notebooks/pagerank.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "445c7ab2-dc22-4f24-bbd1-ec6f3693f8a3",
   "metadata": {},
   "source": [
    "# Purpose\n",
    "1. Understand pagerank algorithm\n",
    "\n",
    "<table align=\"left\">\n",
    "  <td>\n",
    "    <a href=\"https://colab.research.google.com/github/Ankush-Chander/graph-notebooks/blob/main/notebooks/pagerank.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>\n",
    "  </td>\n",
    "  <td>\n",
    "    <a target=\"_blank\" href=\"https://www.kaggle.com/code/latebloomer/pagerank-introduction\"><img src=\"https://kaggle.com/static/images/open-in-kaggle.svg\" /></a>\n",
    "  </td>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "717e88b7-3bb4-4142-bbf8-0130eba58dac",
   "metadata": {},
   "outputs": [],
   "source": [
    "import networkx as nx\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dc67a6e4-86cc-4b6f-b1d6-9df56e7f32e5",
   "metadata": {},
   "source": [
    "## Directed graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "71654555-d7e7-4823-bc06-c7d4eb27a410",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# create an example graph\n",
    "G1 = nx.DiGraph()\n",
    "G1.add_edges_from([(1, 2), (3,2), (3, 4), (4, 1)])\n",
    "\n",
    "# draw the graph\n",
    "nx.draw(G1, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32f700c5-6910-4b84-b4c7-862a97ec6ccb",
   "metadata": {},
   "source": [
    "## Adjacency matrix\n",
    "For a simple graph with vertex set $U = {u1, …, un}$, the adjacency matrix is a square n × n matrix A such that its element $A_{ij}$ is one when there is an edge between vertex $u_i$ to vertex $u_j$, and zero when there is no edge.\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c26cdd34-e6a4-49ec-993e-24e1ce06a707",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1 0 0]\n",
      " [0 0 0 0]\n",
      " [0 1 0 1]\n",
      " [1 0 0 0]]\n"
     ]
    }
   ],
   "source": [
    "A = nx.adjacency_matrix(G1).todense()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca52c3da-2f32-4a39-b050-15eb3149e0a8",
   "metadata": {},
   "source": [
    "### Powers of adjacency matrix\n",
    "If $A$ is the adjacency matrix of the directed or undirected graph $G$, then the matrix $A^n$ (i.e., the matrix product of n copies of A) has an interesting interpretation: the element $(i, j)$ gives the number of (directed or undirected) walks of length $n$ from vertex $i$ to vertex $j$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "fbeb901c-c7d8-40fa-8b54-1b1103c43677",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A_2:\n",
      "[[0 0 0 0]\n",
      " [0 0 0 0]\n",
      " [1 0 0 0]\n",
      " [0 1 0 0]]\n",
      "A_3:\n",
      "[[0 0 0 0]\n",
      " [0 0 0 0]\n",
      " [0 1 0 0]\n",
      " [0 0 0 0]]\n",
      "A_4:\n",
      "[[0 0 0 0]\n",
      " [0 0 0 0]\n",
      " [0 0 0 0]\n",
      " [0 0 0 0]]\n"
     ]
    }
   ],
   "source": [
    "# 2 step reachability\n",
    "A_2 = np.linalg.matrix_power(A, 2)\n",
    "print(f\"A_2:\\n{A_2}\")\n",
    "\n",
    "A_3 = np.linalg.matrix_power(A, 3)\n",
    "print(f\"A_3:\\n{A_3}\")\n",
    "\n",
    "A_4 = np.linalg.matrix_power(A, 4)\n",
    "print(f\"A_4:\\n{A_4}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30fe817c-8f91-409b-b5ec-ff6220dd9b7b",
   "metadata": {},
   "source": [
    "### Connectedness of a graph\n",
    "**Connected Graphs:** A graph is called connected if for any two different nodes i and j there is a directed path\n",
    "either from i to j or from j to i.  \n",
