IX.A. Map equation centrality.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# IX.A. Map equation centrality"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "from infomap import Infomap\n",
    "from typing import Dict, Set\n",
    "\n",
    "import networkx as nx\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "sns.set_palette(sns.color_palette(\"colorblind\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mkPartition( infomap\n",
    "               ) -> Dict[int, Set[str]]:\n",
    "    \"\"\"\n",
    "    Extracts a partition from an infomap instance.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    infomap \n",
    "        An infomap instance. We assume that it has been run so that\n",
    "        we can actually extract a partition from it.\n",
    "\n",
    "    Returns\n",
    "    -------\n",
    "    Dict[int, Set[str]]\n",
    "        The networks partition calculated by the given infomap instance\n",
    "        with community IDs as keys and sets of node labels as values.\n",
    "        (Assuming that nodes were added with strings as IDs to Infomap.)\n",
    "    \"\"\"\n",
    "    communities = dict()\n",
    "    for node_ID, community in infomap.get_modules(depth_level = -1).items():\n",
    "        if not community in communities:\n",
    "            communities[community] = set()\n",
    "        communities[community].add(infomap.get_name(node_ID))\n",
    "\n",
    "    return communities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Many centrality scores take a local or global perspective and derive node centralities either from node features, such as node degrees, or global patterns, such as shortest paths. Map equation centrality takes a modular perspective and derives node centralities from a node's flow and the flow of its module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "G = nx.Graph()\n",
    "G.add_edges_from([ (1,2), (1,3), (1,4), (2,4), (3,4) # left module\n",
    "                 , (4,5)                             # bridge\n",
    "                 , (5,6), (5,7), (5,8), (6,7)        # right module\n",
    "                 ])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# hand-crafted layout\n",
    "pos = { 1 : [1,2]\n",
    "      , 2 : [0,1]\n",
    "      , 3 : [1,0]\n",
    "      , 4 : [2,1]\n",
    "      , 5 : [3,1]\n",
    "      , 6 : [4,2]\n",
    "      , 7 : [4,1]\n",
    "      , 8 : [4,0]\n",
    "      }"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 400x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 1, figsize = (4,2))\n",
    "nx.draw_networkx_nodes(G, pos = pos, node_color = \"grey\")\n",
    "nx.draw_networkx_edges(G, pos = pos)\n",
    "nx.draw_networkx_labels(G, pos = pos, font_color = \"white\")\n",
    "ax.axis(\"off\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some centrality scores, such as degree centrality and PageRank, consider nodes 4 and 5 equally important."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>degree centrality</th>\n",
       "      <th>betweenness centrality</th>\n",
       "      <th>PageRank</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.428571</td>\n",
       "      <td>0.023810</td>\n",
       "      <td>0.149994</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.285714</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.099996</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.285714</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.099996</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.571429</td>\n",
       "      <td>0.595238</td>\n",
       "      <td>0.199995</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.571429</td>\n",
       "      <td>0.666667</td>\n",
       "      <td>0.200007</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.285714</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.100005</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.285714</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.100005</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.142857</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.050002</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   degree centrality  betweenness centrality  PageRank\n",
       "1           0.428571                0.023810  0.149994\n",
       "2           0.285714                0.000000  0.099996\n",
       "3           0.285714                0.000000  0.099996\n",
       "4           0.571429                0.595238  0.199995\n",
       "5           0.571429                0.666667  0.200007\n",
       "6           0.285714                0.000000  0.100005\n",
       "7           0.285714                0.000000  0.100005\n",
       "8           0.142857                0.000000  0.050002"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pd.concat( [ pd.DataFrame(index = sorted(G.nodes))\n",
    "           , pd.Series(nx.algorithms.centrality.degree_centrality(G), name = \"degree centrality\")\n",
    "           , pd.Series(nx.algorithms.centrality.betweenness_centrality(G), name = \"betweenness centrality\")\n",
