Complex_Network_Analysis/Measuring Networks.ipynb

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    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#全局测量\" data-toc-modified-id=\"全局测量-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>全局测量</a></span><ul class=\"toc-item\"><li><span><a href=\"#节点数\" data-toc-modified-id=\"节点数-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>节点数</a></span></li><li><span><a href=\"#连边数\" data-toc-modified-id=\"连边数-1.2\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>连边数</a></span></li><li><span><a href=\"#密度\" data-toc-modified-id=\"密度-1.3\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>密度</a></span></li></ul></li><li><span><a href=\"#探索邻居\" data-toc-modified-id=\"探索邻居-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>探索邻居</a></span><ul class=\"toc-item\"><li><span><a href=\"#使用隐式的字典表征图\" data-toc-modified-id=\"使用隐式的字典表征图-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>使用隐式的字典表征图</a></span></li><li><span><a href=\"#调用函数\" data-toc-modified-id=\"调用函数-2.2\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>调用函数</a></span></li></ul></li><li><span><a href=\"#聚类系数\" data-toc-modified-id=\"聚类系数-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>聚类系数</a></span></li><li><span><a href=\"#考虑-路径(paths)\" data-toc-modified-id=\"考虑-路径(paths)-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>考虑 路径(paths)</a></span></li><li><span><a href=\"#Node-eccentricity\" data-toc-modified-id=\"Node-eccentricity-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>Node eccentricity</a></span></li><li><span><a href=\"#选择正确的中心\" data-toc-modified-id=\"选择正确的中心-6\"><span class=\"toc-item-num\">6&nbsp;&nbsp;</span>选择正确的中心</a></span><ul class=\"toc-item\"><li><span><a href=\"#度中心性\" data-toc-modified-id=\"度中心性-6.1\"><span class=\"toc-item-num\">6.1&nbsp;&nbsp;</span>度中心性</a></span></li><li><span><a href=\"#接近中心性\" data-toc-modified-id=\"接近中心性-6.2\"><span class=\"toc-item-num\">6.2&nbsp;&nbsp;</span>接近中心性</a></span></li><li><span><a href=\"#介数中心性\" data-toc-modified-id=\"介数中心性-6.3\"><span class=\"toc-item-num\">6.3&nbsp;&nbsp;</span>介数中心性</a></span></li><li><span><a href=\"#特征向量中心性\" data-toc-modified-id=\"特征向量中心性-6.4\"><span class=\"toc-item-num\">6.4&nbsp;&nbsp;</span>特征向量中心性</a></span></li></ul></li><li><span><a href=\"#中心性指标分析\" data-toc-modified-id=\"中心性指标分析-7\"><span class=\"toc-item-num\">7&nbsp;&nbsp;</span>中心性指标分析</a></span><ul class=\"toc-item\"><li><span><a href=\"#中心性指标计算\" data-toc-modified-id=\"中心性指标计算-7.1\"><span class=\"toc-item-num\">7.1&nbsp;&nbsp;</span>中心性指标计算</a></span></li><li><span><a href=\"#中心性指标之间关联\" data-toc-modified-id=\"中心性指标之间关联-7.2\"><span class=\"toc-item-num\">7.2&nbsp;&nbsp;</span>中心性指标之间关联</a></span></li></ul></li><li><span><a href=\"#通过协调性评估网络的均匀性\" data-toc-modified-id=\"通过协调性评估网络的均匀性-8\"><span class=\"toc-item-num\">8&nbsp;&nbsp;</span>通过协调性评估网络的均匀性</a></span><ul class=\"toc-item\"><li><span><a href=\"#NetworkX-评估属性的协调性\" data-toc-modified-id=\"NetworkX-评估属性的协调性-8.1\"><span class=\"toc-item-num\">8.1&nbsp;&nbsp;</span>NetworkX 评估属性的协调性</a></span></li></ul></li></ul></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 创建网络之后，分析网络特性\n",
    "\n",
    "* 聚类系数， 中心性等"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 全局测量"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T03:13:31.208074Z",
     "start_time": "2019-10-24T03:13:31.204535Z"
    }
   },
   "outputs": [],
   "source": [
    "import networkx as nx\n",
    "import pprint as pp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T03:14:14.789481Z",
     "start_time": "2019-10-24T03:14:13.864259Z"
    }
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      " ('Kermack-Mckendrick Theory', {'contraction': 0}),\n",
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      " ('Piecewise Deterministic Markov Process', {'contraction': 0}),\n",
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      " ('Netlog', {'contraction': 0}),\n",
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      " ('Tvtag', {'contraction': 0}),\n",
      " ('Population', {'contraction': 0}),\n",