    "**Strongly Connected Graphs:**  A graph is called strongly connected if starting at any node i we can reach any other different node j by walking on its edges.   \n",
    "\n",
    "Matrix interpretation:  \n",
    "In terms of matrices, this means that if there is a finite positive integer k such that the matrix B = $I + A + A^2 + A^3 + … +A^k$ is positive(all values greater than zero), then the graph is strongly connected.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "966d996f-0719-4ad2-aee5-a0df42c74552",
   "metadata": {},
   "source": [
    "# Transition matrix\n",
    "The transition matrix $T$ associated to a directed graph is defined as follows. If there is an edge from $i$ to $j$ and the outdegree of vertex $i$ is $d_i$, then on column i and row j we put $\\frac{1}{d_i}$. Otherwise we mark column i, row j with zero.\n",
    "\n",
    "$$\n",
    "t_{i,j} = \n",
    "    \\begin{cases}\n",
    "        \\frac{1}{d_i} \\text{ if there is an outgoing edge from i to j}\\\\\n",
    "        0 \\text{ otherwise}\n",
    "    \\end{cases}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "1aab8ac6-0f5d-450c-8498-52f566de51af",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Transition matrix:\n",
      "[[0.         0.         1.         0.5       ]\n",
      " [0.33333333 0.         0.         0.        ]\n",
      " [0.33333333 0.5        0.         0.5       ]\n",
      " [0.33333333 0.5        0.         0.        ]]\n"
     ]
    }
   ],
   "source": [
    "# transition matrix\n",
    "def get_transition_matrix(G):\n",
    "    n = len(G.nodes())\n",
    "    transition_matrix = np.zeros((n, n))\n",
    "    for i in range(n):\n",
    "        for j in range(n):\n",
    "            if (i+1, j+1) in G.edges():\n",
    "                transition_matrix[j, i] = 1 / G.out_degree(i+1)\n",
    "    return transition_matrix\n",
    "\n",
    "\n",
    "def draw_graph_with_transistion_probs(G):\n",
    "    edge_labels = {edge: round(1/G.out_degree(edge[0]),2) for edge in G.edges()}\n",
    "    \n",
    "    # draw the graph\n",
    "    pos = nx.spring_layout(G)\n",
    "    plt.figure()\n",
    "    nx.draw(\n",
    "        G, pos, edge_color='black', width=1, linewidths=1,\n",
    "        node_size=500, node_color='pink', alpha=0.9,\n",
    "        labels={node: node for node in G.nodes()}\n",
    "    )\n",
    "    nx.draw_networkx_edge_labels(\n",
    "        G, pos,\n",
    "        edge_labels=edge_labels,\n",
    "        font_color='red'\n",
    "    )\n",
    "    plt.axis('off')\n",
    "    plt.show()\n",
    "    \n",
    "\n",
    "\n",
    "\n",
    "G2 = nx.DiGraph()\n",
    "G2.add_edges_from(\n",
    "    [\n",
    "        (1,2),\n",
    "        (1,4),\n",
    "        (1,3),\n",
    "        (2,3),\n",
    "        (2,4),\n",
    "        (3,1),\n",
    "        (4,3),\n",
    "        (4,1)\n",
    "        \n",
    "    ]\n",
    ")\n",
    "\n",
    "draw_graph_with_transistion_probs(G2)\n",
    "\n",
    "\n",
    "G2_T = get_transition_matrix(G2)\n",
    "\n",
    "\n",
    "print(f\"Transition matrix:\\n{G2_T}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45c07325-05c7-408a-a484-1a8eeaa1c52a",
   "metadata": {},
   "source": [
    "### Properties of transition matrix\n",
    "**Markov property:** A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past.  \n",
    "\n",
    "In a **primitive matrix**, every state in the Markov chain can be reached from any other state with positive probability in a finite number of steps.  \n",
    "\n",
    "Matrix interpretation:  \n",
    "In general, a matrix is called primitive if there is a positive integer $k$ such that $A^k$ is a positive\n",
