15-network-science-intro.ipynb

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    "\n",
    "# 第十章 网络科学简介\n"
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    "\n",
    "\n",
    "**Network Science**, a textbook for network science, is freely available under the Creative Commons license. http://networksciencebook.com/\n",
    "\n",
    "<img src=\"images/network.png\" width = 250 align = 'right'>\n",
    "\n",
    "1. Introduction\n",
    "2. Graph Theory\n",
    "3. Random Networks\n",
    "4. The Scale-Free Property\n",
    "5. The Barabási-Albert Model\n",
    "6. Evolving Networks\n",
    "7. Degree Correlations\n",
    "8. Network Robustness\n",
    "9. Communities\n",
    "10. Spreading Phenomena\n",
    "Preface"
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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**. Invasion that started in March 19,  2003. Many of the regime's high ranking officials, including Saddam Hussein, avoided capture. Hussein was last spotted kissing a baby in Baghdad in April 2003, and then his 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, the ace of spades, i.e. Hussein himself, remained at large.\n",
    "\n",
    "\n",
    "\n"
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    "<img src = './img/saddam.png' width = 300 align = 'right'>\n",
    "\n",
    "It demonstrates the remarkable **stability of these networks**.\n",
    "\n",
    "The capture of Hussein was not based on fresh intelligence, but rather on his **pre-invasion social links**, unearthed from old photos stacked in his family album.\n",
    "\n",
    "It 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, was of no use when it came to Saddam Hussein's whereabouts. "
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    "## 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”。`中情局自此开始了对科威特长达数年的跟踪，最终成功窃听到了他本·拉登之间的移动电话，从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`，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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    "## A SIMPLE STORY (2): August 15, 2003 blackout.\n",
    "\n",
    "<img src='./img/blackout.png' width = 1000>\n"
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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\n",
    "\n",
    "[adj., v. kuh m-pleks, kom-pleks; n. kom-pleks] \n",
    "–adjective \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\tSource: Dictionary.com\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",
    "- society\n",
    "- brain\n",
    "- market\n",
    "- cell\n",
    "\n",
    "Stephen Hawking: I think the next century will be the century of complexity. "
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    "> Behind each complex system there is a network, that defines the interactions between the component. \n"
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    "<img src = './img/facebook.png' width = 1000>"
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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 explosive interest in network science is well documented by the citation pattern of two classic network papers, the 1959 paper by Paul Erdos and Alfréd Rényi that marks the beginning of the study of random networks in graph theory [4] and 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. The explosive growth of citations to these papers in the 21st century documents the emergence of network science, drawing a new, interdisciplinary audience to these classic publications. \n"
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    "<img src = './img/citation.png' width = 800>"
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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 universality of network characteristics: \n",
    "\n",
    "The architecture of networks emerging in various domains of science, nature, and technology are more similar to each other than one would have expected. \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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    "THE IMPACT OF NETWORK SCIENCE \n"
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    "- **Google**\n",
    "Market Cap(2010 Jan 1): \n",
    "$189 billion\n",
    "\n",
    "- **Cisco Systems**\n",
    "networking gear Market cap (Jan 1, 2919): \n",
    "$112 billion\n",
    "\n",
    "- **Facebook**\n",
    "market cap: \n",
    "$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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    "## Security: Fighting Terrorism. \n",
    "Terrorism is one of the maladies of the 21st century, absorbing significant resources to combat it worldwide. \n",
    "\n",
    "- **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. \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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    "## Epidemics: From forecasting to halting deadly viruses. \n",
    "\n",
    "While the **H1N1 pandemic** was not as devastating as it was feared at the beginning of the outbreak in 2009, it gained a special role in the history of epidemics: 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. \n"
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    "### The first major mobile epidemic outbreak\n",
    "\n",
    "In January 2010 network science tools have predicted the conditions necessary for the emergence of viruses spreading through mobile phones. \n",
