notebook/05-01-network-science-intro.ipynb

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    "\n",
    "***\n",
    "***\n",
    "# 网络科学简介\n",
    "***\n",
    "***\n",
    "\n",
    "![image.png](img/chengjun.png)"
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    "## FROM SADDAM HUSSEIN TO NETWORK THEORY \n",
    "\n",
    "- SADDAM HUSSEIN: the fifth President of **Iraq**, serving in this capacity from 16 July 1979 until 9 April **2003**\n",
    "- Invasion started in March 19,  2003. Hussein’s trace went cold. \n",
    "- Designed a deck of cards, each card engraved with the images of the 55 most wanted. \n",
    "    - It worked: by May 1, 2003, 15 men on the cards were captured, and by the end of the month another 12 were under custody. \n",
    "    - Yet, Hussein remained at large.\n",
    "\n",
    "\n",
    "\n"
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    "![image.png](img/net1.png)"
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    "The capture of Saddam Hussein\n",
    "\n",
    "- shows the strong **predictive power** of networks.  \n",
    "\n",
    "- underlies \n",
    "    - the need to obtain **accurate maps of the networks**; \n",
    "    - and the **difficulties of the mapping** process.\n",
    "\n"
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    "The capture of Saddam Hussein\n",
    "\n",
    "- demonstrates the remarkable **stability of these networks**\n",
    "- was not based on fresh intelligence\n",
    "    - but rather on his **pre-invasion social links**, unearthed from old photos stacked in his family album.\n",
    "\n",
    "- shows that the choice of network we focus on makes a huge difference: \n",
    "    - the **hierarchical tree** captured the official organization of the Iraqi government, \n",
    "    - was of no use when it came to Saddam Hussein's whereabouts.  \n"
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   "source": [
    "## How about Osama bin Laden?\n",
    "\n",
    "- the founder of al-Qaeda, the organization that claimed responsibility for the September 11 attacks on the United States.\n",
    "\n",
    "2005年9月1日，中情局内部关于猎杀本·拉登任务的布告栏上贴出了如下信息：由于关押囚犯的强化刑讯已经没有任何意义，“我们只能继续跟踪科威特”。\n",
    "- 中情局自此开始了对科威特长达数年的跟踪\n",
    "- 最终成功窃听到了他本·拉登之间的移动电话，确定了他的位置并顺藤摸瓜找到了本·拉登在巴基斯坦的豪宅\n",
    "- 再经过9个月的证实、部署，于2011年5月1日由海豹突击队发动突袭、击毙本·拉登。"
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   "source": [
    "## A SIMPLE STORY (2): August 15, 2003 blackout.\n",
    "\n",
    "![image.png](img/net2.png)"
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    "### VULNERABILITY \n",
    "DUE TO INTERCONNECTIVITY \n",
    "\n",
    "- The 2003 blackout is a typical example of a cascading failure. \n",
    "- 1997, when the International Monetary Fund pressured the central banks of several Pacific nations to limit their credit. \n",
    "- 2009-2011 financial melt-down"
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    "An important theme of this class:  \n",
    "\n",
    "- we must understand **how network structure affects the robustness of a complex system**. \n",
    "\n",
    "- develop quantitative tools to assess the interplay between network structure and the dynamical processes on the networks, and their impact on failures.  \n",
    "\n",
    "- We will learn that failures reality failures follow reproducible laws, that can be quantified and even predicted using the tools of network science.\n"
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    "NETWORKS AT THE HEART OF \n",
    "## COMPLEX SYSTEMS \n"
   ]
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    "`Complex` Source: Dictionary.com\n",
    "- composed of many interconnected parts; compound; composite: a complex highway system. \n",
    "- characterized by a very complicated or involved arrangement of parts, units, etc.: complex machinery. \n",
    "- so complicated or intricate as to be hard to understand or deal with: a complex problem. \n",
    "\t\t\t\t\n"
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    "\n",
    "## Complexity\n",
    "\n",
    "A scientific theory which asserts that some systems display behavioral phenomena that are completely `inexplicable` by any conventional analysis of the systems’ constituent parts. These phenomena, commonly referred to as `emergent behaviour`, seem to occur in many complex systems involving living organisms, such as a stock market or the human brain.\n",
    " \n",
    "Source: John L. Casti, Encyclopædia Britannica\n",
    " \n"
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    "## COMPLEX SYSTEMS\n",
    "\n",
    "For example,  Society/Market/Brain/Cell\n",
    "\n",
    "Stephen Hawking\n",
    "> I think the next century will be `the century of complexity`. "
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    "Behind each complex system there is a network, \n",
    "- that defines the `interactions between the component`. \n"
   ]
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    "![image.png](img/net3.png)"
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    "- Social graph\n",
    "- Organization\n",
    "- Brain\n",
    "- Finantial network\n",
    "- Business \n",
    "- Internet\n",
    "- Genes"
   ]
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    "Behind each system studied in complexity there is an intricate wiring diagram, or a network, that defines the interactions between the component.   \n",
    "\n",
    "> We will never understand complex system unless we map out and understand the networks behind them.\n",
    "  \n"
   ]
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    "## THE HISTORY OF NETWORK ANALYSIS\n",
    "\n",
    "- Graph theory: 1735, Euler\n",
    "\n",
    "- Social Network Research:  1930s, Moreno\n",
    "\n",
    "- Communication networks/internet: 1960s\n",
    "\n",