    "matrix.\n",
    "\n",
    "**A column-stochastic matrix** is a square matrix where each column sums to 1. In other words, it is a matrix representing transition probabilities of a Markov chain where the probabilities of transitioning from one state to all other states sum up to 1 for each state.  \n",
    "*If a graph is strongly connected then it\"s transition matrix is column-stochastic.*\n",
    "\n",
    "  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "5a6ccd0c8305fc41",
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Graph is primitive as A^3 is positive\n",
      "[[0.33333333 0.375      0.41666667 0.45833333]\n",
      " [0.13888889 0.08333333 0.16666667 0.08333333]\n",
      " [0.30555556 0.33333333 0.25       0.29166667]\n",
      " [0.22222222 0.20833333 0.16666667 0.16666667]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def is_primitive(trans_matrix):\n",
    "    T_n = trans_matrix\n",
    "    for i in range(1, 1000):\n",
    "        T_n = np.dot(trans_matrix, T_n)\n",
    "        if np.all(T_n > 0):\n",
    "            print(f\"Graph is primitive as A^{i} is positive\")\n",
    "            print(T_n)\n",
    "            return True\n",
    "    return False\n",
    "is_primitive(G2_T)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e099a4c9-4199-4de2-837d-00f3ba8e554f",
   "metadata": {},
   "source": [
    "### Stationary distribution\n",
    "The stationary distribution, also known as the steady-state distribution or equilibrium distribution, of a Markov chain is a probability distribution over the states of the chain that remains unchanged over time. In other words, *it represents the long-term behavior of the Markov chain as it evolves through its states*.\n",
    "\n",
    "If the transition matrix is:  \n",
    "1. primitive and\n",
    "2. column-stochastic\n",
    "\n",
    "then it has a\n",
    "unique stationary probability distribution vector $p$ such that $i^{th}$ entry of p is that the surfer visits page i at any given time with probability $p_i$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5f905e6-f3dd-437e-9082-27082f470cd4",
   "metadata": {},
   "source": [
    "## How to find stationary distribution?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "1e28e361-80d6-410f-88b8-d9a212911859",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.        , 0.        , 1.        , 0.5       ],\n",
       "       [0.33333333, 0.        , 0.        , 0.        ],\n",
       "       [0.33333333, 0.5       , 0.        , 0.5       ],\n",
       "       [0.33333333, 0.5       , 0.        , 0.        ]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G2_T"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ed1e8b6-3fef-4b04-b5ac-965935792d87",
   "metadata": {},
   "source": [
    "#### Dynamical systems point of view(Power Method)\n",
    "\n",
    "Suppose that all nodes start with equal importance(1/n) represented by vector $v$ and at each step recieve importance from all the incoming nodes. This operation is denoted by: $$Step 0: v$$ $$Step 1: T.v$$ $$Step 3: T^2.v $$ $$...$$ $$Step k: T^k.v$$ where T is the transition matrix and k is the step where the value converges.  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "b32b8861-31a3-4137-bc11-007a2b1c32c3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P.shape: (4, 4)\n",
      "pi.shape: (4, 1)\n",
      "@iteration0: diff=0.23194827009486405, \n",
      "pi:\n",
      "[[0.25]\n",
      " [0.25]\n",
      " [0.25]\n",
      " [0.25]]\n",
      "**********\n",
