Lesson 10 Directed Graphs.ipynb

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    "+ This Jupyter notebook<sup>[1]</sup> is part of the Klopper Letures on Discrete Matheamtics and covers *directed graphs*\n",
    "+ Created by me, Dr Juan H Klopper\n",
    "    + Head of Acute Care Surgery\n",
    "    + Groote Schuur Hospital\n",
    "    + University Cape Town\n",
    "    + <a href=\"mailto:juan.klopper@uct.ac.za\">Email me with your thoughts, comments, suggestions and corrections</a> \n",
    "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc/4.0/\"><img alt=\"Creative Commons Licence\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-nc/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" href=\"http://purl.org/dc/dcmitype/InteractiveResource\" property=\"dct:title\" rel=\"dct:type\">The Klopper Lectures on Discrete Mathematics</span> <span xmlns:cc=\"http://creativecommons.org/ns#\" property=\"cc:attributionName\"></span> study notes is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc/4.0/\">Creative Commons Attribution-NonCommercial 4.0 International License</a>.\n",
    "\n",
    "+ [1] Fernando Pérez, Brian E. Granger, IPython: A System for Interactive Scientific Computing, Computing in Science and Engineering, vol. 9, no. 3, pp. 21-29, May/June 2007, doi:10.1109/MCSE.2007.53. URL: http://ipython.org"
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    "from IPython.core.display import HTML\n",
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   "source": [
    "#import numpy as np\n",
    "#import scipy as sp\n",
    "import networkx as nx\n",
    "import sympy as sym\n",
    "import matplotlib.pyplot as plt"
   ]
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  {
   "cell_type": "code",
   "execution_count": 3,
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   "source": [
    "from IPython.display import Image"
   ]
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   "cell_type": "code",
   "execution_count": 4,
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   "source": [
    "%matplotlib inline\n",
    "sym.init_printing(use_latex = \"mathjax\")"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Set operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## In this lesson"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Follow these links\n",
    "- [Introduction](#Introduction)\n",
    "- [Degrees](#Degrees)\n",
    "- [Paths, trails, cycles, and others](#Paths,-trails,-cycles,-and-others)\n",
    "- [Connectivity](#Connectivity)\n",
    "- [Rooted trees](#Rooted-trees)\n",
    "- [The adjacency matrix](#The-adjacency-matrix)\n",
    "- [Path matrices](#Path-matrices)\n",
    "- [Warshall shortest path algorithm](#Warshall-shortest-path-algorithm)"
   ]
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   "metadata": {},
   "source": [
    "## Introduction"
   ]
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   "source": [
    "We add a bit of direction to our graphs in this lesson by turning our edges into *one-way* roads.  These are termed **directed graphs** or **digraphs**."
   ]
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   "source": [
    "We still have a set $ V $ of vertices or nodes and a set $ E $ of edges, although we might also refer to them as **directed edges** or **arcs**.  When we denote an edge in the form $ \\left( u,v \\right) $ we state that $ u $ is the **origin** or **initial point** and $ v $ is the **destination** or **terminal point**.  In this case we can also say that $ v $ is the **successor** of $ u $."
   ]
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    "Some textbooks also use the terms **adjacent to** and **adjacent from**, as in $ u $ is adjacent to $ v $ and $ v $ is adjacent from $ u $.  All the nodes that are adjacent from $ u $ are called the **adjacency list** or **successor list** of $ u $."
   ]
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   "source": [
    "In the case where $ u = v $ we have a **directed loop** or simply a **loop**."
   ]
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   "source": [
    "We indicate the direction of an edge with a tuple of the form $ \\left( u,v \\right) $, i.e. (below) $ E \\left( G \\right) = \\left\\{ {e}_{1}, {e}_{2} \\right\\} = \\left\\{ \\left( A, B \\right), \\left( B, B \\right) \\right\\} $."
