jeanpat/graph-tool toy model- Response to pruning-DIP4FISH.ipynb

## graph-tool toy model- Response to pruning-DIP4FISH.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Prune iteratively the leaves of a graph with graph -tool:\n",
    "## A measure of the similarity of two weighted multigraphs?\n",
    "\n",
    "To compare shapes, typically obtained after image segmentation, it has been shown that their skeleton could be used.\n",
    "\n",
    "A weighted graph can represents naturally the skeleton of a shape, assuming that the size of the branches of the skeleton, linking the end-points and the branched points together, can be represented by the weights of the edges in a graph. \n",
    "\n",
    "Previously, pruning of skeletons was performed by removing pixels, one after an other, from the end-points of the skeleton. A response curve of a skeleton to pruning, can be defined by counting the end-points of the skeleton as pruning is iterated. It was observed that the responses from similar shapes were similar. So, such response curves may be used to compute some similarity measures, or distances, between shapes.\n",
    "\n",
    "Since the time necessary to prune a skeleton depends on the number of pixels of the skeleton, it may be worth to convert a skeleton into a graph which can be then  pruned in a way independent from the weight of its edges.\n",
    "\n",
    "The response of a graph to iterative pruning was performed with graph-tool (and additionnal python librairies : mahotas, numpy,scipy,  matplotlib). This response yields a curve \"**R**esponse to ***P**runing **C**urve or RPC\". Then a distance was defined between two RPCs to compute a similarity index between two graphs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import timeit\n",
    "import os\n",
    "import mahotas as mh\n",
    "import graph_tool.all as gt\n",
    "import numpy as np\n",
    "from scipy.spatial.distance import cdist\n",
    "from collections import defaultdict\n",
    "from matplotlib import pyplot as plt\n",
    "import matplotlib.gridspec as gridspec\n",
    "from graph_tool import topology as top\n",
    "#from graph_tool import search"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.11 (commit b6b2152c, Mon Oct 26 10:31:30 2015 +0100)\n"
     ]
    }
   ],
   "source": [
    "print gt.__version__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rough algorithm:\n",
    "    Make an undirectional weighted multigraph\n",
    "    Show it properly\n",
    "    Try to iteratively prune the graph:\n",
    "        Need to get vertices of degree 1 and to count them\n",
    "        Read the weight of edges linked to degree 1 vertices\n",
    "        Get the edge of smallest weight:\n",
    "            Remove the corresponding edge.\n",
    "            Decrease the weight of all other edges by this minimal value.\n",
    "        Do it again until there's no more vertices of degree 1 remaining in the graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Make an undirectional weighted graph\n",
    "By default in graph-tool, the graphs are multigraphs and that's what we need. The edges of the graph must be weighted. An edge weight corresponds to the size of a branch in a skeleton (the number of pixels). The weight of an edge is a property that has to be created:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def make_toy_graph():\n",
    "    T = gt.Graph(directed = False)\n",
    "    edge_weights = T.new_edge_property('double')\n",
    "    T.properties[(\"e\",\"weight\")] = edge_weights\n",
    "\n",
    "    T.add_vertex(n=4)\n",
    "\n",
    "    e_1 = T.add_edge(0,1)\n",
    "    e_2 = T.add_edge(1,2)\n",
    "    e_3 = T.add_edge(0,0)\n",
    "    e_4 = T.add_edge(1,3)\n",
    "\n",
    "    edge_weights[e_1]= 8\n",
    "    edge_weights[e_2]= 15\n",
    "    edge_weights[e_3]= 7\n",
    "    edge_weights[e_4]= 20\n",
    "    return T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Let's have a look to the graph properties:\n",
    "The weight is here!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{('e', 'weight'): <PropertyMap object with key type 'Edge' and value type 'double', for Graph 0xb639770c, at 0xb6397b8c>}\n"
     ]
    }
   ],
   "source": [
    "T = make_toy_graph()\n",
    "print T.properties"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A special case of graph will be useful:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_2vertices_1edge_graph():\n",
    "    T = gt.Graph(directed = False)\n",
    "    edge_weights = T.new_edge_property('double')\n",
    "    T.properties[(\"e\",\"weight\")] = edge_weights\n",
    "\n",
    "    T.add_vertex(n=2)\n",
    "    e_1 = T.add_edge(0,1)\n",
    "    edge_weights[e_1]= 8\n",
    "    return T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Now we need to define several functions:\n",
    "The algorithm will rely on finding vertices of degree 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def vertices_of_degree(Graph, degree):\n",
    "    '''returns the index of vertices of a given degree in the Graph\n",
    "    '''\n",
    "    k = Graph.degree_property_map('out')\n",
    "    if degree == 1:\n",
