Data Science/Graph-Mining-with-Networkx/5-clustering.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SD212: Graph mining\n",
    "## Lab 5: Clustering\n",
    "\n",
    "You will learn how to cluster the nodes of a graph and to estimate the quality of the clustering. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import networkx as nx\n",
    "actors_movies = nx.read_graphml(\"actors_movies.graphml\", node_type = int)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Import"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import networkx as nx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The documentation is available [here](https://networkx.readthedocs.io/en/stable/)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data\n",
    "\n",
    "You will need the following datasets (the same as in previous labs, no need to download them again):\n",
    "* [Les Misérables](http://perso.telecom-paristech.fr/~bonald/graphs/miserables.graphml.gz)<br>  Graph connecting the characters of the [novel of Victor Hugo](https://fr.wikisource.org/wiki/Les_Misérables) when they appear in the same chapter. The graph is undirected and weighted. Weights correspond to the number of chapters in which characters appear together. \n",
    "* [Openflights](http://perso.telecom-paristech.fr/~bonald/graphs/openflights.graphml.gz)<br>\n",
    "Graph of the main international flights. Nodes are airports. The graph is undirected (all flights are bidirectional). Weights correspond to the number of daily flights between airports. Extracted from [Openflights](http://openflights.org).\n",
    "* [Wikipedia for schools](http://perso.telecom-paristech.fr/~bonald/graphs/wikipedia_schools.graphml.gz)<br> Graph of the hyperlinks between a subset of the pages of the English Wikipedia. The graph is directed and unweighted.\n",
    "More information [here](https://en.wikipedia.org/wiki/Wikipedia:Wikipedia_for_Schools).\n",
    "* [Actors-Movies](http://perso.telecom-paristech.fr/~bonald/graphs/actors_movies.graphml.gz)<br>\n",
    "Graph between actors and movies (a link exists if the actor stars in the movie). The graph is bipartite (see the field 'type' of each node). \n",
    "Extracted from [YaGo](http://yago-knowledge.org)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Modularity\n",
    "\n",
    "Consider an undirected, unweighted graph of $n$ nodes and $m$ edges with adjacency matrix $A$, without self-loops.\n",
    "\n",
    "The modularity of clustering $C$ is defined by:\n",
    "$$\n",
    "Q(C) = \\frac 1 {2m} \\sum_{i,j\\in V} \\left(A_{ij} - \\frac {d_id_j}{2m}\\right) \\delta_{C(i),C(j)}\n",
    "$$\n",
    "where $d_i = \\sum_j A_{ij}$ is the degree of node $i$.\n",
    "\n",
    "The modularity can also be written:\n",
    "$$\n",
    "Q(C) = \\sum_k  \\frac{m_k}{m} - \\sum_k  \\left( \\frac{v_k}{v}\\right)^2\n",
    "$$\n",
    "where\n",
    "* $m_k$ is the number of edges in cluster $k$  \n",
    "* $v_k $ is the volume of cluster $k$ (total degree of nodes in the cluster)\n",
    "* $v$ is the volume of the graph (total degree of nodes)\n",
    "\n",
    "The definition extends to a weighted graph as follows:\n",
    "$$\n",
    "Q(C) = \\sum_k  \\frac{w_k}{w} - \\sum_k  \\left( \\frac{v_k}{v}\\right)^2\n",
    "$$\n",
    "where\n",
    "* $w_k$ is the weight of cluster $k$ (total weight of edges in the cluster)\n",
    "* $w$ is the weight of the graph (total weight of edges in the graph)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Toy graph\n",
    "\n",
    "Consider the following graph:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "edges = [(7,5),(5,3),(3,7),(4,5),(8,6),(7,4),(1,6),(9,8),(7,8),(6,1),(2,9),(8,2),(1,9)]\n",
    "graph = nx.Graph()\n",
    "graph.add_edges_from(edges)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "with the following clustering:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "C = {1: 0, 2: 0, 6: 0, 8: 0, 9: 0, 3: 1, 4: 1, 5: 1, 7: 1}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Visualization:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "pos = nx.spring_layout(graph)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "prop_cycle = plt.rcParams['axes.prop_cycle']\n",