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   "source": [
    "G = nx.DiGraph()\n",
    "\n",
    "G.add_nodes_from([\"A\", \"B\"])\n",
    "G.add_edges_from([(\"A\", \"B\"), (\"B\", \"B\")])"
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DfM65fWb2Q+DfzrmLgoxrBrwErHDOZdawz/oHJCLSSDnnLNR91Fgh1CAbGAo8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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10867deb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Will be drawn differently every time\n",
    "nx.draw_networkx(G)\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['B']"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# List nodes with loops\n",
    "nx.DiGraph.nodes_with_selfloops(G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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tbNiwgf/8z/+0Oo5SNXTKgnKqdn//JZdcwtNPP639/S66/PLLyc7OZsiQIQQG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\n/rK5JQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b3364e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# A GN digraph with 5 nodes\n",
    "H = nx.gn_graph(5)\n",
    "nx.draw_networkx(H)\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to the top](#In-this-lesson)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "<h2>Degrees</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since edges are directed every node might have a number of edges that originate from and a number of edges that might terminate in it.  The number of edges that originate from a node is the **indegree** of the node and the number of edges that terminate at a node is the node's **outdegree**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sum total of the indegrees equals the sum total of the outdegrees, which equals the numder of edges of a digraph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A connected node with a zero outdegree (only edges that terminate with it) is termed a **sink**.  A node with a zero indegree (only edges that originate from it) is termed a **source**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to the top](#In-this-lesson)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "<h2>Paths, trails, cycles, and others</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These terms are taken from non-directed graphs.  So, a **path** is an alternating set of nodes and directed edges."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **length** of a path counts the number of edges traversed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A path with distinct nodes is a **simple path**, whereas a path with distinct edges is termed a **trail**.  If a node $ v $ can be reached along directed esges from a node $ u $, then by deleting redundant edges, we can create a simple path from $ u $ to $ v $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the first and last node in a path coincides we have a **closed path**.  If all the nodes vsisted are distinct then we have a **cycle**, also termed a **circuit**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A path that visits all the nodes is termed a **spanning path**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A path that connects nodes, but in which at least on edge is against its direction is termed a **semipath**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to the top](#In-this-lesson)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "<h2>Connectivity</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In a **strongly connected** graph we have that for any pair of nodes, $ u $ and $ v $, node $ v $ is reachable from a node $ u $ along a path, but the same node $ u $ is also reachable from the same node $ v $.  This can occur if and only if the graph has a closed, spanning path."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In a **unilaterally connected** graph we have that for any pair of nodes, $ u $ and $ v $, there is only in path in one of the directions.  This can occur if and only if the graph has a spanning path."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In a **weakly connected** graph we have that for any pair of nodes, we can only find a semipath.  This can occur if and only if the graph has a spanning semipath."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We note that there is an order to this connectivity, i.e. a strongly connected graph implies a unilaterally connected graph which implies a weakly connected graph.  A graph that is weakly connected but not unilaterally connected is termed **strictly weak connected** and a unilaterally connected graph that is not strongly connected is termed **strictly unilaterally connected.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From nondirected graphs we remember that a connected graph has a path between any of its nodes.  If we take a directed graph and alter the edges so that they are not directed and this non-directed version is found to be connected, then the original graph is weakly connected."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If a graph is cycle-free, it contains a source and a sink."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Bakc to the top](#In-this-lesson)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "<h2>Rooted trees</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Frm the previous lesson we remember that a **tree** is a cycle-free connected graph.  We can turn a tree into a **rooted tree** by designating one of the nodes as a root.  From this root, which can be any of the nodes in a connected graph, there is a unique path to any of the other nodes.  If we choose a root this implies a direction to each of the other nodes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The length of the unique path from the root to a node $ v $ is termed the **level** or the **depth** of the node and the maximum level or depth is the level of depth of the tree."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The nodes with degree $ 1 $ (other than the root) are termed the **leaves** of the tree and the edge from the root to these leaves are termed the **branches** of the tree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "G = nx.Graph()\n",