    "        return np.where(k.a == np.ones(k.a.shape, dtype=np.int))\n",
    "    if degree == 0:\n",
    "        return np.where(k.a == np.zeros(k.a.shape, dtype=np.int))\n",
    "    if degree > 1:\n",
    "        mask = np.ones(k.a.shape, dtype=np.int)\n",
    "        mask[:] = degree\n",
    "        return np.where(k.a == mask)   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def number_of_degree1_vertices(graph):\n",
    "    ''' returns the number of degree 1 vertices of the graph\n",
    "    '''\n",
    "    degprop = graph.degree_property_map('out')\n",
    "    #print \"count deg 1\", degprop.a\n",
    "    return np.sum(degprop.a == np.ones(degprop.a.shape,dtype=np.int))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def vertex_degree1_edge_weight(graph):\n",
    "    ''' Yields the weight (this property of edges is supposed to exist) \n",
    "    of an edge bound to a vertex of degree 1, as two dictionaries where :\n",
    "    FIRST:\n",
    "        vertex = keys and weight= values\n",
    "    SECOND:\n",
    "        weight= keys and vertex = values\n",
    "    DEPENDS ON:\n",
    "        vertices_of_degree()\n",
    "    ''' \n",
    "    #TO DO : Have a look to Graphview to filter vertices of degree 1 !!\n",
    "    vertex_edge_weight = defaultdict(list)\n",
    "    edge_weight_vertex = defaultdict(list)\n",
    "    \n",
    "    for v1_index in vertices_of_degree(graph, 1)[0]:\n",
    "        vertex = graph.vertex(v1_index)\n",
    "        #This is weird to make a loop to get the sole element\n",
    "        #of the iterartor vertex.out_edges(), but ...\n",
    "        edge =  [e for e in vertex.out_edges()]\n",
    "        weight = graph.ep.weight[e]#edge_weights[e]\n",
    "        vertex_edge_weight[vertex].append(weight)\n",
    "        edge_weight_vertex[weight].append(vertex)\n",
    "    \n",
    "    return vertex_edge_weight, edge_weight_vertex"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Make a Graph, draw it and check the previous functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "T = make_toy_graph()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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LORRykZN9knWtA8DdGzZsGNK6BWV0qAQZgsmzcq4jppsAoKu7J93T25MqKi7O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\nKIoyEfx/81uzo1Dd2mkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<PropertyMap object with key type 'Vertex' and value type 'vector<double>', for Graph 0xb639bc0c, at 0x9c652f8c>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gt.graph_draw(T,\n",
    "              vertex_text=T.vertex_index,\n",
    "              edge_pen_width=T.ep.weight,\n",
    "              inline=True,\n",
    "              output_size=(200, 200))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Check $$ vertices\\_of\\_degree() $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2 3]\n",
      "2  vertices of degree 1\n"
     ]
    }
   ],
   "source": [
    "print vertices_of_degree(T,1)[0]\n",
    "print number_of_degree1_vertices(T),' vertices of degree 1'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Check $$ vertex\\_degree1\\_edge\\_weight(graph)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "defaultdict(<type 'list'>, {<Vertex object with index '2' at 0x9cbe23ac>: [15.0], <Vertex object with index '3' at 0x9cbe276c>: [20.0]})\n",
      "\n",
      "defaultdict(<type 'list'>, {20.0: [<Vertex object with index '3' at 0x9cbe276c>], 15.0: [<Vertex object with index '2' at 0x9cbe23ac>]})\n"
     ]
    }
   ],
   "source": [
    "v_e_w, e_w_v = vertex_degree1_edge_weight(T)\n",
    "print v_e_w\n",
    "print\n",
    "print e_w_v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Try to write a function taking a graph and pruning it once inplace"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before pruning, it's necessary to check if it remains at most two vertices of degree 1 (and one edge)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def at_least_one_vertex_of_degree_above_one(graph):\n",
    "    ''' check if there's at least one vertex of degree above 1 in the graph.\n",
    "    '''\n",
    "    graph = gt.GraphView(graph, vfilt= lambda v : v.out_degree() > 1)\n",
    "    try:\n",
    "        first = next(graph.vertices())\n",
    "    except StopIteration:\n",
    "        return False\n",
    "    return True\n",
    "\n",
    "def is_VeV(graph):\n",
    "    ''' True if the graph contain only two vertices of degree 1 and possibly vertex of deg=0\n",
    "    '''\n",
    "    return (number_of_degree1_vertices(graph) == 2) and \\\n",
    "            (not(at_least_one_vertex_of_degree_above_one(graph)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "False\n"
     ]
    }
   ],
   "source": [
    "print at_least_one_vertex_of_degree_above_one(T)\n",
    "print is_VeV(T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Now decrease the weight of all edges1 by $edges1_{min}$ . For each vertex of degree 1 (v1):\n",
    "       1. get the bound edge : v1.out-edge (The graph is undirected)\n",
    "       2. decrease the weights :\n",