    "colors = prop_cycle.by_key()['color']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "node_colors = [colors[C[u] % len(colors)] for u in graph.nodes()]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
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l5TZbulpbW5GXl2ezpYsNU2bfjTfeiMsuuww33eTa88LJnk6nQ3Fx8WgQ7927FxERETaj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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f47fba63710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize = (6,3))\n",
    "nx.draw(graph, pos, node_size = 400, node_color = node_colors, with_labels = True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## To do\n",
    "\n",
    "* What is the fraction of edges within clusters?\n",
    "* What is the Simpson's index associated with the volume distribution between clusters?\n",
    "* Deduce the modularity of the clustering. \n",
    "* What is the maximum value of modularity for 2 clusters?\n",
    "* Complete the function `get_modularity` below and check it on the above graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Blue = 2 + 3 + 2 + 3 + 2 = 12            \n",
    "Orange = 3 + 2 + 3 + 2 = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.41319444444444464\n",
      "Maximum modularity of two clusters is 1 - 1/K = 0.5\n"
     ]
    }
   ],
   "source": [
    "Q = 6 / 12 + 5 / 12 - (13/24)**2 - (11/24)**2\n",
    "print(Q)\n",
    "print(\"Maximum modularity of two clusters is 1 - 1/K = 0.5\" )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_modularity(graph,C):\n",
    "    '''\n",
    "    graph: networkx graph\n",
    "        undirected graph (possibly weighted)\n",
    "    C: dictionary\n",
    "        cluster of each node\n",
    "        \n",
    "    Returns: float\n",
    "        modularity\n",
    "    '''\n",
    "    # add weights if none\n",
    "    if nx.get_edge_attributes(graph,'weight') == {}:\n",
    "        for u,v in graph.edges():\n",
    "            graph[u][v]['weight'] = 1\n",
    "            \n",
    "    cluster = set(C.values())\n",
    "    cluster_weight = {k: 0 for k in cluster}\n",
    "    cluster_volume = {k: 0 for k in cluster}\n",
    "    m = float(len(edges))\n",
    "    clusters = list(C.values())\n",
    "    for u,v in graph.edges():\n",
    "        if (clusters[u-1] == clusters[v-1]):\n",
    "            cluster_weight[clusters[u-1]] = cluster_weight[clusters[u-1]] + graph[u][v]['weight']\n",
    "    for u in graph.nodes():\n",
    "        summ = 0\n",
    "        for x,v in graph.edges():\n",
    "            if (x==u or v==u):\n",
    "                summ = summ + graph[x][v]['weight']\n",
    "        cluster_volume[clusters[u-1]] = cluster_volume[clusters[u-1]] + summ\n",
    "    Q = 0\n",
    "    for i in cluster:\n",
    "        Q = Q + (cluster_weight[i] / m) - (cluster_volume[i]/(2*m))**2\n",
    "        #((m1 + m2) / m) - (v1/v)**2 - (v2/v)**2\n",
    "    return Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4171597633136096"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_modularity(graph, C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stochastic block model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The [stochastic block model](https://en.wikipedia.org/wiki/Stochastic_block_model) is a random graph with an underlying structure in clusters (the blocks)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "block_sizes = [10, 15, 20]\n",
    "p_in = 0.4 # probability of connection within blocks\n",
    "p_out = 0.02 # probability of connection across blocks\n",
    "graph = nx.random_partition_graph(sizes = block_sizes, p_in = p_in, p_out = p_out,seed = 0)\n",