      " (\"Prisoner'S Dilemma\", {'contraction': 0}),\n",
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      " ('Qzone', {'contraction': 0}),\n",
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      " ('The New Palgrave Dictionary Of Economics', {'contraction': 0}),\n",
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      " ('Béla H. Bánáthy', {'contraction': 0}),\n",
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      " ('Fax', {'contraction': 0}),\n",
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      " (\"Doob'S Martingale Convergence Theorems\", {'contraction': 0}),\n",
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      " ('Biome', {'contraction': 0}),\n",
      " ('Schreier Coset Graph', {'contraction': 0}),\n",
      " (\"Benford'S Law\", {'contraction': 0}),\n",
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      " ('Manuela M. Veloso', {'contraction': 0}),\n",
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      " ('Climate', {'contraction': 0}),\n",
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      " ('Clique', {'contraction': 0}),\n",
      " (\"Conway'S Game Of Life\", {'contraction': 0}),\n",
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      " ('Primate', {'contraction': 0}),\n",
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      " ('Social Web', {'contraction': 0}),\n",
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      " ('Feminist Technoscience', {'contraction': 0}),\n",
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      " ('Time', {'contraction': 0}),\n",
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      " ('Ivan Pavlov', {'contraction': 0}),\n",
      " ('Encyclopædia Britannica', {'contraction': 0}),\n",
      " ('Richard Lazarus', {'contraction': 0}),\n",
      " ('Pager', {'contraction': 0}),\n",
      " ('Progressively Measurable Process', {'contraction': 0}),\n",
      " ('Log-Normal Distribution', {'contraction': 0}),\n",
      " ('W. W. Norton & Company', {'contraction': 0}),\n",
      " ('Harvard Department Of Social Relations', {'contraction': 0}),\n",
      " ('Galton–Watson Process', {'contraction': 0}),\n",
      " ('Q-Gaussian Distribution', {'contraction': 0}),\n",
      " ('Metaheuristic', {'contraction': 0}),\n",
      " ('Classical Wiener Space', {'contraction': 0}),\n",
      " ('Keek', {'contraction': 0}),\n",
      " ('Ip Address', {'contraction': 0}),\n",
      " ('Kolmogorov Extension Theorem', {'contraction': 0}),\n",
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      " ('Null Model', {'contraction': 0}),\n",
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      " ('Computational Chemistry', {'contraction': 0}),\n",
      " ('Harry Harlow', {'contraction': 0}),\n",
      " ('Neuroimaging', {'contraction': 0}),\n",
      " ('Theorem', {'contraction': 0}),\n",
      " ('Laplace Distribution', {'contraction': 0}),\n",
      " ('Input-Output Model', {'contraction': 0}),\n",
      " ('Pearson Product-Moment Correlation Coefficient', {'contraction': 0}),\n",
      " ('Multiple Edge', {'contraction': 0}),\n",
      " ('Interview (Research)', {'contraction': 0}),\n",
      " ('Case Study', {'contraction': 0}),\n",
      " ('Exponential-Logarithmic Distribution', {'contraction': 0}),\n",
      " ('Forest (Graph Theory)', {'contraction': 0})]\n"
     ]
    }
   ],
   "source": [
    "G = nx.read_graphml(\"Data/cna.graphml\") # wikipedia 网络\n",
    "pp.pprint(list(G.nodes(data=True)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  节点数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T03:20:37.766904Z",
     "start_time": "2019-10-24T03:20:37.761439Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2995\n",
      "2995\n",
      "2995\n",
      "2995\n"
     ]
    }
   ],
   "source": [
    "print(len(G))\n",
    "print(len(G.node))\n",
    "print(len(G.nodes()))\n",
    "print(nx.number_of_nodes(G))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 连边数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T03:21:46.469064Z",