    "           , pd.Series(nx.pagerank(G, alpha = 1), name = \"PageRank\")\n",
    "           ]\n",
    "         , axis = 1\n",
    "         )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Map equation centrality can distinguish between these nodes by considering their modular context."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# map equation centrality becomes a local measure when we assign all nodes to the same module because this gives all nodes the same modular context\n",
    "im_one_level = Infomap(silent = True, no_infomap = True)\n",
    "node_mapping = im_one_level.add_networkx_graph(G)\n",
    "im_one_level.run()\n",
    "\n",
    "one_level_centralities = pd.Series({ node_mapping[node.node_id] : node.modular_centrality for node in im_one_level.iterLeafNodes()}, name = \"$M_1$\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 400x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 1, figsize = (4,2))\n",
    "nodelist = sorted(G.nodes)\n",
    "nx.draw_networkx_nodes(G, pos = pos, node_color = \"grey\", nodelist = nodelist, node_size = [2000*one_level_centralities.loc[node] for node in nodelist])\n",
    "nx.draw_networkx_edges(G, pos = pos)\n",
    "nx.draw_networkx_labels(G, pos = pos, font_color = \"white\")\n",
    "ax.axis(\"off\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4    0.257542\n",
       "5    0.257542\n",
       "1    0.199295\n",
       "2    0.136803\n",
       "3    0.136803\n",
       "6    0.136803\n",
       "7    0.136803\n",
       "8    0.070301\n",
       "Name: $M_1$, dtype: float64"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# nodes 4 and 5 have the same centrality if we ignore modules\n",
    "one_level_centralities.sort_values(ascending = False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# centralities for the \"optimal\" partition as identified by Infomap\n",
    "im = Infomap(silent = True, two_level = True, num_trials = 100)\n",
    "node_mapping = im.add_networkx_graph(G)\n",
    "im.run()\n",
    "\n",
    "optimal_centralities = pd.Series({ node_mapping[node.node_id] : node.modular_centrality for node in im.iterLeafNodes() }, name = \"$M_{opt}$\")\n",
    "optimal_modules = dict(im.modules)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 400x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 1, figsize = (4,2))\n",
    "nodelist = sorted(G.nodes)\n",
    "nx.draw_networkx_nodes(G, pos = pos, node_color = [sns.color_palette()[optimal_modules[node]-1] for node in nodelist], nodelist = nodelist, node_size = [2000*optimal_centralities.loc[node] for node in nodelist])\n",
    "nx.draw_networkx_edges(G, pos = pos)\n",
    "nx.draw_networkx_labels(G, pos = pos, font_color = \"white\")\n",
    "ax.axis(\"off\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4    0.228227\n",
       "5    0.211999\n",
       "1    0.183773\n",
       "2    0.130278\n",
       "3    0.130278\n",
       "6    0.126900\n",
       "7    0.126900\n",
       "8    0.067970\n",
       "Name: $M_{opt}$, dtype: float64"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# if we consider modules, node 4 has higher centrality because it belongs to a module with more flow than node 5\n",
    "optimal_centralities.sort_values(ascending = False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# a sub-optimal partition according to Infomap\n",
    "im_sub = Infomap(silent = True, no_infomap = True)\n",
    "node_mapping = im_sub.add_networkx_graph(G)\n",
    "im_sub.run(initial_partition = {1:1, 2:1, 3:1, 4:2, 5:2, 6:2, 7:2, 8:2})\n",
    "\n",
    "suboptimal_centralities = pd.Series({ node_mapping[node.node_id] : node.modular_centrality for node in im_sub.iterLeafNodes() }, name = \"$M_{sub}$\")\n",
    "suboptimal_modules = dict(im_sub.modules)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 400x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 1, figsize = (4,2))\n",
    "nodelist = sorted(G.nodes)\n",
    "nx.draw_networkx_nodes(G, pos = pos, node_color = [sns.color_palette()[suboptimal_modules[node]-1] for node in nodelist], nodelist = nodelist, node_size = [2000*suboptimal_centralities.loc[node] for node in nodelist])\n",
    "nx.draw_networkx_edges(G, pos = pos)\n",
    "nx.draw_networkx_labels(G, pos = pos, font_color = \"white\")\n",
    "ax.axis(\"off\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4    0.238732\n",
       "5    0.238732\n",
       "1    0.161471\n",
       "6    0.132554\n",
       "7    0.132554\n",
       "2    0.121357\n",
       "3    0.121357\n",
       "8    0.069286\n",
       "Name: $M_{sub}$, dtype: float64"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# if 4 and 5 belong to the same module, we cannot identify this difference anymore\n",
    "suboptimal_centralities.sort_values(ascending = False)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": ".venv",
   "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": "3.11.6"
  },
  "orig_nbformat": 4
 },
 "nbformat": 4,
 "nbformat_minor": 2
}