    "pos = nx.spring_layout(graph)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "pos = nx.spring_layout(graph)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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CoRBRUVF49eoVPn78CAMDA7i5udEjMYnY2FjExsZi0qRJ6NChA4RCIcLDwxvkOmZVpvPZ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\naBabnHMDuQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f47fa6734e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize = (6,4))\n",
    "nx.draw(graph, pos, node_size = 400, node_color = 'gray', with_labels = True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## To do\n",
    "\n",
    "* Propose a clustering for this graph and compute the modularity.\n",
    "* What is the strongest cluster? \n",
    "* Complete the function `get_cluster_strength` below and check your guess.\n",
    "* Comment the results.\n",
    "\n",
    "**Hint:** Recall that the strength of cluster $k$ is defined by: $$\\frac{2w_k }{ v_k}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 0,\n",
       " 1: 0,\n",
       " 2: 0,\n",
       " 3: 0,\n",
       " 4: 0,\n",
       " 5: 0,\n",
       " 6: 0,\n",
       " 7: 0,\n",
       " 8: 0,\n",
       " 9: 0,\n",
       " 10: 1,\n",
       " 11: 1,\n",
       " 12: 1,\n",
       " 13: 1,\n",
       " 14: 1,\n",
       " 15: 1,\n",
       " 16: 1,\n",
       " 17: 1,\n",
       " 18: 1,\n",
       " 19: 1,\n",
       " 20: 1,\n",
       " 21: 1,\n",
       " 22: 1,\n",
       " 23: 1,\n",
       " 24: 1,\n",
       " 25: 2,\n",
       " 26: 2,\n",
       " 27: 2,\n",
       " 28: 2,\n",
       " 29: 2,\n",
       " 30: 2,\n",
       " 31: 2,\n",
       " 32: 2,\n",
       " 33: 2,\n",
       " 34: 2,\n",
       " 35: 2,\n",
       " 36: 2,\n",
       " 37: 2,\n",
       " 38: 2,\n",
       " 39: 2,\n",
       " 40: 2,\n",
       " 41: 2,\n",
       " 42: 2,\n",
       " 43: 2,\n",
       " 44: 2}"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = dict([(el,0) for el in [i for i in range(10)]]+[(el,1) for el in [i for i in range(10,25)]]+[(el,2) for el in [i for i in range(25,45)]])\n",
    "C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_cluster_strength(graph,C):\n",
    "    '''\n",
    "    graph: networkx graph\n",
    "        undirected graph (possibly weighted)\n",
    "    C: dictionary\n",
    "        cluster of each node\n",
    "        \n",
    "    Returns: dictionary\n",
    "        strength of each cluster\n",
    "    '''\n",
    "    \n",
    "    # add weights if none\n",
    "    if nx.get_edge_attributes(graph,'weight') == {}:\n",
    "        for u,v in graph.edges():\n",
    "            graph[u][v]['weight'] = 1\n",
    "    \n",
    "    cluster = set(C.values())\n",
    "    cluster_weight = {k: 0 for k in cluster}\n",
    "    cluster_volume = {k: 0 for k in cluster}\n",
    "    cluster_strength = {k: 0 for k in cluster}\n",
    "    # to be completed\n",
    "    \n",
    "\n",
    "    m = float(len(edges))\n",
    "    clusters = list(C.values())\n",
    "    for u,v in graph.edges():\n",
    "        print(u,v)\n",
    "        if (clusters[u-1] == clusters[v-1]):\n",
    "            cluster_weight[clusters[u-1]] = cluster_weight[clusters[u-1]] + graph[u][v]['weight']\n",
    "    for u in graph.nodes():\n",
    "        summ = 0\n",
    "        for x,v in graph.edges():\n",
    "            if (x==u or v==u):\n",
    "                summ = summ + graph[x][v]['weight']\n",
    "        cluster_volume[clusters[u-1]] = cluster_volume[clusters[u-1]] + summ\n",
    "    \n",
    "    for i in cluster_strength.keys():\n",
    "        cluster_strength[i] = 2*cluster_weight[i] / cluster_volume[i]\n",
    "    return cluster_strength"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 8\n",
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     ]
    },
    {
     "data": {
      "text/plain": [
       "{0: 0.7142857142857143, 1: 0.8043478260869565, 2: 0.8902439024390244}"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_cluster_strength(graph,C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. The Louvain algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Louvain algorithm attempts to maximize modularity as follows:\n",