     "start_time": "2019-10-24T03:21:46.455319Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11817\n",
      "11817\n"
     ]
    }
   ],
   "source": [
    "print(len(G.edges()))\n",
    "print(nx.number_of_edges(G))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T03:23:45.474826Z",
     "start_time": "2019-10-24T03:23:45.471751Z"
    }
   },
   "source": [
    "## 密度\n",
    "\n",
    "* 计算已经存在的连边和所有可能连边的比\n",
    "\n",
    "* n个节点 m 个连边的有向网络， m/n(n-1)\n",
    "\n",
    "* n个节点 m 个连边的无向网络 2m/n(n-1)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T03:27:29.531822Z",
     "start_time": "2019-10-24T03:27:29.518209Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.001317827641928264"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.density(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 探索邻居\n",
    "\n",
    "* 上面的节点，连边属于宏观特性\n",
    "\n",
    "* 单个节点和他的邻居属于微观级别\n",
    "\n",
    "**邻居代表网络图的局部特性** \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T06:23:46.512978Z",
     "start_time": "2019-10-24T06:23:46.486814Z"
    }
   },
   "source": [
    "## 使用隐式的字典表征图\n",
    "\n",
    "* 节点的名字的 是 key\n",
    "\n",
    "* 邻居节点的字典的是 value\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T06:24:37.198676Z",
     "start_time": "2019-10-24T06:24:37.192084Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['Degree (Graph Theory)',\n",
      " 'Isolated Vertex',\n",
      " 'Chordal Graph',\n",
      " 'Complete Graph',\n",
      " 'Turán Graph',\n",
      " 'Loop (Graph Theory)',\n",
      " 'Linear Time',\n",
      " 'Gerhard Ringel',\n",
      " 'Adjacency List',\n",
      " 'Clustering Coefficient',\n",
      " 'Independent Set (Graph Theory)',\n",
      " 'Planar Graph',\n",
      " 'Discrete Mathematics (Journal)',\n",
      " 'Chromatic Number',\n",
      " 'Vertex Figure',\n",
      " 'Adjacency Matrix',\n",
      " 'Graph Theory',\n",
      " 'Independence Number',\n",
      " 'Graph Isomorphism',\n",
      " 'Dense Graph',\n",
      " 'Cycle Graph',\n",
      " 'Triangle-Free Graph',\n",
      " 'Perfect Graph',\n",
      " 'Induced Subgraph',\n",
      " 'Vertex (Graph Theory)',\n",
      " 'Octahedron',\n",
      " 'Graph (Discrete Mathematics)',\n",
      " 'Edge (Graph Theory)',\n",
      " 'Modular Decomposition',\n",
      " 'Complement Graph',\n",
      " 'Journal Of The Acm',\n",
      " 'Neighbourhood (Mathematics)']\n",
      "32\n"
     ]
    }
   ],
   "source": [
    "ego = \"Neighbourhood (Graph Theory)\"\n",
    "\n",
    "alters1 = G[ego]\n",
    "pp.pprint(list(alters1))\n",
    "print(len(alters1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 调用函数\n",
    "\n",
    "* 返回一个 generator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T06:27:08.345313Z",
     "start_time": "2019-10-24T06:27:08.336798Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['Interdependent Networks',\n",
      " 'Semantic Network',\n",
      " 'Scientific Collaboration Network',\n",
      " 'Biological Network',\n",
      " 'Clustering Coefficient',\n",
      " 'Community Structure',\n",
      " 'Modularity (Networks)',\n",
      " 'Boolean Network',\n",
      " 'Network Motif',\n",
      " 'Exponential Random Graph Model',\n",
      " 'Complex Network',\n",
      " 'Hyperbolic Geometric Graph',\n",
      " 'Computer Network',\n",
      " 'Watts And Strogatz Model',\n",
      " 'Efficiency (Network Science)',\n",
      " 'Hierarchical Network Model',\n",
      " 'Random Graph',\n",
      " 'Social Network',\n",
      " 'Scale-Free Network',\n",
      " 'Erdős–Rényi Model',\n",
      " 'Network Theory',\n",
      " 'Telecommunications Network',\n",
      " 'Network Science',\n",
      " 'Transport Network',\n",
      " 'Barabási–Albert Model',\n",
      " 'Spatial Network',\n",
      " 'Small-World Network',\n",
      " 'Reciprocity In Network',\n",
      " 'Reciprocity (Network Science)',\n",
      " 'Degree (Graph Theory)',\n",
      " 'Isolated Vertex',\n",
      " 'Chordal Graph',\n",
      " 'Complete Graph',\n",
      " 'Turán Graph',\n",
      " 'Loop (Graph Theory)',\n",
      " 'Linear Time',\n",
      " 'Gerhard Ringel',\n",