    "\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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    "## 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. \n",
    "\n",
    "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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    "## The Bridges of Konigsberg\n"
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   "source": [
    "<img src = './img/konigsberg.png' width = 500>"
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    "Can one walk across the seven bridges and never cross the same bridge twice and get back to the starting place?     \n",
    "\n",
    "<img src ='./img/euler.png' width = 600>"
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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\tG(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) **G(N, L)**"
   ]
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   "source": [
    "<img src = './img/net.png' width = 800>"
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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"
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    "If you connect individuals that work with each other, you will explore the professional network. http://www.theyrule.net\n",
    "\n",
    "If you connect those that have a romantic and sexual relationship, you will be exploring the sexual networks.\n",
    "\n",
    "\n",
    "If you connect individuals based on their first name (all Peters connected to each other), you will be exploring what? 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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ObrDgOnTo4LTl1ph3330XOTk5WLp0qegozcLiI/qN0WhEamoq9Ho9CgoKsGDBAsyaNQseHh6io1ELybKMK1euNDgtmZ2dDbVa3eCUpE6ng5eXlyLLrTGjR4/GrFmzMGHCBNFRmoXFR9SAjIwM6PV6fPnll3jyySexYMECdO/eXXQsaoTZbMbly5cbnJLMzs5GmzZtGpySDAoKQvv27UXHdxi1tbXw8vJCTk4OvLy8RMdpFhYfUSMuXbqEDz/8ECtXrsTw4cORmJiIoUOHcgQgiNlsRn5+foPTkufPn4e7u3uDU5JBQUHw9PQUHd8pHDp0CPPmzcMPP/wgOkqzsfiI7kFFRQVWr16NpKQkeHh4IDExEZMmTUKrVq1ER3M6JpOp3qbJN78uXLiA9u3bNzglGRQUhHbt7Gc/SGf1xhtv4Nq1a3jvvfdER2k2Fh9RE5jNZnz++efQ6/U4e/Ys5s2bhzlz5jjslI8oRqMReXl5DU5L5ubmokOHDg2uuXXv3p3PXgoWHR2N5557Do899pjoKM3G4iNqphMnTiApKQk7duxo0fFIxRXVSDmWj6zCMpQZjHDXqhHS0R2T+tn/beGNqampQW5uboPTkhcvXoSfn1+D05Ldu3dvdNNkEsdgMMDHxwcFBQVwd3cXHafZWHxELVRYWIhly5ZhxYoVTToe6eSlUixNz8a+s0UAUGf3+5sPAkf29EFchA5hAZ5W/B00X3V1db1Nk28WXEFBATp37tzg7iSBgYEt2jSZxEhLS8NLL72Ew4cPi47SIiw+IgupqqrCunXrkJSUdNfjkRxp66eqqqpb5fbHacnCwkJ06dKlwTW3rl27cg3Uybzyyiswm8148803RUdpERYfkYXJsow9e/ZAr9fj+PHj9Y5HssfNfisrK+ucCHB7wRUVFTW4aXJwcLDQTZPJ9oYMGYLXX38d0dHRoqO0CIuPyIoyMzORlJSEzZs3IyYmBmOejMPLaSV3PN6l9moBfl75LNqGDIH3n5+v872WHu9SXl7e4J2S586dw7Vr125tmvzHr4CAALvdNJlsp7y8HH5+figqKnL4NVgWH5ENFBcXY8WKFfjXaTNUXfoCUsMbFF/Z+ApkYzXUHh3qFd+9HOh5/fr1Oz7AXV5efscTAfz9/R1202Syjd27d2PRokVIT08XHaXFOEdBZAPe3t6IjX8eqxftRY2x4SnOyp/2wUXbFhqvEBhLL9f7viwDaWeKcD7/Ckp+zmtwWtJgMNQptGHDhuHpp5+GTqeDn58fH7ynZtu7dy8efvhh0TEsgsVHZCMpx/Jxp9oxV99A6YF18J36JipOfnXHaxiqqjBoWiK6VJ69VW7R0dGIjY1FcHCwYjdNJuvbu3cv3n//fdExLILFR2QjWYVldR5ZuF3p/k/hFvYnqN19Gr2GpHHFtLgXoJ/8gDUiEjXo6tWrOHfunNMc28VJfSIbKTMYG3y95soFGPJOwv2hsS26DpG17Nu3D4MHD3aax1M44iOyEXdtw/+7GS7+COP1K8hfNhMAINcYANmMy8Xx8JtZf2rJXauxak6iP0pLS3Oa9T2AxUdkMyEd3eGqLqw33enWdyTa9hp+69dlGdtgvH4F7UfOq3cNrdoFIX7ciJlsa+/evVi1apXoGBbDqU4iG5nYr3ODr7totFC53XfrS9JoIalbQdWm/gG4MoCJDzZ8HSJruHLlCgoKCvDAA86zrswRH5GNeLu5IqKHD/ZkXml0mzLPYdMafF2SgKiePg69cTU5nrS0NERERDjVJgYc8RHZ0LxIHbTq5v0FolWrEBeps3AiosY50/N7N7H4iGwoLMATC0eHoLWmaf/rtXIBFo4OafZ2ZUTNtXfvXkRFRYmOYVEsPiIbmx7eDQtH90JrjQp3e9Zckn4tvYoDazDIp+H9PYmsJS8vD2VlZQgNDRUdxaK4xkckwPTwbujT2RPL0rORdqYIEgBDA+fxRfX0QVykDgc65SImJgaHDx+Gm5ubsNykLGlpaYiKinK6fVy5STWRYCUV1Ug5no+sy+UoM9TCXatBiF87THzw9xPYZVnGM888g2vXrmHLli1O9xcR2acnn3wSQ4YMQWxsrOgoFsXiI3IQ1dXViIyMxGOPPYaXX35ZdBxycrIsIyAgAGlpaQgODhYdx6I41UnkIFxdXbF161YMGDAAYWFh+POf/yw6Ejmxc+fOQZIk6HTOdycx50uIHEinTp2wdetWzJo1C5mZmaLjkBO7uU2ZM572weIjcjADBw7EokWLEBMTg9LSUtFxyEk54/N7N3GNj8hBzZ8/HxcuXMDOnTudalcNEs9sNqNjx474/vvv0aVLF9FxLI4jPiIHtXjxYty4cQOvvPKK6CjkZE6fPg0PDw+nLD2AxUfksDQaDTZv3oz169dj8+bNouOQE3HmaU6AxUfk0Hx8fLB9+3bMmzcPJ0+eFB2HnIQzblN2O67xETmBDRs2YOHChcjIyIC3t7foOOTAjEYjvL29cebMGfj6+oqOYxUc8RE5galTp2LSpEmYPHkyjEaj6DjkwH744Qd07tzZaUsPYPEROY233noLGo0GL7zwgugo5MCcfX0PYPEROQ2VSoUNGzbgs88+w5o1a0THIQelhOLjGh+Rkzl9+jQiIyOxa9cuPPTQQ6LjkAOpqamBl5cXLl68iPvuu090HKvhiI/IyYSGhiI5ORkTJkxAYWGh6DjkQDIyMtCzZ0+nLj2AxUfklGJiYvD0009j4sSJqKmpER2HHIQSpjkBFh+R0/r73/8Ob29vLFiwQHQUchBKKT6u8RE5sbKyMoSHhyM+Pt7pDhMly7px4wY6dOiAwsJCuLm5iY5jVTyPj8iJubu7Y8eOHRg6dChCQ0MxdOhQ0ZHITh06dAh9+/Z1+tIDONVJ5PSCg4OxevVqPP7448jPzxcdh+yUs29TdjsWH5ECPProo4iPj8e4ceNQVVUlOg7ZIaWs7wFc4yNSDFmW8cQTT0Cj0WD16tVOebI2Nc/169fh7++P4uJiaLVa0XGsjiM+IoWQJAkrV67Ejz/+iPfff190HLIjBw4cwMCBAxVRegBvbiFSlDZt2mD79u0YNGgQevfujejoaNGRyA4oaZoT4IiPSHG6deuGDRs2YNq0abhw4YLoOGQHlFZ8XOMjUqgPPvgAycnJOHz4MNq2bSs6DglSUlKC7t27o7i4GBqNRnQcm+CIj0ihnn32WfTr1w8zZ84E//2rXOnp6Rg6dKhiSg9g8REpliRJWL58OfLy8vD222+LjkOCKG2aE2DxESmaVqvF9u3bsWzZMnz++eei45AASiw+rvEREQ4fPoyxY8fiwIED6Nmzp+g4ZCM///wzevfujaKiIri4KGccpJzfKRHd0aBBg/DWW29h7NixuH79uug4ZCNpaWmIiIhQVOkBLD4i+s3s2bMRHR2N6dOnw2w2i45DNqDEaU6AxUdEt0lKSkJZWRleffVV0VHIBlh8RKR4Go0GW7ZswZo1a7B161bRcciKcnJyUFVVhV69eomOYnMsPiKqo0OHDti2bRvmzp2LH3/8UXQcspKboz0lblbO4iOievr164ekpCTExMTg6tWrouOQFSh1mhPg4wxE1Ijnn38eJ0+exO7du6FWc097ZyHLMvz9/XHw4EF0795ddByb44iPiO7onXfegSRJePHFF0VHIQs6c+YMWrVqhcDAQNFRhGDxEdEdqdVqbNy4EampqVi3bp3oOGQhSl7fA1h8RHQX7du3R2pqKhISEnDs2DHRccgClLy+B3CNj4ju0bZt25CYmIiMjAz4+vqKjkPNZDab0aFDB5w8eRL+/v6i4wjBER8R3ZPx48fjqaeewqRJk1BTUyM6DjXTqVOn4OXlpdjSA1h8RNQEr732Gjw9PZGQkCA6CjWT0qc5ARYfETWBi4sL1q5di7S0NCQnJ4uOQ83A4uMaHxE1w9mzZzF06FCkpqZi8ODBouPQPaqtrYW3tzeys7Ph4+MjOo4wHPERUZP16NEDq1atwqRJk1BQUCA6Dt2jY8eOoVu3boouPYDFR0TNNHr0aDz77LMYP348DAaD6Dh0DzjN+SsWHxE124svvoiuXbvir3/9K7hqYv/S0tJYfOAaHxG1UGVlJQYPHoxZs2ZhwYIFouPQHVRXV8Pb2xv5+fnw8PAQHUco7jpLRC3Stm1bpKamYtCgQejduzeioqJER6IGHDlyBPfff7/iSw/gVCcRWUBgYCDWrVuHqVOnIjc3V3QcagDX937H4iMii4iOjsaLL76ImJgYVFZWio5Df8Di+x3X+IjIYmRZxsyZM2EwGLBhwwbF7v5vbyorK+Hr64tffvkFbdq0ER1HOI74iMhiJEnCv/71L1y4cAH//Oc/Rceh3xw8eBAPPvggS+83vLmFiCxKq9Vi27ZtGDhwIPr06YNRo0aJjqR4nOasiyM+IrK4zp07Y/PmzZgxYwbOnTsnOo7isfjq4hofEVnNRx99hKSkJBw5cgTu7u6i4yjStWvX0KVLFxQXF8PV1VV0HLvAER8RWc2cOXMwfPhw/OUvf4HZbBYdR5H279+PQYMGsfRuw+IjIqtasmQJrl69itdff110FEXiNmX1sfiIyKpatWqFlJQUfPzxx9i+fbvoOIrD9b36uMZHRDbx/fffY9SoUUhPT0doaKjoOIrwyy+/oEePHiguLoZazZv4b+KIj4hson///li8eDHGjh2La9euiY6jCOnp6Rg+fDhL7w9YfERkM3/5y1/w3//935gyZQpMJpPoOE6P05wNY/ERkU3985//hMlkwt/+9jfRUZze3r17eVpGA1h8RGRTarUamzZtQkpKCjZs2CA6jtO6dOkSrl69it69e4uOYnc48UtENufl5YXU1FRER0cjJCQEDzzwgOhITictLQ1RUVFwceH45o/4J0JEQvTp0wfLli3DuHHj8Msvv4iO43S4vndnfJyBiIRauHAhvv32W+zZswcajUZ0HKcgyzK6du2KPXv2oGfPnqLj2B2O+IhIqNdffx1ubm547rnnREdxGufPn4fJZEKPHj1ER7FLLD4iEkqlUmHdunX46quv8PHHH4uO4xRublPGg4AbxptbiEg4Dw8P7NixA8OHD8f999+P8PBw0ZEc2t69e/GnP/1JdAy7xTU+IrIbn332GebOnYuMjAx06tRJdByHJMsyOnbsiIyMDHTt2lV0HLvEqU4ishtjxozB3LlzMWHCBFRXV4uO45B++uknuLm5sfQaweIjIruycOFC+Pv7Iy4uDpyQajo+xnB3LD4isiuSJGHVqlXIyMjAsmXLRMdxONym7O64xkdEdun8+fMYPHgwNm/ejIiICNFxHILJZIKPjw9Onz4NPz8/0XHsFkd8RGSXgoKCsG7dOkyZMgUXL14UHcchnDhxAh07dmTp3QWLj4js1ogRI/DCCy8gJiYGN27cEB3H7nF9796w+IjIriUmJiI0NBSzZ8/mzS53weK7N1zjIyK7V1VVhWHDhmHKlCl4/vnnRcexSzU1NfD29kZOTg68vLxEx7Fr3LmFiOxe69atsX37dgwcOBC9e/fGyJEjRUeyO0ePHoVOp2Pp3QNOdRKRQwgICMCmTZvw5JNPIjs7W3Qcu3Nzf066OxYfETmMYcOG4bXXXkNMTAzKy8tFx7ErXN+7d1zjIyKHIssyYmNjUVxcjJSUFJ4wjl/XQH18fHD58mW0a9dOdBy7x/9iiMihSJKEDz74AIWFhXjjjTdEx7ELhw8fRp8+fVh694jFR0QOx9XVFVu3bkVycjJ27twpOo5w3KasaVh8ROSQ/Pz8sHXrVsyePRuZmZmi4wjF9b2m4RofETm0VatW4a233kJGRgY8PT1Fx7G58vJy+Pn5oaioCK1btxYdxyFwxEdEDm3GjBkYNWoUnnjiCZhMJtFxbO7AgQN46KGHWHpNwOIjIof37rvvwmAw4OWXXxYdxeY4zdl0LD4icngajQabN2/Gxo0bsWnTJtFxbIrF13Rc4yMip3HixAk88sgj+PrrrxEWFiY6jtVdvXoV3bp1Q3FxMVq1aiU6jsPgiI+InEbfvn3x4YcfIiYmBsXFxaLjWN2+ffswZMgQll4TsfiIyKlMnjwZU6ZMweOPP47a2lrRcayK05zNw+IjIqfzxhtvQKvV4oUXXhAdxapYfM3D4iMip6NSqbB+/Xrs2rULq1evFh3HKgoLC3H58mX07dtXdBSHw/P4iMgpeXp6YseOHYiIiECvXr0wYMAA0ZEsKi0tDcOHD4dKpRIdxeFwxEdETqtXr15ITk7GhAkTUFhYKDqORXGas/lYfETk1MaOHYtnnnkGEyZMQHV1teg4FsPiaz4+x0dETs9sNmPixInw9vbGihUrIEmS6Egtkpubi4EDB6KwsNDhfy8icMRHRE7PxcUFq1evxqFDh7BixQrRcVosLS0NUVFRLL1m4s0tRKQI7dq1Q2pqKoYMGYLQ0FAMGzZMdKRm4zRny3DER0SKodPpsGbNGkyePBmXLl0SHadZZFlm8bUQi4+IFGXkyJFITEzEuHHjUFVVJTpOk507dw4qlQpBQUGiozgsFh8RKc7zzz+PHj16YM6cOXC0+/tujva4vtd8LD4iUhxJkvDvf/8bp0+fhl6vFx2nSTjN2XJ8nIGIFCsvLw/h4eH49NNPMWLECNFx7spsNsPX1xfHjx9HQECA6DgOiyM+IlKsrl27YuPGjZg+fTouXLggOs5d/d///R88PT1Zei3E4iMiRYuIiMArr7yCsWPHoqKiQnScRnGa0zJYfESkeHFxcRgwYABmzJhh1ze7sPgsg2t8REQAqqurERkZiTFjxmDhwoWi49RjNBrh7e2NM2fOwNfXV3Qch8adW4iIALi6umLr1q0YMGAAwsLCMGbMGNGR6rh5QwtLr+U41UlE9JtOnTohJSUFTz/9NLKyskTHqYPTnJbD4iMiuk14eDjeeecdxMTE4Pr166Lj3MLisxyu8RERNWD+/PnIycnBjh07hJ9yXlNTAy8vL1y6dAmenp5CszgDjviIiBqwePFiVFRU4NVXXxUdBd999x1CQkJYehbC4iMiaoBGo8GWLVuwdu1apKSkCM3CaU7LYvEREd2Bj48Ptm/fjr/+9a84deqUsBwsPsviGh8R0V1s2LABCxcuxNGjR+Hl5WXTz75x4wY6dOiAwsJCuLm52fSznRVHfEREdzF16lRMnDgRkydPhtFotOlnf/vtt+jbty9Lz4JYfERE9+Dtt9+GSqXC//zP/9j0cznNaXksPiKie6BSqbBx40b85z//waeffmqzz2XxWR7X+IiImuD06dOIiorCrl270L9/f6t+1vXr1+Hv74/i4mJotVqrfpaScMRHRNQEoaGhWLFiBcaPH48rV65Y9bP279+P8PBwlp6FsfiIiJpo3LhxmDlzJiZOnIiamhqrfQ6nOa2DxUdE1AyvvvoqvLy8EB8fb7XPSEtLY/FZAdf4iIiaqaysDOHh4UhISMCcOXMseu3i4mIEBQWhpKQEajVPkLMk/mkSETWTu7s7UlNTMXToUISGhmLIkCEWu3Z6ejqGDRvG0rMCTnUSEbVAjx49sGbNGjz++OPIz8+32HW5vmc9LD4iohZ69NFHsWDBAowfPx4Gg8Ei12TxWQ/X+IiILECWZUydOhVarRaffPIJJElq9rUKCgrQp08fFBUVwcWF4xNL458oEZEFSJKElStX4sSJE1iyZEmLrpWWlobIyEiWnpVw1ZSIyELatm2L1NRUhIeH47/+678QHR3drOtwmtO6+M8JIiIL6tatGzZs2IBp06YhJyenyT8vyzK++eYbFp8VsfiIiCwsKioKL730EmJiYlBZWdmkn83JyUFNTQ1CQkKslI5YfEREVjB//nw8+OCDmDlzJppyD+HNac6W3BxDjWPxERFZgSRJWL58OfLy8rBo0aJ7/jluU2Z9fJyBiMiKCgoKMGDAACQnJ2P06NGNvleWZXTq1AmHDh1CYGCgjRIqD0d8RERW5O/vjy1btmDGjBk4e/Zso+/NysqCVqtl6VkZR3xERDaQnJyMxYsX47vvvoO7uzsAoLiiGinH8pFVWIYygxGX8y6g9pccbHonEV5uroITOy8WHxGRjcTFxSE/Px//+HAVlu+7gH1niwAA1UbzrfeoJRkqlQqRPX0QF6FDWICnoLTOi8VHRGQjNTU1GDjteZTpRsAsqdDY376SBGjVKiwcHYLp4d1sllEJuMZHRGQjm4//jKqQUTCh8dIDAFkGqmpNeHNXJtYeybVJPqXgiI+IyAZOXirFlOQjqKo13XpNNtai5KtlMOSegNlQAbWnH+6LeBKtg/rX+dnWGhU2zQlHn86eNk7tnDjiIyKygaXp2TAYTXVek80mqNt5o+MT7yAgcRM8h09H0Y5FMJZeqfM+g9GEZenZtozr1Fh8RERWVlxRjX1ni+pNb7q00sJz2DSoPX0hSS5ooxsAtYcvqgvrlpwsA2lnilBSUW3D1M6LxUdEZGUpx+7tZHZT5TXUXi1AK58u9b4nAUg5brkT3pWMxUdEZGVZhWV1HlloiGwyonjnu3DrHQ2NV0C97xuMZmRdLrdWREVh8RERWVmZwdjo92XZjOLP3gNUarR/ZG4j16m1dDRFYvEREVmZu/bOZ37LsoySXUtgqiyFz7iXIKnu/F53rcYa8RSHxUdEZGUhHd3hqm74r9urXy5FbckldJj4d7ho7rxNmVbtghC/dtaKqCh8jo+IyMqKK6oxZNHeeut8xuu/oGD504BKA8lFdev19o/Og1toVJ33uqpdcOh/H+YenhZw5zE1ERFZhLebKyJ6+GBP5pU6jzSoPTqg64uf3fXnJQmI6unD0rMQTnUSEdnAvEgdtGrV3d/YAK1ahbhInYUTKReLj4jIBsICPLFwdAhaa5r2125rjQsWjg7hdmUWxKlOIiIbuXnKwpu7smAwmng6gyC8uYWIyMZO5ZdiWXo20s4UQcKvD6ffpFW7QMava3pxkTqO9KyAxUdEJEhJRTVSjucj63I5ygy1cNdqEOLXDhMf7MwbWayIxUdERIrCm1uIiEhRWHxERKQoLD4iIlIUFh8RESkKi4+IiBSFxUdERIrC4iMiIkVh8RERkaKw+IiISFFYfEREpCgsPiIiUhQWHxERKQqLj4iIFIXFR0REisLiIyIiRWHxERGRorD4iIhIUVh8RESkKCw+IiJSFBYfEREpCouPiIgU5f8Btb3PJ5BdyiAAAAAASUVORK5CYII=\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": {