      "@iteration1: diff=0.1063014581273465, \n",
      "pi:\n",
      "[[0.38]\n",
      " [0.08]\n",
      " [0.33]\n",
      " [0.21]]\n",
      "**********\n",
      "@iteration2: diff=0.09643650760992954, \n",
      "pi:\n",
      "[[0.44]\n",
      " [0.13]\n",
      " [0.27]\n",
      " [0.17]]\n",
      "**********\n",
      "@iteration3: diff=0.053851648071345064, \n",
      "pi:\n",
      "[[0.36]\n",
      " [0.15]\n",
      " [0.3 ]\n",
      " [0.21]]\n",
      "**********\n",
      "@iteration4: diff=0.014142135623730963, \n",
      "pi:\n",
      "[[0.4 ]\n",
      " [0.12]\n",
      " [0.3 ]\n",
      " [0.19]]\n",
      "**********\n",
      "@iteration5: diff=0.022360679774997918, \n",
      "pi:\n",
      "[[0.4 ]\n",
      " [0.13]\n",
      " [0.29]\n",
      " [0.19]]\n",
      "**********\n",
      "@iteration6: diff=0.014142135623730963, \n",
      "pi:\n",
      "[[0.38]\n",
      " [0.13]\n",
      " [0.29]\n",
      " [0.2 ]]\n",
      "**********\n",
      "@iteration7: diff=0.010000000000000009, \n",
      "pi:\n",
      "[[0.39]\n",
      " [0.13]\n",
      " [0.29]\n",
      " [0.19]]\n",
      "**********\n",
      "Stationary Distribution (Power Method):\n",
      "[[0.38]\n",
      " [0.13]\n",
      " [0.29]\n",
      " [0.19]]\n"
     ]
    }
   ],
   "source": [
    "def power_method(P, tol=1e-2, max_iter=10000):\n",
    "    P = np.around(P, 2)\n",
    "    n = len(P)\n",
    "    pi = np.ones((n,1)) / n  # Initial guess for stationary distribution\n",
    "    print(f\"P.shape: {P.shape}\")\n",
    "    print(f\"pi.shape: {pi.shape}\")\n",
    "    for _ in range(max_iter):\n",
    "        pi_next = np.dot(P,pi)\n",
    "        pi_next = np.around(pi_next, 2)\n",
    "        if np.linalg.norm(pi_next - pi) < tol:\n",
    "            return pi_next\n",
    "        if _%1==0:\n",
    "            print(f\"@iteration{_}: diff={np.linalg.norm(pi_next - pi)}, \\npi:\\n{pi}\")\n",
    "            print(f\"*\"*10)\n",
    "        pi = pi_next\n",
    "\n",
    "    raise ValueError(\"Power method did not converge.\")\n",
    "\n",
    "# Example usage\n",
    "stationary_distribution = power_method(G2_T)\n",
    "print(f\"Stationary Distribution (Power Method):\\n{stationary_distribution}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd124834-532b-43c9-9651-214d4aa02f1c",
   "metadata": {},
   "source": [
    "### Linear Algebraic Method\n",
    "This method involves solving a linear system of equations: $$\\pi{P} = \\pi$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "78052285-aecd-445a-a5ae-848809b03123",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Stationary probability vector: [0.38709677 0.12903226 0.29032258 0.19354839]\n"
     ]
    }
   ],
   "source": [
    "def calculate_stationary_probability_via_eigen_vectors(transition_matrix):\n",
    "    # Calculate eigenvalues and eigenvectors\n",
    "    eigenvalues, eigenvectors = np.linalg.eig(transition_matrix)\n",
    "    \n",
    "    # Find the index of the eigenvalue closest to 1\n",
    "    index = np.argmin(np.abs(eigenvalues - 1))\n",
    "    # Extract the eigenvector corresponding to the eigenvalue 1\n",
    "    stationary_vector = np.real(eigenvectors[:, index])\n",
    "    \n",
    "    # Normalize the stationary vector\n",
    "    stationary_vector /= np.sum(stationary_vector)\n",
    "    \n",
    "    return stationary_vector\n",
    "\n",
    "stationary_probability = calculate_stationary_probability_via_eigen_vectors(G2_T)\n",