      " 'Adjacency List',\n",
      " 'Clustering Coefficient',\n",
      " 'Independent Set (Graph Theory)',\n",
      " 'Planar Graph',\n",
      " 'Discrete Mathematics (Journal)',\n",
      " 'Chromatic Number',\n",
      " 'Vertex Figure',\n",
      " 'Adjacency Matrix',\n",
      " 'Graph Theory',\n",
      " 'Independence Number',\n",
      " 'Graph Isomorphism',\n",
      " 'Dense Graph',\n",
      " 'Cycle Graph',\n",
      " 'Triangle-Free Graph',\n",
      " 'Perfect Graph',\n",
      " 'Induced Subgraph',\n",
      " 'Vertex (Graph Theory)',\n",
      " 'Octahedron',\n",
      " 'Graph (Discrete Mathematics)',\n",
      " 'Edge (Graph Theory)',\n",
      " 'Modular Decomposition',\n",
      " 'Complement Graph',\n",
      " 'Journal Of The Acm',\n",
      " 'Neighbourhood (Mathematics)']\n",
      "61\n"
     ]
    }
   ],
   "source": [
    "alters2 = list(nx.all_neighbors(G, ego)) \n",
    "pp.pprint(alters2) \n",
    "print(len(alters2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "注意到两种方式返回的结果数目不相同（网络是有向网络）\n",
    "\n",
    "*  第一种只包含 中心节点指向的邻居 （out- neighborhood）\n",
    "\n",
    "*  第二种 包含所有的邻居节点\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "注意到两种方式返回的结果数目不相同（网络是有向网络）\n",
    "\n",
    "* 第一种只包含 中心节点指向的邻居 （out- neighborhood）\n",
    "\n",
    "![](https://i.loli.net/2019/10/24/9WIutpDh4MC6K1E.png)\n",
    "\n",
    "* 第二种 包含所有的邻居节点\n",
    "\n",
    "![](https://i.loli.net/2019/10/24/kVRg1qUHvxa8AsW.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 聚类系数\n",
    "\n",
    "> 在你的朋友关系网络中， 你的两个朋友也可能是朋友， 这种可能性的大小反映了朋友圈的紧密程度。 聚类系数： 定量刻画任意两个朋友之间也相互为朋友的概率。 \n",
    "\n",
    "* 聚类系数： 包含中心节点的三角闭合数目和 以中心节点为中心的连通三元组的数目的比\n",
    "\n",
    "* 平均聚类系数： 所有节点聚类系数的平均值\n",
    "\n",
    "* 整个网络的聚类系数 ：网络中所有的三角闭合和可能存在的三角闭合之间的比值 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T06:47:24.749629Z",
     "start_time": "2019-10-24T06:47:24.660437Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.39378531073446327\n"
     ]
    }
   ],
   "source": [
    "cc = nx.clustering(nx.Graph(G), ego) # 中心网络的聚类系数\n",
    "print(cc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T14:16:08.940814Z",
     "start_time": "2019-10-24T14:16:08.412521Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7155239923349956\n"
     ]
    }
   ],
   "source": [
    "acc = nx.average_clustering(nx.Graph(G)) # 平均网络聚类系数\n",
    "print(acc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T14:17:21.174322Z",
     "start_time": "2019-10-24T14:17:20.939219Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.03278041468514661\n"
     ]
    }
   ],
   "source": [
    "trans = nx.transitivity(G) # 整个网络的聚类系数  ： 横截性\n",
    "print(trans)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 上述考虑的都是无权网络，对于加权网络， 分析较为复杂"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 考虑 路径(paths)\n",
    "\n",
    "\n",
    "* walk: 连边的任意序列\n",
    "\n",
    "* trail: 不使用重复连边的 walk \n",
    "\n",
    "* path：  不访问重复节点的 trail\n",
    "\n",
    "---\n",
    "\n",
    "* 对于无权的网络， 路径的长度就是连边的数目\n",
    "\n",
    "---\n",
    "\n",
    "* path 是将两个不相邻的节点连接在一起，允许他们相互影响\n",
    "\n",
    "----\n",
    "\n",
    "* 两个节点之间可能不只一条 path， 最短的路径通常最重要"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T14:53:17.939330Z",
     "start_time": "2019-10-24T14:53:17.920673Z"
    }
   },
   "outputs": [],
   "source": [
    "path_generator = nx.shortest_simple_paths(G, ego, \"Graph Theory\") # 返回一个生成器"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T14:53:18.480019Z",
     "start_time": "2019-10-24T14:53:18.473699Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['Neighbourhood (Graph Theory)', 'Graph Theory']"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "next(path_generator)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T14:53:19.210072Z",
     "start_time": "2019-10-24T14:53:19.187211Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['Neighbourhood (Graph Theory)', 'Clustering Coefficient', 'Graph Theory']"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "next(path_generator)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T14:53:24.912976Z",
