [v for k, v in nx.degree_centrality(G).items()]\n",
    "plt.hist(degree_centralities)\n",
    "plt.title('Degree Centralities')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10bfab6d8>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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sjO5gNkZ4szG6m41jMgs1wKwcE3W254hbkrSdGNyS1BiDW5IaY3BLUmMMbklqjMEtSY0x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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109e78f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(1)\n",
    "neighbors = [len(G.neighbors(node)) for node in G.nodes()]\n",
    "plt.hist(neighbors)\n",
    "plt.title('Number of Neighbors')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10bf8de48>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c130208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(2)\n",
    "plt.scatter(degree_centralities, neighbors)\n",
    "plt.xlabel('Degree Centralities')\n",
    "plt.ylabel('Number of Neighbors')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Paths in a Network\n",
    "\n",
    "Graph traversal is akin to walking along the graph, node by node, restricted by the edges that connect the nodes. Graph traversal is particularly useful for understanding the local structure (e.g. connectivity, retrieving the exact relationships) of certain portions of the graph and for finding paths that connect two nodes in the network. \n",
    "\n",
    "Using the synthetic social network, we will figure out how to answer the following questions:\n",
    "\n",
    "1. How long will it take for a message to spread through this group of friends? (making some assumptions, of course)\n",
    "2. How do we find the shortest path to get from individual A to individual B?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Shortest Path"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cLs7zeino25dfNDczhUP/8D7pfOYx2GnwNc7PF4ArsNPRnwC3A3/DBvxZ2Gnr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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c1381d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw(G, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let's say we wanted to find the shortest path between two nodes. How would we approach this? One approach is what one would call a **breadth-first search** (http://en.wikipedia.org/wiki/Breadth-first_search). While not necessarily the fastest, it is the easiest to conceptualize. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The approach is essentially as such:\n",
    "\n",
    "1. Begin with a queue of the starting node.\n",
    "2. Add the neighbors of that node to the queue.\n",
    "    1. If destination node is present in the queue, end.\n",
    "    2. If destination node is not present, proceed.\n",
    "3. For each node in the queue:\n",
    "    1. Remove node from the queue.\n",
    "    2. Add neighbors of the node to the queue. Check if destination node is present or not.\n",
    "    3. If destination node is present, end. Credit: @cavaunpeu for finding bug in pseudocode.\n",
    "    4. If destination node is not present, continue."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Try implementing this algorithm in a function called `path_exists(node1, node2, G)`.\n",
    "\n",
    "The function should take in two nodes, `node1` and `node2`, and the graph `G` that they belong to, and return a Boolean that indicates whether a path exists between those two nodes or not. For convenience, also print out whether a path exists or not between the two node.s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def path_exists(node1, node2, G):\n",
    "    \"\"\"\n",
    "    This function checks whether a path exists between two nodes (node1, node2) in graph G.\n",
    "    \"\"\"\n",
    "    queue = [node1]\n",
    "    destination = node2\n",
    "    \n",
    "    for node in queue:\n",
    "        neighbors = G.neighbors(node)\n",
    "        if destination in neighbors:\n",
    "            print('Path exists between nodes {0} and {1}'.format(node1, node2))\n",
    "            return True\n",
    "            break\n",
    "        else:\n",
    "            queue.remove(node)\n",
    "            queue.extend(neighbors)\n",
    "        if len(queue) == 0:\n",
    "            print('Path does not exist between nodes {0} and {1}'.format(node1, node2))\n",
    "            return False\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "And testing the function on a few test cases:\n",
    "\n",
    "- 18 and any other node (should return `False`)\n",
    "- 29 and 26 (should return `True`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Path does not exist between nodes 18 and 5\n",