    "1. **(Initialization)** $C \\gets \\text{identity}$ \n",
    "2.  **(Maximization)** While modularity increases, update $C$ by moving one node  from its cluster to one of its neighbors\n",
    "3. **(Aggregation)** Merge all nodes belonging to the same cluster into a single node, update the weights accordingly and apply step 2 to the  aggregate graph\n",
    "\n",
    "In step 2, nodes are considered in a cyclic way. The target cluster of a node is chosen among  its neighbors as that providing the **largest gain** in modularity, if any."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## To do\n",
    "\n",
    "* Complete the function `maximize_modularity` corresponding to step 2.\n",
    "* Test your `louvain` function on the stochastic block model.\n",
    "* Test different resolution parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "def maximize_modularity(graph, resolution):\n",
    "    '''\n",
    "    graph: networkx graph\n",
    "        undirected graph \n",
    "    resolution: float\n",
    "        resolution parameter (positive)\n",
    "        \n",
    "    Returns: dictionary\n",
    "        cluster of each node\n",
    "    '''\n",
    "    m = len(list(graph.edges()))\n",
    "    node_weight = {u: 0. for u in graph.nodes()}\n",
    "    w = 0\n",
    "    for u,v in graph.edges():\n",
    "        node_weight[u] += graph[u][v]['weight']\n",
    "        node_weight[v] += graph[u][v]['weight']\n",
    "        w += graph[u][v]['weight']\n",
    "    # init the clustering\n",
    "    C = {u:u for u in graph.nodes()}\n",
    "    # cluster weights\n",
    "    cluster_weight = {u: node_weight[u] for u in graph.nodes()}\n",
    "    # node-cluster weights (between each node and the clusters)\n",
    "    node_cluster_weight = {u:{v: 1. * graph[u][v]['weight'] for v in graph.neighbors(u) if v != u} \n",
    "                           for u in graph.nodes()}\n",
    "    increase = True\n",
    "    while increase:\n",
    "        increase = False\n",
    "        for u in graph.nodes():\n",
    "            k = C[u]            \n",
    "            if k not in node_cluster_weight[u]:\n",
    "                node_cluster_weight[u][k] = 0\n",
    "            variation = {k: 0}\n",
    "            for l in C.keys():\n",
    "                if (l!=k and (l in node_cluster_weight[u])):\n",
    "                    variation[l] = (1/float(w)) * (node_cluster_weight[u][l] - node_cluster_weight[u][k]) - 2*resolution*node_weight[u]*(cluster_weight[l] - cluster_weight[k] + node_weight[u])/(4 * (m**2))\n",
    "            l = max(variation, key = variation.get)\n",
    "            if variation[l] > 0:\n",
    "                increase = True\n",
    "                # move node u from cluster k to cluster l\n",
    "                C[u] = l\n",
    "                cluster_weight[k] -= node_weight[u]\n",
    "                cluster_weight[l] += node_weight[u]\n",
    "                for v in graph.neighbors(u):\n",
    "                    if v != u:\n",
    "                        node_cluster_weight[v][k] -= graph[u][v]['weight']\n",
    "                        if node_cluster_weight[v][k] <= 0:\n",
    "                            node_cluster_weight[v].pop(k)\n",
    "                        if l not in node_cluster_weight[v]:\n",
    "                            node_cluster_weight[v][l] = 0\n",
    "                        node_cluster_weight[v][l] += graph[u][v]['weight']\n",
    "    return C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "def aggregate_graph(graph,C):\n",
    "    '''\n",
    "    graph: networkx graph\n",
    "        undirected graph \n",
    "    C: dictionary\n",
    "        cluster of each node\n",
    "        \n",
    "    Returns: networkx graph\n",
    "        aggregate graph with one node per cluster\n",