    "- Ecological Networks: May, 1979.\n"
   ]
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    "\n",
    "While the study of networks has a long history from graph theory to sociology, **the modern chapter of network science emerged only during the first decade of the 21st century, following the publication of two seminal papers in 1998 and 1999**. \n",
    "\n",
    "The citation pattern of two classic network papers\n",
    "- the 1959 paper by Paul Erdos and Alfréd Rényi that marks the beginning of the study of random networks in graph theory [4] and \n",
    "- the 1973 paper by Mark Granovetter, the most cited social network paper [5]. \n",
    "\n",
    "Both papers were hardly or only moderately cited before 2000. \n"
   ]
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   "source": [
    "![image.png](img/net4.png)"
   ]
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    "### THE EMERGENCE OF NETWORK SCIENCE\n",
    "- Movie Actor Network,  1998;\n",
    "- World Wide Web,  1999.\n",
    "- C elegans neural wiring diagram 1990\n",
    "- Citation Network,  1998\n",
    "- Metabolic Network, 2000; \n",
    "- PPI network, 2001\n"
   ]
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    "### THE CHARACTERISTICS OF NETWORK SCIENCE \n",
    "- Interdisciplinary \n",
    "- Empirical\n",
    "- Quantitative and Mathematical \n",
    "- Computational \n"
   ]
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   "source": [
    "# THE IMPACT OF NETWORK SCIENCE \n"
   ]
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    "- Google\n",
    "    - Market Cap(2010 Jan 1): $189 billion\n",
    "    \n",
    "- Cisco Systems\n",
    "    - networking gear Market cap (Jan 1, 2009): $112 billion\n",
    "\n",
    "- Facebook\n",
    "    - market cap: $50 billion\n"
   ]
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    "### Health: From drug design to metabolic engineering. \n",
    "The human genome project, completed in 2001, offered the first comprehensive list of all human genes. \n",
    "\n",
    "- Yet, to fully understand how our cells function, and the origin of disease, \n",
    "    - we need accurate maps that tell us how these genes and other cellular components interact with each other. "
   ]
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   "source": [
    "### Security: Fighting Terrorism. \n",
    "Terrorism is one of the maladies of the 21st century, absorbing significant resources to combat it worldwide. \n",
    "    \n",
    "- Using social networks to capture Saddam Hussein\n",
    "- Capturing of the individuals behind the March 11, 2004 Madrid train bombings **through the examination of the mobile call network**. \n"
   ]
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    "**Network thinking** is increasingly present in the arsenal of various law enforcement agencies in charge of limiting terrorist activities. \n",
    "- To disrupt the financial network of terrorist organizations\n",
    "- to map terrorist networks\n",
    "- to uncover the role of their members and their capabilities. "
   ]
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   "source": [
    "### Epidemics: From forecasting to halting deadly viruses. \n",
    "\n",
    "While the **H1N1 pandemic** was not so devastating in 2009, it was **the first pandemic whose course and time evolution was accurately predicted months before the pandemic reached its peak**. \n",
    "\n",
    "- Before 2000 epidemic modeling was dominated by **compartment models**, assuming that everyone can infect everyone else one word the same socio-physical compartment. \n",
    "- The emergence of a network-based framework has fundamentally changed this, offering a new level of predictability in epidemic phenomena. "
   ]
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    "In January 2010 network science tools have predicted the conditions necessary for the emergence of viruses spreading through mobile phones. \n",
    "\n",
    "**The first major mobile epidemic outbreak**\n",
    " in the fall of 2010 in China, infecting over 300,000 phones each day, closely followed the predicted scenario. \n"
   ]
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   "source": [
    "### Brain Research: Mapping neural network. \n",
    "The human brain, consisting of hundreds of billions of interlinked neurons, is one of the least understood networks from the perspective of network science. The reason is simple: \n",
    "- we lack maps telling us which neurons link to each other. \n",
    "- The only fully mapped neural map available for research is that of the C.Elegans worm, with only 300 neurons. \n",
    "\n",
    "Driven by the potential impact of such maps, in 2010 the **National Institutes of Health** has initiated the Connectome project, aimed at developing the technologies that could provide an accurate neuron-level map of mammalian brains. \n"
   ]
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   "source": [
    "# The Bridges of Konigsberg\n"
   ]
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   "source": [
    "<div><img src=img/net5.png width='500px' \\></div>"
   ]
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   "source": [
    "> Can one walk across the seven bridges and never cross the same bridge twice and get back to the starting place?     \n",
    "\n",
    "![image.png](img/net6.png)"
   ]
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    "### Euler’s theorem (1735):\n",
    "\n",
    "- If a graph has more than two nodes of odd degree, there is no path. \n",
    "- If a graph is connected and has no odd degree nodes, it has at least one path.\n"
   ]
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    "COMPONENTS OF A COMPLEX SYSTEM\n",
    "\n",
    "# Networks and Graphs\n",
    " - components: nodes, vertices\t\t  N\n",