    "\n",
    "G.add_nodes_from([\"a\", \"b\", \"c\", \"d\", \"e\", \"f\", \"g\", \"h\", \"i\", \"j\", \"r\"])\n",
    "G.add_edges_from([(\"r\", \"a\"), (\"r\", \"b\"), (\"r\", \"c\"), (\"a\", \"d\"), (\"a\", \"e\"),\n",
    "                  (\"e\", \"h\"), (\"b\", \"f\"), (\"c\", \"g\"),\n",
    "                 (\"g\", \"i\"), (\"g\", \"j\")])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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xbNkykpOTY27dAhWcJgELaU1AWc3KpRcrKip47733yM/P55VXXqFly5aMGDGC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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b57f2b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw_networkx(G)\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Checking of G is a tree\n",
    "nx.is_tree(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that ever node except the root is preceded by a unique node, but that every node can have more than one successor."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use these trees to look at all the possible. logical outcomes of a sequence of events.  Imagine that we have a binary outcome (either *success* of *failure*) and that an expreminet will rerun until we have two successes or failures in a row or a total of three of either.  The image below shows the logical outcomes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Rt90xi/RF0pMiP8uILqSnFuC67FkAQHADAKCo4NHW961rx0pa185rWKdtJROALBX4sqW2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\nAAAAAEC4AQAAAAAAINwAAAAAAACgJP4/Keu/xRmR2QcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 12,
     "metadata": {
      "image/png": {
       "width": 400
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(\"StartExperiment.png\", width = 400)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nodes in a rooted tree can be ordered, giving every node an address.  The image below shows the addresses of each node."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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GAMi/Qh6KVjTc/BnpSrqb+fG5kpY2QhsAENYAAI4r4NNa0S51dd+tiL0EQhsAENYAAMVU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\nVxgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 13,
     "metadata": {
      "image/png": {
       "width": 400
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(\"OrderedRootedTrees.png\", width = 400)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to the top](#In-this-lesson)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "<h2>The adjacency matrix</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We complete an adjacency matrix as we did for non-directed graphs.  This can be kept as a sparse, scipy matrix or a numpy matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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WrEnnzp355JNPXBSZEI7x6quvcuLECRYtWuTwBADw7rvvUq9ePfr27UtOTo7D+y/LJAkY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\n8GKSBIQQwov9f12eoW+rq4O3AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b6af710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# G from above\n",
    "nx.draw_networkx(G)\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  (0, 2)\t1\n",
      "  (1, 3)\t1\n",
      "  (2, 0)\t1\n",
      "  (2, 7)\t1\n",
      "  (3, 1)\t1\n",
      "  (3, 4)\t1\n",
      "  (4, 3)\t1\n",
      "  (4, 7)\t1\n",
      "  (4, 10)\t1\n",
      "  (5, 6)\t1\n",
      "  (6, 5)\t1\n",
      "  (6, 9)\t1\n",
      "  (6, 10)\t1\n",
      "  (7, 2)\t1\n",
      "  (7, 4)\t1\n",
      "  (7, 8)\t1\n",
      "  (8, 7)\t1\n",
      "  (9, 6)\t1\n",
      "  (10, 4)\t1\n",
      "  (10, 6)\t1\n"
     ]
    }
   ],
   "source": [
    "# Sparse scipy matrix\n",
    "print(nx.adj_matrix(G))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "        [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "        [ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],\n",
       "        [ 0.,  1.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "        [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  1.,  0.,  0.,  1.],\n",
       "        [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],\n",
       "        [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  1.,  1.],\n",
       "        [ 0.,  0.,  1.,  0.,  1.,  0.,  0.,  0.,  1.,  0.,  0.],\n",
       "        [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],\n",
       "        [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],\n",
       "        [ 0.,  0.,  0.,  0.,  1.,  0.,  1.,  0.,  0.,  0.,  0.]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.to_numpy_matrix(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This matrix depends on the order of the nodes.  Different orders can be represented by interchanging rows or columns."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can calculate the number of paths between vertices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "G = nx.DiGraph()\n",
    "\n",
    "G.add_nodes_from([1, 2, 3, 4])\n",
    "G.add_edges_from([(1, 4), (2, 1), (2, 3), (2, 4),\n",
    "                  (3, 1), (3, 4), (4, 3), (4, 1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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sZiofyeshhL0WtDvIamfPnpWy7u6ZngMYDeKXRbfQRRB3kCJFikiXLl3ku+++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\nmgSUUsqF/T9hBR2ZPZF2BAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b61e630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw_networkx(G)\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.  0.  0.  1.]\n",
      " [ 1.  0.  1.  1.]\n",
      " [ 1.  0.  0.  1.]\n",
      " [ 1.  0.  1.  0.]]\n"
     ]
    }
   ],
   "source": [
    "adjacency_matrix_G = nx.to_numpy_matrix(G)\n",