    "$$weight(v1-v')_{new} = weight(v1-v')_{old}-edge1_{min}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def prune_VeV(graph):\n",
    "    '''prune a graph with two vertices of degree 1 (V1) and one weighted edge\n",
    "        return the the weight of the edge and 0, the number of V1 (0).\n",
    "    '''\n",
    "    index_degree1_vertices = vertices_of_degree(graph, 1)[0]\n",
    "    v1_index = index_degree1_vertices[0]\n",
    "    vertex = graph.vertex(v1_index)\n",
    "    bound_edge = vertex.out_edges()\n",
    "    BE = [be for be in bound_edge][0]\n",
    "    weight = graph.ep.weight[BE]\n",
    "    graph.remove_edge(BE)\n",
    "    assert(number_of_degree1_vertices(graph) == 0)\n",
    "    return weight, number_of_degree1_vertices(graph)\n",
    "\n",
    "def prune_graph_once(graph):\n",
    "    '''Prune \"graph\" once and inplace and return the:\n",
    "        number of vertex of degree 1\n",
    "        the weight of the shortest\n",
    "    '''\n",
    "    #print number_of_degree1_vertices(graph)\n",
    "    test_degree = is_VeV(graph)\n",
    "    if test_degree == False:\n",
    "        v1_ew, ew_v1 = vertex_degree1_edge_weight(graph)\n",
    "\n",
    "        min_weights = np.min(v1_ew.values())\n",
    "        #print 'min weight of v1:', min_weights\n",
    "\n",
    "        # Decrease all edges weight by the smallest weight\n",
    "        index_degree1_vertices = vertices_of_degree(graph, 1)[0]\n",
    "        #print index_degree1_vertices\n",
    "        for v1_index in index_degree1_vertices:\n",
    "            vertex = graph.vertex(v1_index)\n",
    "            bound_edge = vertex.out_edges()\n",
    "            BE = [be for be in bound_edge][0]\n",
    "            graph.ep.weight[BE] = graph.ep.weight[BE]-min_weights\n",
    "            #print graph.ep.weight[BE]\n",
    "    \n",
    "            if graph.ep.weight[BE] == 0.0:\n",
    "                graph.remove_edge(BE)\n",
    "        return min_weights, number_of_degree1_vertices(graph)\n",
    "    \n",
    "    elif test_degree == true:\n",
    "        # Get the two bound vertices of degree 1 Vertex_edge_Vertex\n",
    "        return prune_VeV(graph)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 loops, best of 3: 94.5 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit prune_graph_once(make_toy_graph())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A better pruning function?\n",
    "\n",
    "The previous function succeeds in pruning a graph, but it probably not use the full possibility of the graph-tool library. \n",
    "It is possible to extract the vertices of degree 1 directly with the function GraphView() as follow:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(8.0, 0)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def prune_once_faster(graph):\n",
    "    \n",
    "    #filter the vertices of degree 1\n",
    "    subg = gt.GraphView(graph, vfilt= lambda v : v.out_degree() == 1)\n",
    "    weight_vertex_edge = defaultdict(list)\n",
    "    \n",
    "    test_degree = is_VeV(graph)\n",
    "    if test_degree == False:\n",
    "        for v in subg.vertices():\n",
    "            #print type(v)\n",
    "            #print graph.vertex(v)\n",
    "            #GET THE ONLY EDGE BOUND TO THE VERTEX (SINCE DEGREE=1)\n",
    "            edge = next(graph.vertex(v).out_edges())\n",
    "            weight = graph.ep.weight[edge]\n",
    "            #print'edge', edge\n",
    "            #print edge, ' weight ',weight\n",
    "            weight_vertex_edge[weight].append((v, edge))\n",
    "\n",
    "        shortest_edge = min(weight_vertex_edge.keys())\n",
    "        #print 'shortest', shortest_edge\n",
    "        #print 'decrease weight'\n",
    "        #loop = 1\n",
    "        for v in subg.vertices():\n",
    "            vertex = graph.vertex(v)\n",
    "            bound_edge = vertex.out_edges()\n",
    "            BE = next(bound_edge)\n",
    "            #BE = [be for be in bound_edge][0]\n",
    "            #print vertex, BE, graph.ep.weight[BE]\n",
    "            graph.ep.weight[BE] = graph.ep.weight[BE]-shortest_edge\n",
    "            #print 'after', graph.ep.weight[BE]\n",
    "            if graph.ep.weight[BE] == 0.0:\n",
    "                graph.remove_edge(BE)\n",
    "        return shortest_edge,number_of_degree1_vertices(graph)  \n",
    "    elif test_degree == True:\n",
    "            # Get the two bound vertices of degree 1 Vertex_edge_Vertex\n",
    "            #V_e_V = gt.GraphView(graph, vfilt= lambda v : v.out_degree() == 1)\n",
    "            #edge_of_VEV = next(V_e_V.edges()) # ONLY ONE EDGE\n",
    "            #weight = graph.ep.weight[edge_of_VEV]\n",
    "            #graph.remove_edge(edge_of_VEV)\n",
    "            return prune_VeV(graph)  \n",
    "\n",
    "TT = make_toy_graph()\n",
    "prune_once_faster(TT)\n",
    "prune_once_faster(TT)\n",
    "prune_once_faster(TT)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The slowest run took 12.66 times longer than the fastest. This could mean that an intermediate result is being cached \n",
      "1000000 loops, best of 3: 226 ns per loop\n",
      "10 loops, best of 3: 33 ms per loop\n",