    " - interactions:  links, edges\t\t\t   L\n",
    " - system:  \t  network, graph\t\t(N,L)\n"
   ]
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    "Network often refers to real systems\n",
    "- www, \n",
    "- social network\n",
    "- metabolic network. \n",
    "\n",
    "Language: (Network, node, link)\n"
   ]
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    "Graph: mathematical representation of a network\n",
    "- Web graph, \n",
    "- Social graph (a Facebook term)\n",
    "\n",
    "Language: (Graph, vertex, edge)\n",
    "\n",
    "\n",
    "G(N, L)"
   ]
  },
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   "source": [
    "![image.png](img/net7.png)"
   ]
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    "### CHOOSING A PROPER REPRESENTATION\n",
    "\n",
    "The choice of the proper network representation determines our ability to use network theory successfully.\n",
    "\n",
    "In some cases there is a unique, unambiguous representation. \n",
    "In other cases, the representation is by no means unique.\n",
    " \n",
    "For example, the way we assign the links between a group of individuals will determine the nature of the question we can study.\n",
    "\n"
   ]
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    "**Maximilian Schich**, an art historian at the University of Texas at Dallas, and his colleagues used the Google-owned knowledge base, Freebase, to find 120,000 individuals who were notable enough in their life-times that the dates and locations of their births and deaths were recorded.\n",
    "\n",
    "\n",
    "<div><img src=img/net8.png width ='500px'></div>"
   ]
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   "source": [
    "![image.png](img/net9.png)"
   ]
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    "\n",
    "![image.png](img/net10.png)\n",
    "\n",
    "**A network framework of cultural history**. Maximilian Schich et al. Science 345, 558 (2014) Video: **Charting Culture**"
   ]
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   "source": [
    "Homework:\n",
    "\n",
    "1. 下载12万名人出生死亡地理位置数据\n",
    "Data S1 [SchichDataS1_FB.xlsx](https://science.sciencemag.org/content/suppl/2014/07/30/345.6196.558.DC1) (Freebase.com)\n",
    "\n",
    "2. 构建上图D名人出生死亡的城市网络\n",
    "3. 绘制度分布曲线\n",
    "4. 尝试绘制上图C"
   ]
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   "source": [
    "![image.png](img/net11.png)\n",
    "\n",
    "**The Product Space Conditions the Development of Nations** C. A. Hidalgo et al. Science 317, 482 (2007)"
   ]
  },
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   "source": [
    "If you connect individuals that work with each other, you will explore the professional network. http://www.theyrule.net"
   ]
  },
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    "If you connect those that have a romantic and sexual relationship, you will be exploring the sexual networks.\n"
   ]
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   "source": [
    "If you connect individuals based on their first name (all Peters connected to each other), you will be exploring what? \n",
    "\n",
    "It is a network, nevertheless.\n"
   ]
  },
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   "source": [
    "## UNDIRECTED VS. DIRECTED NETWORKS\n"
   ]
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    "### Undirected\n",
    "Links: undirected\n",
    "- Co-authorship \n",
    "- Actor network\n",
    "- Protein interactions"
   ]
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import networkx as nx\n",
    "Gu = nx.Graph()\n",
    "for i, j in [(1, 2), (1, 4), (4, 2), (4, 3)]:\n",
    "    Gu.add_edge(i,j)\n",
    "nx.draw(Gu, with_labels = True)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Directed\n",
    "Links: directed\n",
    "- Urls on the www\n",
    "- Phone calls\n",
    "- Metabolic reactions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-07-30T02:00:12.136673Z",
     "start_time": "2024-07-30T02:00:12.023499Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import networkx as nx\n",
    "Gd = nx.DiGraph()\n",
    "for i, j in [(1, 2), (1, 4), (4, 2), (4, 3)]:\n",
    "    Gd.add_edge(i,j)\n",
    "nx.draw(Gd, with_labels = True, pos=nx.circular_layout(Gd))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "![image.png](img/net12.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Degree, Average Degree and Degree Distribution\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-08-05T05:40:39.403271Z",
     "start_time": "2020-08-05T05:40:39.256906Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw(Gu, with_labels = True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Undirected network: \n",
    "Node degree: the number of links connected to the node.\n",
    "### $k_1 = k_2 = 2, k_3 = 3, k_4 = 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-08-05T05:40:40.681915Z",
     "start_time": "2020-08-05T05:40:40.536419Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw(Gd, with_labels = True, pos=nx.circular_layout(Gd))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Directed network\n",
    "In directed networks we can define an in-degree and out-degree. The (total) degree is the sum of in-and out-degree.\n",
    "\n",
    "###  $k_3^{in} = 2, k_3^{out} = 1, k_3 = 3$\n",
    "\n",
    "Source: a node with $k^{in}= 0$; Sink: a node with $k^{out}= 0$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For a sample of N values: $x_1, x_2, ..., x_N$:\n",
    "\n",
    "**Average(mean)**:\n",
    "\n",
    "$<x> = \\frac{x_1 +x_2 + ...+x_N}{N} = \\frac{1}{N}\\sum_{i = 1}^{N} x_i$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For a sample of N values: $x_1, x_2, ..., x_N$:\n",
    "\n",
    "**The nth moment**:\n",
    "\n",