    "collapsed": true,
    "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": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-06-14T01:53:09.866905Z",
     "start_time": "2020-06-14T01:53:09.605369Z"
    },
    "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": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "<img src = './img/networks.png' width = 1000>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Degree, Average Degree and Degree Distribution\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "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": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Undirected network: \n",
    "Node degree: the number of links connected to the node.\n",
    "\n",
    "$k_1 = k_2 = 2, k_3 = 3, k_4 = 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-05-05T10:48:27.253744Z",
     "start_time": "2018-05-05T10:48:27.170334Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw(Gd, with_labels = True, pos=nx.circular_layout(Gd))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Directed network\n",
    "In directed networks we can define an in-degree and out-degree. The (total) degree is the sum of in-and out-degree.\n",
    "\n",
    "$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": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "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": "2020-06-14T02:04:34.191676Z",
     "start_time": "2020-06-14T02:04:34.178665Z"
    },
    "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": {
    "collapsed": true,
    "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": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-20T22:52:22.051941",
     "start_time": "2017-05-20T22:52:21.864640"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "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",
    "plt.hist(x)\n",
    "plt.show() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-20T22:53:12.228948",
     "start_time": "2017-05-20T22:53:12.222577"
    },
    "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": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-20T22:53:16.580337",
     "start_time": "2017-05-20T22:53:16.575781"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "freq_sum = np.sum(list(freq.values()))\n",
    "freq_sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-20T22:53:17.449486",
     "start_time": "2017-05-20T22:53:17.444514"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.5, 0.3333333333333333, 0.16666666666666666]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "px = [float(i)/freq_sum for i in freq.values()]\n",
    "px"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-05-20T22:53:30.898701",
     "start_time": "2017-05-20T22:53:30.717662"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+ibuPcfeEuyfKysoilCVSQ4cdFmbmnHsu3HprWL928uS4qxLJuihBvxxolfK6JbCiqp3d/SVgBzNrnny9Ivn7SmAC4VaQSDyaNAkh//LL0LBhCP9TT4VPPom7MpGsiRL0M4H2ZtbOzEqB/sDE1B3MbEczs+TXXYFS4CMza2xmTZLbGwM9gQWZ/AFENsu++8KcOXDZZaF9Qnk5PPpo3FWJZEW1Qe/uG4AhwCTgDeBBd19oZoPNbHByt+OABWY2hzBD54Tk4Oy2wFQzmwvMAJ5092ey8HOI1FzDhnDttTBjBmy3HRx3XBi8/fe/465MJKPM87ARVCKR8IoKTbmXHPrqK7jhBrj6amjUCG68EX7+8zBzR6QAmNmsqqaw68lYEQhN0i67LNzOKS8P9+179YKlS+OuTKTWFPQiqXbeGV56KQzY/vOfYWbOrbfCX/4CbdtCvXrh93Hj4q5UJDLduhGpyrJlcNZZMGlSCPjUjpiNGsGYMeFBLJE8oFs3IpujTRt4+unwwFXltsfr1sEVV8RTl0gNKehFNsUMPv44/Xta5EQKhIJepDqtW6ff3qQJfPFFbmsR2QwKepHqXHNNuCefqqQE1qyBLl1g6tRYyhKJSkEvUp2BA8PAa5s24VZOmzZw332hT8769bDffjBkCHz2WdyViqSlWTcitbF2bRiUvfVWaNUK7rgjzL8XyTHNuhHJli23DAuTT50abu/07h2eqP3oo7grE/mWgl4kE/beG157Laxi9cAD4enahx6CPPwfs9Q9CnqRTGnYEEaOhIqKcBvn+OPh2GPhgw/irkzqOAW9SKZ17gzTp8N114VVrXbZBe6+W1f3EhsFvUg21K8Pl14Kc+dCp05w+unQsye8807clUkdpKAXyaaddoIXXoDRo8NVfseOYfD266/jrkzqEAW9SLbVqwdnnw0LF8IBB8AFF4QVrl5/Pe7KpI5Q0IvkSuvW8OSToeXxW2/BbruFwdv16+OuTIqcgl4kl8zCk7avvw7HHAMjRsAee4SZOiJZoqAXicM228D48fDYY7BqFXTvHgZv1SRNskBBLxKnPn3C1f1pp8H114cZOi++GHdVUmQU9CJx23pruPNOeO65MBunR48weLtmTdyVSZGIFPRm1svMFpnZYjMbnub9PmY2z8zmmFmFme0b9VgRSTr4YJg/Hy66KHTL7NAhDN6K1FK1QW9mJcAooDdQDgwws/JKu00BOrt7F+A0YGwNjhWRjRo3hj/8AaZNg6ZN4cgj4aST4MMP465MCliUK/puwGJ3X+Lu64HxQJ/UHdx9rX/X77gx4FGPFZE0uneH2bPhyivhb38LbRTGj1cbBdksUYK+BfBeyuvlyW3/w8yOMbN/AU8SruojH5s8flDytk/FqlWrotQuUtx+8AO46qoQ+G3bwoAB0LcvvP9+zIVJoYkS9JZm2/cuK9x9grvvDPQFRtbk2OTxY9w94e6JsrKyCGWJ1BG77gqvvAI33BBWtSovD4O3urqXiKIE/XKgVcrrlsCKqnZ295eAHcyseU2PFZEq1K8PF18cBmu7doVBg8Lg7dtvx12ZFIAoQT8TaG9m7cysFOgPTEzdwcx2NDNLft0VKAU+inKsiNTAjjvClClhycJZs8LV/o03qkmabFK1Qe/uG4AhwCTgDeBBd19oZoPNbHByt+OABWY2hzDL5gQP0h6bhZ9DpO6oVy9c0S9cGK7qL744rHC1YEHclUme0uLgIoXMPczGOf98WL06LFR+2WVQWhp3ZZJjWhxcpFiZhdk4r78OP/tZmKWz++4wY0bclUkeUdCLFIOyMhg3DiZOhE8+gb32gqFDYd26uCuTPKCgFykmRx0V7t2feWZ4wnbXXeH55+OuSmKmoBcpNlttBbffHgLeDA46KAzerl4dd2USEwW9SLHq0QPmzQu3cO66Kzxo9fe/x12VxEBBL1LMGjUKfe6nT4dmzeDoo8PgrdqM1CkKepG6YONyhVdfDY88EpqkPfCA2ijUEQp6kbqitDSsUfvaa+EJ24EDw+Dte+9Vf6wUNAW9SF3ToQP885/wxz+GAdsOHcLg7TffxF2ZZImCXqQuKimBCy4ITdK6dQtLFx50ELz1VtyVSRYo6EXqsu23h2efhbFjYc6csDj59dfDhg1xVyYZpKAXqevM4PTTQxuFww6DSy8NT9bOmxd3ZZIhCnoRCX7yE5gwISxduGxZ6JkzYgR8+WXclUktKehF5DtmcPzx8MYbYb79yJFhoZPp0+OuTGpBQS8i39esGdx/Pzz1FHz2Weh3f+GF8PnncVcmm0FBLyJV6907