{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 第十二讲：图和网络\n", "\n", "## 图和网络" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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CBx98gMDAQJw+fdog9RFi7KKjoyGVSqFUKlmtQ6lUQiqVIjo6mtU6NEWXg/6G\nYRhERUXh4MGDCA8P1+spplwux65duzBs2DCkpqY26VJPXV0dtmzZgsWLF+O1117D0qVL4e3trbca\nCTEFxtTTozWGTRyPx0NqairCw8MhFoshk8n0sh+ZTAaxWIzw8PAmBwAAtGrVClOmTMHly5cxcOBA\nDBkyBO+99x6uX7+ulzoJMQXJycmQSqV6+319EZlMhsLCQqxfv56V/TcHhUAjeDwekpOTsWHDBmRk\nZODQoUM6O9VUKpU4dOgQMjIysHHjRiQnJ2vV7G3dujU++eQTFBYWwsPDA35+fvjwww9RWlqqkzoJ\nMRWVlZWYMGECampqsHv3bqPp6ZkKCoHnCA0NRX5+Pjp16oRvvvkGx48f1/rJYrlcjuPHj+Obb77B\nSy+9hPz8fJ3M1962bVskJCQgPz8flpaWeOWVVzB//nzcv3+/2WMTYuzKy8sRGBjYMPd/TU0Nvv32\nW6Pp6ZkC6gk0UU5ODlJSUrB//3507doVnp6ecHZ2hq2tbaNH8gzDoKKiomEh64KCAoSFhSE6Ohp+\nfn56q7OoqAhLlizBDz/8gFmzZiE6Otoobp0jRNfu3LmDwMBAnD9//onXO3ToAEtLS4wdO9boenrG\niEJAQ2VlZRCLxZBIJJBKpVAoFHB1dYVAIIC5uXnDKlvFxcUQCAQQiUQICgpCRESEQW8bu3z5MuLj\n45GdnY3Y2FhMnTrVZGY1JORFiouLERAQgIKCgide79u3LzIzM7F8+XKIxWIMHz5cL8/9yGQyZGZm\nIiIiAitXrjTZAAAoBJqttLQUeXl5qKqqQm1tLSwtLWFjYwMfHx+jmDskNzcXCxYswKVLl5CQkIB3\n332Xnj4mJk0mkyEgIADXrl174nV/f3+kp6c3zN6Znp6OyMhICIVC+Pv762QKFqVSiezsbBQWFmLT\npk0mewnorygEOOLYsWOIjY3FvXv3sHTpUowcOdKkj14IN12/fh1vvPEGioqKnng9KCgIe/fufWql\nsfLyckyfPh1HjhyBSCSCr6+vVpeI5HI5cnNzIZVKERgYiHXr1sHe3r5ZfxdjQSHAIQzD4ODBg4iN\njYWFhQWSkpIQGBhIYUBMxsOHDzF48GCcPXu24bWwsDDs3LnzuZc7TaWnxwYKAQ5Sq9XYs2cP4uLi\n4OLiguXLl6Nv375sl0XICxUXF2Pw4MF4+PAhSktLMXbsWHz33XdNXlPaVHp6hkQhwGH19fUQi8VI\nTEyESCTCOmHwAAARw0lEQVTC0qVL0aNHD7bLIqRRj3sBU6ZMwZQpU/Dll18iLi6uWT0uY+/pGQKF\nAIFCoUBqaipWrFiBN998E4mJifD09GS7LEIaFBYWIjAwELNnz8ZHH33EdjktCj0sRiAQCDBr1ixc\nuXIFXbp0QZ8+fRAVFYWSkhK2SyMEFy9ehL+/PxYuXEgBoAcUAqSBjY0NFi1ahPz8fFhbW6NHjx6I\niYlBeXk526URjpJKpQgICMDKlSsxZcoUtstpkSgEyFPat2+PVatWIS8vDxUVFejSpQuWLl2Khw8f\nsl0a4ZBTp05h2LBhWLduHd599122y2mxKATIM7m6uuLrr7/GqVOncOnSJbz88stISUnReu1VQprq\n6NGjCAkJwbfffotRo0axXU6LRiFAXujll1/Gtm3bIJFIkJWVhS5duuDbb79FfX0926WRFujQoUMY\nPXo0duzYgeDgYLbLafHo7iCisePHjyM2NhZ37tzBkiVLMGrUKHrgjOjE46ke9u7diwEDBrBdDidQ\nCBCtMAyDQ4cOITY2FjweD8uWLcPQoUMpDIjWdu3ahZkzZ+LHH39scU/lGjMKAdIsarUae/fuxcKF\nC+Hk5ISkpCT0799f43Hq6upw9+5duLi46KFKYuzEYjHmzZuHgwcPomfPnmyXwynUEyDNYmZmhtGj\nR+PChQt47733MH78eISEhCAvL6/JYzAMg61bt2LEiBF4++23sX79eqhUKj1WTYzJV199hYULF+Ln\nn3+mAGABhQDRCQsLC7z//vu4fPkyAgMDMXToUEyYMAFXrlx54bY8Hg/29vawsbHB7du38eGHH2LQ\noEHUeOaA8+fPY8WKFcjOzka3bt3YLoeT6HIQ0YuqqiqkpKSgqKgIKSkpEAgETdouLS0NkZGR+Pe/\n/42ZM2fquUqibwzDPLdPVF1dDYVC0WKmZTZFFAJEr1Qq1Qsn+FKr1TAzM0NFRQXmz5+PPXv24PLl\nyya5aDfXFRUVQSaT4f79+xg4cCD9G5oAuhxE9KopMzyamT36Mbx48SJ2796N999/n748TFBaWhre\nfPNNTJw4ESNGjECfPn1w4MAB6u8YOQoBYhSqq6uRkZEBuVyOqKgotsshGvrss88wc+ZMTJo0CZs3\nb8ahQ4fQtm1bzJs3D3fu3GG7PPIcFmwXQAjwaKrgHTt2YMyYMXpZGJzoT3Z2Nv7zn/8gISEBUVFR\nDf2ftWvXYtCgQZBIJJg0aRK7RZJnohAgrKutrUVWVhZkMhl2797NdjlEQ4+nHA8LC3viBgAnJyfY\n2Ng0nAm8qElM2EEhQFhTX18PCwsLlJSUYNu2bQgMDESvXr3YLotoaNy4cejRowc8PDwaXmMYBm5u\nbrC0tERVVRUAUAAYKQoBwooLFy7g8OHDGD58OH777TdIpVJIJBIAdMRoaszMzJ5alpTH4zWs10tP\ngRs3CgHCiqysLMyePRurV6+GWq1G9+7d0adPHwB0xNgSqFQqlJeXo66uDu3bt294vaqqCrm5uejd\nuzcsLS1ZrJA8RncHEVZ8/PHHqKysxKRJk1BXV4c//vgDsbGxOHHiBJRK5VOfr6mpQWlpKQuVEm2Y\nm5tDoVCgtrYWfD4fAPDnn39i5MiRmD9/Ph48eMByheQxCgHCGhsbGyxevBi3b9/Gl19+iR9//BHz\n5s1rdLoIlUqFnj17YurUqbh58yYL1RJNPXz4ECqVCnZ2dsjLy8OwYcNQVFSEPXv24B//+Afb5ZH/\noRAgRmH69OmQyWTYvHkz2rRp89T71tbWyM/Ph729PXx8fDB79mzcvXuXhUpJU92+fRsODg745Zdf\nMHjwYHh6eqKgoABOTk5sl0b+gkKAGBVPT89nvmdvb48VK1bg4sWLUCgU8PLyQkJCAl1aMFJ1dXW4\nd+8eEhMTMW3aNBw4cIDtkkgjKASIyXF2dsa6deuQk5ODa9euQSgU4vPPP0dNTQ3bpXFSTU1No+tO\n9+3bFx07dsT+/fuRlJTEQmWkKSgEiMny8PBAWloajhw5gl9//RVdunTBhg0bUFdXx3ZpnPHHH3/g\ntddea/T/uY2NDa5cuYKQkBAWKiNNRSFATN4rr7yCffv2Yc+ePdixYwe8vb2xY8cOqNVqtktr0c6d\nO4chQ4YgJiYG1tbWjX7m8Z1BxHjRVNKkxcnKykJsbCyUSiWWLVuG4cOH07MHOnbmzBmEhIRg7dq1\nGDNmDNvlkGagECAtEsMw2LdvHxYuXAh7e3skJSVh4MCBbJfVIhw7dgxvv/02Nm3aRJd6WgAKAdKi\nqVQq/Oc//0F8fDy8vLyQlJQEkUjEdlkmKysrC+PHj8e2bdvw5ptvsl0O0QHqCZAWzdzcHO+99x7y\n8/MRHByM4cOHIzw8HAUFBWyXZnIyMjIwfvx4fP/99xQALQiFAOEES0tLfPjhh7hy5QpEIhEGDBiA\nyMhIFBUVsV2aSfj+++/x/vvv48cff8SgQYPYLofoEIUA4RQrKyvMnz8fly9fhqOjI3x9fTFr1ix6\n+vg5tm7dig8//BAHDx5smOSPtBwUAoST7OzskJSUhIsXL6K+vh5eXl5YtGgRKisr2S7NqGzYsAEx\nMTHIysqiXkoLRSFAOM3JyQlffvklzp49ixs3bkAoFOKzzz6jp48BfPHFF1i6dCmys7Ph7e3NdjlE\nTygECAHg7u4OsViMX375BSdOnIBQKMTXX3+Nuro65ObmYsGCBaioqGC7TINZsWIFUlJScPToUQiF\nQrbLIXpEt4gS0ogzZ85gwYIFuH79OqytrXHu3DnY2dkhJiYGM2fObHSm05aAYRjEx8dj9+7dyMrK\ngqurK9slET2jECDkOdasWYNZs2Y98ZqTkxPi4uIQGRnZoqZFYBgGc+bMweHDh3H48GE4OjqyXRIx\nAAoBQp5jxIgR2L9/f6Pvubu7IzExERMmTIC5ubnBaiopKUFeXh6qqqqgVCrB5/NhY2ODnj17wtnZ\nWasx1Wo1PvzwQ+Tk5EAikcDe3l7HVRNjRSFAyHM8fPgQX3zxBZKTk59555C3tzeWLl2KsLAwvcxR\nVFZWhi1btkAikUAqlUKpVMLV1RUCgQBmZmZQq9VQKBQoLi4Gn8+HSCRCUFAQJk2aBAcHhxeOr1Kp\nEBkZicLCQmRkZKBdu3Y6/zsQ40UhQEgTlJeXY9WqVUhJSXnmnUN9+vRBUlIShgwZopN9njlzBikp\nKUhPT4eXlxc8PDzg4uKCdu3aNRo2DMOgsrISJSUluHbtGvLz8xEaGoro6Gj07t270X3U1dVh4sSJ\nuHv3LtLT02FlZaWT2onpoBAgRAOlpaVYunQpvvnmm0bXQgaAwMBALFu27JlfvC9SXl6O6dOn48iR\nIxCJRPD19dXqy1kulyM3NxdSqRQBAQFYv379E5d5amtrMXbsWNTV1WHPnj1o3bq1VvUS00YhQIgW\nrl27hoSEBGzduhXP+hUaOXIklixZotE99unp6YiMjIRQKIS/v79OGs9KpRLZ2dkoLCxsmPmzuroa\nb7/9NqysrLB9+/YW1eAmmqEQIKQZLly4gLi4OOzbt6/R93k8HiZOnIiEhAS4u7s/cxyGYRATEwOx\nWIzhw4fDzc1N57XKZDJkZmZ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import networkx as nx\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "\n", "dg = nx.DiGraph()\n", "dg.add_edges_from([(1,2), (2,3), (1,3), (1,4), (3,4)])\n", "edge_labels = {(1, 2): 1, (1, 3): 3, (1, 4): 4, (2, 3): 2, (3, 4): 5}\n", "\n", "pos = nx.spring_layout(dg)\n", "nx.draw_networkx_edge_labels(dg,pos,edge_labels=edge_labels, font_size=16)\n", "nx.draw_networkx_labels(dg, pos, font_size=20, font_color='w')\n", "nx.draw(dg, pos, node_size=1500, node_color=\"gray\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "该图由4个节点与5条边组成，\n", "\n", "$$\n", "\\begin{array}{c | c c c c}\n", " & node_1 & node_2 & node_3 & node_4 \\\\\n", "\\hline\n", "edge_1 & -1 & 1 & 0 & 0 \\\\\n", "edge_2 & 0 & -1 & 1 & 0 \\\\\n", "edge_3 & -1 & 0 & 1 & 0 \\\\\n", "edge_4 & -1 & 0 & 0 & 1 \\\\\n", "edge_5 & 0 & 0 & -1 & 1 \\\\\n", "\\end{array}\n", "$$\n", "\n", "我们可以建立$5 \\times 4$矩阵\n", "$\n", "A=\n", "\\begin{bmatrix}\n", "-1 & 1 & 0 & 0 \\\\\n", "0 & -1 & 1 & 0 \\\\\n", "-1 & 0 & 1 & 0 \\\\\n", "-1 & 0 & 0 & 1 \\\\\n", "0 & 0 & -1 & 1 \\\\\n", "\\end{bmatrix}\n", "$\n", "\n", "观察前三行，易看出这三个行向量线性相关，也就是这三个向量可以形成回路（loop）。\n", "\n", "现在，解$Ax=0$：\n", "$\n", "Ax=\n", "\\begin{bmatrix}\n", "-1 & 1 & 0 & 0 \\\\\n", "0 & -1 & 1 & 0 \\\\\n", "-1 & 0 & 1 & 0 \\\\\n", "-1 & 0 & 0 & 1 \\\\\n", "0 & 0 & -1 & 1 \\\\\n", "\\end{bmatrix}\n", "\\begin{bmatrix}\n", "x_1\\\\x_2\\\\x_3\\\\x_4\\\\\n", "\\end{bmatrix}\n", "$。\n", "\n", "展开得到：\n", "$\\begin{bmatrix}x_2-x_1 \\\\x_3-x_2 \\\\x_3-x_1 \\\\x_4-x_1 \\\\x_4-x_3 \\\\ \\end{bmatrix}=\\begin{bmatrix}0\\\\0\\\\0\\\\0\\\\0\\\\ \\end{bmatrix}$\n", "\n", "引入矩阵的实际意义：将$x=\\begin{bmatrix}x_1 & x_2 & x_3 & x_4\\end{bmatrix}$设为各节点电势（Potential at the Nodes）。\n", "\n", "则式子中的诸如$x_2-x_1$的元素，可以看做该边上的电势差（Potential Differences）。\n", "\n", "容易看出其中一个解$x=\\begin{bmatrix}1\\\\1\\\\1\\\\1\\end{bmatrix}$，即等电势情况，此时电势差为$0$。\n", "\n", "化简$A$易得$rank(A)=3$，所以其零空间维数应为$n-r=4-3=1$，即$\\begin{bmatrix}1\\\\1\\\\1\\\\1\\end{bmatrix}$就是其零空间的一组基。\n", "\n", "其零空间的物理意义为，当电位相等时，不存在电势差，图中无电流。\n", "\n", "当我们把图中节点$4$接地后，节点$4$上的电势为$0$，此时的\n", "$\n", "A=\n", "\\begin{bmatrix}\n", "-1 & 1 & 0 \\\\\n", "0 & -1 & 1 \\\\\n", "-1 & 0 & 1 \\\\\n", "-1 & 0 & 0 \\\\\n", "0 & 0 & -1 \\\\\n", "\\end{bmatrix}\n", "$，各列线性无关，$rank(A)=3$。