    "$<x^n> = \\frac{x_1^n +x_2^n + ...+x_N^n}{N} = \\frac{1}{N}\\sum_{i = 1}^{N} x_i^n$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For a sample of N values: $x_1, x_2, ..., x_N$:\n",
    "\n",
    "**Standard deviation**:\n",
    "\n",
    "$\\sigma_x = \\sqrt{\\frac{1}{N}\\sum_{i = 1}^{N} (x_i - <x>)^2}$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-07-30T02:04:09.010503Z",
     "start_time": "2024-07-30T02:04:09.003892Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.6666666666666667, 10, 0.7453559924999299)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "x = [1, 1, 1, 2, 2, 3]\n",
    "np.mean(x), np.sum(x), np.std(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For a sample of N values: $x_1, x_2, ..., x_N$:\n",
    "\n",
    "**Distribution of x**:\n",
    "\n",
    "$p_x = \\frac{The \\: frequency \\: of \\: x}{The\\: Number \\:of\\: Observations}$\n",
    "\n",
    "其中，$p_x 满足 \\sum_i p_x = 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T02:11:55.904909Z",
     "start_time": "2021-08-05T02:11:55.689796Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 1, 1, 2, 2, 3]\n"
     ]
    },
    {
     "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": [
    "# 直方图\n",
    "print(x) \n",
    "plt.hist(x)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T02:12:37.925584Z",
     "start_time": "2021-08-05T02:12:37.920524Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "defaultdict(int, {1: 3, 2: 2, 3: 1})"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from collections import defaultdict, Counter\n",
    "freq = defaultdict(int)\n",
    "for i in x:\n",
    "    freq[i] +=1\n",
    "\n",
    "freq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T02:12:50.194024Z",
     "start_time": "2021-08-05T02:12:50.189788Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "freq_sum = np.sum(list(freq.values()))\n",
    "freq_sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T02:12:56.426318Z",
     "start_time": "2021-08-05T02:12:56.421222Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.5, 0.3333333333333333, 0.16666666666666666]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "px = [float(i)/freq_sum for i in freq.values()]\n",
    "px"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T02:13:31.127940Z",
     "start_time": "2021-08-05T02:13:30.960413Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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gbJEk7LdfeAr2iivgscdC87Jnn426KpEqk0zANyMEd7FPgMbFL8yst5mtAF4Hhrl78eOFPwGvmdl7ZjawtD/Y3ce4e8zdY7m5uaW9RWTr1KkT+tkUFIQlA489NgzKrloVdWUilS6ZgK8NJD4Tvgn49XaLu89w92ZAd+A+M2sZ357v7nsAA4Cbzax16soWqaCOHUOHyr//HV5+OVzNjxun5mWS1ZIJ+FVAk4TXTYEVJd/k7kuBfxPuxSduXwLMAvK2vEyRFKhVCy67LPScb9cOTjstDMoWFUVdmUilSCbgpwD5ZtbQzBoBPYCpAGZWx8y6xH9uCHQD5sVft4p/bw50Ld4uErnWrcPqUXfdFRb7btcuNDL7RfMAJLuUG/DuvgYYDswmXIkPBfqa2TDAgHvMrAh4Dbja3Qvjh040s2XAZOAidy9KffkiWygnJ0ylXLQozJk///ywotTSpVFXJpIy5mlyDzIWi3lBQUHUZUh15B5m2VxwQXgwasQIuOSScEtHJM2Z2Vx3j5W2T0+yipjBSSeFdgdHHAFXXhn62rz7LowfHzpY5uSE7+PHR12tSNJqRl2ASNrYdVd4+umw0Pdf/gKxWOhDv3Fj2L98OQwZEn4eMCC6OkWSpCt4kZKOOipczdet+99wL7Z+fehYKZIBFPAipdlppxDmpdGasJIhFPAiZdl999K366lryRAKeJGyjBwZbtMkMoPPPoOTT4YvvoimLpEkKeBFyjJgAIwZA82bh2Bv3jysBXvVVfDEE5CXFwZl02SqsUhJmgcvsiXmz4fBg2HuXDjySLj77tDMTKSKaR68SKp16ABvvQU33RRWkWrTBsaO1dW8pBUFvMiWqlkTLr4YFiwIgX/66XDQQfDRR1FXJgIo4EW23v/8D7z2Gtx7L8yZA+3bw+23q3mZRE4BL5IKOTlw1lmweDH07g0XXhhWlFq8OOrKpBpTwIukUrNm8NJLoXlZYSF06gTXXgsbNkRdmVRDCniRVDMLUyyXLAnLA44YEfrazJkTdWVSzSjgRSpLw4ZhvvwLL4SHorp1C4OyZbVAEEkxBbxIZTv88HA1P3gw3HJLmHEzY0bUVUk1oIAXqQo77BCeip02DTZtggMOCIOy33wTdWWSxRTwIlXpwANh4UK46CJ44AFo2xZefjnqqiRLKeBFqlrdujBqFLz5Juy4Ixx6aBiUXbs26sokyyjgRaLStWtYFvDqq+GZZ0K7gyeeULsDSZmkAt7M+pvZMjMrNLNBJfbdYGb/a2Yfm9klCdt7mdkH8eO0BI5IaWrXhr/+NTQt22MPOPHEMCj7ySdRVyZZoNyAN7N6wCigZ/zrejNLXPHgTnffC+gMXGVm9czMgAeBY4F2wClm1jHl1Ytki/btYfbscOtm2rRwb37MmDAgK7KFkrmCzwdmuvtKd18NTAf6FO9090/jP+4GLAfWAV2ANe6+wN3XAROAfimtXCTb1KgRBl8XLoQuXeDMM6FPn/BErMgWSCbgmxGCu9gnQOPiF2bW28xWAK8Dw9x9U3nHJBw7xMwKzKxgrQaYRILf/S5cxY8ZE+7R7713uLJX8zKpoGQCvjaQ+P+Jm4BfzzR3n+HuzYDuwH1m1rK8YxKOHePuMXeP5WqdS5H/MoMzzggPSB10EAwbBt27w6JFUVcmGSSZgF8FNEl43RRYUfJN7r4U+DfhXnxSx4hIOZo0Ca0OnngCli2Dzp3DoOxPP0VdmWSAZAJ+CpBvZg3NrBHQA5gKYGZ1zKxL/OeGQDdgHvAW0NrMWpvZdsDRwHOV8RcQyXpmcPzxsHQp9O8P11wT7tG//XbUlUmaKzfg3X0NMByYDcwChgJ9zWwYYMA9ZlYEvAZc7e6F7r4BGAxMAhYDo919eWl/vogkaZddQhvil14KLQ66dw+DsuvWRV2ZpCktui2Sib79Fi69FO67D1q2DG0PDjww6qokAlp0WyTb1K8flgicMSOsJtWnTxiU/frrqCuTNKKAF8lkvXrB/Pmhz/xDD4UHpF58MeqqJE0o4EUyXd26cNNNYdC1QQM44ogwKPvZZ1FXJhFTwItki1gMCgrCGrATJ0JeXhiUTZNxNql6CniRbFK7Nlx5Jbz3Huy5J5x0UmhHvEKPoVRHCniRbNSmDfz733D77WEgtm3bMCir5mXVigJeJFvVqAHnnx+al+27L/zlL2GpwA8/jLoyqSIKeJFs17IlvPoqjB0bZtzsvXcYlN24MerKpJIp4EWqAzMYNCg0L+vXLzwk1a1bCHzJWgp4kepkt93guefg6afDwGssBlddpeZlWUoBL1LdmMGf/hSu5k88Ea67Djp1CitKSVZRwItUVw0awMMPw+TJ8P33sN9+cMEF4WfJCgp4keruj3+ExYvDLJs77gjrw776atRVSQoo4EUE6tWDu+6C118PD0v17QuDB8NXX0VdmWwFBbyI/Nf++4eZNZddFm7ftGkT2h5IRlLAi8hv1akDf/87vPMONGoERx8dBmVXr466MqkgBbyIlK5z5xDyI0eGFsRt2sAjj6h5WQZRwItI2WrVgiuuCLdt8vLglFPCoOxyrcCZCRTwIlK+vfaCN96A0aNDE7N27eDuu9W8LM0p4EUkOTk5cM45sGgR9OgRfu7VC95/P+rKpAwKeBGpmBYt4JVXYNy4MH++Qwe44Qb4+eeoK5MSkgp4M+tvZsvMrNDMBpXYd76ZLTWzIjN71MxqxrfPiG8rjH/VqIy/gIhEwCzcj1+yJCwocvnl0LVrWGhE0ka5AW9m9YBRQM/41/Vmlpvwlu+ADkArYFfguIR9vd29Vfzrl9SVLSJpoVEjmDAhfH36KeyzTxiU/fHHqCsTkruCzwdmuvtKd18NTAf6FO9094fcfYO7bwQWADsn+8vNbIiZFZhZwdq1aytau4iki2OOCVfzJ58c5tB37AizZkVdVbWXTMA3AxLnRH0CNC75JjOrCxwCTIpv+gl4zczeM7OBpf3B7j7G3WPuHsvNzS3tLSKSKXbeGR56CKZMCVfw++8P554L330XdWXVVjIBXxtInAu1CfjN7RYzywEeAUa7exGAu+e7+x7AAOBmM2udkopFJL317Rtm2px7bphK2a5dCH2pcskE/CqgScLrpsCvS7SbmQEPAkvc/Z6SB7v7EmAWkLd1pYpIxth++9CZ8o03oG7dsIrUKafAl19GXVm1kkzATwHyzayhmTUCegBTE/bfA6x29xGJB5lZq/j35kBXYF5qShaRjLHffmFmzfDhMH58eBp2woSoq6o2yg14d18DDAdmE67EhwJ9zWyYmfUEzgT6J0yHPCF+6EQzWwZMBi4qvnUjItVMnTph1aiCAmjaNDQuO+YYWLUq6sqynnmaNA6KxWJeUFAQdRkiUpk2boRRo+Dqq2HbbeHWW+HUU8O8etkiZjbX3WOl7dOTrCJSdWrWhEsvhQULwspRgwZBfj4UFUVdWVZSwItI1dtzT5gxI8yymT07zLS58074Rc9DppICXkSikZMT1oFdvDjMmT///PB96dKoK8saCngRidbuu8PkyWExkfffD0/Bjhyp5mUpoIAXkeiZwUknhXYHRx4JV14JsRjMnRt1ZRlNAS8i6WPXXeGpp8JC32vXhg6Vl10GP/wQdWUZSQEvIunnyCPD1fypp8KNN4ae86+/HnVVGUcBLyLpaccd4cEH4V//CvPne/UKg7Lffht1ZRlDAS8i6a1PH1i4EC64AO67L0ypnDw56qoyggJeRNLfdtvBbbfBm29CvXpwyCFhUPbzz6OuLK0p4EUkc3TrBu++CyNGwJNPQps28PTTkCYtV9KNAl5EMss228A114QplM2bw3HHwVFHhSUD5TcU8CKSmfbeO7Q5uPnmsKBImzYwdqyu5hMo4EUkc9WsCcOGhUHYjh3h9NPhoIPgo4+iriwtKOBFJPO1agXTp4dZNnPmhJk2t91W7ZuXKeBFJDvk5MCZZ4YHpA48EC66KKwotXhx1JVFRgEvItmlaVOYNAkefxz+8x/o1An+9jfYsCHqyqqcAl5Eso8ZnHBCuJo/9tiwglQsFm7fVCMKeBHJXrm54Ur+xRfhyy/DPPqLL4b166OurEoo4EUk+x12WLgXf/rpcMstYYrljBlRV1Xpkgp4M+tvZsvMrNDMBpXYd76ZLTWzIjN71Mxqxrf3MrMP4scNr4ziRUSStsMOcP/9YbYNwAEHhEHZb76Jtq5KVG7Am1k9YBTQM/51vZnlJrzlO6AD0ArYFTjOzAx4EDgWaAecYmYdU1y7iEjFHXBAWPR72LDQrbJtW3jppairqhTJXMHnAzPdfaW7rwamA32Kd7r7Q+6+wd03AguAnYEuwBp3X+Du64AJQL/Uly8isgXq1g1PwM6eDTvtFG7hnHhiWGQkiyQT8M2A5QmvPwEal3yTmdUFDgEmVeCYIWZWYGYFa7PsgxWRDLDvvqGnzTXXwIQJod3BE09kTbuDZAK+NrAp4fUm4DePh5lZDvAIMNrdi5I5BsDdx7h7zN1jubm5JXeLiFS+2rVDd8p334WWLcOV/OGHwyefRF3ZVksm4FcBTRJeNwVWFL9IuN++xN3vSeYYEZG0065d6Dd/660wbVq4mr//fti0qfxj01QyAT8FyDezhmbWCOgBTE3Yfw+w2t1HJGx7C2htZq3NbDvgaOC5VBUtIlIpatSACy+ERYtgn33grLPCilKFhVFXtkXKDXh3XwMMB2YDs4ChQF8zG2ZmPYEzgf7xKZSFZnaCu28ABhPuxy8m3LpZXsavEBFJLy1bhrVgH3gg3Lpp3z7Mn9+4MerKKsQ8TQYTYrGYFxQURF2GiMhvrVwZFvt+8cVwVT92bAj8NGFmc909Vto+PckqIrI5TZrA88+HJQKLiqBz59Db5qefoq6sXAp4EZHymIWlAZcsgeOPD90pu3SBt9+OurLNUsCLiCRrl13g0Ufh5ZdDi4Pu3UPf+XXroq6sVAp4EZGKOvjg0LzsrLPCylHt24eplWlGAS8isiXq14d77oGZM8PasAcdBGecAV9/HXVlv1LAi4hsjd//HubPh0sugYceCg9IvfBC1FUBCngRka237bZw441h0DU3F448MgzGfvZZpGUp4EVEUiUWg4ICuO46mDgR8vLgsccia16mgBcRSaVatWD4cJg3D1q3hpNOgkMOgY8/rvJSFPAiIpUhLw/eeAPuuCMMxLZtC/feW6XNyxTwIiKVpUYNOO+80LysW7fQ8qB3b/jggyr59Qp4EZHKtsceMHVqmGWzcCF06AA33RQemmrRAnJywvfx41P6a2um9E8TEZHSmcFpp0G/fnD22XDppWFb8QDs8uUwZEj4ecCAlPxKXcGLiFSlxo3huedC24OSs2vWrw8DtCmigBcRicIXX5S+PYWzbRTwIiJR2H33im3fAgp4EZEojBwJdev+dlvdumF7iijgRUSiMGAAjBkDzZuHwdbmzcPrFA2wgmbRiIhEZ8CAlAZ6SbqCFxHJUikJeDPLMbP0WYVWRESSC3gz629my8ys0MwGldh3M7ACmFhi+wwzK4ofU2hmNVJXtoiIlKfce/BmVg8YBXQDfgHmmdkkd18bf8urwCOUCPi43u5elKJaRUSkApK5gs8HZrr7SndfDUwH+hTvdPepwHeVVJ+IiGyhZGbRNAOWJ7z+BGicxHE/Aa+Z2dfAKHd/rOQbzGwIEG++wPdm9n4Sf25pdgE+38JjK1O61gXpW5vqqhjVVTHZWFfzsnYkE/C1gcQGxpsIt2o2y93zAcysDTDNzOa4+/sl3jMGGJNEDZtlZgXuHtvaPyfV0rUuSN/aVFfFqK6KqW51JXOLZhXQJOF1U8KgalLcfQkwC8irWGkiIrI1kgn4KUC+mTU0s0ZAD2BqeQeZWav49+ZAV2De1hQqIiIVU+4tGndfY2bDgdnxTUOBvmb2O3e/xcxeBPYGdjOzQuBBd78BmGhm2wPrgYsqeTbNVt/mqSTpWhekb22qq2JUV8VUq7rMI1rtW0REKpdaFYiIZCkFvIhIlsqYgDezbc1sz6jrKEl1VUy61iWSjdI+4M2svpk9D6wBLillfzszm29my81stJnlxLf3MrMP4j10UrfIYfJ1nW9mS+P9eB41s5rx7ZXaoyeJusaZ2cqE3797fHtkn5eZHTOb5wsAAAOJSURBVJBQT6GZfWVmN8X3VfbnVcfMxpjZ+/Fz6MIS+6M6v8qrK6rzq7y6ojq/yqwr4vMrx8xejf/d3zez/BL7K/f8cve0/gK2J7RGOJ0wQ6fk/teBPwI1gJnAkYABHxJm92wHfAB0rOK6BhEeEqtJmFY6IL59BtAiws9rHKFHUOK2yD+vEu+dBbSvos+rAXBM/DPYhfAfoGZpcH6VV1dU51d5dUV1fm22rgjPLwMax3/uBxSU2F+p51faX8G7+/fuPg3YWHKfmeUCe7j7P939F2A84UPsAqxx9wXuvg6YEN9eJXXF9z/k7hvcfSOwANg5lb9/S+sqQ+SfVzEzywNqufvCVP7+zdT1hbs/68HnhIf4dozXEuX5VWZd8f1RnV+brasMkX9exSI4v9zdV8VfNgfmJ9RS6edX2gd8OZoCiUuQF/fJ2dL+OSlnZnWBQ4BJ8U3FPXreM7OBEZT0M/CwmS02s6HxbWnzeQFnAGMTXlfZ52Vm7YA6wKL4prQ4v0qpK3FfZOdXGXVFfn5t7vMigvPLzC4xsy+AC4G/Jeyq9PMr05fsK6tPzhb1z0m1+P20R4DRHn/Qy5Po0VOZ3P2M+O9vBrxqZvNJn89rG+BY4NfFY6rq8zKzXYBHgdM8/v/OpMH5VUZdxfsiO7/Kqivq86uczyuS88vdbwJuMrOjgSlmlhevrdLPr0y/gi+rT85W9c9JBTMz4EFgibvfU3K/R9yjx91XAC8B7UiDzyvuKGCGu39Tckdlfl5mthPhCvgKd5+TsCvS82szdUV6fm2uroTfX+XnVxJ1RXJ+JfyO5whjUQ3imyr//NqSG/dRfAGnUvqg4UKgN/8dpOhJ+C/gSqA1YZBiCdC8iuu6F7i+lO2t4t+bx/+htajiuop/fwPC/8Lulw6fV3zfNKBXVX5eQH3gDeDQMvZHcn4lUVck51cSdUVyfpVXV4TnV0ugUfzn7kBhVZ5fKfuLVNYXUA8oJIyKfxP/+ShgWHx/5/iHtAK4NuG4foTR5yLgz1VZV/wf0qb4tuKvExL+gS4DFgN/iuDzmhz/TN4HzkmHzyvhX4QPibfPKPEvQGV+XlcC60r8sxqaBudXmXVFfH6V93lFdX6VV1dU51fn+N/7P4R+Xl2owvxSLxoRkSyV6ffgRUSkDAp4EZEspYAXEclSCngRkSylgBcRyVIKeBGRLKWAFxHJUgp4EZEs9X/iSVIJ2hxQFwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(list(freq.keys()), px, 'r-o')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Average Degree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Undirected\n",
    "\n",
    "$<k> = \\frac{1}{N} \\sum_{i = 1}^{N} k_i  = \\frac{2L}{N}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Directed\n",
    "\n",
    "$<k^{in}> = \\frac{1}{N} \\sum_{i=1}^N k_i^{in}= <k^{out}> = \\frac{1}{N} \\sum_{i=1}^N k_i^{out} = \\frac{L}{N}$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Degree distribution \n",
    "P(k): probability that a randomly selected node  has degree k\n",
    "\n",
    "\n",
    "$N_k = The \\:number\\: of \\:nodes\\:with \\:degree\\: k$\n",
    "\n",
    "$P(k) = \\frac{N_k}{N}$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Adjacency matrix\n",
    "$A_{ij} =1$ if there is a link between node i and j\n",
    "\n",
    "$A_{ij} =0$ if there is no link between node i and j"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T02:16:54.024329Z",
     "start_time": "2021-08-05T02:16:53.714500Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(1)\n",
    "plt.subplot(121)\n",
    "pos = nx.nx.circular_layout(Gu)  #定义一个布局，此处采用了spring布局方式\n",
    "nx.draw(Gu, pos, with_labels = True)\n",
    "plt.subplot(122)\n",
    "nx.draw(Gd, pos, with_labels = True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Undirected\n",
    "$A_{ij} =1$ if there is a link between node i and j\n",
    "\n",
    "$A_{ij} =0$ if there is no link between node i and j\n",
    "\n",
    "$A_{ij}=\\begin{bmatrix} 0&1  &0  &1 \\\\ 1&0  &0  &1 \\\\ 0 &0  &0  &1 \\\\ 1&1  &1  & 0 \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Undirected\n",
    "\n",
    "无向网络的矩阵是对称的。\n",
    "\n",
    "$A_{ij} = A_{ji} , \\: A_{ii} = 0$\n",
    "\n",
    "$k_i = \\sum_{j=1}^N A_{ij}, \\: k_j = \\sum_{i=1}^N A_{ij} $\n",
    "\n",
    "网络中的链接数量$L$可以表达为：\n",
    "\n",