      "10 loops, best of 3: 73.8 ms per loop\n"
     ]
    }
   ],
   "source": [
    "#print 'vertex', h.vertex(2), 'edges:', \\next(h.vertex(2).out_edges()),\\'weight ', h.ep.weight[next(h.vertex(2).out_edges())]\n",
    "#print #h.ep.weight[]\n",
    "%timeit make_toy_graph\n",
    "%timeit number_of_degree1_vertices(make_toy_graph()) \n",
    "%timeit prune_once_faster(make_toy_graph())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graph Response to Pruning\n",
    "%timeit shows clearly that the second version is faster.\n",
    "Now, it remains to define a function which count the number of vertices of degree 1 as the graph is pruned iteratively"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 loops, best of 3: 228 ms per loop\n"
     ]
    }
   ],
   "source": [
    "def total_length_graph(graph):\n",
    "    v_ew, _ =vertex_degree1_edge_weight(T)\n",
    "    return np.sum(v_ew.values())    \n",
    "    \n",
    "def response_to_iterative_pruning(graph):\n",
    "    '''return the number of vertices of degree 1 as a function of the number of pruning\n",
    "    '''\n",
    "    response = []\n",
    "    response.append((0,number_of_degree1_vertices(graph)))\n",
    "    #print number_of_degree1_vertices(graph)\n",
    "    while number_of_degree1_vertices(graph)>0:\n",
    "        #response.append(prune_graph_once(graph))\n",
    "        response.append(prune_once_faster(graph))\n",
    "        #print number_of_degree1_vertices(graph)\n",
    "    data = np.array(response)\n",
    "    #print data\n",
    "    x = np.cumsum(data[:,0])\n",
    "    y = data[:,1]\n",
    "    return x, y\n",
    "\n",
    "#%timeit \n",
    "%timeit response_to_iterative_pruning(make_toy_graph())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Check some things again:\n",
    "The number of edges should be decreased by one after pruning. One cas see that the pruning algorithm is destructive:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<Graph object, undirected, with 4 vertices and 4 edges at 0xb639bc0c>\n",
      "<Graph object, undirected, with 4 vertices and 3 edges at 0xb639b88c>\n"
     ]
    }
   ],
   "source": [
    "g = T.copy()\n",
    "prune_graph_once(g)\n",
    "print T\n",
    "print g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check if a graph made of two linked vertices is recognized (**V**ertex-**E**dge-**V**ertex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "False\n",
      "<Graph object, undirected, with 2 vertices and 1 edge at 0x9c65ceec>\n",
      "(8.0, 0)\n",
      "<Graph object, undirected, with 2 vertices and 0 edges at 0x9c65ceec>\n"
     ]
    }
   ],
   "source": [
    "t = make_2vertices_1edge_graph()\n",
    "#t=T\n",
    "print is_VeV(t)\n",
    "print is_VeV(T)\n",
    "print t\n",
    "print prune_VeV(t)\n",
    "print t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Check iterative pruning by hand"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(15.0, 1)\n",
      "(5.0, 1)\n",
      "(8.0, 0)\n",
      "<Graph object, undirected, with 4 vertices and 4 edges at 0x9c65c1ac> 2\n",
      "<Graph object, undirected, with 4 vertices and 3 edges at 0x9c65cc2c> 1\n",
      "<Graph object, undirected, with 4 vertices and 2 edges at 0x9c654dec> 1\n",
      "<Graph object, undirected, with 4 vertices and 1 edge at 0x9c65424c> 0\n"
     ]
    }
   ],
   "source": [
    "G0 = make_toy_graph()\n",
    "g0 = G0.copy()\n",
    "print prune_graph_once(g0)\n",
    "g1 = g0.copy()\n",
    "print prune_graph_once(g0)\n",
    "g2 = g0.copy()\n",
    "print prune_graph_once(g0)\n",
    "g3 = g0.copy()\n",
    "\n",
    "print G0, number_of_degree1_vertices(G0)\n",
    "print g1, number_of_degree1_vertices(g1)\n",
    "print g2, number_of_degree1_vertices(g2)\n",
    "print g3, number_of_degree1_vertices(g3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## With a graphic, it's better:\n",
    "I still don't get how to include graphic representation of graph with the function **gt.graphdraw()** in a subplot. So the graphic will be saved on disk and reloaded after."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plt.subplot(121)\n",
    "graphs = [G0,g1,g2,g3]\n",
    "name =['G0','g1','g2','g3']\n",
    "for g,n in zip(graphs, name):\n",
    "    gt.graph_draw(g, vertex_text=g.vertex_index, edge_pen_width=T.ep.weight,inline=True,\n",
    "                  output_size=(150, 150), output= n+'.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generate the response curve of the graph to iterative pruning\n",
    "Let's count the number of vertices of degree 1 as the graph is submitted to pruning.\n",
    "The toy graph used to check its response to pruning has two vertices of degree 3 and two vertices of degree 1 (vertices 2 an 3).\n",
    "The initial values of the weight of the edges bound to vertices 2 and 3, are:\n",
    "\n",
    "                                    15, 20\n",
    "The smallest weight is 15 So after one pruning, the weights becomes:\n",
    "\n",
    "                            15-15 = 0, 20-15 = 5\n",
    "The edge 1-2 of weight 0 is removed (vertex 2 is now of degree 0). The edge 1-3 has the smallest weight (5). So after a second round of pruning, the edge 1-3 is removed.\n",
