sandbox.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction\n",
    "\n",
    "The taxi-drive is defined in paper: \n",
    "\n",
    "The code for simulating the taxi-drive is found in funcs.py. It is pretty straightforward. We show some examples below for clarity.\n",
    "\n",
    "### Ring graph cover times\n",
    "\n",
    "The taxi-drive covers the ring graph more efficiently than the random walk"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'Cover time $T$')"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import funcs as f\n",
    "import osmnx as ox\n",
    "import networkx as nx\n",
    "from multiprocessing import Pool\n",
    "from datetime import datetime\n",
    "import time as time\n",
    "from scipy import optimize\n",
    "import pickle\n",
    "%matplotlib inline\n",
    "\n",
    "num_nodes = 10\n",
    "G = nx.cycle_graph(num_nodes)\n",
    "\n",
    "num_trials, m = 100, 1  # m = number of times each node has to be covered\n",
    "Ts_rw = f.random_walk_covertime(G,m,num_trials=num_trials)\n",
    "Ts_td = f.taxi_drive_covertime(G,m,num_trials=num_trials)\n",
    "plt.hist(Ts_rw);plt.hist(Ts_td);plt.legend(['Random walk','Taxi-drive'])\n",
    "plt.xlabel('Cover time $T$', fontsize=18)\n",
    "\n",
    "#np.savetxt('data/covertimes_ring_urban_explorer.txt',Ts_ue)\n",
    "#np.savetxt('data/covertimes_ring_random_walker.txt',Ts_rw)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ring graph looks like"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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t2+Pq1avw8PAQXU6Dcedsh4EDB+L1119HTEwMrl+/LrocIiL6DwcOHECPHj10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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G = nx.cycle_graph(num_nodes)\n",
    "nx.draw(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear graph stationary densities\n",
    "\n",
    "For the taxi-drive \n",
    "\n",
    "$ p_i = b_i'$\n",
    "\n",
    "where $p_i$ is the stationary density, and $b_i'$ is the adjusted betweeness (adjusted betweeness is a minor variation of regular betweeness; see eqn 2 in paper)\n",
    "\n",
    "This constrats with the random walk relation\n",
    "\n",
    "$ p_i = d_i$\n",
    "\n",
    "where $d_i$ is a node's degree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fad343e2790>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Xrx+dO3fmpJNOYty4cUyaNIkOHTrQvn37ysNDf/jDH2jQoAHnnnsu5eXlHHPMMTz11FMc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\nSJIyGRSSpEwGhSQpk0EhScpkUEiSMv1/uD24MHa2ntcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "num_nodes, T = 25, 10**6\n",
    "G = nx.path_graph(num_nodes)\n",
    "bs_path = np.array(nx.betweenness_centrality(G, endpoints=True, normalized=False).values())\n",
    "bs_path = bs_path / sum(bs_path)\n",
    "counts = np.zeros(G.number_of_nodes())\n",
    "counts = f.urban_explorer_stationary_densities(G,counts,T)\n",
    "counts_path = counts / sum(counts)\n",
    "plt.xlabel('b', fontsize=18);plt.ylabel('p', fontsize=18)\n",
    "#np.savetxt('data/covertimes_path_urban_explorer.txt',Ts_ue)\n",
    "#np.savetxt('data/covertimes_path_random_walker.txt',Ts_rw)\n",
