4. Cliques, Triangles and Squares (Instructor).ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "import networkx as nx\n",
    "import matplotlib.pyplot as plt\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Cliques, Triangles and Squares\n",
    "\n",
    "Let's pose a problem: If A knows B and B knows C, would it be probable that A knows C as well? In a graph involving just these three individuals, it may look as such:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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nAgA/R3wBALAYL7gCAMBixBcAAIsRXwAALEZ8AQCwGPEFAMBixBcAAIsRXwAA\nLEZ8AQCwGPEFAMBixBcAAIsRXwAALEZ8AQCwGPEFAMBixBcAAIsRXwAALEZ8AQCwGPEFAMBixBcA\nAIsRXwAALEZ8AQCwGPEFAMBixBcAAIsRXwAALEZ8AQCwGPEFAMBixBcAAIsRXwAALEZ8AQCwGPEF\nAMBixBcAAIsRXwAALEZ8AQCwGPEFAMBixBcAAIsRXwAALEZ8AQCwGPEFAMBixBcAAIsRXwAALEZ8\nAQCwGPEFAMBixBcAAIsRXwAALEZ8AQCwGPEFAMBixBcAAIsRXwAALEZ8AQCwGPEFAMBixBcAAIsR\nXwAALEZ8AQCwGPEFAMBixBcAAIsRXwAALEZ8AQCwGPEFAMBixBcAAIv9f+zhlfjYSfUuAAAAAElF\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109d26f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G = nx.Graph()\n",
    "G.add_nodes_from(['a', 'b', 'c'])\n",
    "G.add_edges_from([('a','b'), ('b', 'c')])\n",
    "nx.draw(G, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let's think of another problem: If A knows B, B knows C, C knows D and D knows A, is it likely that A knows C and B knows D? How would this look like?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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xI77fZOOzb74hLS0Nm83G8ePHmTZtGqGhodx+++1VEVlEbsLZs2fp06cPw4YN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\nRETEYCpfERERg/1/kroo0hLs8VoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10bd2aef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G.add_node('d')\n",
    "G.add_edge('c', 'd')\n",
    "G.add_edge('d', 'a')\n",
    "nx.draw(G, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The set of relationships involving A, B and C, if closed, involves a triangle in the graph. The set of relationships that also include D form a square.\n",
    "\n",
    "You may have observed that social networks (LinkedIn, Facebook, Twitter etc.) have friend recommendation systems. How exactly do they work? Apart from analyzing other variables, closing triangles is one of the core ideas behind the system. A knows B and B knows C, then A probably knows C as well."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "If all of the triangles in the two small-scale networks were closed, then the graph would have represented **cliques**, in which everybody within that subgraph knows one another.\n",
    "\n",
    "In this section, we will attempt to answer the following questions:\n",
    "\n",
    "1. Can we identify cliques?\n",
    "2. Can we identify *potential* cliques that aren't captured by the network?\n",
    "3. Can we model the probability that two unconnected individuals know one another?\n",
    "\n",
    "As usual, let's start by loading the synthetic network."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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GRmLvwe7D/ge2Oer4rFCIMSNGsHz58tbwBZhaUsLsBQsY7/WyNs57noFd1AXwY+xe4CfZ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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1040ba908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Load the network.\n",
    "G = nx.read_gpickle('Synthetic Social Network.pkl')\n",
    "nx.draw(G, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Cliques\n",
    "\n",
    "In a social network, cliques are groups of people in which everybody knows everybody. Triangles are a simple example of cliques. Let's try implementing a simple algorithm that finds out whether a node is present in a triangle or not.\n",
    "\n",
    "The core idea is that if a node is present in a triangle, then its neighbors' neighbors' neighbors should include itself."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Example code that shouldn't be too hard to follow.\n",
    "def in_triangle(G, node):\n",
    "    neighbors1 = G.neighbors(node)\n",
    "    neighbors2 = []\n",
    "    for n in neighbors1:\n",
    "        neighbors = G.neighbors(n)\n",
    "        if node in neighbors2:\n",
    "            neighbors2.remove(node)\n",
    "        neighbors2.extend(G.neighbors(n))\n",
    "    \n",
    "    neighbors3 = []\n",
    "    for n in neighbors2:\n",
    "        neighbors = G.neighbors(n)\n",
    "        neighbors3.extend(G.neighbors(n))\n",
    "        \n",
    "    return node in neighbors3\n",
    "\n",
    "in_triangle(G, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "In reality, NetworkX already has a function that *counts* the number of triangles that any given node is involved in. This is probably more useful than knowing whether a node is present in a triangle or not, but the above code was simply for practice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.triangles(G, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Can you write a function that takes in one node and its associated graph as an input, and returns a list or set of itself + all other nodes that it is in a triangle relationship with?\n",
    "\n",
    "Hint: The neighbor of my neighbor should also be my neighbor, then the three of us are in a triangle relationship.\n",
    "\n",
    "Hint: Python Sets may be of great use for this problem. https://docs.python.org/2/library/stdtypes.html#set\n",
    "\n",
    "Verify your answer by drawing out the subgraph composed of those nodes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1, 2, 3, 6, 23}"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Possible answer\n",