LGhy9tlw003QsSM891zcVUkNKehFZNOaNoVRo+DFF6FBAzj00DB4++mncVcmESnoRSSa/feHuXNh2DC4777QJO2xx+KuSiJQ0ItIdFtsAb/7Hbz6KmyzDRxzTBi8/c9/4q5MNkFBLyI1t/vuMHMmXHMNPP54uLr/85/VJC1PRQp6M+tlZovMbLGZDU/z/kAzm5f8Nc3MOqe8t9TM5pvZHDPTit8ixaJBA7j88vBE7U9/CqecAocfDu++G3dlUkm1QW9mJcAooDdQDgwws/JKu70DHODunYCRwJhK7x/o7l2qWqFcRArYLrvAyy/DLbeE3zt0CIO3apKWN6Jc0XcDFrv7EndfD4wH+qTu4O7T3P2T5MvpQMvMlikiea2kBM47L0zF3GsvGDIEDjgAFi2KuzIhWtC3AFIbVi9PbqvK6cDTKa8dmGxms8xsUFUHmdkgM6sws4pVWv1GpDC1bQuTJsE994TQ79w5DN6qSVqsogS9pdmWdsTFzA4kBP2wlM37uHtXwq2fc81s/3THuvsYd0+4e6KsrCxCWSKSl8zgF78IbRSOOAIuuwy6dw/38iUWUYJ+OdAq5XVLYEXlncysEzAW6OPuH23c7u4rkr+vBCYQbgWJSLHbbruwbOHDD8P770MiAVdcAf/9b9yV1TlRgn4m0N7M2plZKdAfmJi6g5m1Bh4FTnb3N1O2NzazJhu/BnoCCzJVvIgUgOOOCy2QTzoJrr0WunQJK1xJzlQb9O6+ARgCTALeAB5094VmNtjMBid3GwE0A0ZXmka5LTDVzOYCM4An3f2ZjP8UIpLffvQjuPdeeOYZ+OIL2G8/OP98WLs27srqBPM8fMAhkUh4RYWm3IsUpc8+C7dwbrsNWreGMWOgZ8+4qyp4ZjarqinsejJWRHKrSZPv5tw3bBhWtTr1VPj447grK1oKehGJxz77hJk4l18e2ieUl4fBW8k4Bb2IxKdhw9Avp6IiLGXYr1/49e9/x11ZUVHQi0j8unQJHTF/+1t44onQVuHee9UkLUMU9CKSHxo0gOHDQ8/7jh3DffvDDoOlS+OurOAp6EUkv/z0p2E1q1Gj4JVXQujfequapNWCgl5E8k+9enDOOaFfzsY59/vtF9oqSI0p6EUkf7VpA089BfffD//6V7iXf+218NVXcVdWUBT0IpLfzODkk0MbhT59wsNWe+wBs2fHXVnBUNCLSGHYdlt48EF49NGwRm23bmHw9osv4q4s7ynoRaSwHHNMuLr/xS/guuvC7ZyXX467qrymoBeRwvPDH8LYsfDss7B+Pey/P5x7buijI9+joBeRwnXIIWFmzgUXwJ/+FNarffrpag+raxT0IlLYGjeGP/4x9Ljfcks4/HA45RT46KPqj60jFPQiUhz22gteew1+/Wv4619Dk7SHHlIbBRT0IlJMfvAD+M1vYNYsaNUKjj8ejj0WVnxv9dM6RUEvIsWnUyeYPh1+//uwqlV5Odx1V529ulfQi0hxql8fLrkE5s2Dzp3hjDPg0ENhyZK4K8s5Bb2IFLf27eH558OsnBkzYNdd4aab4Ouv464sZxT0IlL86tWDwYNh4ULo0QMuvBD23Tc8eFUHKOhFpO5o1SosbPKXv8Bbb8Fuu8HIkeGhqyIWKejNrJeZLTKzxWY2PM37A81sXvLXNDPrHPVYEZGcMoOBA8PV/LHHwogRkEjAzJlxV5Y11Qa9mZUAo4DeQDkwwMzKK+32DnCAu3cCRgJjanCsiEjubbNNmG//+OPh4ao994RLL4V16+KuLOOiXNF3Axa7+xJ3Xw+MB/qk7uDu09z9k+TL6UDLqMeKiMTq6KPD1f3pp8P114cZOi++GHdVGRUl6FsA76W8Xp7cVpXTgY3NJiIfa2aDzKzCzCpWrVoVoSwRkQzZaisYMwamTAlLFvboAWefDWvWxF1ZRkQJekuzLe1TB2Z2ICHoh9X0WHcf4+4Jd0+UlZVFKEtEJMMOOgjmz4eLLgrB36EDPPlk3FXVWpSgXw60SnndEvje88Rm1gkYC/Rx949qcqyISN5o1Aj+8AeYNi1c6R95ZBi8LeA7DVGCfibQ3szamVkp0B+YmLqDmbUGHgVOdvc3a3KsiEhe6t49LFd41VWhOVp5OYwfX5BtFKoNenffAAwBJgFvAA+6+0IzG2xmg5O7jQCaAaPNbI6ZVWzq2Cz8HCIimVdaCldeGQJ/++1hwICwbu3778ddWY2Y5+G/TolEwisqKuIuQ0TkO19/DTffDL/6FTRoADfcEPrnWLqhyNwzs1nunkj3np6MFRGJoqQkDNLOnw+77w6DBsHBB8Pbb8ddWbUU9CIiNbHDDmEa5pgxoe/9rruGwds8bpKmoBcRqSkzOPPM8KDVIYfA0KFhhasFC+KuLC0FvYjI5mrRIrRQGD8eli6Frl3DLJ08a5KmoBcRqQ0zOOGEcHV//PFw9dUh8GfMiLuybynoRUQyoXnz0P74iSdg9epwK+fii/OiSZqCXkQkk444IixwMmgQ3HhjGKx9/vlYS1LQi4hkWtOmYenCF14Iq1sddFAI/k8/jaUcBb2ISLYccEBYnPzSS+Guu0KTtIm57wKjoBcRyaYttoDrroNXX4VmzUILhf79YeXKnJWgoBcRyYVEAioqwhq1EyaEJmnjxuWkSZqCXkQkV0pLQ6+c116D9u3hpJPgqKPgllugbdtwP79t2/APQAbVz+ifJiIi1Ssvh6lT4bbb4JJL/ndxk2XLwsAthD74GaArehGROJSUwC9/CelW1Fu3Dq64ImPfSkEvIhKnDz5Iv/3ddzP2LRT0IiJxat26Zts3g4JeRCRO11wT1qlN1ahR2J4hCnoRkTgNHBh627dpExqktWkTXmdoIBY060ZEJH4DB2Y02CvTFb2ISJFT0IuIFLlIQW9mvcxskZktNrPhad7f2cxeMbMvzWxopfeWmtl8M5tjZhWZKlxERKKp9h69mZUAo4BDgeXATDOb6O6vp+z2MXA+0LeKP+ZAd/+wlrWKiMhmiHJF3w1Y7O5L3H09MB7ok7qDu69095nAV1moUUREaiHKrJsWwHspr5cD3WvwPRyYbGYO3OHuY9LtZGaDgGSDB9aa2aIafI9UzYF8/N+D6qoZ1VUzqqtmirGuNlW9ESXoLc22mvTV3MfdV5jZNsCzZvYvd3/pe39g+Acg7T8CNWFmFe6eqO2fk2mqq2ZUV82orpqpa3VFuXWzHGiV8rolsCLqN3D3FcnfVwITCLeCREQkR6IE/UygvZm1M7NSoD8QaS0sM2tsZk02fg30BBZsbrEiIlJz1d66cfcNZjYEmASUAHe7+0IzG5x8/3Yz2w6oAJoC35jZBUA54X7TBDPb+L0ecPdnsvKTfKfWt3+yRHXVjOqqGdVVM3WqLvMcLGMlIiLx0ZOxIiJFTkEvIlLkCibozexuM1tpZmkHcy24JdmmYZ6ZdU15b5MtHLJc18BkPfPMbJqZdU55L2vtISLU1cPMVie/9xwzG5HyXpyf1yUpNS0ws6/N7EfJ97L5ebUys+fN7A0zW2hmv0yzT87PsYh15fwci1hXzs+xiHXl/Bwzs4ZmNsPM5ibrujrNPtk7v9y9IH4B+wNdgQVVvH848DRh3v+ewKvJ7SXA28D2QCkwFyjPYV17Az9Mft17Y13J10uB5jF9Xj2AJ9Jsj/XzqrTvUcA/cvR5/Rjomvy6CfBm5Z87jnMsYl05P8ci1pXzcyxKXXGcY8lzZsvk1w2AV4E9c3V+FcwVvYeHrD7exC59gPs9mA5sbWY/JkILh2zW5e7T3P2T5MvphOcQsi7C51WVWD+vSgYAf83U994Ud//A3Wcnv/4MeIPwVHiqnJ9jUeqK4xyL+HlVJdbPq5KcnGPJc2Zt8mWD5K/KM2Gydn4VTNBHkK5VQ4tNbI/D6YR/sTfa2B5iloUWELm2V/K/kk+bWYfktrz4vMysEdALeCRlc04+LzNrC+xGuOpKFes5tom6UuX8HKumrtjOseo+r1yfY2ZWYmZzgJXAs+6es/OrmFaYqqpVQ21bOGSEmR1I+Eu4b8rmSO0hsmQ20Mbd15rZ4cBjQHvy5PMi/Jf6n+6eevWf9c/LzLYk/MW/wN3XVH47zSE5OceqqWvjPjk/x6qpK7ZzLMrnRY7PMXf/GuhiZlsTni/q6O6pY1VZO7+K6Yq+qlYNtWrhkAlm1gkYC/Rx9482bvcY20O4+5qN/5V096eABmbWnDz4vJL6U+m/1Nn+vMysASEcxrn7o2l2ieUci1BXLOdYdXXFdY5F+byScn6OJf/sT4EXCP+bSJW98ytTgw25+AW0perBxSP434GMGcnt9YElQDu+G8jokMO6WgOLgb0rbW8MNEn5ehrQK4d1bcd3D8x1A95Nfnaxfl7J97ci3MdvnKvPK/mz3w/ctIl9cn6ORawr5+dYxLpyfo5FqSuOcwwoA7ZOfr0F8DJwZK7Or4K5dWNmfyWM4jc3s+XAlYQBDdz9duApwqj1YmAdcGryvbQtHHJY1wigGTDaQiuIDR66021LFttDRKirH3C2mW0AvgD6ezir4v68AI4BJrv75ymHZvXzAvYBTgbmJ++jAlxOCNE4z7EodcVxjkWpK45zLEpdkPtz7MfAfRYWcqoHPOjuT1hKKxmyeH6pBYKISJErpnv0IiKShoJeRKTIKehFRIqcgl5EpMgp6EVEipyCXkSkyCnoRUSK3P8Dx7i4vwNonDQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(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",
    "\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": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-05-05T10:49:16.096063Z",
     "start_time": "2018-05-05T10:49:15.969452Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "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": "slide"
    }
   },
   "source": [
    "### Directed\n",
    "\n",
    "$A_{ij} \\neq A_{ji}, \\: A_{ii} = 0$\n",
    "\n",
    "$k_i^{in} = \\sum_{j=1}^N A_{ji}, \\: k_i^{out} = \\sum_{j=1}^N A_{ij} $\n",
    "\n",