      "Path exists between nodes 29 and 26\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "path_exists(18, 5, G)\n",
    "path_exists(29, 26, G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "If you write an algorithm that runs breadth-first, the recursion pattern is likely to follow what we have done above. If you do depth-first, the recursion pattern is likely to look a bit different. We can figure that out together later, if there is time.\n",
    "\n",
    "Meanwhile... thankfully, NetworkX has a function for us to use, titled `has_path`, so we don't have to implement this on our own. :-)\n",
    "\n",
    "http://networkx.lanl.gov/reference/generated/networkx.algorithms.shortest_paths.generic.has_path.html#networkx.algorithms.shortest_paths.generic.has_path"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.has_path(G, 18, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "NetworkX also has other shortest path algorithms implemented. \n",
    "\n",
    "https://networkx.github.io/documentation/latest/reference/generated/networkx.algorithms.shortest_paths.unweighted.predecessor.html#networkx.algorithms.shortest_paths.unweighted.predecessor\n",
    "\n",
    "We can build upon these to build our own graph query functions. Let's see if we can trace the shortest path from one node to another."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "`nx.shortest_path(G, source, target)` gives us a list of nodes that exist within one of the shortest paths between the two nodes. (Not all paths are guaranteed to be found.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4, 19, 17, 14]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.shortest_path(G, 4, 14)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Incidentally, the node list is in order as well - we will travel through 19 and 17 in that order to get from 14 from 4."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Write a function that extracts the edges in the shortest path between two nodes and puts them into a new graph, and draws it to the screen. It should also return an error if there is no path between the two nodes.\n",
    "\n",
    "Hint: You may want to use `G.subgraph(iterable_of_nodes)` to extract just the nodes and edges of interest from the graph `G`. You might want to use the following lines of code somewhere:\n",
    "\n",
    "    newG = G.subgraph(nodes_of_interest)\n",
    "    nx.draw(newG)\n",
    "    \n",
    "newG will be comprised of the nodes of interest and the edges that connect them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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MjKAj6Sh5DViSWoD09HSeeOIJevXqRb9+/XjnnXeCjqSjZAFLUgvRtm1bbrnl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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c21eb00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Possible Answer:\n",
    "\n",
    "def extract_path_edges(G, source, target):\n",
    "    # Check to make sure that a path does exists between source and target.\n",
    "    if nx.has_path(G, source, target) == True:\n",
    "        nodes = nx.shortest_path(G, source, target)\n",
    "        newG = G.subgraph(nodes)\n",
    "        return newG\n",
    "\n",
    "    if nx.has_path(G, source, target) == False:\n",
    "        raise Exception('Path does not exist between nodes {0} and {1}.'.format(source, target))\n",
    "        \n",
    "newG = extract_path_edges(G, 1, 14)\n",
    "nx.draw(newG, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Since we've been drawing some graphs to screen, we might as well draw a few other things while we're on a roll.\n",
    "\n",
    "Write a function that extracts only node, its neighbors, and the edges between that node and its neighbors as a new graph. Then, draw the new graph to screen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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HjhGe57lvAqf5JXxfBa4BM3J+PgfUBY4Cu4C/N2nCd4cPa/GUiIE0pyTihcqV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\nRwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c46b048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Possible Answer\n",
    "\n",
    "def extract_neighbor_edges(G, node):\n",
    "    neighbors = G.neighbors(node)\n",
    "    newG = nx.Graph()\n",
    "    \n",
    "    for n1, n2 in G.edges():\n",
    "        if (n1 == node and n2 in neighbors) or (n1 in neighbors and n2 == node):\n",
    "            newG.add_edge(n1, n2)\n",
    "            \n",
    "    return newG\n",
    "\n",
    "fig = plt.figure(0)\n",
    "newG = extract_neighbor_edges(G, 19)\n",
