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   "cell_type": "markdown",
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   "source": [
    "### PageRank算法\n",
    "以下图所示的有向图为例，计算每个结点的PR：\n",
    "<img style=\"float: center;\" src=\"directed_graph.png\" width=\"20%\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "\n",
    "n = 7  #有向图中一共有7个节点\n",
    "d = 0.85  #阻尼因子根据经验值确定，这里我们随意给一个值\n",
    "M = np.array([[0, 1/4, 1/3, 0, 0, 1/2, 0],\n",
    "              [1/4, 0, 0, 1/5, 0, 0, 0],\n",
    "              [0, 1/4, 0, 1/5, 1/4, 0, 0],\n",
    "              [0, 0, 1/3, 0, 1/4, 0, 0],\n",
    "              [1/4, 0, 0, 1/5, 0, 0, 0],\n",
    "              [1/4, 1/4, 0, 1/5, 1/4, 0, 0],\n",
    "              [1/4, 1/4, 1/3, 1/5, 1/4, 1/2, 0]])  #根据有向图中各节点的连接情况写出转移矩阵\n",
    "R0 = np.full((7, 1), 1/7)  #设置初始向量R0，R0是一个7*1的列向量，因为有7个节点，我们把R0的每一个值都设为1/7\n",
    "eps = 0.000001  #设置计算精度"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. PageRank的迭代算法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = 0  #用来累计迭代次数\n",
    "R = R0  #对R向量进行初始化\n",
    "judge = False  #用来判断是否继续迭代\n",
    "while not judge:\n",
    "    next_R = d * np.matmul(M, R) + (1 - d) / n * np.ones((7, 1))  #计算新的R向量\n",
    "    diff = np.linalg.norm(R - next_R)  #计算新的R向量与之前的R向量之间的距离，这里采用的是欧氏距离\n",
    "    if diff < eps:  #若两向量之间的距离足够小\n",
    "        judge = True  #则停止迭代\n",
    "    R = next_R  #更新R向量\n",
    "    t += 1  #迭代次数加一\n",
    "R = R / np.sum(R)  #对R向量进行规范化，保证其总和为1，表示各节点的概率分布"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "迭代次数： 24\n",
      "PageRank: \n",
      " [[0.17030305]\n",
      " [0.10568394]\n",
      " [0.11441021]\n",
      " [0.10629792]\n",
      " [0.10568394]\n",
      " [0.15059975]\n",
      " [0.24702119]]\n"
     ]
    }
   ],
   "source": [
    "print('迭代次数：', t)\n",
    "print('PageRank: \\n', R)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. PageRank的幂法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = 0  #用来累计迭代次数\n",
    "x = R0  #对x向量进行初始化\n",
    "judge = False  #用来判断是否继续迭代\n",
    "A = d * M + (1 - d) / n * np.eye(n)  #计算A矩阵，其中np.eye(n)用来创建n阶单位阵E\n",
    "while not judge:\n",
    "    next_y = np.matmul(A, x)  #计算新的y向量\n",
    "    next_x = next_y / np.linalg.norm(next_y)  #对新的y向量规范化得到新的x向量\n",
    "    diff = np.linalg.norm(x - next_x)  #计算新的x向量与之前的x向量之间的距离，这里采用的是欧氏距离\n",
    "    if diff < eps:  #若两向量之间的距离足够小\n",
    "        judge = True  #则停止迭代\n",
    "        R = x  #得到R向量\n",
    "    x = next_x  #更新x向量\n",
    "    t += 1  #迭代次数加一\n",
    "R = R / np.sum(R)  #对R向量进行规范化，保证其总和为1，表示各节点的概率分布"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "迭代次数： 25\n",
      "PageRank: \n",
      " [[0.18860772]\n",
      " [0.09038084]\n",
      " [0.0875305 ]\n",
      " [0.07523049]\n",
      " [0.09038084]\n",
      " [0.15604764]\n",
      " [0.31182196]]\n"
     ]
    }
   ],
   "source": [
    "print('迭代次数：', t)\n",
    "print('PageRank: \\n', R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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