     "start_time": "2019-10-24T14:53:24.900978Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['Neighbourhood (Graph Theory)', 'Loop (Graph Theory)', 'Graph Theory']"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "next(path_generator)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T14:54:52.093050Z",
     "start_time": "2019-10-24T14:54:52.084981Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['Neighbourhood (Graph Theory)', 'Graph Theory']\n"
     ]
    }
   ],
   "source": [
    "path = nx.shortest_path(G, ego, \"Graph Theory\") # 只返回一个最短路径\n",
    "print(path)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T14:57:58.785546Z",
     "start_time": "2019-10-24T14:57:58.778898Z"
    }
   },
   "source": [
    "* 如果参数中省略目标节点， 则返回从起始节点到所有其他节点的最短路径\n",
    "\n",
    "* 如果参数中省略起始节点， 则返回从所有其他节点到目标节点的最短路径\n",
    "\n",
    "* 如果全部省略 ： 返回所有最短路径"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Node eccentricity\n",
    "\n",
    "* 一个节点到其他所有节点的距离"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T15:04:48.360675Z",
     "start_time": "2019-10-24T15:04:10.410529Z"
    }
   },
   "outputs": [],
   "source": [
    "ecc = nx.eccentricity(nx.Graph(G))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T15:05:08.399298Z",
     "start_time": "2019-10-24T15:05:08.367024Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n"
     ]
    }
   ],
   "source": [
    "print(ecc[ego])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**网络的直径就是最大的 eccentricity**\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T15:06:49.505984Z",
     "start_time": "2019-10-24T15:06:49.500259Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4\n"
     ]
    }
   ],
   "source": [
    "print(nx.diameter(G,ecc))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 网络的半径是最小偏心量。这个定义并不直观，但它就是这样。更违反直觉的是，一般来说，半径不是直径的一半。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T15:07:49.993525Z",
     "start_time": "2019-10-24T15:07:49.985941Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n"
     ]
    }
   ],
   "source": [
    "print(nx.radius(G,ecc))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**网络的中心是 离心距等于半径的节点的集合**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T15:09:23.108314Z",
     "start_time": "2019-10-24T15:09:23.102520Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['Complex Network']"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nx.center(G,ecc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**网络的外围是所有离心距等于直径的节点的集合。复杂网络中的外围节点集通常很大。**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T15:11:24.476626Z",
     "start_time": "2019-10-24T15:11:24.470322Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2875"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(nx.periphery(G,ecc))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 选择正确的中心\n",
    "\n",
    "* 选出最重要的节点\n",
    "\n",
    "* 一个节点的重要性取决于该节点在网络中的结构位置，可以通过邻近、最短路径或其他一些结构元素来计算。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  度中心性\n",
    "\n",
    "* 一个节点的度越大就意味着这个节点越重要\n",
    "\n",
    "* 为了便于不同网络之间比较， 对中心性指标进行归一化处理 （不然大网络中 小度节点比小网络中大度节点还多）\n",
    "\n",
    "$$DC= \\frac{k_{i}}{N-1}$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 接近中心性\n",
    "\n",
    "* 一个节点到网络中所有及诶单的距离的平均值 的倒数\n",
    "\n",
    "* 值越大说明节点越接近其他的节点\n",
    "\n",
    "* 接近数大的节点对于网络中信息的流动有着最佳的观察视野"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 介数中心性\n",
    "\n",
    "* 经过某个节点的最短路径的数目\n",
    "\n",
    "* 刻画了节点对网络中信息传输的控制能力\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 特征向量中心性\n",
    "\n",
    "* 节点的重要性既取决于邻居节点的数目，也取决于邻居节点的重要性\n",
    "\n",
    "* 基于这种想法有两种重要的算法： HITS 和 PageRank"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 中心性指标分析"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 中心性指标计算"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T16:37:49.214894Z",
     "start_time": "2019-10-24T16:37:30.132163Z"
    },
    "collapsed": true