    "nx.draw(newG, with_labels=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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pBk6fPs3KlSvN9nSDBg2IiYnB29ub1PffZ2VeHouA2sAyoID/hm8y8Pi5P+8B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\nesZeAgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c6c0198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def extract_neighbor_edges2(G, node):\n",
    "    neighbors = G.neighbors(node)\n",
    "    newG = nx.Graph()\n",
    "    \n",
    "    for neighbor in neighbors:\n",
    "        if (node, neighbor) in G.edges() or (neighbor, node) in G.edges():\n",
    "            newG.add_edge(node, neighbor)\n",
    "\n",
    "    return newG\n",
    "\n",
    "fig = plt.figure(1)\n",
    "newG = extract_neighbor_edges2(G, 19)\n",
    "nx.draw(newG, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Challenge Exercises (optional)\n",
    "\n",
    "Let's try some other problems that build on the NetworkX API.\n",
    "\n",
    "Refer to the following for the relevant functions:\n",
    "\n",
    "https://networkx.github.io/documentation/latest/reference/algorithms.shortest_paths.html\n",
    "\n",
    "1. If we want a message to go from one person to another person, and we assume that the message takes 1 day for the initial step and 1 additional day per step in the transmission chain (i.e. the first step takes 1 day, the second step takes 2 days etc.), how long will the message take to spread from any two given individuals? Write a function to compute this.\n",
    "2. What is the distribution of message spread times from person to person? What about chain lengths?\n",
    "3. Are there certain individuals who consistently show up in the chain? (Hint: you might wish to use the following functions/objects:\n",
    "    1. `Counter` object from the `collections` module \n",
    "    2. `combinations` function from the `itertools` module.\n",
    "    3. `all_shortest_paths(G, node1, node2)` which is part of the networkX algorithms.\n",
    "4. As a bonus, if you were able to compute the answer to question 3, can you plot a histogram of the number of times each node shows up in a connecting path?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Possible answer to Question 1:\n",
    "# All we need here is the length of the path.\n",
    "\n",
    "def compute_transmission_time(G, source, target):\n",
    "    \"\"\"\n",
    "    Fill in code below.\n",
    "    \"\"\"\n",
    "    length = nx.shortest_path_length(G, source, target)\n",
    "    \n",
    "    time = sum([i for i in range(1, length+1)])\n",
    "    \n",
    "    return time\n",
    "\n",
    "compute_transmission_time(G, 14, 4)       "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 7 artists>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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D4E6GU6qzYAFYWPTO7naGIT9rLgYe6x7/d7SRof4ocGaS07tXnCuAuzdw/+9qGZ6udiOw\nv6qu67uecSU5Ocnmbvlngd9leFxg6lXVZ6tqe1WdwfDt8z9V1R/3Xdc4khyf5MRu+b3AhcBMfBKs\nqg4BLyQ5qxu6AHi6x5ImdRXDpmBZG/YjGbN+UlKSvcD5wC8keQH4XFXd3HNZ4zgP+AjwRJLHu7Hd\nVfWNHmsaxynAnu5TVMcAX6+qf+y5pknN4lTkFuDO7lT2TcAtVXVfvyWN5Rrglq6h/AHw0Z7rGUv3\nQnoBsOKxDE8+kqSG+BulktQQQ12SGmKoS1JDDHVJaoihLkkNMdQlqSGGuiQ1xFCXpIb8H1ENyxyx\nUCUeAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c6d62b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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3tIPvIEmuZjJrh8nFZl/fTX2R5ATwXuAtTNZUPw38C3ASeBtwBri1qn69XW1cljX64igw\nYvISvICngY9MrUG3leQ9wPeAxzi/DHOEydXwu2psXKAvPgXcxhxjwwudJKkp/yerJDVlwEtSUwa8\nJDVlwEtSUwa8JDVlwEtSUwa8JDVlwEtSU/8PK088kkI80T0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c6d2ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Possible answer to Question 2:\n",
    "# We need to know the length of every single shortest path between every pair of nodes.\n",
    "# If we don't put a source and target into the nx.shortest_path_length(G) function call, then\n",
    "# we get a dictionary of dictionaries, where all source-->target-->lengths are shown.\n",
    "\n",
    "lengths = []\n",
    "times = []\n",
    "for source, sink_length in nx.shortest_path_length(G).items():\n",
    "    for sink, length in sink_length.items():\n",
    "        times.append(sum(range(1, length+1)))\n",
    "        lengths.append(length)\n",
    "        \n",
    "plt.figure(0)\n",
    "plt.bar(Counter(lengths).keys(), Counter(lengths).values())\n",
    "\n",
    "plt.figure(1)\n",
    "plt.bar(Counter(times).keys(), Counter(times).values())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({0: 24,\n",
       "         1: 68,\n",
       "         2: 79,\n",