    "\n",
    "ds = np.array([1.0*d[1] for d in nx.degree(G)])\n",
    "ds /= (1.0*sum(ds))\n",
    "plt.hist(bs_path);plt.hist(ds);plt.hist(counts_path)\n",
    "plt.legend(['Betweeness', 'Degree', '$p_i$ taxi-drive'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The path graph looks like"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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t2rVLiYmJpssBAMQgZsInkZCQoEsvvVTvvPOO6VIAADGKEG4GS9IAgFAihJvB\n5iwAQCgRws1gJgwACCVCuBnnnHOOfD6fvvzyS9OlAABiECHcDIfDofT0dJakAQAhQQi3gCVpAECo\nEMItYHMWACBUaNbRgqqqKl1wwQX65ptv1K4dn1kAAPZJMF1ApEuR9EdJ311/vU7z+6XERCktTZoy\nRerWzXR5AIAoxkz4ZMrLpTlzpLIyHT5yRJ3q64+/5nRKfr+UkSHNmiWlp5urEwAQtQjhE1m4UMrL\nk3w+K2xPxuGwAnnePCk3N3z1AQBiAsvRTTUEcG1ty+/1+6335eVZXxPEAIA2YCbcWHm5NGJE6wK4\nKZdLWrNGGjjQ9rIAALGJ7b6NzZljLUEHwuezrgcAoJWYCTeoqpJSU6VDhwIfo3Nnaft2dk0DAFqF\nmXCD/Pzgx3A47BkHABAXCOEGFRXBzYIla0m6stKeegAAMY8QblBTY884+/bZMw4AIOYRwg0SE+0Z\nJynJnnEAADGPEG6QlmZtrAqG0yn1729PPQCAmMfu6AbsjgYAhBkz4QYpKVYvaIcjsOsdDikzkwAG\nALQaM+HG6JgFAAgjZsKNpadbhzG4XG27zuWyriOAAQBtwEz4RFp5itIxSfUdO6rD3//O4Q0AgDZj\nJnwiubnW0nJWlrXZyun88etOp9S5s779r//SmC5d9M2ECWbqBABENWbCLamutlpRVlZajTiSkqzH\nkHJypG7dNG3aNNXX12vBggWmKwUARBlCOEh79+7VRRddpNLSUg0YMMB0OQCAKMJydJCSk5N1//33\na/r06eLzDACgLQhhG0ydOlVHjhzR4sWLTZcCAIgiLEfb5O2331ZWVpY+/PBDnXbaaabLAQBEAULY\nRlOnTlVycrLmzZtnuhQAQBQghG309ddf6+KLL9Ybb7yhfv36mS4HABDhuCdso+7du+vPf/6zfve7\n37FJCwDQIkLYZrfffrt2796tpUuXmi4FABDhWI4Ogddff105OTnaunWrXG3tQw0AiBvMhENgxIgR\nGjJkiObOnWu6FABABGMmHCI7d+7UpZdeqg0bNujcc881XQ4AIAIxEw6RXr166c4779Qf/vAH06UA\nACIUIRxCd9xxh7Zu3aqysjLTpQAAIhAhHEKdOnXSww8/rBkzZujw4cOmywEARBhCOMQyMzN14YUX\n6qGHHjJdCgAgwrAxKww+/fRTDR48WO+++6569epluhwAQIQghMPk7rvv1ieffKKioiLTpQAAIgQh\nHCa1tbXq16+fCgoKdNVVV5kuBwAQAbgnHCYul0sPPPCApk2bprq6OtPlAAAiACEcRuPHj1f37t21\nYMEC06UAACIAy9FhtnXrVl2iVBS0AAADLElEQVR11VV6//33lZKSYrocAIBBhLABd955p/bv36+n\nnnrKdCkAAIMIYQMOHDigCy+8UMuWLdOgQYNMlwMAMIR7wgacdtppmjt3rqZNm6b6+nrT5QAADCGE\nDZk8ebISEhL09NNPmy4FAGAIy9EGbdq0SZmZmfrggw+UlJRkuhwAQJgRwobl5uYqISFB//jHP0yX\nAgAIM0LYsG+++Ub9+vXTq6++qrS0NNPlAADCiHvChp1xxhmaPXu2pk2bJj4PAUB8YSYcAY4dO6b0\n9HTNnDlTN998s1RVJeXnSxUVUk2NlJgopaVJU6ZI3bqZLhcAYBNCOEKsW7dOf8nK0vIhQ5TwyivW\nNw8dOv4Gp1Py+6WMDGnWLCk93UyhAADbEMKRYuFCHZ4+XR2OHWv+HoHDYQXyvHlSbm64qgMAhECC\n6QIgaeFCKS9PnY4da/m9fr9UWyvl5VlfE8QAELWYCZtWXi6NGGEFa1u5XNKaNdLAgbaXBQAIPXZH\nmzZnjuTzBXatz2ddDwCISsyETaqqklJTf7wBq606d5a2b2fXNABEIWbCJuXnBz+Gw2HPOACAsCOE\nTaqoCG4WLFlL0pWV9tQDAAgrQtikmhp7xtm3z55xAABhRQiblJhozzicwAQAUYkQNiktzdpYFQyn\nU+rf3556AABhxe5ok9gdDQBxjZmwSSkpVi9ohyOw6x0OKTOTAAaAKMVM2DQ6ZgFA3GImbFp6unUY\ng8vVtutcLus6AhgAohYHOESChkMY8vKs536bW5zgFCUAiBksR0eSjRutXtClpVbYNu4p3XCecGam\ndZ4wM2AAiHqEcCSqrrZaUVZWWo04kpKsx5ByctiEBQAxhBAGAMAQNmYBAGAIIQwAgCGEMAAAhhDC\nAAAYQggDAGAIIQwAgCGEMAAAhhDCAAAYQggDAGAIIQwAgCGEMAAAhhDCAAAYQggDAGAIIQwAgCGE\nMAAAhhDCAAAYQggDAGAIIQwAgCGEMAAAhhDCAAAYQggDAGAIIQwAgCGEMAAAhhDCAAAYQggDAGAI\nIQwAgCGEMAAAhhDCAAAYQggDAGAIIQwAgCGEMAAAhhDCAAAY8v8Bk/j5g97Z0F0AAAAASUVORK5C\nYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G = nx.path_graph(num_nodes)\n",
    "nx.draw(G)"
   ]
  }
 ],
 "metadata": {
  "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}