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'pd' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-42-1f3b6944056e>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      7\u001b[0m \u001b[0mhits\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mhits\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      8\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 9\u001b[0;31m centralities = pd.concat(\n\u001b[0m\u001b[1;32m     10\u001b[0m     \u001b[0;34m[\u001b[0m\u001b[0mpd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mSeries\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mc\u001b[0m \u001b[0;32min\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mhits\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0meig\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpgr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhar\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclo\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhits\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdgr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbet\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     11\u001b[0m     axis=1)\n",
      "\u001b[0;31mNameError\u001b[0m: name 'pd' is not defined"
     ]
    }
   ],
   "source": [
    "dgr = nx.degree_centrality(G)\n",
    "clo = nx.closeness_centrality(G)\n",
    "har = nx.harmonic_centrality(G)\n",
    "eig = nx.eigenvector_centrality(G)\n",
    "bet = nx.betweenness_centrality(G)\n",
    "pgr = nx.pagerank(G)\n",
    "hits = nx.hits(G)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T16:39:26.590556Z",
     "start_time": "2019-10-24T16:39:24.761483Z"
    }
   },
   "outputs": [],
   "source": [
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T16:39:27.446981Z",
     "start_time": "2019-10-24T16:39:27.416169Z"
    }
   },
   "outputs": [],
   "source": [
    "centralities = pd.concat(\n",
    "    [pd.Series(c) for c in (hits[1], eig, pgr, har, clo, hits[0], dgr, bet)],\n",
    "    axis=1)\n",
    "\n",
    "centralities.columns = (\"Authorities\", \"Eigenvector\", \"PageRank\",\n",
    "                        \"Harmonic Closeness\", \"Closeness\", \"Hubs\",\n",
    "                        \"Degree\", \"Betweenness\")\n",
    "centralities[\"Harmonic Closeness\"] /= centralities.shape[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T16:39:34.979494Z",
     "start_time": "2019-10-24T16:39:34.962297Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Authorities</th>\n",
       "      <th>Eigenvector</th>\n",
       "      <th>PageRank</th>\n",
       "      <th>Harmonic Closeness</th>\n",
       "      <th>Closeness</th>\n",
       "      <th>Hubs</th>\n",
       "      <th>Degree</th>\n",
       "      <th>Betweenness</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Military Communications</th>\n",
       "      <td>0.000296</td>\n",
       "      <td>0.005190</td>\n",
       "      <td>0.000321</td>\n",
       "      <td>0.014719</td>\n",
       "      <td>0.013766</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.000668</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Robin Thomas (Mathematician)</th>\n",
       "      <td>0.000113</td>\n",
       "      <td>0.010767</td>\n",
       "      <td>0.000334</td>\n",
       "      <td>0.014802</td>\n",
       "      <td>0.013480</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.001002</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Food Engineering</th>\n",
       "      <td>0.000307</td>\n",
       "      <td>0.005892</td>\n",
       "      <td>0.000323</td>\n",
       "      <td>0.015164</td>\n",
       "      <td>0.014115</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.001002</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Wrapped Normal Distribution</th>\n",
       "      <td>0.000010</td>\n",
       "      <td>0.000609</td>\n",
       "      <td>0.000323</td>\n",
       "      <td>0.010690</td>\n",
       "      <td>0.009788</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.000668</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>U. S. R. Murty</th>\n",
       "      <td>0.000105</td>\n",
       "      <td>0.011778</td>\n",
       "      <td>0.000349</td>\n",
       "      <td>0.014969</td>\n",