    "\n",
    "The vertex 1 was of degree 2, now it is of 1. The weight of the edge 0-1 is 8.\n",
    "\n",
    "A third pruning removes the last edge 0-1 of weight equal to 8.\n",
    "\n",
    "The degree of vertex 0 is 2 (there's a loop), so the edge 0-0 can't be pruned.\n",
    "\n",
    "The different weights are cumulated as follow:\n",
    "\n",
    "                                    0, 15, 15+5, 20+8\n",
    "Applied to a skeleton made of pixels, this would mean that:\n",
    "\n",
    "        15 prunings of 1 pixel would remove one branche of size 15\n",
    "        20 prunings would remove the two branches of size 15 and 20\n",
    "        28 prunings would remove the three branches of sizes 15, 20 and 8 (initially between two vertices of degree 3).\n",
    "      \n",
    "The weights are store in a vector **X** and the corresponding counts of degree 1 vertices are store in a vector **Y**, both returned by the function **response_to_iterative_pruning()**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "T = make_toy_graph()\n",
    "TL = total_length_graph(T)\n",
    "X, Y= response_to_iterative_pruning(T.copy())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    " %matplotlib inline\n",
    "\n",
    "path = os.getcwd()\n",
    "im0 = mh.imread(path+'/'+'G0.png')\n",
    "im1 = mh.imread(path+'/'+'G1.png')\n",
    "im2 = mh.imread(path+'/'+'G2.png')\n",
    "im3 = mh.imread(path+'/'+'G3.png')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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nETnOgdE4d3QEb2yUdD7P4uKiqBM+w2RZZmRkhBs3bnA/kaBTLFLPF3Dbk9id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\n/wO+GIdYV81feQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9b84ed0c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gs = gridspec.GridSpec(3, 4)\n",
    "\n",
    "fig = plt.figure(figsize=(12,10))\n",
    "\n",
    "ax1 = plt.subplot(gs[0,0])\n",
    "ax1.set_xticks([])\n",
    "ax1.set_yticks([])\n",
    "ax1.imshow(im0)\n",
    "\n",
    "ax2 = plt.subplot(gs[0,1])\n",
    "ax2.set_xticks([])\n",
    "ax2.set_yticks([])\n",
    "ax2.imshow(im1)\n",
    "\n",
    "ax3 = plt.subplot(gs[0,2])\n",
    "ax3.set_xticks([])\n",
    "ax3.set_yticks([])\n",
    "ax3.imshow(im2)\n",
    "\n",
    "ax4 = plt.subplot(gs[0,3])\n",
    "ax4.set_xticks([])\n",
    "ax4.set_yticks([])\n",
    "ax4.imshow(im3)\n",
    "\n",
    "ax5 = plt.subplot(gs[1,:])\n",
    "ax5.set_ylabel('number  of degree 1 vertices')\n",
    "ax5.set_xlabel('pruning size')\n",
    "#print np.arange(0,35,5)\n",
    "ax5.set_xticks(np.arange(0, 45,5))\n",
    "ax5.set_ylim(-0.1,3.1)\n",
    "ax5.set_yticks([0, 1, 2, 3])\n",
    "ax5.step(X, Y, lw = 3)\n",
    "ax5.set_title('Response to iterative pruning of a weighted graph')\n",
    "ax5.scatter(X,Y,s=80,c='red')\n",
    "\n",
    "ax6 = plt.subplot(gs[2,:])\n",
    "ax6.set_ylabel('number  of degree 1 vertices')\n",
    "ax6.set_xlabel('pruning size (%size total weight)')\n",
    "#print np.arange(0,35,5)\n",
    "ax6.set_xticks(np.arange(0, 100,5))\n",
    "ax6.set_ylim(-0.1,3.1)\n",
    "ax6.set_yticks([0, 1, 2, 3])\n",
    "ax6.step(100*X/TL, Y, lw = 3)\n",
    "ax6.scatter(100*X/TL,Y,s=80,c='red')\n",
    "\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Make a feature vector from the pruning curve?\n",
    "Is it possible to make a feature vector fom the step curve in order to compute a similarity index between two curves?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's make three different graphs and generate their respective pruning curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def make_toy_graph2():\n",
    "    T = gt.Graph(directed = False)\n",
    "    edge_weights = T.new_edge_property('double')\n",
    "    T.properties[(\"e\",\"weight\")] = edge_weights\n",
    "\n",
    "    T.add_vertex(n=4)\n",
    "\n",
    "    e_1 = T.add_edge(0,1)\n",
    "    e_2 = T.add_edge(1,2)\n",
    "    e_3 = T.add_edge(0,0)\n",
    "    e_4 = T.add_edge(1,3)\n",
    "\n",
    "    edge_weights[e_1]= 5\n",
    "    edge_weights[e_2]= 11\n",
    "    edge_weights[e_3]= 9\n",
    "    edge_weights[e_4]= 7\n",
    "    return T\n",
    "\n",
    "def make_toy_graph3():\n",
    "    T = gt.Graph(directed = False)\n",
    "    edge_weights = T.new_edge_property('double')\n",
    "    T.properties[(\"e\",\"weight\")] = edge_weights\n",
    "\n",
    "    T.add_vertex(n=5)\n",
    "\n",
    "    e_1 = T.add_edge(0,1)\n",
    "    e_2 = T.add_edge(1,2)\n",
    "    #e_3 = T.add_edge(0,0)\n",
    "    e_4 = T.add_edge(1,3)\n",
    "    e_5 = T.add_edge(3,4)\n",
    "\n",
    "    edge_weights[e_1]= 6\n",
    "    edge_weights[e_2]= 13\n",
    "    #edge_weights[e_3]= 7\n",
    "    edge_weights[e_4]= 8\n",
    "    edge_weights[e_5]= 2\n",
    "    return T\n",
    "def make_toy_graph4():\n",
    "    T = gt.Graph(directed = False)\n",
    "    edge_weights = T.new_edge_property('double')\n",
    "    T.properties[(\"e\",\"weight\")] = edge_weights\n",
    "\n",
    "    T.add_vertex(n=5)\n",
    "\n",
    "    e_1 = T.add_edge(0,1)\n",
    "    e_2 = T.add_edge(1,2)\n",
    "    e_3 = T.add_edge(1,3)\n",