       "         3: 59,\n",
       "         4: 80,\n",
       "         6: 8,\n",
       "         7: 2,\n",
       "         8: 40,\n",
       "         9: 54,\n",
       "         10: 39,\n",
       "         11: 36,\n",
       "         12: 32,\n",
       "         13: 42,\n",
       "         14: 75,\n",
       "         15: 6,\n",
       "         16: 70,\n",
       "         17: 136,\n",
       "         19: 203,\n",
       "         20: 11,\n",
       "         21: 28,\n",
       "         22: 8,\n",
       "         23: 11,\n",
       "         24: 46,\n",
       "         25: 41,\n",
       "         26: 8,\n",
       "         27: 54,\n",
       "         28: 58,\n",
       "         29: 24})"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Possible answer to Question 3:\n",
    "from collections import Counter\n",
    "from itertools import combinations\n",
    "\n",
    "totals = Counter()\n",
    "for n1, n2 in combinations(G.nodes(), 2):\n",
    "    if nx.has_path(G, n1, n2):\n",
    "        for path in nx.all_shortest_paths(G, n1, n2):\n",
    "            path.pop(0)\n",
    "            path.pop(-1)\n",
    "            totals = totals + Counter(path)\n",
    "            \n",
    "totals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 28 artists>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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9WlVXApcAf5jk99e8vuH4bWfAPwvsHXq+l8EsvhlV9Vz33/8EvsWgLNWa1a7+SZKLgBfm\n3J+pqqoXqgN8iSUewyTnMgj3f6qqu7rmZsZv6Pi+eub4Whq/M6rqf4AHgD9ni+O3nQH/MPDOJJcm\nOQ94P3D3Nu5/ppK8Kclbu+U3A9cAT4zeaindDdzSLd8C3DVi3aXTvWnOeB9LOoYZfCnRYeBEVd0x\n9FIT43e242to/C48U15K8tvAnzE4z7Cl8dvW6+CTXAfcwes3QP39tu18xpJcxmDWDoMbyL627MeX\n5CjwXuBCBvW+TwH/AhwD3g6cAm6qql/Oq4+TWOf4DgE9Bh/vC3gG+OhQzXNpJHkP8D3gcV7/GH+Q\nwZ3lSz9+Zzm+24CbaWP8rmBwEvWNDCbiX6+qv0tyAVsYP290kqRG+Y9uS1KjDHhJapQBL0mNMuAl\nqVEGvCQ1yoCXpEYZ8JLUKANekhr1/9MpCWlvZx4TAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10baf8860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Possible answer to Question 4:\n",
    "plt.bar(totals.keys(), totals.values())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Hubs Revisited\n",
    "\n",
    "It looks like individual 19 is an important person of some sorts - if a message has to be passed through the network in the shortest time possible, then usually it'll go through person 19. Such a person has a high **betweenness centrality**. This is implemented as one of NetworkX's centrality algorithms. Check out the Wikipedia page for a further description.\n",
    "\n",
    "http://en.wikipedia.org/wiki/Betweenness_centrality"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 30 artists>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ym2R3Va100yvA7llWZhO8O8nXk3xiUYcv1kpyLvBy4HYabL9Vx/dvXVETbZjkaUnuYtRO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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cbf0a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "btws = nx.betweenness_centrality(G, normalized=False)\n",
    "plt.bar(btws.keys(), btws.values())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Plot betweeness centrality against degree centrality for the synthetic social network above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x10cf6b160>"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cdc46d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Possible answer:\n",
    "\n",
    "deg_centrality = nx.degree_centrality(G)\n",
    "btw_centrality = nx.betweenness_centrality(G)\n",
    "\n",
    "deg_cent_sorted = [i[1] for i in sorted(zip(deg_centrality.keys(), deg_centrality.values()))]\n",
    "btw_cent_sorted = [i[1] for i in sorted(zip(btw_centrality.keys(), btw_centrality.values()))]\n",
    "\n",
    "plt.scatter(deg_cent_sorted, btw_cent_sorted)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Think about it...**\n",
    "\n",
    "From the scatter plot, we can see that the dots don't all fall on the same line. Degree centrality and betweenness centrality don't necessarily correlate. Can you think of a reason why?\n",
    "\n",
    "What would be the degree centrality and betweenness centrality of the middle connecting node in the **barbell graph** below?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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AmDFjcPPmTezbt6/cLSD1Ycg0dNEGHEUHNFQ0guYUtOkxgImIjODYsWO4du0axowZU+rz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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cd0e9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw(nx.barbell_graph(5, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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