{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "0e4847aa",
   "metadata": {},
   "source": [
    "# Network Measures\n",
    "\n",
    "Network structure can be quantified using different network measures. These parameters can be classified into two: centrality measures or node-level measures and network-level measures.  \n",
    "\n",
    "## Centrality Measures (node-level measures)\n",
    "\n",
    "In network analysis, centrality measures identify the most \"important\" nodes inside a graph; in other words, how \"central\" a node is within the whole network. However, depending on how \"importance\" is defined, different nodes may be considered important. Because of its vast application and relatively easy computation, centrality is frequently the first measurement done in network analysis. Calculating it can help identify the influencers in a social media network or the disease super-spreaders during a pandemic. We discuss here some of the centrality measures used in network analysis.\n",
    "\n",
    "In weighted networks, we can also calculate the **node strength**, which is the sum of the weights of edges connected to the node. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1608f98f",
   "metadata": {},
   "source": [
    "**Degree centrality** is the number of edges connected to a given node. In a social network, this might mean the number of friends an individual has. It can be computed using the equation $\\frac{m}{n-1}$ , where m is the number of edges connected to a node and n is the total number of nodes. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "edb5d0c3",
   "metadata": {},
   "source": [
    "### Example 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1309c1d2",
   "metadata": {},
   "source": [
    "Let us consider the network of some of the characters from Noli Me Tangere shown in {numref}`noli`. \n",
    "\n",
    "```{figure} ./images/noli2.png\n",
    ":name: noli\n",
    ":width: 500px\n",
    "\n",
    "Network of characters from Noli Me Tangere\n",
    "```\n",
    "\n",
    "We can convert this to a networkx graph as shown here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b782fcae",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-01T13:10:07.338889Z",
     "start_time": "2022-04-01T13:10:05.873275Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import networkx as nx\n",
    "from alis.network_tools.plots import plot_degree_distribution\n",
    "from alis.network_tools.metrics import (\n",
    "    degree_centrality, betweenness_centrality, closeness_centrality,\n",
    "    eigenvector_centrality, average_degree\n",
    ")\n",
    "\n",
    "G = nx.Graph()\n",
    "G.add_nodes_from([\"A\",\"B\",\"C\",\"D\",\"E\",\"F\",\"G\"])\n",
    "G.add_edges_from([(\"A\",\"B\"),(\"A\",\"C\"),(\"A\",\"D\"),(\"C\",\"F\"),\n",
    "(\"D\",\"E\"),(\"D\",\"F\"),(\"D\",\"G\"),(\"E\",\"F\"),(\"F\",\"G\")])\n",
    "nx.draw(G, node_size=400, node_color='red', with_labels=True, font_weight='bold')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4dc8fd0",
   "metadata": {},
   "source": [
    "And then calculate the degree centrality of each node."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "1999a577",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-01T13:10:07.342102Z",
     "start_time": "2022-04-01T13:10:07.340320Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Degree centrality:\n",
      "D:0.667 F:0.667 A:0.5 C:0.333 E:0.333 G:0.333 B:0.167 \n",
      "\n"
     ]
    }
   ],
   "source": [
    "degree_centrality(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e06c510",
   "metadata": {},
   "source": [
    "**Betweenness centrality** is the number of geodesic paths (shortest paths) that go through a given node. Nodes with high betweenness may be influential in a network because information tends to flow through them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "5d0f5fd5",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-01T13:10:07.346513Z",
     "start_time": "2022-04-01T13:10:07.342755Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Betweenness centrality:\n",
      "A:0.367 D:0.367 F:0.2 C:0.0667 B:0.0 E:0.0 G:0.0 \n",
      "\n"
     ]
    }
   ],
   "source": [
    "betweenness_centrality(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c869afc",
   "metadata": {},
   "source": [
    "**Closeness centrality** is the number of steps required to access every other node from a given node. It is given by the equation inverse($\\frac{number of steps}{n-1}$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "89b9aa21",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-01T13:10:07.350891Z",
     "start_time": "2022-04-01T13:10:07.347983Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Closeness centrality:\n",
      "D:0.75 A:0.667 F:0.667 C:0.6 E:0.545 G:0.545 B:0.429 \n",
      "\n"
     ]
    }
   ],
   "source": [
    "closeness_centrality(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1edecbe2",
   "metadata": {},
   "source": [
    "**Eigenvector centrality** is determined from the values of the first eigenvector of the graph adjacency matrix. The values are high for vertices that are connected to many other vertices that are, in turn, connected many others and so on."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "615ea91f",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-01T13:10:07.355693Z",
     "start_time": "2022-04-01T13:10:07.352286Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvector centrality:\n",
      "D:0.529 F:0.522 E:0.358 G:0.358 A:0.314 C:0.285 B:0.107 \n",
      "\n"
     ]
    }
   ],
   "source": [
    "eigenvector_centrality(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8ad5425",
   "metadata": {},
   "source": [
    "{numref}`centrality` illustrates the various centrality measures in a sample graph.\n",
    "\n",
    "```{figure} ./images/centrality1.png\n",
    ":name: centrality\n",
    ":width: 500px\n",
    "\n",
    "Illustration of various centrality measures\n",
    "```\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5f46735",
   "metadata": {},
   "source": [
    "### Network-level Measures\n",
    "\n",
    "The network measures tell us about the structure of the graph. Presented here are some of the network that are often calculated.\n",
    "\n",
    "1. Network size - number of nodes.\n",
    "2. Density - number of edges that exist / number of possible edges. If n is the number of nodes and m is the number of edges then density = $\\frac{2m}{n(n-1)}$ for an undirected graph and density = $\\frac{m}{n(n-1)}$ for a directed graph.\n",
    "3. Degree distribution - statistical distribution of node degrees in a network.\n",
    "4. Average path length - the fewest number of edges that you would have to go on to get from one node to another. \n",
    "5. Diameter - maximum path length.\n",
    "6. Clustering coefficient (Transitivity) - probability of two nodes that are connected to a common node being connected themselves. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "0f8ea9eb",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-01T13:10:07.363330Z",
     "start_time": "2022-04-01T13:10:07.358346Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "number of nodes =  7\n",
      "density =  0.42857142857142855\n",
      "average path length =  1.7142857142857142\n",
      "diameter =  3\n",
      "average_degree =  2.5714285714285716\n",
      "average clustering coefficient =  0.38095238095238093\n"
     ]
    }
   ],
   "source": [
    "print(\"number of nodes = \", G.number_of_nodes())\n",
    "print(\"density = \", nx.density(G))\n",
    "print(\"average path length = \", nx.average_shortest_path_length(G))\n",
    "print(\"diameter = \", nx.diameter(G))\n",
    "print(\"average_degree = \", average_degree(G))\n",
    "\n",
    "local_clustering_coefficient = nx.algorithms.cluster.clustering(G)\n",
    "av_local_clustering_coefficient = sum(local_clustering_coefficient.values())/len(local_clustering_coefficient)\n",
    "print(\"average clustering coefficient = \", av_local_clustering_coefficient)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "da2a0507",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-04-01T13:10:07.442711Z",
     "start_time": "2022-04-01T13:10:07.370788Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_degree_distribution(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "10974dfe",
   "metadata": {},
   "source": [
    "## Exercise"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ab75744",
   "metadata": {},
   "source": [
    "Calculate the centrality and network measures of the graph in {numref}`centrality` . Determine the nodes with the highest centrality measures. "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.8.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}