    "    e_4 = T.add_edge(1,3)\n",
    "    e_5 = T.add_edge(3,4)\n",
    "\n",
    "    edge_weights[e_1]= 6\n",
    "    edge_weights[e_2]= 13\n",
    "    edge_weights[e_3]= 7\n",
    "    edge_weights[e_4]= 8\n",
    "    edge_weights[e_5]= 2\n",
    "    return T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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m3utNN3RdnNi8uUdxJrG0hQvteTeu8gxMIxSF9KKiFGGzdwshohbrPJGJ+QlS\nAPD5fNXo6rrF8taOiOWttzvO/rtz1gzL0v96etaF6Q/9n1+3HXryqc6zn9XlyzK1JYsl7nE/u27d\num9Mlu54Jh4mSKFp2s8Ul2uBtqRsDusKqGeXdXqPHFNfrrju/ZHy6sVFHn1RqVO4nO9LkvSLyVEc\n/8TFIvTLL7/cedPNNx8nq2WxkZ+XA1mR2Knm0IgL0YpC2rLrMrSl1yZzj/s4Z+whr9e7d/JUxzdx\n0RSeZeB+LDbCfix99iwXLy5ONlKTQ8LlOgZJ+o7X6912ObXHG3FlLODMDlIhxP1MN5ZSKOSBqjpJ\n1yUKRwTsNuKyrJ+JdW7vJquyTdf1n69fv/7o5dYdb8Sdsc5SXV1dDGAFzqz9ZRPnbkhSN4DTRLRT\nCLHV6/U2XF6VJjGPuUxjYmJiYmJiYmJiYmIyBP8NWpb/uMw0QKgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<PropertyMap object with key type 'Vertex' and value type 'vector<double>', for Graph 0x9b993f4c, at 0x9b81f78c>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T1 = make_toy_graph()\n",
    "T2 = make_toy_graph2()\n",
    "T3 = make_toy_graph3()\n",
    "T4 = make_toy_graph4()\n",
    "\n",
    "gt.graph_draw(T1, vertex_text=T1.vertex_index, edge_pen_width=T1.ep.weight,inline=True,\n",
    "                  output_size=(150, 150))\n",
    "gt.graph_draw(T2, vertex_text=T2.vertex_index, edge_pen_width=T2.ep.weight,inline=True,\n",
    "                  output_size=(150, 150))\n",
    "gt.graph_draw(T3, vertex_text=T3.vertex_index, edge_pen_width=T3.ep.weight,inline=True,\n",
    "                  output_size=(150, 150))\n",
    "gt.graph_draw(T4, vertex_text=T4.vertex_index, edge_pen_width=T4.ep.weight,inline=True,\n",
    "                  output_size=(150, 150))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.   7.  11.  16.] [ 2.  1.  1.  0.]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x988d8e6c>]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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FbgZurmp3F/A08Bjw9gZ99fzvzINHH300rpuYiJUrV8bKlSvjuomJmJ+fH4jxRPnbfR9P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\nuGA2MzMzM2vCBbOZmZmZWRP/HwI2MElJiYZpAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9b8d2eac>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X1, Y1= response_to_iterative_pruning(T2.copy())\n",
    "X2, Y2= response_to_iterative_pruning(T3.copy())\n",
    "X3, Y3= response_to_iterative_pruning(T4.copy())\n",
    "print X1, Y1\n",
    "plt.figure(figsize=(12,3))\n",
    "plt.subplot(141)\n",
    "plt.scatter(X,Y)\n",
    "plt.step(X,Y)\n",
    "plt.subplot(142)\n",
    "plt.scatter(X1,Y1, c='r')\n",
    "plt.step(X1,Y1)\n",
    "plt.subplot(143)\n",
    "plt.scatter(X2,Y2, c='pink', s=50)\n",
    "plt.step(X2,Y2,c='pink')\n",
    "\n",
    "plt.subplot(144)\n",
    "plt.scatter(X1,Y1)\n",
    "plt.step(X1,Y1)\n",
    "plt.scatter(X2,Y2, c='pink', s=50)\n",
    "plt.step(X2,Y2,c='pink')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "graph-tool provides a similarity measure, but it doesn't seem to take the weight of the edges into account"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Similarity between the four graphs\n",
      "0.8\n",
      "0.75   0.6\n",
      "1.0   1.0   0.75   0.6\n"
     ]
    }
   ],
   "source": [
    "print \"Similarity between the four graphs\"\n",
    "\n",
    "print top.similarity(T4, T3)\n",
    "print top.similarity(T2, T3), ' ',top.similarity(T4, T2)\n",
    "print top.similarity(T1,T1),  ' ',\\\n",
    "      top.similarity(T1,T2), ' ',\\\n",
    "      top.similarity(T1, T3), ' ',top.similarity(T1, T4)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Distance between two pruning curves:\n",
    "\n",
    "   We could try to compute the area between two step curves. For two identical curves the area would be null, as the curves differe the area may increase. Fine! However, since the curves are just defined from discrete data, it may not be so easy to define.\n",
    "   \n",
    "   Let's forget the curves and just look at the points from the scatter plot.When looking at two pruning curves obtained from toy-graphs 2 and 3, we could just compute the distances matrix between the points from the two curves. The more the curves are different, the more the distances should increase ... why not, give it a try:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### scipy provides just what we need!\n",
    "\n",
    "The response curve is given as X, Y:\n",
    "\n",
    "    X : The pruning values\n",
    "    Y : the corresponding number of leaf in the graph (vertices of degree 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The X and Y arrays have to zipped so that we have couples of\n",
    "\n",
    "                     (pruning size, number of leafs)\n",
    "\n",
    "First compute the distances between the points of the same curve:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.  15.  20.  28.] [ 2.  1.  1.  0.]\n",
      "[(0.0, 2.0), (15.0, 1.0), (20.0, 1.0), (28.0, 0.0)]\n"
     ]
    }
   ],
   "source": [
    "T=make_toy_graph()\n",
    "#print T.edge_properties\n",