    "print(adjacency_matrix_G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Considering the first row, we note that node $ 1 $ does not loop to itself, nor does it have directed edges to nodes $ 2 $ or $ 3 $.  It does have to node $ 4 $, though.  This results in a first row of $ \\left[ 0,0,0,1 \\right] $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we multiply the adjacency matrix by itself we get a new matrix (see below)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 1.,  0.,  1.,  0.],\n",
       "        [ 2.,  0.,  1.,  2.],\n",
       "        [ 1.,  0.,  1.,  1.],\n",
       "        [ 1.,  0.,  0.,  2.]])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "adjacency_matrix_G * adjacency_matrix_G"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The square in `adjacency_matrix_G`<sup>2</sup> indicates paths of length $ 2 $.  Again, if we consider row $ 1 $ we see that there is $ 1 $ path from node $ 1 $ back to node $ 1 $.  It is via node $ 4 $ if we consider the image of the graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we cube the adjacency matrix we get the number of paths of length $ 3 $ from a node to another and so on.  Remember that we have set up the graph so that we start at the row level and go to each colum for that row to read of the edge from node $ u $ to node $ v $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we add the matrices formed by successive powers, we get a matrix termed $ {B}_{r} $ such that if the adjacency matrix is termed $ A $, we have\n",
    "$$ {B}_{r} = {A} + {A}^{2} + {A}^{2} + \\dots + {A}^{r} $$\n",
    "such that the $ ij $<sup>th</sup> entry of $ {B}_{r} $ gives the number of paths of length $ r $ or less from node $ {v}_{i} $ to node $ {v}_{j} $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to the top](#In-this-lesson)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "<h2>Path matrices</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "A **path matrix**, also termed a **reachability matrix** is created in mucg the same way as an indentity matrix.  Here we state a $ 0 $ if one node is not reachable by from another and $ 1 $ if it is."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In our successive power matrix $ {B}_{r} $ above, let's consider a graph with $ m $ nodes and state that $ {B}_{m} = {A} + {A}^{2} + \\cdots + {A}^{m} $.  If we now have a graph with $ m $ nodes and suppose that there is a path from $ {v}_{i} $ to $ {v}_{j} $.  Such a simple path must have a length of $ m - 1 $ or less.  If $ {v}_{i} = {v}_{j} $ then we will have a cycle of length $ m $ or less.  This means that the path matrix and $ {B}_{m} $ have the same non-zero entries."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we make a path matrix from $ G $ above (not available in `networkx`) we get\n",
    "$$ { P }_{ G }=\\begin{bmatrix} 1 & 0 & 1 & 1 \\\\ 1 & 0 & 1 & 1 \\\\ 1 & 0 & 1 & 1 \\\\ 1 & 0 & 1 & 1 \\end{bmatrix} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 0.,  0.,  0.,  1.],\n",
       "        [ 1.,  0.,  1.,  1.],\n",
       "        [ 1.,  0.,  0.,  1.],\n",
       "        [ 1.,  0.,  1.,  0.]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Agacency matrix of G\n",
    "ag_mat = adjacency_matrix_G\n",
    "ag_mat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's do $ {B}_{3} $\n",
    "$$ {B}_{3} = {A} + {A}^{2} + {A}^{3} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 2.,  0.,  1.,  3.],\n",
       "        [ 6.,  0.,  4.,  6.],\n",
       "        [ 4.,  0.,  2.,  4.],\n",
       "        [ 4.,  0.,  3.,  3.]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ag_mat + ag_mat**2 + ag_mat**3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Node $ 2 $ was a source.  Note then how all the entries other than those in column $ 2 $ remain non-zero."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Btw, remember how a strongly connected digraph has a path from $ {v}_{i} $ to $ {v}_{j} $ and the otehr way around.  This means that for a digraph to be strongly connected, its path matrix cannot have any zero entries."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to the top](#In-this-lesson)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "<h2>Warshall shortest path algorithm</h2>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The warshall algorith finds teh shortest path in a digraph and can also deal with weighted digraphs.  It is more efficient than calculating the pwers of adjacency matrices."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a digraph with $ m $ nodes we need to construct $ m $ square Boolean matrices $ {P}_{0}, {P}_{1}, {P}_{2}, \\cdots, {P}_{m} $.  If we take any of of the path matrices, say $ {P}_{k} $ then entry $ {P}_{k} \\left[ i,j \\right] $ is $ 1 $ if there is a simply path from $ {v}_{i} $ to $ {v}_{j} $ which does not use any nodes except possible $ {v}_{1}, {v}_{2}, \\cdots, {v}_{k} $ and $ 0 $ if there isn't."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So if $ k = 4 $, then $ {p}_{4} \\left[ i,j \\right] = 1 $ if there is a simple path from $ {v}_{i} $ to $ {v}_{j} $ which does not use any other nodes, except possibly $ {v}_{1}, {v}_{2}, {v}_{3} $ and $ {v}_{4} $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first path matrix $ {P}_{0} = A $, the adjacency matrix.  The last path matrix $ {P}_{m} = P $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ {P}_{k} \\left[ i,j \\right] = 1 $ can only occur if there is a simple path from $ {v}_{i} $ to $ {v}_{j} $ which does not use any other nodes excepts perhaps $ {v}_{1}, {v}_{2}, \\cdots , {v}_{k-1} $ or if there is a simple path from $ {v}_{i} $ to $ {v}_{k} $ and a simple path from $ {v}_{k} $ to $ {v}_{j} $ where each simple path does not use any other nodes except perhaps $ {v}_{1}, {v}_{2}, \\cdots , {v}_{k-1} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[  0.,  inf,   2.,   1.],\n",
       "        [  1.,   0.,   1.,   1.],\n",
       "        [  1.,  inf,   0.,   1.],\n",
       "        [  1.,  inf,   1.,   0.]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.floyd_warshall_numpy(G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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",
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