    "$ L = \\frac{1}{2}\\sum_{i=1}^N k_i = \\frac{1}{2}\\sum_{ij}^N A_{ij} $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Directed\n",
    "$A_{ij} =1$ if there is a link between node i and j\n",
    "\n",
    "$A_{ij} =0$ if there is no link between node i and j\n",
    "\n",
    "$A_{ij}=\\begin{bmatrix} 0&0  &0  &0 \\\\ 1&0  &0  &1 \\\\ 0 &0  &0  &1 \\\\ 1&0  &0  & 0 \\end{bmatrix}$\n",
    "\n",
    "Note that for a directed graph the matrix is not symmetric.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Directed\n",
    "$A_{ij} \\neq A_{ji}, \\: A_{ii} = 0$\n",
    "\n",
    "$k_i^{in} = \\sum_{j=1}^N A_{ji}, \\: k_j^{out} = \\sum_{i=1}^N A_{ji} $\n",
    "\n",
    "$ L = \\sum_{i=1}^N k_i^{in} = \\sum_{j=1}^N k_j^{out}= \\sum_{i,j}^N A_{ij} $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### WEIGHTED AND UNWEIGHTED NETWORKS\n",
    "\n",
    "$A_{ij} = W_{ij}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# BIPARTITE NETWORKS \n",
    "\n",
    "\n",
    "`bipartite graph` (or bigraph) is a graph whose nodes can be divided into two disjoint sets U and V such that every link connects a node in U to one in V; that is, U and V are independent sets. \n",
    "\n",
    "- Hits Algorithm\n",
    "- Recommendation System\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "![image.png](img/net13.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-07-30T02:17:38.320361Z",
     "start_time": "2024-07-30T02:17:38.317053Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "edges = [('甲', '新辣道'), ('甲', '海底捞'), ('甲', '五方院'), \n",
    "         ('乙', '海底捞'), ('乙', '麦当劳'), ('乙', '俏江南'),\n",
    "         ('丙', '新辣道'), ('丙', '海底捞'),\n",
    "         ('丁', '新辣道'), ('丁', '五方院'), ('丁', '俏江南')]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T02:49:24.465895Z",
     "start_time": "2021-08-05T02:49:24.456103Z"
    },
    "code_folding": [],
    "scrolled": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 steps\n",
      "{'新辣道': 3, '海底捞': 3, '五方院': 2, '麦当劳': 1, '俏江南': 2}\n",
      "{'甲': 8, '乙': 6, '丙': 6, '丁': 7}\n",
      "1 steps\n",
      "{'新辣道': 21, '海底捞': 20, '五方院': 15, '麦当劳': 6, '俏江南': 13}\n",
      "{'甲': 56, '乙': 39, '丙': 41, '丁': 49}\n",
      "2 steps\n",
      "{'新辣道': 146, '海底捞': 136, '五方院': 105, '麦当劳': 39, '俏江南': 88}\n",
      "{'甲': 387, '乙': 263, '丙': 282, '丁': 339}\n",
      "3 steps\n",
      "{'新辣道': 1008, '海底捞': 932, '五方院': 726, '麦当劳': 263, '俏江南': 602}\n",
      "{'甲': 2666, '乙': 1797, '丙': 1940, '丁': 2336}\n",
      "4 steps\n",
      "{'新辣道': 6942, '海底捞': 6403, '五方院': 5002, '麦当劳': 1797, '俏江南': 4133}\n",
      "{'甲': 18347, '乙': 12333, '丙': 13345, '丁': 16077}\n"
     ]
    }
   ],
   "source": [
    "h_dic = {i:1 for i,j in edges}\n",
    "for k in range(5):\n",
    "    print(k, 'steps')\n",
    "    a_dic = {j:0 for i, j in edges}\n",
    "    for i,j in edges:\n",
    "        a_dic[j]+=h_dic[i]\n",
    "    print(a_dic)\n",
    "    h_dic = {i:0 for i, j in edges}\n",
    "    for i, j in edges:\n",
    "        h_dic[i]+=a_dic[j]\n",
    "    print(h_dic) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T02:49:56.020993Z",
     "start_time": "2021-08-05T02:49:56.004898Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'新辣道': 0.2861876476571226, '海底捞': 0.2636544282409802, '五方院': 0.20624908818768029, '麦当劳': 0.07379187216602481, '俏江南': 0.17011696374819205}\n"
     ]
    }
   ],
   "source": [
    "def norm_dic(dic):\n",
    "    sumd = np.sum(list(dic.values()))\n",
    "    return {i : dic[i]/sumd for i in dic}\n",
    "    \n",
    "h = {i for i, j in edges}\n",
    "h_dic = {i:1/len(h) for i in h}\n",
    "for k in range(100):\n",
    "    a_dic = {j:0 for i, j in edges}\n",
    "    for i,j in edges: \n",
    "            a_dic[j]+=h_dic[i]\n",
    "    a_dic = norm_dic(a_dic)\n",
    "    h_dic = {i:0 for i, j in edges}\n",
    "    for i, j in edges:\n",
    "        h_dic[i]+=a_dic[j]\n",
    "    h_dic = norm_dic(h_dic)\n",
    "print(a_dic)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T02:50:31.735260Z",
     "start_time": "2021-08-05T02:50:31.727674Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'海底捞': 0.13182721414423623, '五方院': 0.10312454403576715, '麦当劳': 0.036895936147322755, '新辣道': 0.14309382376113222, '俏江南': 0.0850584819115415}\n"
     ]
    }
   ],
   "source": [
    "B = nx.Graph()\n",
    "users, items = {i for i, j in edges}, {j for i, j in edges}\n",
    "for i, j in edges:\n",
    "    B.add_edge(i,j)\n",
    "h, a = nx.hits(B) \n",
    "print({i:a[i] for i in items} )\n",
    "# {j:h[j] for j in users}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-05-05T12:45:45.265498Z",
     "start_time": "2018-05-05T12:45:45.261530Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "![image.png](img/net14.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2024-07-30T02:26:17.697357Z",
     "start_time": "2024-07-30T02:26:10.850946Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import networkx as nx\n",
    "Gp = nx.DiGraph()\n",
    "edges = [('a', 'b'), ('a', 'c'), ('b', 'd'), ('b', 'e'), ('c', 'f'), ('c', 'g'),\n",
    "        ('d', 'h'), ('d', 'a'), ('e', 'a'), ('e', 'h'), ('f', 'a'), ('g', 'a'), ('h', 'a')]\n",
    "for i, j in edges:\n",
    "    Gp.add_edge(i,j)\n",
    "nx.draw(Gp, with_labels = True, font_size = 25, font_color = 'blue', alpha = 0.5,\n",
    "        pos = nx.kamada_kawai_layout(Gp))\n",
    "        #pos=nx.spring_layout(Gp, iterations = 5000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T03:03:15.212036Z",
     "start_time": "2021-08-05T03:03:15.177323Z"
    },
    "scrolled": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 : 0.5 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.125\n",
      "2 : 0.3125 0.25 0.25 0.03125 0.03125 0.03125 0.03125 0.0625\n",
      "3 : 0.15625 0.15625 0.15625 0.125 0.125 0.125 0.125 0.03125\n",
      "4 : 0.40625 0.078125 0.078125 0.078125 0.078125 0.078125 0.078125 0.125\n",