    "X, Y = response_to_iterative_pruning(T.copy())\n",
    "#TL = total_length_graph(T)\n",
    "print X, Y\n",
    "pruning_curve =  zip(X,Y)\n",
    "print pruning_curve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the distances between the points of the same curve (toy graph 1). The matrix is symetric, let's just see the upper elements of the matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  0.          15.03329638  20.02498439  28.0713377 ]\n",
      " [ 15.03329638   0.           5.          13.03840481]\n",
      " [ 20.02498439   5.           0.           8.06225775]\n",
      " [ 28.0713377   13.03840481   8.06225775   0.        ]]\n",
      "\n",
      "[[  0.          15.03329638  20.02498439  28.0713377 ]\n",
      " [  0.           0.           5.          13.03840481]\n",
      " [  0.           0.           0.           8.06225775]\n",
      " [  0.           0.           0.           0.        ]]\n",
      "sum of distances :  89.2302810268\n"
     ]
    }
   ],
   "source": [
    "D = cdist( pruning_curve, pruning_curve)\n",
    "print D\n",
    "\n",
    "print np.triu(D)\n",
    "print 'sum of distances : ', np.sum(np.triu(D))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This triangular matrix can be displayed as an image:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x9b99350c>"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO0AAADtCAYAAABTTfKPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nooUY0UKMaCHmN8MRJy9gzhAgAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9b99318c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.subplot(111, xticks=[],yticks=[])\n",
    "plt.imshow(np.triu(D),interpolation='nearest')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Similarity between two graphs:\n",
    "#### compute the distances between the points of their response to pruning curves \n",
    "\n",
    "For each of the three graphs $T, T_1, T_2$ , one can define $\\frac{3^2-3}{2}+3={6}$ matrices of distances.\n",
    "\n",
    "Those matrices are elements of another matrix (a matrix of matrices). The elements of the diagonal are squared matrices corresponding to \"intra-distances\" (distances between the points of the same response curve). The elements outside the diagonal correspond to \"inter-distances\" matrices (distances computed from points belonging to two different response curves), the \"interdistance matrices\", here, are rectangular since the corresponding pairs of graphs do not have the same number of vertices.  \n",
    "\n",
    "#### computing the matrix of distances between two response curves\n",
    "\n",
    "Response curves of a graph to pruning, **prc**, are generated. Then the matrix of distances between the two response curves, is generated (Pruning Curve Matrix or **PCM**).\n",
    "    \n",
    "    PCM_00, PCM_11, PCM_22 are are matrices of 'intra-distances'\n",
    "    Other matrices are 'inter-distances' matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def distance_matrix(prc_a, prc_b):\n",
    "    #print len(prc_a) == len(prc_b)\n",
    "    return cdist(prc_a, prc_b)\n",
    "    #return np.triu(D)\n",
    "\n",
    "prc0 = pruning_curve\n",
    "prc1 = zip(X1,Y1)\n",
    "prc2 = zip(X2,Y2)\n",
    "prc3 = zip(X3,Y3)\n",
    "\n",
    "PCM_00=distance_matrix(prc0, prc0)\n",
    "PCM_01=distance_matrix(prc0, prc1)\n",
    "PCM_02=distance_matrix(prc0, prc2)\n",
    "PCM_03=distance_matrix(prc0, prc3)\n",
    "\n",
    "PCM_11=distance_matrix(prc1, prc1)\n",
    "PCM_12=distance_matrix(prc1, prc2)\n",
    "PCM_13=distance_matrix(prc1, prc3)\n",
    "\n",
    "PCM_22=distance_matrix(prc2, prc2)\n",
    "PCM_23=distance_matrix(prc3, prc2)\n",
    "\n",
    "PCM_33=distance_matrix(prc3, prc3)\n",
    "#distance total\n",
    "dt00 = np.sum(PCM_00)\n",
    "dt01 = np.sum(PCM_01)\n",
    "dt02 = np.sum(PCM_02)\n",
    "\n",
    "dt11 = np.sum(PCM_11)\n",
    "dt12 = np.sum(PCM_12)\n",
    "\n",
    "dt22 = np.sum(PCM_22)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plotting matrices of distances"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x98ef750c>"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO0AAADtCAYAAABTTfKPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nYUQLYUQLYUQLYUQLYf4DNBYV2oBlVvIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9ba104cc>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gs = gridspec.GridSpec(3, 3)\n",
    "\n",
    "fig = plt.figure(figsize=(4,4))\n",
    "\n",
    "ax00 = plt.subplot(gs[0,0])\n",
    "ax00.set_xticks([])\n",
    "ax00.set_yticks([])\n",
    "ax00.imshow(PCM_00, interpolation='nearest')\n",
    "\n",
    "ax01 = plt.subplot(gs[0,1])\n",
    "ax01.set_xticks([])\n",
    "ax01.set_yticks([])\n",
    "ax01.imshow(PCM_01, interpolation='nearest')\n",
    "\n",
    "ax02 = plt.subplot(gs[0,2])\n",
    "ax02.set_xticks([])\n",
    "ax02.set_yticks([])\n",
    "ax02.imshow(PCM_02, interpolation='nearest')\n",
    "\n",
    "ax11 = plt.subplot(gs[1,1])\n",
    "ax11.set_xticks([])\n",
    "ax11.set_yticks([])\n",
    "ax11.imshow(PCM_11, interpolation='nearest')\n",
    "\n",
    "ax12 = plt.subplot(gs[1,2])\n",
    "ax12.set_xticks([])\n",
    "ax12.set_yticks([])\n",