    "    '''\n",
    "\n",
    "    aggregate_graph_ = nx.Graph()\n",
    "    aggregate_graph_.add_nodes_from(set(C.values()))\n",
    "    for u,v in graph.edges():\n",
    "        if aggregate_graph_.has_edge(C[u],C[v]):\n",
    "            aggregate_graph_[C[u]][C[v]]['weight'] += graph[u][v]['weight']\n",
    "        else:\n",
    "            aggregate_graph_.add_edge(C[u],C[v])\n",
    "            aggregate_graph_[C[u]][C[v]]['weight'] = graph[u][v]['weight']\n",
    "    return aggregate_graph_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "def louvain(graph,resolution = 1):\n",
    "    '''\n",
    "    graph: networkx graph\n",
    "        undirected graph \n",
    "    resolution: float\n",
    "        resolution parameter (positive)\n",
    "        \n",
    "    Returns: dictionary\n",
    "        cluster of each node\n",
    "    '''\n",
    "\n",
    "    if nx.get_edge_attributes(graph,'weight') == {}:\n",
    "        for u,v in graph.edges():\n",
    "            graph[u][v]['weight'] = 1\n",
    "\n",
    "    C = maximize_modularity(graph, resolution)\n",
    "    n = len(C)\n",
    "    k = len(set(C.values()))\n",
    "    while k < n:\n",
    "        aggregate_graph_ = aggregate_graph(graph,C) \n",
    "        C_new = maximize_modularity(aggregate_graph_, resolution)\n",
    "        C = {u: C_new[C[u]] for u in graph.nodes()}\n",
    "        n = k\n",
    "        k = len(set(C_new.values()))\n",
    "    # reindex clusters in decreasing order of size\n",
    "    clusters = list(set(C.values()))\n",
    "    cluster_size = {k: 0 for k in clusters}\n",
    "    for u in C:\n",
    "        cluster_size[C[u]] += 1\n",
    "    cluster_index = sorted(cluster_size, key = cluster_size.get, reverse = True)\n",
    "    reindex = {k:i for i,k in enumerate(cluster_index)}\n",
    "    C = {u:reindex[C[u]] for u in C}\n",
    "    return C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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     ]
    },
    {
     "data": {
      "text/plain": [
       "{0: 0.8043478260869565,\n",
       " 1: 0.5052631578947369,\n",
       " 2: 0.2318840579710145,\n",
       " 3: 0.46153846153846156,\n",
       " 4: 0.4}"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = louvain(graph)\n",
    "get_cluster_strength(graph,C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Real data\n",
    "\n",
    "The function `get_clusters` below returns the clusters as lists of nodes in decreasing order of weight."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_clusters(graph, C):\n",
    "    '''\n",
    "    graph: networkx graph\n",
    "        undirected graph \n",
    "    C: dictionary\n",
    "        cluster of each node\n",
    "        \n",
    "    Returns: dictionary of lists\n",
    "        for each cluster, nodes are listed in decreasing order of weight\n",
    "    '''\n",
    "\n",
    "    if nx.get_edge_attributes(graph,'weight') == {}:\n",
    "        for u,v in graph.edges():\n",
    "            graph[u][v]['weight'] = 1\n",
    "\n",
    "    node_weight = {u: 0 for u in graph.nodes()}\n",
    "    for u,v in graph.edges():\n",
    "        node_weight[u] += graph[u][v]['weight']\n",
    "        node_weight[v] += graph[u][v]['weight']\n",
    "        \n",
    "    clusters = list(set(C.values()))\n",
    "    cluster_nodes = {k:{} for k in clusters}\n",
    "    for u in graph.nodes():\n",
    "        cluster_nodes[C[u]][u] = node_weight[u]\n",
    "\n",
    "    cluster_nodes = {k: sorted(cluster_nodes[k], key = cluster_nodes[k].get, reverse = True)\n",
    "                             for k in clusters}\n",
    "        \n",
    "    return cluster_nodes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: [14, 15, 23, 16, 18, 19, 17, 20, 21, 11, 22, 13, 24, 10, 12],\n",
       " 1: [29, 32, 38, 39, 25, 27, 34, 43, 37, 31, 40],\n",
       " 2: [35, 36, 44, 41, 26, 28, 33, 42, 30],\n",