      "5 : 0.359375 0.203125 0.203125 0.0390625 0.0390625 0.0390625 0.0390625 0.078125\n",
      "6 : 0.1953125 0.1796875 0.1796875 0.1015625 0.1015625 0.1015625 0.1015625 0.0390625\n",
      "7 : 0.34375 0.09765625 0.09765625 0.08984375 0.08984375 0.08984375 0.08984375 0.1015625\n",
      "8 : 0.37109375 0.171875 0.171875 0.048828125 0.048828125 0.048828125 0.048828125 0.08984375\n",
      "9 : 0.236328125 0.185546875 0.185546875 0.0859375 0.0859375 0.0859375 0.0859375 0.048828125\n",
      "10 : 0.306640625 0.1181640625 0.1181640625 0.0927734375 0.0927734375 0.0927734375 0.0927734375 0.0859375\n"
     ]
    }
   ],
   "source": [
    "steps = 11\n",
    "n = 8\n",
    "a, b, c, d, e, f, g, h = [[1.0/n for i in range(steps)] for j in range(n)]\n",
    "for i in range(steps-1):\n",
    "    a[i+1] = 0.5*d[i] + 0.5*e[i] + h[i] + f[i] + g[i]\n",
    "    b[i+1] = 0.5 *a[i]\n",
    "    c[i+1] = 0.5*a[i]\n",
    "    d[i+1] = 0.5*b[i]\n",
    "    e[i+1] = 0.5*b[i] \n",
    "    f[i+1] = 0.5*c[i]\n",
    "    g[i+1] = 0.5*c[i]\n",
    "    h[i+1] = 0.5*d[i] + 0.5*e[i]\n",
    "    print(i+1,':', a[i+1], b[i+1], c[i+1], d[i+1], e[i+1], f[i+1], g[i+1], h[i+1])  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-05-05T11:09:53.460039Z",
     "start_time": "2018-05-05T11:09:53.455822Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "![image.png](img/net15.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-08-05T03:07:07.625650Z",
     "start_time": "2021-08-05T03:07:07.610507Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 0.06181813973848581,\n",
       " 1: 0.11515299000945314,\n",
       " 2: 0.1100340840842764,\n",
       " 3: 0.10714737750289352,\n",
       " 4: 0.10584740866489117,\n",
       " 5: 0.10584740866489117,\n",
       " 6: 0.10714737750289352,\n",
       " 7: 0.11003408408427638,\n",
       " 8: 0.11515299000945314,\n",
       " 9: 0.061818139738485794}"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G = nx.DiGraph(nx.path_graph(10))\n",
    "pr = nx.pagerank(G, alpha=0.9)\n",
    "pr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Ingredient-Flavor Bipartite Network\n",
    "\n",
    "![image.png](img/net16.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "<div><img src=img/net17.png align='left'></div>\n",
    "\n",
    "**The building blocks of economic complexity**\n",
    "\n",
    "Cesar A. Hidalgo and Ricardo Hausmann. 2009. PNAS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Path 路径\n",
    "A path is a sequence of nodes in which  each node is adjacent to the next one\n",
    "    \n",
    "   - In a directed network, the path can follow only the direction of an arrow. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Distance  距离\n",
    "\n",
    "The distance (shortest path, geodesic path) between two nodes is defined as the number of edges along the shortest path connecting them.\n",
    "\n",
    "> If the two nodes are disconnected, the distance is **infinity**.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Diameter 直径\n",
    "\n",
    "**Diameter $d_{max}$** is the maximum distance between any pair of nodes in the graph. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Shortest Path  最短路径\n",
    "The path with the shortest length between two nodes (distance). \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Average path length/distance, $<d>$ 平均路径长度\n",
    "\n",
    "\n",
    "The average of the shortest paths for all pairs of nodes.\n",
    "\n",
    "\n",
    "- for a **directed graph**: where $d_{ij}$ is the distance from node i to node j\n",
    "\n",
    "$<d> = \\frac{1}{2 L }\\sum_{i, j \\neq i} d_{ij}$ \n",
    "\n",
    "> 有向网络当中的$d_{ij}$数量是链接数量L的2倍\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The average of the shortest paths for all pairs of nodes.\n",
    "\n",
    "- In an **undirected** graph $d_{ij} =d_{ji}$ , so we only need to count them once \n",
    "\n",
    "> 无向网络当中的$d_{ij}$数量是链接数量L\n",
    "\n",
    "\n",
    "$<d> = \\frac{1}{L }\\sum_{i, j > i} d_{ij}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Cycle  环\n",
    "A path with the same start and end node. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# CONNECTEDNESS\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Connected (undirected) graph\n",
    "\n",
    "- In a connected **undirected** graph, any two vertices can be joined by a path. \n",
    "- A disconnected graph is made up by two or more connected components.   \n",
    "    - Largest Component:  Giant Component\n",
    "    - The rest: Isolates\n",
    "\n",
    "### Bridge 桥\n",
    "if  we erase it, the graph becomes disconnected. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The adjacency matrix of a network with several components can be written in a block-diagonal form, so that nonzero elements are confined to squares, with all other elements being zero:\n",
    "\n",
    "![image.png](img/net18.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# 结构洞\n",
    "\n",
    "洞在哪里？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "洞在桥下！\n",
    "\n",
    "结构“坑”"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Strongly connected *directed* graph 强连通有向图\n",
    "\n",
    "has a path from each node to every other node and vice versa (e.g. AB path and BA path).\n",
    "\n",
    "### Weakly connected directed graph  弱连接有向图\n",
    "it is connected if we disregard the edge directions.\n",
    "\n",
    "Strongly connected components can be identified, but not every node is part of a nontrivial strongly connected component.   \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# In-component -> SCC ->Out-component\n",
    "\n",
    "- In-component: nodes that can reach the **scc** (strongly connected component 强连通分量或强连通子图) \n",
    "- Out-component: nodes that can be reached from the scc. \n",
    "\n",