       "      <td>0.013574</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.001002</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Multiple Edge</th>\n",
       "      <td>0.000101</td>\n",
       "      <td>0.021774</td>\n",
       "      <td>0.000378</td>\n",
       "      <td>0.016110</td>\n",
       "      <td>0.014482</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.001670</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Interview (Research)</th>\n",
       "      <td>0.000043</td>\n",
       "      <td>0.001559</td>\n",
       "      <td>0.000323</td>\n",
       "      <td>0.012276</td>\n",
       "      <td>0.011195</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.000668</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Case Study</th>\n",
       "      <td>0.000054</td>\n",
       "      <td>0.002260</td>\n",
       "      <td>0.000325</td>\n",
       "      <td>0.013267</td>\n",
       "      <td>0.012136</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.001002</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Exponential-Logarithmic Distribution</th>\n",
       "      <td>0.000010</td>\n",
       "      <td>0.000609</td>\n",
       "      <td>0.000323</td>\n",
       "      <td>0.010690</td>\n",
       "      <td>0.009788</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.000668</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Forest (Graph Theory)</th>\n",
       "      <td>0.000036</td>\n",
       "      <td>0.006774</td>\n",
       "      <td>0.000337</td>\n",
       "      <td>0.014079</td>\n",
       "      <td>0.012943</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.000668</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>2995 rows × 8 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                                      Authorities  Eigenvector  PageRank  \\\n",
       "Military Communications                  0.000296     0.005190  0.000321   \n",
       "Robin Thomas (Mathematician)             0.000113     0.010767  0.000334   \n",
       "Food Engineering                         0.000307     0.005892  0.000323   \n",
       "Wrapped Normal Distribution              0.000010     0.000609  0.000323   \n",
       "U. S. R. Murty                           0.000105     0.011778  0.000349   \n",
       "...                                           ...          ...       ...   \n",
       "Multiple Edge                            0.000101     0.021774  0.000378   \n",
       "Interview (Research)                     0.000043     0.001559  0.000323   \n",
       "Case Study                               0.000054     0.002260  0.000325   \n",
       "Exponential-Logarithmic Distribution     0.000010     0.000609  0.000323   \n",
       "Forest (Graph Theory)                    0.000036     0.006774  0.000337   \n",
       "\n",
       "                                      Harmonic Closeness  Closeness  Hubs  \\\n",
       "Military Communications                         0.014719   0.013766   0.0   \n",
       "Robin Thomas (Mathematician)                    0.014802   0.013480   0.0   \n",
       "Food Engineering                                0.015164   0.014115   0.0   \n",
       "Wrapped Normal Distribution                     0.010690   0.009788   0.0   \n",
       "U. S. R. Murty                                  0.014969   0.013574   0.0   \n",
       "...                                                  ...        ...   ...   \n",
       "Multiple Edge                                   0.016110   0.014482   0.0   \n",
       "Interview (Research)                            0.012276   0.011195   0.0   \n",
       "Case Study                                      0.013267   0.012136   0.0   \n",
       "Exponential-Logarithmic Distribution            0.010690   0.009788   0.0   \n",
       "Forest (Graph Theory)                           0.014079   0.012943   0.0   \n",
       "\n",
       "                                        Degree  Betweenness  \n",
       "Military Communications               0.000668          0.0  \n",
       "Robin Thomas (Mathematician)          0.001002          0.0  \n",
       "Food Engineering                      0.001002          0.0  \n",
       "Wrapped Normal Distribution           0.000668          0.0  \n",
       "U. S. R. Murty                        0.001002          0.0  \n",
       "...                                        ...          ...  \n",
       "Multiple Edge                         0.001670          0.0  \n",
       "Interview (Research)                  0.000668          0.0  \n",
       "Case Study                            0.001002          0.0  \n",
       "Exponential-Logarithmic Distribution  0.000668          0.0  \n",
       "Forest (Graph Theory)                 0.000668          0.0  \n",
       "\n",
       "[2995 rows x 8 columns]"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "centralities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 中心性指标之间关联"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T16:42:43.476339Z",
     "start_time": "2019-10-24T16:42:43.412621Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Authorities         Authorities           0.000000\n",
       "PageRank            Hubs                  0.000000\n",
       "                    Degree                0.000000\n",
       "                    Betweenness           0.000000\n",
       "Harmonic Closeness  Harmonic Closeness    0.000000\n",
       "                                            ...   \n",
       "                    Eigenvector           0.825148\n",
       "Betweenness         Degree                0.841139\n",
       "PageRank            Eigenvector           0.890543\n",
       "Eigenvector         Authorities           0.938011\n",
       "Closeness           Harmonic Closeness    0.978282\n",
       "Length: 64, dtype: float64"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_df = centralities.corr()\n",
    "ll_triangle = np.tri(c_df.shape[0], k=-1)\n",
    "c_df *= ll_triangle\n",
    "c_series = c_df.stack().sort_values()\n",
    "c_series"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 通过协调性评估网络的均匀性"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 进行节点属性分析， 关注于协调性。\n",
    "\n",
    "* 协调性指的是 连边的两端的两个节点的关联性是正向的还是负向的\n",
    "\n",
    "* 正向的属性关联称为协调网络， 协调网络倾向于和有相同属性的节点呢连接"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最简单的协调性是 节点的度（出度和入度）： 节点的度和她他的邻居的度之间的相互关系。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T16:57:08.647036Z",
     "start_time": "2019-10-24T16:57:08.598735Z"
    }
   },
   "outputs": [],
   "source": [
    "my_degree, their_degree = zip(*nx.average_degree_connectivity(G).items())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> nx.average_degree_connectivity 返回一个字典： key是 自身的度， value 是邻居节点的度的平均值。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T16:59:04.316151Z",
     "start_time": "2019-10-24T16:59:04.021445Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-24T17:02:39.203474Z",
     "start_time": "2019-10-24T17:02:38.988478Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x10b005f10>"
      ]
     },
     "execution_count": 58,
     "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": [
    "plt.scatter( my_degree, their_degree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 如果网络是均匀的， 分布应该是 $y = x$\n",
    "\n",
    "* 可以看到对于度大的节点， 不太均匀"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## NetworkX 评估属性的协调性"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nx.attribute_mixing_matrix(G, \"gender\", mapping={\"M\": 0, \"F\": 1})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    " array([[0.22771058, [0.24064205, 0.24064205], 0.29100532]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 返回一个二维的矩阵： 0.22771058 (≈23%)表示 女性之间的连接占 23， 男男之间占 0.29100532"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-----"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nx.attribute_assortativity_coefficient(G, \"gender\") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> The function returns the assortativity coefficient—the correlation between the values of an attribute across edges."
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