       " 3: [8, 5, 6, 1, 2, 3],\n",
       " 4: [0, 7, 9, 4]}"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_clusters(graph,C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## To do\n",
    "\n",
    "* List the top-3 characters of each cluster of Les Miserables (resolution 1).\n",
    "* List the top-3 airports of each cluster of Openflights (resolutions 1 and 2); visualize the clusters, with the associate top airports.\n",
    "* List the top-5 pages of the 20 largest clusters of Wikipedia for Schools (resolution 2). Compare the results obtained when the graph is considered as undirected and bipartite.\n",
    "* List the top-5 actors in the cluster of **Marion Cotillard** in the actors-movies graph (resolution 10).\n",
    "\n",
    "**Note:** You may compute the size and strength of each cluster."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "miserables = nx.read_graphml(\"miserables.graphml\", node_type = int)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "openflights = nx.read_graphml(\"openflights.graphml\", node_type = int)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "wikipedia = nx.read_graphml(\"wikipedia_schools.graphml\", node_type = int)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "wikipedia_undirected = wikipedia.to_undirected()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bipartite_from_directed(graph):\n",
    "    '''\n",
    "    graph: networkx graph\n",
    "        directed graph \n",
    "        \n",
    "    Returns: networkx graph\n",
    "        undirected, bipartite graph\n",
    "    '''    \n",
    "    new_graph = nx.Graph()\n",
    "    nodes = list(graph.nodes())\n",
    "    if nodes == list(np.arange(len(nodes))):\n",
    "        new_graph.add_nodes_from(nodes)\n",
    "        new_graph.add_nodes_from(list(np.arange(-len(nodes),0)))\n",
    "        edges = [(u,-v-1) for u,v in graph.edges()]\n",
    "        new_graph.add_edges_from(edges)\n",
    "    else:\n",
    "        print('Nodes must be indexed from 0 to n-1.')\n",
    "    return new_graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "wikipedia_bipartite = bipartite_from_directed(wikipedia)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "actors_movies = nx.read_graphml(\"actors_movies.graphml\", node_type = int)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "marion_cotillard = 5172"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "def louvain_resolution(graph,resolution):\n",
    "    '''\n",
    "    graph: networkx graph\n",
    "        undirected graph \n",
    "    resolution: float\n",
    "        resolution parameter (positive)\n",
    "        \n",
    "    Returns: dictionary\n",
    "        cluster of each node\n",
    "    '''\n",
    "\n",
    "    if nx.get_edge_attributes(graph,'weight') == {}:\n",
    "        for u,v in graph.edges():\n",
    "            graph[u][v]['weight'] = 1\n",
    "\n",
    "    C = maximize_modularity(graph, resolution)\n",
    "    n = len(C)\n",
    "    k = len(set(C.values()))\n",
    "    while k < n:\n",
    "        aggregate_graph_ = aggregate_graph(graph,C) \n",
    "        C_new = maximize_modularity(aggregate_graph_, resolution)\n",
    "        C = {u: C_new[C[u]] for u in graph.nodes()}\n",
    "        n = k\n",
    "        k = len(set(C_new.values()))\n",
    "    # reindex clusters in decreasing order of size\n",
    "    clusters = list(set(C.values()))\n",
    "    cluster_size = {k: 0 for k in clusters}\n",
    "    for u in C:\n",
    "        cluster_size[C[u]] += 1\n",
    "    cluster_index = sorted(cluster_size, key = cluster_size.get, reverse = True)\n",
    "    reindex = {k:i for i,k in enumerate(cluster_index)}\n",
    "    C = {u:reindex[C[u]] for u in C}\n",
    "    return C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: [0, 3, 2, 8, 1, 4, 5, 6, 7, 9],\n",
       " 1: [34, 35, 36, 37, 38, 29],\n",
       " 2: [69, 68, 70, 75, 71],\n",
       " 3: [49, 51, 54, 56, 53],\n",
       " 4: [19, 17, 18],\n",