    "print(\"Stationary probability vector:\", stationary_probability)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee61b726-384b-4105-aa47-6150992796ff",
   "metadata": {},
   "source": [
    "# Webgraph\n",
    "Webgraph is a directed graph in which web pages are represented as vertices and the\n",
    "links between them are represented as edges."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "341a4484-fa3c-4d4c-8278-b31702baeb90",
   "metadata": {},
   "source": [
    "## Problems with WebGraph\n",
    "**Dangling nodes**: Nodes that have no outgoing edges(also known as sinks). Such nodes are not able to transfer their importance value to any other node.  \n",
    "**Disconnected components**:  \n",
    "Since Web is huge and heterogenous in nature, it\"s common to have disconnected components in graph. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "87f80362-1e9f-4fac-a648-db9e2415d0ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# dangling nodes\n",
    "G_dangling = nx.DiGraph()\n",
    "G_dangling.add_edges_from(((1,3),(2,3)))\n",
    "nx.draw(G_dangling,with_labels=True)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "d9e0d4cd-9ced-4c81-a187-39dcf7947c7d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0. 0. 0.]\n",
      " [0. 0. 0.]\n",
      " [1. 1. 0.]]\n",
      "P.shape: (3, 3)\n",
      "pi.shape: (3, 1)\n",
      "@iteration0: diff=0.5792811637423287, \n",
      "pi:\n",
      "[[0.33333333]\n",
      " [0.33333333]\n",
      " [0.33333333]]\n",
      "**********\n",
      "@iteration1: diff=0.67, \n",
      "pi:\n",
      "[[0.  ]\n",
      " [0.  ]\n",
      " [0.67]]\n",
      "**********\n",
      "[[0.]\n",
      " [0.]\n",
      " [0.]]\n"
     ]
    }
   ],
   "source": [
    "# get transition matrix from graph\n",
    "transition_matrix = get_transition_matrix(G_dangling)\n",
    "print(transition_matrix)\n",
    "\n",
    "\n",
    "stationary_prob = power_method(transition_matrix)\n",
    "print(stationary_prob)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "a9edc531-724f-4c2f-9fbd-2d3acaee6943",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.  0.  0.  0.  0. ]\n",
      " [1.  0.  0.  0.  0. ]\n",
      " [0.  0.  0.  0.5 0.5]\n",
      " [0.  0.  0.5 0.  0.5]\n",
      " [0.  0.  0.5 0.5 0. ]]\n",
      "[0.         0.         0.33333333 0.33333333 0.33333333]\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G_disconnected = nx.DiGraph()\n",
    "G_disconnected.add_edges_from(\n",
    "    [\n",
    "        (1,2),\n",
    "        (3,4),(4,3),\n",
    "        (4,5),(5,4),\n",
    "        (3,5), (5,3)\n",
    "    ]\n",
    ")\n",
    "nx.draw(G_disconnected, with_labels=True)\n",
    "transition_matrix = get_transition_matrix(G_disconnected)\n",
    "print(transition_matrix)\n",
    "stationary_distribution = calculate_stationary_probability_via_eigen_vectors(transition_matrix)\n",
    "print(stationary_distribution)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44140b2f-76eb-4163-ba38-805dcd89605c",
   "metadata": {},
   "source": [
    "## Solution: Google matrix\n",
    "1. To solve the problem of dangling nodes, each dangling node is connected with all the other nodes with equal probabilty($1/n$). Modified transition matrix now looks like:\n",
    "   $$\n",
    "t'_{i,j} = \n",
    "    \\begin{cases}\n",
    "        \\frac{1}{d_i} \\text{ if there is an outgoing edge from i to j}\\\\\n",
    "        \\frac{1}{n} \\text{ otherwise}\n",
    "    \\end{cases}\n",
    "$$\n",
    "3. To solve disconnected components issue, we add another matrix(random jump matrix) to modified transition matrix(T').\n",
    "   $$\n",
