{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Необязательное ДЗ по лекции \"Спектральная теория графов\". Масляков Глеб, группа 517."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import networkx as nx\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Изоспектральность деревьев"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Будем генерировать случайные деревья и сравнивать спектр по метрике MSE для разного числа вершин."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Сначала маленькие деревья."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "lst = []\n",
    "for n in range(10, 101):\n",
    "    mse = 0\n",
    "    for _ in range(10):\n",
    "        G_1 = nx.generators.trees.random_tree(n)\n",
    "        G_2 = nx.generators.trees.random_tree(n)\n",
    "        s_1 = nx.adjacency_spectrum(G_1).astype(np.float64)\n",
    "        s_2 = nx.adjacency_spectrum(G_2).astype(np.float64)\n",
    "        s_1.sort()\n",
    "        s_2.sort()\n",
    "        mse += np.mean(np.square(s_1 - s_2))\n",
    "    lst.append(mse / 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11ef94a20>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Yc1PjyU6Oo/609jEqpYZ58EVErgNqjTFbB7rWGPO4MabEGFOSk5MT9neEGpW2+Ve+DLz+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\nqSAajEopFUSDUSmlgvx/BLRTnSDRYjYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11eed5ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(5,5))\n",
    "plt.plot(np.arange(10,101), lst)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Потом побольше."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lst = []\n",
    "for n in range(100, 3001, 100):\n",
    "    mse = 0\n",
    "    for _ in range(10):\n",
    "        G_1 = nx.generators.trees.random_tree(n)\n",
    "        G_2 = nx.generators.trees.random_tree(n)\n",
    "        s_1 = nx.adjacency_spectrum(G_1).astype(np.float64)\n",
    "        s_2 = nx.adjacency_spectrum(G_2).astype(np.float64)\n",
    "        s_1.sort()\n",
    "        s_2.sort()\n",
    "        mse += np.mean(np.square(s_1 - s_2))\n",
    "    lst.append(mse / 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1212bacf8>]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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TKokt7P8RzrkOM1sCrASSgQedc5vMbLG3fhmwApgPVALNwC299QUws+uB/wTy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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x121777eb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(5,5))\n",
    "plt.plot(np.arange(100, 3001, 100), lst)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Видно, что MSE спектров двух случайно выбранных деревьев в среднем стремится к нулю при росте $n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Для деревьев большего размера уже вычислительно сложно считать спектр."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Спектр Лапласа не распознаёт двудольность"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Пример коспектральных в смысле Лапласа графов. Один из этих графов двудольный, другой нет."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "G1 = nx.Graph()\n",
    "G2 = nx.Graph()\n",
    "G1.add_nodes_from([1,2,3,4,5,6])\n",
    "G2.add_nodes_from([1,2,3,4,5,6])\n",
    "G1.add_edges_from([(1,2), (2,3), (3,4), (4,1), (2,5), (5,6), (6,2)])\n",
    "G2.add_edges_from([(1,2), (2,3), (3,4), (4,1), (2,5), (5,4), (5,6)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "В первом графе есть цикл нечётной длины $2 \\rightarrow 5 \\rightarrow 6 \\rightarrow 2$. Значит он не является двудольным."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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k6o4aFcCAIvZw/PhxwsPDefHFFxk4cKDpOI6hcg6MkydPkti8Oc8VFVG3tPS8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\nnz9A68AR9FEqCawpU7yn8Ko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      "text/plain": [
       "<matplotlib.figure.Figure at 0x105ad44e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw_shell(G1, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Во втором графе вершины $2,4,6$ и $3,5,1$ попарно несоединены. Значит он двудольный."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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G6TgiHud0OunatSu33XYbs2fPNh3Hb+zevZvw8HDtiXATTc5+7NixY0RHR5OYmKhiFsua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\nIiLiZf4/p5czxEO2VL0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11f81ef98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw_shell(G2, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Матрица Лапласа первого графа"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2 -1  0 -1  0  0]\n",
      " [-1  4 -1  0 -1 -1]\n",
      " [ 0 -1  2 -1  0  0]\n",
      " [-1  0 -1  2  0  0]\n",
      " [ 0 -1  0  0  2 -1]\n",
      " [ 0 -1  0  0 -1  2]]\n"
     ]
    }
   ],
   "source": [
    "print(nx.laplacian_matrix(G1).toarray())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Характеристический многочлен имеет вид: $(\\lambda-3)^2(\\lambda-2)\\lambda (\\lambda^2 - 6\\lambda + 4)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Матрица Лапласа второго графа."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2 -1  0 -1  0  0]\n",
      " [-1  3 -1  0 -1  0]\n",
      " [ 0 -1  2 -1  0  0]\n",
      " [-1  0 -1  3 -1  0]\n",
      " [ 0 -1  0 -1  3 -1]\n",
      " [ 0  0  0  0 -1  1]]\n"
     ]
    }
   ],
   "source": [
    "print(nx.laplacian_matrix(G2).toarray())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Характеристический многочлен также имеет вид: $(\\lambda-3)^2(\\lambda-2)\\lambda (\\lambda^2 - 6\\lambda + 4)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "То есть их спектры Лапласа совпадают!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "На всякий случай численно вычисленные собственные значения."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-9.69399495e-16,  7.63932023e-01,  2.00000000e+00,  3.00000000e+00,\n",
       "        3.00000000e+00,  5.23606798e+00])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eigvalsh(nx.laplacian_matrix(G1).toarray())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.21920032e-15,  7.63932023e-01,  2.00000000e+00,  3.00000000e+00,\n",
       "        3.00000000e+00,  5.23606798e+00])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eigvalsh(nx.laplacian_matrix(G2).toarray())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [Root]",
   "language": "python",
   "name": "Python [Root]"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}