    "ax12.imshow(PCM_12, interpolation='nearest')\n",
    "\n",
    "ax22 = plt.subplot(gs[2,2])\n",
    "ax22.set_xticks([])\n",
    "ax22.set_yticks([])\n",
    "ax22.imshow(PCM_22, interpolation='nearest')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Distance between two graphs:\n",
    "For each pair of graphs, there is a matrix of distance**s**. So for such **a** matrix, a distanc**e**, a total distance (**dt**) can be computed as the sum of all distances.\n",
    "\n",
    "So, for three graphs, we have 6 different total distances $dt_{00}, dt_{01},..., dt_{22}$ since :\n",
    "\n",
    "$$distance(graph_1, graph_2) = distance(graph_2, graph_1)$$\n",
    "\n",
    "Those distances can be arranged in 3x3 symetric square matrix. The matrix of total distances (**DTM**):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "DTM = np.zeros((3,3))\n",
    "DTM[0,0]=dt00\n",
    "DTM[0,1]=dt01\n",
    "DTM[0,2]=dt02\n",
    "DTM[1,1]=dt11\n",
    "DTM[1,2]=dt12\n",
    "DTM[2,2]=dt22"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plotting DTM\n",
    "We can see that some upper element of **DTM** are smaller than elements of the diagonal. This isn't too good, a kind of normalization should be done to fix this issue."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 178.  187.  256.]\n",
      " [   0.  105.  135.]\n",
      " [   0.    0.  140.]]\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO0AAADtCAYAAABTTfKPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAAzlJREFUeJzt2TFNBFEYRtF5ZAXghBYJKKBHCWCDBhtIoMUCCnDwMLA7\nhAR43M055fxbfM3NTLJjzrkBHRerBwDfI1qIES3EiBZiRAsxooWYw95xjOH/IFhkzjmOPd+Ndtu2\nbb7+/Jiz8756QMfD7eoFDY87N5/HECNaiBEtxIgWYkQLMaKFGNFCjGghRrQQI1qIES3EiBZiRAsx\nooUY0UKMaCFGtBAjWogRLcSIFmJECzGihRjRQoxoIUa0ECNaiBEtxIgWYkQLMaKFGNFCjGghRrQQ\nI1qIES3EiBZiRAsxooUY0UKMaCFGtBAjWogRLcSIFmJECzGihRjRQoxoIUa0ECNaiBEtxIgWYkQL\nMaKFGNFCjGghRrQQI1qIES3EiBZiRAsxooUY0UKMaCFGtBAjWogRLcSIFmJECzGihRjRQoxoIebw\n1Q/G9f1f7Ei7mjerJ2TczefVExrG08mTNy3EiBZiRAsxooUY0UKMaCFGtBAjWogRLcSIFmJECzGi\nhRjRQoxoIUa0ECNaiBEtxIgWYkQLMaKFGNFCjGghRrQQI1qIES3EiBZiRAsxooUY0UKMaCFGtBAj\nWogRLcSIFmJECzGihRjRQoxoIUa0ECNaiBEtxIgWYkQLMaKFGNFCjGghRrQQI1qIES3EiBZiRAsx\nooUY0UKMaCFGtBAjWogRLcSIFmJECzGihRjRQoxoIUa0ECNaiBEtxIgWYkQLMaKFGNFCjGghRrQQ\nI1qIES3EHFYPOAdv42X1hIzL+bF6Qp43LcSIFmJECzGihRjRQoxoIUa0ECNaiBEtxIgWYkQLMaKF\nGNFCjGghRrQQI1qIES3EiBZiRAsxooUY0UKMaCFGtBAjWogRLcSIFmJECzGihRjRQoxoIUa0ECNa\niBEtxIgWYkQLMaKFGNFCjGghRrQQI1qIES3EiBZiRAsxooUY0UKMaCFGtBAjWogRLcSIFmJECzGi\nhRjRQoxoIUa0ECNaiBEtxIgWYkQLMaKFGNFCjGghRrQQI1qIES3EiBZiRAsxooUY0UKMaCFGtBAj\nWogRLcSIFmLGnPP0cYzTR+BXzTnHsee70QL/j89jiBEtxIgWYkQLMaKFmE/s0h8orI8JaQAAAABJ\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9897fd6c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.subplot(111,xticks=[], yticks=[])\n",
    "plt.imshow(DTM, interpolation='nearest')\n",
    "print np.round(DTM)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Defining a normalized distance between two graphs\n",
    "The total inter distances is normalized by the sum of total intra-distances. The total inter distances is multiplied by two in order to have a normalised distance $nD$ $$ nD(response curve_i, response curve_i) =1$$ when conputed against the pair of same response curve. Finally $nD$ is computed as follow:\n",
    "\n",
    "$$nD(r_i,r_j)=\\frac{1}{\\frac{2distance(r_i,r_j)}{r_i+r_j}}=\\frac{r_i+r_j}{2distance(r_i,r_j)} $$ such that $$nD(r_i,r_j)<=1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def normalized_total_distance(response1, response2):\n",
    "    intra_distances_1 = cdist(response1, response1)\n",
    "    intra_distances_2 = cdist(response2, response2)\n",
    "    inter_distances = cdist(response1, response2)\n",
    "    total_intra_1 = np.sum(intra_distances_1)\n",
    "    total_intra_2 = np.sum(intra_distances_2)\n",
    "    total_inter = np.sum(inter_distances)\n",
    "    ntd = 2.0*total_inter/(total_intra_1+total_intra_2)\n",
    "    return 1/ntd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ntd_00 = normalized_total_distance(prc0, prc0)\n",
    "ntd_01 = normalized_total_distance(prc0, prc1)\n",
    "ntd_02 = normalized_total_distance(prc0, prc2)\n",
    "ntd_11 = normalized_total_distance(prc1, prc1)\n",
    "ntd_12 = normalized_total_distance(prc1, prc2)\n",
    "ntd_22 = normalized_total_distance(prc2, prc2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Similarity of three graphs\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.          0.75751403  0.62186779]\n",
      " [ 0.          1.          0.90636131]\n",
      " [ 0.          0.          1.        ]]\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO0AAADtCAYAAABTTfKPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x98e0220c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nDTM = np.zeros((3,3))\n",
    "nDTM[0,0]=ntd_00\n",
    "nDTM[0,1]=ntd_01\n",
    "nDTM[0,2]=ntd_02\n",
    "nDTM[1,1]=ntd_11\n",
    "nDTM[1,2]=ntd_12\n",
    "nDTM[2,2]=ntd_22\n",
    "\n",
    "plt.subplot(111,xticks=[], yticks=[])\n",
    "plt.imshow(nDTM, interpolation='nearest')\n",
    "print (nDTM)"
   ]
  }
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