    "   T'' = \\alpha{T'} + (1- \\alpha){E}\n",
    "   $$\n",
    "where   \n",
    "$\\alpha$ is the is a damping factor ie. the probability, at any step, that the person will continue following links.  \n",
    "1-$\\alpha$ denotes the probability that surfer will make a random jump. \n",
    "   \n",
    "\n",
    "   This matrix is by defination stochastic(column propabilities adds up to one) and primitive(every node is reachable from other node)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "0b4594bd-785a-4ed4-9842-a161ffede24a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "transition_matrix:\n",
      "[[0.  0.  0.  0.  0. ]\n",
      " [1.  0.  0.  0.  0. ]\n",
      " [0.  0.  0.  0.5 0.5]\n",
      " [0.  0.  0.5 0.  0.5]\n",
      " [0.  0.  0.5 0.5 0. ]]\n",
      "modified_transition_matrix(solves dangling nodes):\n",
      "[[0.  0.2 0.  0.  0. ]\n",
      " [1.  0.2 0.  0.  0. ]\n",
      " [0.  0.2 0.  0.5 0.5]\n",
      " [0.  0.2 0.5 0.  0.5]\n",
      " [0.  0.2 0.5 0.5 0. ]]\n",
      "google_matrix:\n",
      "[[0.03  0.2   0.03  0.03  0.03 ]\n",
      " [0.88  0.2   0.03  0.03  0.03 ]\n",
      " [0.03  0.2   0.03  0.455 0.455]\n",
      " [0.03  0.2   0.455 0.03  0.455]\n",
      " [0.03  0.2   0.455 0.455 0.03 ]]\n"
     ]
    }
   ],
   "source": [
    "def get_google_matrix(G, alpha=0.85):\n",
    "    n = len(G.nodes())\n",
    "    transition_matrix = get_transition_matrix(G)\n",
    "    print(f\"transition_matrix:\\n{transition_matrix}\")\n",
    "    # handle dangling nodes\n",
    "    dangling_nodes = np.where(transition_matrix.sum(axis=0) == 0)[0]\n",
    "    modified_transition_matrix = transition_matrix.copy()\n",
    "    modified_transition_matrix[:, dangling_nodes] = 1/n\n",
    "    print(f\"modified_transition_matrix(solves dangling nodes):\\n{modified_transition_matrix}\")\n",
    "    # handle disconnected components\n",
    "    E = np.ones((n, n)) / n\n",
    "    google_matrix = alpha * modified_transition_matrix + (1 - alpha) * E \n",
    "    return google_matrix\n",
    "\n",
    "\n",
    "google_matrix = get_google_matrix(G_disconnected)\n",
    "print(f\"google_matrix:\\n{google_matrix}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "30b5f7e2-67da-498d-9469-e0cd2ba9e6d8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pagerank: {1: 0.04376425171891564, 2: 0.08096440572344879, 3: 0.29175711418587863, 4: 0.29175711418587863, 5: 0.29175711418587863}\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# can be directly calculated for any graph\n",
    "nx.google_matrix(G_disconnected)\n",
    "\n",
    "nx.pagerank(G_disconnected)\n",
    "\n",
    "\n",
    "\n",
    "# Draw the graph with pagerank values\n",
    "pagerank = nx.pagerank(G_disconnected)\n",
    "print(f\"pagerank: {pagerank}\")\n",
    "nx.draw(\n",
    "    G_disconnected, with_labels=True,\n",
    "    node_size=[v * 10000 for v in pagerank.values()],\n",
    "    node_color=list(pagerank.values())\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c18e9e7-b96e-4901-a869-4e656d59de6a",
   "metadata": {},
   "source": [
    "# References\n",
    "1. [Adjacency matrix - wiki](https://en.wikipedia.org/wiki/Adjacency_matrix)\n",
    "2. [Pagerank - wiki](https://en.wikipedia.org/wiki/PageRank)\n",
    "3. [pagerank - networkx](https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.link_analysis.pagerank_alg.pagerank.html#networkx.algorithms.link_analysis.pagerank_alg.pagerank)"
   ]
  }
 ],
 "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": "3.11.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}