       " 5: [20, 21, 22],\n",
       " 6: [28, 44, 45],\n",
       " 7: [48, 47, 46],\n",
       " 8: [26, 72],\n",
       " 9: [66, 76],\n",
       " 10: [57, 67],\n",
       " 11: [23, 12],\n",
       " 12: [73, 74],\n",
       " 13: [31, 30],\n",
       " 14: [27, 33],\n",
       " 15: [25, 40],\n",
       " 16: [41, 42],\n",
       " 17: [24, 50],\n",
       " 18: [39, 52],\n",
       " 19: [64],\n",
       " 20: [65],\n",
       " 21: [10],\n",
       " 22: [11],\n",
       " 23: [13],\n",
       " 24: [14],\n",
       " 25: [15],\n",
       " 26: [16],\n",
       " 27: [32],\n",
       " 28: [43],\n",
       " 29: [55],\n",
       " 30: [58],\n",
       " 31: [59],\n",
       " 32: [60],\n",
       " 33: [61],\n",
       " 34: [62],\n",
       " 35: [63]}"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = louvain_resolution(miserables,1)\n",
    "get_clusters(miserables,C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## 3. Soft clustering"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have so far considered **hard clustering** where the objective is to find a partition of the set of nodes. In practice, it may be interesting to allow some nodes to belong to several clusters. This is **soft clustering**. A simple approach consists in identifying so-called flipping nodes, for which a change of cluster has a negligible impact on modularity (for some given threshold). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## To do\n",
    "\n",
    "* Complete the function `get_flipping_nodes` below.\n",
    "* Test your function on the bow-tie graph.\n",
    "* Display the airports belonging to several clusters in Openflights (resolutions 1 and 2)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_flipping_nodes(graph, C, resolution = 1, threshold = 1e-5):\n",
    "    '''\n",
    "    graph: networkx graph\n",
    "        undirected graph \n",
    "    C: dictionary\n",
    "        cluster of each node\n",
    "    resolution: float\n",
    "        resolution parameter \n",
    "    threshold: float\n",
    "        threshold on modularity \n",
    "        \n",
    "    Returns: dictionary of lists\n",
    "        list of clusters of each flipping node (i.e., with more than 1 cluster)\n",
    "    '''\n",
    "\n",
    "    flipping_nodes = {}\n",
    "\n",
    "    node_weight = {u: 0. for u in graph.nodes()}\n",
    "    number_cluster = len(set(C.values()))\n",
    "    for u in graph.nodes():\n",
    "        k = C[u] #cluster dont le point u appartient            \n",
    "        variation = {k: 0} #modularity variation\n",
    "        \n",
    "        #Calculer la modularite pour chaque nouvelle config quand le node u passe dans le cluster i\n",
    "        for i in range(1, number_cluster+1): #Premier node=1\n",
    "            new_clustering = C.copy()\n",
    "            new_clustering[u] = i #Essai avec le node u dans le cluster i\n",
    "            variation[i], a, b = get_modularity(graph, new_clustering)\n",
    "            print(variation[i])\n",
    "            \n",
    "            #C'est un flipping node si la modularite est inferieure au seuil\n",
    "            if abs(variation[i]) < threshold:\n",
    "                flipping_nodes[u] = i\n",
    "    \n",
    "    return flipping_nodes "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "bowtie = nx.star_graph(4)\n",
    "bowtie.add_edges_from([(1,2),(3,4)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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npaWxdOlSZs2axalTp+jXrx/dunWjTJkyN/+9RApQ0aJF+eOPP/A0yXGsWo7W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\nIiJikP8Ha7n/EwT9BJIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f47fa2ef400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize = (6,3))\n",
    "nx.draw(bowtie, node_size = 400, with_labels = True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "hide_input": false,
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}