    "def get_triangles(G, node):\n",
    "    neighbors = set(G.neighbors(node))\n",
    "    triangle_nodes = set()\n",
    "    \"\"\"\n",
    "    Fill in the rest of the code below.\n",
    "    \"\"\"\n",
    "    for n in neighbors:\n",
    "        neighbors2 = set(G.neighbors(n))\n",
    "        neighbors.remove(n)\n",
    "        neighbors2.remove(node)\n",
    "        triangle_nodes.update(neighbors2.intersection(neighbors))\n",
    "        neighbors.add(n)\n",
    "    triangle_nodes.add(node)\n",
    "    return triangle_nodes\n",
    "\n",
    "# Verify your answer with the following funciton call. Should return:\n",
    "# {1, 2, 3, 6, 23}\n",
    "get_triangles(G, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ffJLNmzcXe/le1aVLFz799FOGDh1KTExMibymCGgELDfRuHFjNm/eTJcuXXB2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\nKgNSFAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c054f98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Then, draw out those nodes.\n",
    "nx.draw(G.subgraph(get_triangles(G, 3)), with_labels=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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WkpKSiI6MZOcPP+Bjs1GyZEnO2Ww0qleP8IgIunfvbuq1qrdakvH372nNWpLR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\nO6+99hrffvst69evx8fHx+x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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c059320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compare for yourself that node 8 isn't in a triangle relationship with the other nodes.\n",
    "neighbors3 = G.neighbors(3)\n",
    "neighbors3.append(3)\n",
    "nx.draw(G.subgraph(neighbors3), with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Friend Recommendation: Open Triangles\n",
    "\n",
    "Now that we have some code that identifies closed triangles, we might want to see if we can do some friend recommendations by looking for open triangles.\n",
    "\n",
    "Open triangles are like those that we described earlier on - A knows B and B knows C, but C's relationship with A isn't captured in the graph. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "Can you write a function that identifies, for a given node, the other two nodes that it is involved with in an open triangle, if there is one? \n",
    "\n",
    "Hint: You may still want to stick with set operations. Suppose we have the A-B-C triangle. If there are neighbors of C that are also neighbors of B, then those neighbors are in a triangle with B and C; consequently, if there are nodes for which C's neighbors do not overlap with B's neighbors, then those nodes are in an open triangle. The final implementation should include some conditions, and probably won't be as simple as described above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Possible Answer, credit Justin Zabilansky (MIT) for help on this.\n",
    "def get_open_triangles(G, node):\n",
    "    \"\"\"\n",
    "    There are many ways to represent this. One may choose to represent only the nodes involved \n",
    "    in an open triangle; this is not the approach taken here.\n",
    "    \n",
    "    Rather, we have a code that explicitly enumrates every open triangle present.\n",
    "    \"\"\"\n",
    "    open_triangle_nodes = []\n",
    "    neighbors = set(G.neighbors(node))\n",
    "    \n",
    "    for n in neighbors:\n",
    "        neighbors2 = set(G.neighbors(n))\n",
    "        neighbors2.remove(node)\n",
    "        \n",
    "        overlaps = set()\n",
    "        for n2 in neighbors2:\n",
    "            if n2 in neighbors:\n",
    "                overlaps.add(n2)\n",
    "                \n",
    "        difference = neighbors.difference(overlaps)\n",
    "        difference.remove(n)\n",
    "        \n",
    "        for n2 in difference:\n",
    "            if set([node, n, n2]) not in open_triangle_nodes:\n",
    "                open_triangle_nodes.append(set([node, n, n2]))\n",
    "    return open_triangle_nodes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[{0, 2, 19}, {0, 19, 4}, {0, 16, 19}, {0, 17, 19}, {0, 19, 22}, {0, 27, 19}, {2, 19, 4}, {2, 19, 12}, {17, 2, 19}, {2, 19, 22}, {27, 2, 19}, {19, 4, 12}, {16, 19, 4}, {17, 19, 4}, {19, 4, 22}, {27, 19, 4}, {16, 19, 12}, {17, 19, 12}, {19, 12, 22}, {27, 19, 12}, {16, 19, 22}, {16, 27, 19}, {17, 19, 22}, {27, 17, 19}, {27, 19, 22}]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "25"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# # Uncomment the following code if you want to draw out each of the triplets.\n",
    "# nodes = get_open_triangles(G, 2)\n",
    "# for i, triplet in enumerate(nodes):\n",
    "#     fig = plt.figure(i)\n",
    "#     nx.draw(G.subgraph(triplet), with_labels=True)\n",
    "print(get_open_triangles(G, 19))\n",
    "len(get_open_triangles(G, 19))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "If you remember the previous section on hubs and paths, you will note that node 19 was involved in a lot of open triangles.\n",
    "\n",
    "Triangle closure is also the core idea behind social networks' friend recommendation systems; of course, it's definitely more complicated than what we've implemented here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Please Network With One Another!\n",
    "\n",
    "I would like to become a hub node in our PyData network :).\n",
    "\n",
    "I also wish to create many triangle cliques with each of you. \n",
    "\n",
    "Please exchange your cards with a neighbor. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": []
  }
 ],
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