    "例如，万维网的蝴蝶结模型🎀 Bowtie Model\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Clustering coefficient 聚集系数\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Clustering coefficient 聚集系数\n",
    "What fraction of your neighbors are connected? Watts & Strogatz, Nature 1998.\n",
    "\n",
    "Node i with degree $k_i$  节点i有k个朋友\n",
    "- $e_i$ represents the number of links between the $k_i$ neighbors of node i.\n",
    "- 节点i的k个朋友之间全部是朋友的数量  $\\frac{k_i(k_i -1)}{2}$\n",
    "\n",
    "$$C_i = \\frac{2e_i}{k_i(k_i -1)}, \\space  \\space  \\space C_i  \\space in  \\space [0,1] $$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 节点的**聚集系数**\n",
    "\n",
    "![image.png](img/net19.png)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Global Clustering Coefficient 全局聚集系数（Transtivity 传递性)\n",
    "\n",
    "triangles 三角形 vs. triplets 三元组\n",
    "\n",
    "- A triplet consists of three connected nodes. \n",
    "    - A `connected triplet` is defined to be a connected subgraph consisting of three vertices and **two edges**. \n",
    "- A triangle includes three `closed triplets`\n",
    "- A triangle forms three `connected triplets` \n",
    "\n",
    "  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "center",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$C  = \\frac{\\mbox{number of closed triplets}}{\\mbox{number of connected triplets}}= \\frac{3 \\times \\mbox{number of triangles}}{\\mbox{number of connected triplets}}$$\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-20T23:58:35.189052",
     "start_time": "2017-05-20T23:58:35.014423"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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HdHS06UgiKmIpPTIyMpgxYwZeXl4kJibyzTffsHjxYmrXrm06ml1xtKnpS7m7u7NkyRJe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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f19ef90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G1 = nx.complete_graph(4)\n",
    "pos = nx.spring_layout(G1)  #定义一个布局，此处采用了spring布局方式\n",
    "nx.draw(G1, pos = pos, with_labels = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-20T23:58:35.785051",
     "start_time": "2017-05-20T23:58:35.781593"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0\n"
     ]
    }
   ],
   "source": [
    "print(nx.transitivity(G1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-20T23:58:36.706567",
     "start_time": "2017-05-20T23:58:36.550505"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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yecOGDSu113RBQQH3NGnCd6WlF/xOuBjwBvYAzX+/LRJozJnJ+KyjQHNXV/L2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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10efb3ad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G2 = nx.Graph()\n",
    "for i, j in [(1, 2), (1, 3), (1, 0), (3, 0)]:\n",
    "    G2.add_edge(i,j)\n",
    "nx.draw(G2,pos = pos, with_labels = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-21T00:00:07.271947",
     "start_time": "2017-05-21T00:00:07.268756"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6\n"
     ]
    }
   ],
   "source": [
    "print(nx.transitivity(G2))\n",
    "# 开放三元组有5个，闭合三元组有3个"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-20T23:58:41.789976",
     "start_time": "2017-05-20T23:58:41.629511"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f19ebd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G3 = nx.Graph()\n",
    "for i, j in [(1, 2), (1, 3), (1, 0)]:\n",
    "    G3.add_edge(i,j)\n",
    "nx.draw(G3, pos =pos, with_labels = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-21T00:00:29.398434",
     "start_time": "2017-05-21T00:00:29.394775"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n"
     ]
    }
   ],
   "source": [
    "print(nx.transitivity(G3))\n",
    "# 开放三元组有3个，闭合三元组有0个"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "THREE CENTRAL QUANTITIES IN NETWORK SCIENCE\n",
    "- A. Degree distribution: \t\t\t  $p_k$\n",
    "- B. Path length: \t\t\t\t\t\t$<d>$\t\t\t\n",
    "- C. Clustering coefficient: $C_i$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Typical Network Science Research\n",
    "\n",
    "- Discovering, Modeling, Verification\n",
    "    - WATTSDJ,STROGATZSH.Collective dynamics of‘small-world’ networks. Nature, 1998, 393(6684): 440–442.\n",
    "    - BARABÁSI A-L, ALBERT R. Emergence of scaling in random networks. Science, 1999, 286(5439): 509-512."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Typical Math Style\n",
    "Fan Chung & Linyuan Lu, The average distance in random graphs with given expected degree,. PNAS, 19, 15879-15882 (2002)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Typical Physical Style\n",
    "A.-L.Barabási,R.Albert,H.Jeong Mean-field theory for scale-free random networks. Physica A 272, 173–187 (1999)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Typical Computer Science Style\n",
    "\n",
    "- Community detection\n",
    "- Link prediction\n",
    "- Recommendation algorithms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Typical control sytle\n",
    "Controllability of Complex Networks\n",
    "\n",
    "Liu Y Y, Slotine J J, Barabási A L. Nature, 2011, 473(7346): 167-173."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "![image.png](./img/disruption.png)\n",
    "\n",
    "\n",
    "https://doi.org/10.1038/s41586-019-0941-9\n",
    "Large teams develop and small teams disrupt science and technology"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 阅读材料\n",
    "- Barabasi 2016 Network Science. Cambridge\n",
    "- 汪小帆、李翔、陈关荣 2012 《网络科学导论》. 高等教育出版社\n",
    "- 梅拉妮·米歇尔 2011 《复杂》,湖南科学技术出版社\n",
    "- 菲利普-鲍尔 2004 《预知社会：群体行为的内在法则》，当代中国出版社\n",
    "- 巴拉巴西 2007 《链接：网络新科学》 湖南科技出版社"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "![image.png](img/chengjun2.png)"
   ]
  }
 ],
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