\n", "\n", "现在看看$A^Ty=0$（这是应用数学里最常用的式子）：\n", "\n", "$A^Ty=0=\\begin{bmatrix}-1 & 0 & -1 & -1 & 0 \\\\1 & -1 & 0 & 0 & 0 \\\\0 & 1 & 1 & 0 & -1 \\\\0 & 0 & 0 & 1 & 1 \\\\ \\end{bmatrix}\\begin{bmatrix}y_1\\\\y_2\\\\y_3\\\\y_4\\\\y_5\\end{bmatrix}=\\begin{bmatrix}0\\\\0\\\\0\\\\0\\end{bmatrix}$，对于转置矩阵有$dim N(A^T)=m-r=5-3=2$。\n", "\n", "接着说上文提到的的电势差，矩阵$C$将电势差与电流联系起来，电流与电势差的关系服从欧姆定律：边上的电流值是电势差的倍数，这个倍数就是边的电导（conductance）即电阻（resistance）的倒数。\n", "\n", "$\n", "电势差\n", "\\xrightarrow[欧姆定律]{矩阵C}\n", "各边上的电流y_1, y_2, y_3, y_4, y_5\n", "$，而$A^Ty=0$的另一个名字叫做“基尔霍夫电流定律”（Kirchoff's Law, 简称KCL）。\n", "\n", "再把图拿下来观察：" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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0bNq0SafXeignJwdeXl44efIkzp8/j3nz5tHiZjoilUrh6elJ//t2EoUA0Yv9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\nZkldFnUHdRG1tbXw9/fHe++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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import networkx as nx\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "\n", "dg = nx.DiGraph()\n", "dg.add_edges_from([(1,2), (2,3), (1,3), (1,4), (3,4)])\n", "edge_labels = {(1, 2): 1, (1, 3): 3, (1, 4): 4, (2, 3): 2, (3, 4): 5}\n", "\n", "pos = nx.spring_layout(dg)\n", "nx.draw_networkx_edge_labels(dg,pos,edge_labels=edge_labels, font_size=16)\n", "nx.draw_networkx_labels(dg, pos, font_size=20, font_color='w')\n", "nx.draw(dg, pos, node_size=1500, node_color=\"gray\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "将$A^Ty=0$中的方程列出来：\n", "$\n", "\\left\\{\n", "\\begin{aligned}\n", "y_1 + y_3 + y_4 &= 0 \\\\\n", "y_1 - y_2 &= 0 \\\\\n", "y_2 + y_3 - y_5 &= 0 \\\\\n", "y_4 - y_5 &= 0 \\\\\n", "\\end{aligned}\n", "\\right.\n", "$\n", "\n", "对比看$A^Ty=0$的第一个方程，$-y_1-y_3-y_4=0$，可以看出这个方程是关于节点$1$上的电流的，方程指出节点$1$上的电流和为零，基尔霍夫定律是一个平衡方程、守恒定律，它说明了流入等于流出，电荷不会在节点上累积。\n", "\n", "对于$A^T$，有上文得出其零空间的维数是$2$，则零空间的基应该有两个向量。\n", "\n", "* 现在假设$y_1=1$，也就是令$1$安培的电流在边$1$上流动；\n", "* 由图看出$y_2$也应该为$1$；\n", "* 再令$y_3=-1$，也就是让$1$安培的电流流回节点$1$；\n", "* 令$y_4=y_5=0$；\n", "\n", "得到一个符合KCL的向量$\\begin{bmatrix}1\\\\1\\\\-1\\\\0\\\\0\\end{bmatrix}$，代回方程组发现此向量即为一个解，这个解发生在节点$1,2,3$组成的回路中，该解即为零空间的一个基。\n", "\n", "根据上一个基的经验，可以利用$1,3,4$组成的节点求另一个基：\n", "\n", "* 令$y_1=y_2=0$；\n", "* 令$y_3=1$；\n", "* 由图得$y_5=1$；\n", "* 令$y_4=-1$；\n", "\n", "得到令一个符合KCL的向量$\\begin{bmatrix}0\\\\0\\\\1\\\\-1\\\\1\\end{bmatrix}$，代回方程可知此为另一个解。\n", "\n", "则$N(A^T)$的一组基为$\\begin{bmatrix}1\\\\1\\\\-1\\\\0\\\\0\\end{bmatrix}\\quad\\begin{bmatrix}0\\\\0\\\\1\\\\-1\\\\1\\end{bmatrix}$。\n", "\n", "看图，利用节点$1,2,3,4$组成的大回路（即边$1,2,5,4$）：\n", "\n", "* 令$y_3=0$；\n", "* 令$y_1=1$；\n", "* 则由图得$y_2=1, y_5=1, y_4=-1$；\n", "\n", "得到符合KCL的向量$\\begin{bmatrix}1\\\\1\\\\0\\\\-1\\\\1\\end{bmatrix}$，易看出此向量为求得的两个基之和。\n", "\n", "接下来观察$A$的行空间，即$A^T$的列空间，方便起见我们直接计算\n", "$\n", "A^T=\n", "\\begin{bmatrix}\n", "-1 & 0 & -1 & -1 & 0 \\\\\n", "1 & -1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 1 & 0 & -1 \\\\\n", "0 & 0 & 0 & 1 & 1 \\\\\n", "\\end{bmatrix}\n", "$\n", "的列空间。\n", "\n", "易从基的第一个向量看出前三列$A^T$的线性相关，则$A^T$的主列为第$1,2,4$列，对应在图中就是边$1,2,4$，可以发现这三条边没有组成回路，则在这里可以说**线性无关等价于没有回路**。由$4$个节点与$3$条边组成的图没有回路，就表明$A^T$的对应列向量线性无关，也就是节点数减一（$rank=nodes-1$）条边线性无关。另外，没有回路的图也叫作树（Tree）。\n", "\n", "再看左零空间的维数公式：$dim N(A^T)=m-r$，左零空间的维数就是相互无关的回路的数量，于是得到$loops=edges-(nodes-1)$，整理得：\n", "\n", "$$\n", "nodes-edges+loops=1\n", "$$\n", "\n", "此等式对任何图均有效，任何图都有此拓扑性质，这就是著名的欧拉公式（Euler's Formula）。$零维（节点）-一维（边）+二维（回路）=1$便于记忆。\n", "\n", "总结：\n", "\n", "* 将电势记为$e$，则在引入电势的第一步中，有$e=Ax$；\n", "* 电势差导致电流产生，$y=Ce$；\n", "* 电流满足基尔霍夫定律方程，$A^Ty=0$；\n", "\n", "这些是在无电源情况下的方程。\n", "\n", "电源可以通过：在边上加电池（电压源），或在节点上加外部电流 两种方式接入。\n", "\n", "如果在边上加电池，会体现在$e=Ax$中；如果在节点上加电流，会体现在$A^Ty=f$中，$f$向量就是外部电流。\n", "\n", "将以上三个等式连起来得到$A^TCAx=f$。另外，最后一个方程是一个平衡方程，还需要注意的是，方程仅描述平衡状态，方程并不考虑时间。最后，$A^TA$是一个对称矩阵。" ] } ], "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": "3.5.1" } }, "nbformat": 4, "nbformat_minor": 0 }