    "$ L = \\sum_{i=1}^N k_i^{in} = \\sum_{j=1}^N k_j^{out}= \\sum_{i,j}^N A_{ij} $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## WEIGHTED AND UNWEIGHTED NETWORKS\n",
    "\n",
    "$A_{ij} = W_{ij}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ingredient-Flavor Bipartite Network\n",
    "\n",
    "<img src = './img/bipartite.png' width = 800>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Link Analysis\n",
    "\n",
    "餐馆推荐问题\n",
    "\n",
    "![image.png](images/hits.png)\n",
    "\n",
    "Credit: 李晓明"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "<img src = './img/hits2.png' width = 800>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-05-23T03:12:03.821396Z",
     "start_time": "2021-05-23T03:12:03.428264Z"
    },
    "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": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-05-23T03:12:13.114856Z",
     "start_time": "2021-05-23T03:12:13.106073Z"
    },
    "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": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-06-14T02:42:20.127177Z",
     "start_time": "2020-06-14T02:42:20.114696Z"
    },
    "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": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-06-14T02:42:17.724970Z",
     "start_time": "2020-06-14T02:42:17.715690Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'俏江南': 0.10473813736328773, '海底捞': 0.16232757223681854, '麦当劳': 0.045432407638426445, '新辣道': 0.1762008943232534, '五方院': 0.12698407527205638}\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": "2018-05-05T11:09:53.460039Z",
     "start_time": "2018-05-05T11:09:53.455822Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### PageRank\n",
    "\n",
    "<img src = './img/pagerank.png' width = 800>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-06-14T02:52:16.209972Z",
     "start_time": "2020-06-14T02:52:15.813347Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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qrzwmmThffPEFjEYjnnrqKbFL8VuMMVy5cgXHjx/H5MmTsWLFCrpK4iMo+PzM0OkMHMchLkzh6e+7lZsXYDQ7wRhD6bxM5MbIsWfPHiQmJuKJJ54Yt1mLjDF0d3ePCEO1Wj0iDOn+h+9ijOG1117DunXr6JgeL7Db7Th58iT0ej2WLFmC2bNn09l/IqPg80MdJjtO1xhx7uZ5fFKpBFKOA88Y+FvO4yvMifNc3nQ6nXjnnXfAGMPmzZsnrO9IEIQRYdje3o6wsLARYUiX1HxDQ0MDDh8+jO9///t0f8qLOjs7cfToUVitVhQXFyMzM1PskoIWBZ8fu/UEdruLh1J+7xPYBUHA4cOH0d7ejpKSEtF2agqCAKPROCwMOzo6EB4ePiwMk5KSKAxFcPDgQSQnJ2PBggVilxJwGGOoqKjABx98gMzMTBQVFVEvrQgo+IIMYwynTp3ClStXsH37dp8ZuSQIArq6ukaEYWRk5IgwpCkZ48dms+F3v/sdfvSjH9FGpXHkdDrx2Wef4auvvsLChQsxf/78gD/uyZdQ8AWpixcv4sSJE9i6dSvS0tLELueOeJ5HV1fXsBPfOzs7ER0d7TldYigMabKId5w/fx6NjY3YuHGj2KUEhZ6eHhw9ehQ9PT1Ys2YNcnNzxS4pKFDwBbGqqiq8++67eOqppzBlyhSxyxkVnufR2dk5LAy7uroQExMzLAwTExMpDMeIMYbXX38dRUVFdPjqBKuqqsKxY8eQkJCAVatWITo6WuySAhoFX5BraWnBvn37sHTpUsyePVvsch6I2+0eEYZGoxGxsbEjwpDOUru71tZWvP322/jxj39Mm1pE4Ha7cfr0aZw5cwbz5s3DwoUL6cnbOKHgI+jp6UFZWRl0Op1Xe/3E5Ha70dHRMSwMu7u7ERcXNywMExISKAxvOnz4MCIiIrB48WKxSwlq/f39+PDDD9HS0oJVq1YhPz8/IL4nfQkFHwEwOGppInr9xORyuTxhOBSIPT09iI+P97RUDIVhIH7+9+J0OvHqq6/ihRdeQEREhNjlEAC1tbU4evQoIiIisGbNGsTFxYldUsCg4CMeYvX6icnlcqG9vX1YGPb29iIhIWFYGMbHx/ttGO7eDdTXA0uXDv66k8uXL6OiogIlJSUTVxi5L57ncf78eXz22WeYOXMmFi9eTC0+XkDBR4bxlV4/MTmdzhFh2NfXh8TExBFh6A8TOEYTfG+88QYWLlyI/Pz8iSuMjJrZbMbx48dRW1uLlStXQqfT0eXPh0DBR0bw1V4/MTkcjmFh2NraioGBAU8YDgViXFycz4Xh/YKvq6sLf/3rX/GTn/zEb1e1waKpqQlHjhyBQqHAmjVrkJSUJHZJfomCj9yVP/T6iclut48IQ7PZjKSkpGFhGBsbK2oY3hp8cwu/nvZjc/JQKaSov3YF2dFyFBetFK1GMnqCIODixYv45JNPoNFosGzZMhoEP0YUfOSe/LHXT0x2u33EWYZWq/WOYThRl6p27wYM192IyOmFNbYN7pvzXCUcBzfP46q+HPlTJmNJQTIWZMchKZIGiPsDq9WKTz75BBUVFVi+fDlmzpzpc1cbfBUFH7mvQOj1E5PNZhsRhjabbVhbRUpKCqKjo8clDP/tFSs+OGtCptaC6bNdaK4KRVtdCKwmKWw2M9zoxIKiWHCRFs+JHtPTo7xeBxkfbW1tOHLkCHieR3FxMV2dGQUKPjIqgdjrJyar1ToiDB0Ox4gwjIqKeqh/6ytNfXjpv0zg+1WYPN2Ovk45ejrl4DgGmZyhubELYWFqqMPCkDvdirSpZnQOOPCtBZMo/PwIYwxXr17F8ePHkZubixUrVgTlxrTRouAjoxYMvX5islgsI8LQ5XKNCMPIyMhRhWF7vx2vfFSJ2tPxGDCGQK4QIPAc8mdbkJLrgMtpw9nTekRIlqGzafDy5qxlJkQk2dFrceJnRVM8x1oR/+BwOHDq1ClcvnwZixcvxpw5c+j79A4o+MiYBGOvn5jMZvOIMOR5fkQYRkREjAjDg5eacb6uBw1nEtDTMTj6SltoRlquAwBQX1cHt9uNnNw8XPggAj2dcoRFurHoqX6099sxPzsG62fSZTN/1NXVhaNHj8JsNmPNmjXIysoSuySfQsFHxox6/cQ1MDAwbBRba2srGGPDR7GlpOPXx2sQq1bg4vEo9HTIoVLzWLyhDxw3eGns7NmzmKbTQR0WBmOrHF8eH5zYsvDJPqgiXOi2OPHLJ6be8WxH4vsYY7h27Ro+/PBDpKWloaioiKby3ETBRx4I9fr5DsbYiDB0R2dAbwlHanQozn8QgZ4OOVKyHZi2yAwA6O7uRkN9A2Y9MgsAwPPA8b0xYIzzrApbeq3YMjsDc7JixPz0yENyuVz4/PPPceHCBSxYsAALFiwI+vm0tPeVPBCO47B06VI8+uijePPNN9Hc3Cx2SUGL4zhEREQgPz8fy5YtQ2lpKWLTciCTDv/2VoYKnj+3tbUhOfnr5mepFFAoB58DO+3czZdJ0DFgn4DPgIwnuVyOZcuW4Tvf+Q6am5vxhz/8AdXV1WKXJargjn3y0GbNmoWwsDDs3buXev18iM3JQzJiA8zNYHM60dfXh4J7jCdzOp2oqa6C0F4F1hCC0NDQu/5SKBS0y9cPREdHY9u2baiursaxY8dw4cIFrF69GjEx3l3Rv/sucPkyMGMGsG6dVz+011DwkYc2efJklJaWYt++fRgYGKBePx+gUkgh3HYXw24d3N3X3t4+OHT7lstdPP/1Sk+hZFDI5UhKTkFOnAzJCRJYrVb09/ejra0NVqsVFosFVqsVVqsVjLF7BqNarR7xMtppKJ68vDxkZWXh7Nmz+POf/4zZs2dj0aJFQbVRjYKPeEVqaiqeffZZlJWVwWQyYdmyZbQKEFFiuBJuXhj2st4OGZjA0NbWhoKCgmGv6+uUg7HB/6/IWDfAcVCqQvFIQQZm3+cen8vl8oTgrb8sFgs6Ozvv+Dq5XH7fcLz1l0qloq8nL5LJZFi0aBGmTZuGDz/8EDt37sSqVatQUFAQFP/OFHzEa2JiYvDcc89h7969MJlMePLJJ+mZvUg0qRGQXZIMCz+bRYrKKy5IJBJEhId7Xs4YUKsfnPUYFulGeDTvGWumSb3/LkC5XI7IyEhERkaOqjbGGBwOx4iV49Cv7u7uEQHqdDqhUqnGtKqUy+VB8UP8YURERGDjxo2or6/HkSNHcOHCBRQXFyM+Pv6+72t1jpz7mhiuhMMVCcC3v+9pVyfxOur18w239/HJFQLa243In23GjAWRkEoBm0WCyi9D0d4weMbbzCUDSMx0+lwfH8/zsNlsd1w93i1Ab78Ee79VZbBfghUEARcuXMCpU6cwffp0LFmyBErlyAEG7f12nK414nxdz7C5rwJjcPMCar+MgrwvBqsXK/FMqW9+71PwkXEx1OvX1taG0tJS6vUTwe2TWzILBvDV2SbEhE2BTC6BVMbgcn698zNHZ0XeTBvsLj4gJrfcfgn2TuF4+y+FQjGmVaVSqQy4VaXFYsHHH3+M6upqrFy5EtOmTfN8jlea+rDnXAOMjSoMNIXD2i8HOAZ1pIC0XDvS8hy48nkoblyTI2GSFf/fD6N9cvQdBR8ZN9TrJ75bZ3XGpTchLKkRoewRtNUpYB2QQiJliIh1Y1KBHfFpLthdfNDO6mSMwW633zMYbw/P2y/B3m9VqVarIZfLxf5UR6WlpQVHjhyBRCLBE088gXZnCN48XY/u8lh01ocCgGfmq9vFgTEOyZMckEgZWmqUSMi0Ik7X7ZNfSxR8ZNzRuX7iutLUh7JzDai8fg3avEmIjx25WcXNCzCanXQ6wxjdfgn2fqtKi8UCjuPGtKpUqVSiXYJljOHSpUto6bHgs74ImJsiUHNp8L5vxhQ7cqdboVAyuJwcGq4pUXNVBZl88EpCao4deXNNPnn1gDa3kHFHvX7imp4ehe6WWrRd7gAvK0BLrxVSqQRSjgPPGPib92nmZ8egMCfOp35A+TqpVIqwsLBRX8pnjN11F6zVakV7e/uI8LTZbJ5LsKO5T6lWqxESEjLmS7C1tbU4duwYNmzY4DnZneM4zJo1C/WXmoHeHjSWD36eKdkOTJ1n8byvXMGQO90GgedQa/j6UFylXAqO43C6xugz94sBCj4yQajXTzydnZ34lxd/gKlTp+L//eUPPDvx7C4eSvngTjxNagTN5JwAHMdBoVBAoVAgKipqVO9z+yXYW4PRYrGgq6trxKrS5XKNaVUZGhqKjo4OHD9+HF9++SXWrl2L1atXQ6lUwup043xdD1h/mOeecM406x1rzdLa0HBNCZ7/OnTjwhQ4V9eDVZokn/ka840qSFCgXr+J19jYiP/8z/9EXV0dnnjiCYQqZDR7089wHAeVSgWVSjXq++Q8z991Vdnd3Y2mpqYRL79x4waamppgt9vx61//Gm+88Qb+z//5P+CjJ8HNC7D2Dd6bVKl5qCOEOz6uXDF4z7i38+v7mDLpYFtNeYvJZ772KPjIhKJev4lTVVWFl19+GU1NTUhMTIQg3PmHFQk8UqkU4eHhCL+lX/NeGGPYu3cvBgYGPO87c+ZMJCQk4HKPHTKpBE774GovRHXvr6NbZ8J+XY9vzX2l4CMTTq1W45vf/Cbeeecd7Nu3j3r9xonRaIRCoUB1dTVyc3PR19cndknER3EcB5lMBo1Gg9WrV2PmzJlQqQbv1Z1pbxo+9/UBLtJIOQ52F++lah8eBR8RhUKhwLZt23D48GG8+eab1Os3DgoLC6FWq2EwGLB8+XL09/eLXRLxYZs3bwaAEbcfhua+KpSDKzmH9d6H+tjv8HqeMSjlvnNlh4KPiEYikeDJJ5/EqVOn8MYbb1Cv3zg4fPgwnnjiCbz00ktwu91il0N82K2B53A4cP36dZSXl8PQboMr/ZHBGa4YHH1nMUnueJ/P7eRg6h4ZKzwvIDHcd3YL03l8RFR0rt/4sdvt+OKLL7BhwwYACPrDR8m9uVwulJeXY//+/XjllVdQUVEBnU6Hl76zDXKpBJEJDsgVg2E3NNv1dnXlw3d0AhjT3NeJQt8JxCdQr5/3HT16FGlpacjJyRG7FOKj3G43ampqYDAYUF1djdTUVGi1Wqxdu9Zzjw8A5mbF4HxdD3Km2XD9SzVaapSQKRhydDYolAxuJ4f6a0rUGlSQK4Rho/CMZifmZ8f4TCsDQMFHfAj1+nnX0aNHsWbNGrHLID6G53nU1dXBYDCgsrISiYmJnk0tarX6ju9TmB2HszXdSMy1wNQjQ2ttCBquqdB4XXnXkWUAYHfxYIyhMCduIj/F+6KRZcTn9PT0oKysDFqtlnr9HlBdXR1+8IMf4ODBg3ecsE+CiyAIaGxshMFgwLVr1xAdHQ2tVoupU6ciImJ0lyCvNPVh95l6JISHoLtRhaZKJcx9sq+HVOfYkTbZAcNpNc3qJORBWCwW7N27F/Hx8dTr9wB+85vfwGaz4V//9V/FLoWIhDGG5uZmGAwGVFRUQK1WQ6vVQqPRIDo6+oE+5tDpDBzHIS5MAZl05DYRf5j7SsFHfBad6/dg3G431q9fj1/96lfQaDRil0MmEGMM7e3tMBgMKC8vh0wmg06ng0ajQVycdy43dpjsOF1jxLmb5/Hdae7rvCzfnvtKwUd8Gp3rN3YffPAB9u7di7/85S9il0ImSFdXFwwGAwwGAwRBgFarhVarRUJCwrjdKrj1BHZ/m/tKwUd8Hp3rNzY/+tGPMG/ePJSWlopdChlHPT09nrCz2+2ey5gpKSl0X/w+KPiI36Bz/e6vtbUVzz77LA4cOECr4wDU398/2FRuMMBkMmHq1KnQarVIT0+nsBsD316PEnIL6vW7v4MHD2L27NkUegHEbDajvLwc5eXl6OrqQkFBAVauXIlJkyZBIqEZJA+Cgo/4Fer1uztBEPDJJ5/g5z//udilkIdktVpx7do1GAwGtLW1YcqUKXj00UeRnZ1NO5y9gIKP+B061+/OPv/8cygUCjzyyCNil0IegN1u98zHbGxsRG5uLubOnYvc3FzI5fL7fwAyanSPj/gt6vUb7qWXXkJ+fj6ee+45sUsho+R0OlFVVQWDwYC6ujpkZWVBo9FgypQp1L4zjij4iF9zOp04cOAABEEI6l4/o9GI7du3Y+/evYiJ8Y1Trsmdud1u3LhxwzMfMz09HVqtFvn5+TRlZ4JQ8BG/JwgC3n//fbS2tgZtr98bb7yB69ev4+WXXxa7FHIHPM+jtrYWBoMBVVVVSExM9IwMCw0NFbu8oEPBRwICYwyffvopLl++jNLSUq9NqfAHgiCgpKQE3//+97F48WKxyyE3CYKAhoYGz3zM2NhYT9iFh4eLXV5Qo80tJCBwHIclS5YgPDwcu3fvxpYtW5Ceni52WRPiq6++gsvlwqJFi8QuJegxxtDU1ORpPwgPD4dWq8Xzzz+PqKgoscsjN1HwkYAya9YshIeHY9++fVi7di3y8/PFLmncvfvuu1i+fDn1dImEMYa2tjbPfEyFQgGdTodnnnmGpgz5KLrUSQJSS0sL3nrrLSxZsiSge/1MJhM2b96MXbt2ISUlRexygkpnZ6dnZBiAYfMxiW+jFR8JSKmpqXjmmWcCvtfvvffew+TJkyn0Jkh3d7dnZedwOKDRaLBx40YkJycH5NdXoKIVHwlogd7r981vfhMlJSVYtWqV2KUErL6+Ps98zIGBAWg0Gmi1WqSlpVHY+SkKPhLwbu3127RpE0JCQsQuySsMBgN+8Ytf4ODBg5DJ6OKNNw0MDKCiogIGgwHd3d0oKCiAVqtFZmYm3UsNABR8JCgEYq/ff/zHf0ClUuFnP/uZ2KUEBKvV6gm79vZ25OfnQ6PR0HzMAETBR4JGIPX62e12bNiwATt37kRWVpbY5fitofmYBoMBTU1NyMvLg1arRW5uLq2iAxj9z5KgEUi9fkeOHEFaWhqF3gNwOp2orKyEwWBAfX09srKyMGPGjKAeeRdsKPhI0AmEXr+jR4+iuLhY7DL8hsvl8szHvHHjBjIyMqDVarF+/XqajxmEKPhIUMrLy0NpaSneeustmM1mv+r1q6mpQUtLC9asWSN2KT6N53nU1NSgvLwclZWVSE5OhlarxeOPP07zMYMc3eMjQa2npwdlZWXQarV+0+v38ssvw+l04l/+5V/ELsXnCIKA+vp6GAwGXL9+HXFxcdBoNDQfkwxDwUeCnj/1+rndbqxfvx6/+tWvoNFoxC7HJwzNxzQYDKioqEBERAS0Wi00Gg0iIyPFLo/4IAo+QuA/vX5Hjx7F/v37sXv3brFLERVjDK2trZ4pKiqVytNYTucRkvuh4CPkJn/o9fvhD3+IwsJCbNu2TexSJhxjbNh8TIlE4lnZ0XxMMhYUfITcwpd7/Zqbm/Htb38bBw4c8MlQHi9Go9EzMszpdHqGQSclJfnFPVnie2hXJyG3GOr1i4iI8Llev3fffRdz5swJitDr6+vzrOwsFgs0Gg3Wrl1L8zGJV1DwEXIHM2fORFhYmM/0+gmCgBMnTuAXv/iFqHWMJ5PJ5BkZ1tPTg6lTp2L16tXIyMig+ZjEqyj4CLmLvLw8bN++Hfv27RO91+/TTz9FSEgIZs2aJVoN48FisXjCrrOzE1OmTMHSpUuRlZXl07triX+j4CPkHlJSUnziXL/33nsPRUVFAbHysdlsnvmYLS0tyMvLQ2FhIXJycmg+JpkQtLmFkFEQs9fPaDRi+/bt2Lt3r99u1Xc4HKiqqvLMx8zOzoZWq8XkyZMhl8vFLo8EGQo+QkZJrF6/P//5z6iqqsKvf/3rCXk8b3G5XKiurobBYEBNTQ0yMzOh0WiQn5/vs32SJDhQ8BEyBhPd6ycIAkpKSvBP//RPWLRo0bg+ljcMzcc0GAyoqqpCSkoKtFotCgoKoFKpxC6PEAAUfISM2UT2+l24cAG/+tWv8M477/js/T1BEFBXV+eZjxkfHw+tVoupU6cGResF8T90J5mQMZrIXr93330XK1as8LnQY4yhsbHRMx8zKioKWq0WS5cupfmYxOfRio+Qh1BdXY1Dhw6NS6+fyWTC5s2bsXv3biQlJXn1Yz8IxhhaWlo8YadSqTwjw/x10w0JTrTiI+QhjEevX1tbG0JCQnD06FFMmTJF1NBjjKGjo8MzRUUmk0Gr1WLHjh2Ij48XrS5CHgYFHyEPydu9focOHcLp06dRWVmJ73znO3C73RPe39bV1eWZj+l2u6HVarF161YkJibSyDDi9yj4CPGCmJgYPPfcc9i7dy9MJtND9fpFRkbCbrejq6sLx48fR39/P1566SUvVzxSb2+vZ2VntVqh0Wiwbt06pKamUtiRgEL3+AjxoqFeP57nsXnzZigUCly/fh2TJ08edRB+8MEH+PnPf47Q0FDMmDEDP/vZz5CdnT0u9ZpMJs/Krq+vD1OnToVWq0VGRgaFHQlYFHyEeNmtvX4ZGRkoKyvDP//zP2PatGmjev+TJ09iy5YtWLt2Lf7t3/4NycnJXq3PbDajoqIC5eXl6OzsRH5+PrRaLbKysnxu9ygh44EudRLiZRKJBE888QR27dqFX/7yl5gyZQo+/vjjUQff0JSTX//614iOjvZKTTabDdeuXYPBYEBraysmT56MhQsXIicnh4ZBk6BDwUfIOGhsbMSpU6eQm5uLjo4OnDlzBjt27BjR7G51ulHeYkLHgB02Jw+VQopuSTT+8Kc/P3ToORwOVFZWwmAwoKGhATk5OZg9ezby8vJoPiYJahR8hIwDmUyGRYsW4fLly3C5XDAYDDhw4AC+973vAQDa++04XWvE+boeuHkBMqkEEo6DwBic6iwcqAWaWDMWZMchKVI54uO7XC5wHDdit6fL5fIMg66trUVmZia0Wi2efvppmo9JyE10j4+QcSQIAlpbW3HmzBmYTCY899xzOF/dh/99rwN9bSGATQmXXQqB56BQCohKcCMz346wGCeMZicYYyidl4np6VGej2m1WvHKK69Aq9Vi3bp1cLvdnvmY1dXVSE1NhVarRX5+Ps3HJOQOKPgImUBXmvrw7382ovdGJGTSwY0kMhkDYwDPD+6i5DiG/NlWZBbYYXfx6Bxw4FsLJmF6ehSsVit++9vfwmAwICQkBGvXrkVVVRUSEhI88zHVarWYnyIhPo8udRIyQdr77dhzrgGJMRGI1TqQNMkJdSQPuWIw+GxmCRquK9F4XYnrX4YiOsGFiFggITwEe841IEIu4H9f+RUuXrwIYHDDCmMML7zwAiIiIkT+7AjxHxR8hEyQ07VGSCQccjUuAK5hr+M4IDRcQMEcK5jAobFSicZKJbSFFijlUnAch//adRBXTp1CXFwcoqKiYDabwfM8hR4hY0TBR8gEsDrdOF/Xg1i14r5vG5/qRGOlEr2dX++8jAtTAFPm4P8e2QiHxYTe3l50d3fTSQiEPAAKPkImQHmLybN7EwCsAxI0VSrR3S6HzSyB28WBseGTUhzWr5vJZVIJeIGh0SRgTlYGMjIyJrR+QgIJBR8hE6BjwO4JvY4GBa5+HubZzAIAMrkAiZSB4wCBB1xOCdzu4UEolUrQMWCf0LoJCUQUfIRMAJuTh4Tj4LRz0J9Wg+c5xCa5kDPNisg4N6S3fCd2t8lx4aOR9+2kHAe7i5/AqgkJTBR8hHiR2+2Gw+FAaGjosCHPKoUUAmPoapHD7ZJArhAwa7lpWOANcdjuPByaZwxKOY0XI+RhUfAR4kVffPEF3njjDSiVSsTFxUEqlWLSpEmYtnw93LwAh3UwuNQR/B1DDxhc8d0JzwtIDB85xYUQMjY0ip0QL8rJyYHFYkFvby9OnTqFo0ePQiqVQpMaMXiPTzp4qdJikoK/w1VLU48UbXUjR4sNbYzRpFLrAiEPi1Z8hDwkxhhaW1uh1+uh1+ths9mgVquRlpaGb33rW1i9ejU4jsPcrBicGugHx4XB5ZTg6mdhKJhjhVItQOCBziYFKs6rIZMzOB3DL3cazU7Mz45BqIK+ZQl5WPRdRMgD6urqgl6vh8FgAMdx0Ol0eOaZZzBr1izs3LkT3/jGNzyhBwCF2XE4W9ON1CkWNF8PQ0djCDoaQyBXCODdHASBQ2gYj9yZVlz9LNzzON19JjicDIU5cXcrhRAyBhR8hIxBf38/DAYD9Ho9LBYLtFotNm7ciOTkZE/AzZ07F0qlEoWFhcM2uCRFKlE6LxO7hXpMieTRXqOGuXdwQHVoOI+EdCeytHYM9Hy9gcXu4nFBfx1dXxxAxPVpWLhwIfLz85GWlkbn6BHygGhINSH3YbVaUVFRAb1ej87OTkydOhU6nQ4ZGRkPdGL5laY+/PVMHdrb2jAjP9vT33crNy94TmdYMSkEL//8n8DzPFJSUiAIAr797W9jxYoV3vj0CAk6tOIj5A6cTieuX78OvV6PxsZG5OXlobCwELm5uQ+90pqeHoXVDS4c77ai2+KEmxcglUog5TjwjIG/uZFlfnYMCnPikBAegi8KC1FZWYmamhosX74cCxcu9NJnSkjwoeAj5Cae53Hjxg3o9XrcuHEDGRkZ0Ol02LRpExSK+8/YHIv6a1fw7ZVLkZqZ5TmB3e7ioZRLkRiuhCY1YthGllWrVsFgMECn00GlUqGrqwvp6elerYmQYEHBR4KaIAhoaGiAwWBARUUFEhISoNPpUFxcjNDQ0HF5zPb2dpjNZuTk5EAikWBOVsx932fGjBlYtmwZSkpK0Nvbi3379mHdunWYPHnyuNRISCCje3wk6DDG0NbW5tmRqVarodPpoNVqJ+S0gyNHjiA0NBRLly594I/R3NyMt956CytXrsSMGTO8VhshwYCCjwQNo9Ho2ZHJGPOEXXx8/ITV4HK58Morr+C73/0uoqKiHupjGY1GlJWVYfbs2Vi4cOGwHaSEkLuj4CMBzWQywWAwwGAwwGQyQavVQqfTISUlRZSguHr1Kq5evYrt27d75eOZTCaUlZUhJycHRUVFFH6EjAIFHwk4NpvN037Q0dGB/Px86HQ6TJo06YHaD7xp9+7dmDt3LqZOneq1j2mz2bBv3z5ERkZi3bp11N9HyH1Q8JGA4HQ6UVVVBb1ej/r6euTm5kKn0yE3NxcymW/s4eru7sauXbvw05/+1Ovh5HK5cODAAfA8j82bN3t9FyohgYSCj/gtnudRU1MDvV6P6upqpKWlQafTIT8/HyEhIwc9i+348eMQBAFFRUXj8vEFQcB7772Hzs5OlJSUQK1Wj8vjEOLvKPiIX2GMobGxEXq9HhUVFYiLi4NWq4VGo/HpH/Q8z+PVV1/Ft771LcTFjd/MTcYYTpw4gYqKCuzYseOhN9AQEoh84xoQIffAGEN7e7tnR6ZKpYJOp8Pzzz/vNz/Yq6urERMTM66hBwAcx2HFihVQq9XYtWsXSktLkZiYOK6PSYi/oRUf8Vk9PT2eo37cbjd0Oh10Oh0SEhLELm3M9uzZA41GM6E9d3q9HseOHcOWLVuQkZExYY9LiK+jFR/xKQMDAygvL4der0dfXx+0Wi3WrVuH1NRUv92q39/fj+bmZmzevHlCH1en0yE0NBT79+/H2rVrMWXKlAl9fEJ8Fa34iOhsNhuuXbsGvV6PtrY2T/tBVlaW6O0H3nDq1CmYzWY8/vjjojx+S0sL9u3bh+XLl2PWrFmi1ECIL6EVHxGFy+XytB/U1dUhOzsbc+bMQV5eHuRyudjleQ1jDJcuXcKWLVtEqyE1NRXPPPMM/va3v8FisWDRokV+u3omxBso+MiE4XkedXV10Ov1qKysRGpqKnQ6HdatWwelUil2eeOitrYWKpUKycnJotYRGxuL5557DmVlZTCbzcNOhick2NClTjKuGGNoamrytB9ER0dDp9NBo9EgLCxM7PLG3TvvvINJkyZhzpw5YpcCALDb7di3bx/Cw8Oxfv16mvJCghIFH/E6xhg6Ozs9px/I5XJMmzYNWq0W0dHRYpc3YSwWC37/+9/jJz/5iU+taF0uF/7+97/D6XRiy5YtPtnsT8h4ouAjXtPb2+vptXM4HMPaD4LxstqZM2fQ3t6O9evXi13KCIIg4P3330dbWxtKS0t9uvmfEG+j4CMPxWw2e9oPent7MXXqVOh0OqSnpwdl2A1hjGHnzp148sknkZmZKXY5d8QYw8mTJ2EwGLB9+/agWo2T4EabW8iY2e12XL9+HXq9Hi0tLZgyZQqWLl2KrKwsumd0U1NTEwD4dOM4x3FYtmwZ1Go13nzzTZSUlCApKUnssggZd7TiI6PidrtRVVUFg8GAmpoaZGVlQafTYfLkyQHVfuAt7777LhISElBYWCh2KaNSXl6OI0eOYNOmTZg0aZLY5RAyrmjFR+5KEIRh7QfJycnQarV48sknoVKpxC7PZw2tiB977DGxSxk1jUYDlUqFt99+G08++SQKCgrELomQcUPBR4ZhjKGlpQV6vR7l5eWIjIyETqfDihUrEB4eLnZ5fsFgMCA7O9vvNoxkZ2dj+/bt2LdvH6xWKx555BGxSyJkXNClTgIAw9oPpFIpdDodtFotYmNjxS7N7/zpT3/C8uXLkZubK3YpD6S7uxtlZWWYMWMGFi9eHNSblEhgohVfEOvr6/O0H9hsNuh0OmzevBlJSUn0w+4BtbW1wWKxIDs7W+xSHlhsbCyeffZZ7NmzBxaLBatXrw6ImamEDKEVX5CxWCyoqKiAXq+H0Wj0tB9kZGRQ2HnBkSNHEBoaiqVLl4pdykOz2+146623oFarsX79eshk9DyZBAYKviDgcDg87QfNzc3Iy8uDTqdDTk4OtR94kcvlwiuvvILvfe97iIyMFLscr3C73fj73/8Ou92OrVu30pQXEhAo+AKU2+3GjRs3oNfrcePGDUyaNMnTfqBQKMQuLyBduXIFer0e27dvF7sUrxIEAUeOHEFLSwtKS0uDYsYqCWwUfAFEEATU19fDYDDg2rVrSExMhE6nQ0FBAUJDQ8UuL+C9+eabmD9/fkC2AjDGcOrUKVy9ehU7duygKS/Er9FFez/HGENra6un/SAsLAw6nQ4vvPACIiIixC4vaHR3d6O7uxuTJ08Wu5RxwXEcli5dCrVajV27dqG0tJSmvBC/RSs+P9XV1eXZkclxnKf9IC4uTuzSgtJHH30EAH7VtP6gKioq8P7772Pjxo3IysoSuxxCxoxWfH6kv78fBoMBBoMBZrMZWq0WGzduRHJyMu3IFBHP87h8+TKeeeYZsUuZEFOnToVKpcKBAwfw+OOPY+rUqWKXRMiYUPD5OKvV6mk/6OzsREFBAYqKipCZmUm9VT6iqqoKcXFxQbXazsrKwvbt27F3715YrVbMnj1b7JIIGTUKPh/kdDpRWVkJvV6PhoYG5OXlobCwEDk5OdRL5YMuXryIWbNmiV3GhEtOTsYzzzyDsrIymM1mLFmyhK48EL9A9/h8BM/znvaD6upqZGRkQKfTYcqUKdQ75cP6+/vxxz/+ET/96U+D9pQKs9mMPXv2IDU1FcXFxXQlgvg8Cj4RMcbQ0NAAvV6Pa9euIT4+HlqtFlOnTvW7AcfB6uTJk7BYLHj88cfFLkVUDocDb731FpRKJZ5++mm6MkF8GgXfBGOMoa2tzbNJJTQ01LMjM1CmfQQLQRDwu9/9Dtu2baOt/RgcmnDw4EFYrVZs3boVSqVS7JIIuSMKvgnS3d0NvV4PvV4PQRCg0+mg0+kQHx8vdmnkAd24cQMnTpzA888/L3YpPkMQBBw9ehRNTU3Yvn07TXkhPomCbxyZTCaUl5dDr9fDZDJBq9VCq9UiNTWVNgEEgLfffhvZ2dm0o/E2jDF8+umnuHz5Mnbs2IGYmBixSyJkGLoQ72U2mw0VFRUwGAxob29Hfn4+Vq5ciUmTJtFN/wBisVhQW1uLtWvXil2Kz+E4DkuWLEFYWBjefPNNlJSUIDk5WeyyCPGg4PMCp9OJqqoq6PV61NfXIycnB3PnzkVeXh7d5A9QV65cQX5+Pt3HuodHHnkEoaGhKCsrw9NPP+3XZxSSwEI/lR8Qz/Oora2FXq9HVVUV0tLSoNPpsGHDBmo/CHCMMVy8eJFWe6NQUFAAlUqFd955B8XFxdBoNGKXRAgF31gwxtDY2Ai9Xo+KigrExsZCp9OhqKiIbuIHkcbGRgBAenq6yJX4h0mTJmHHjh2eE93nzp0rdkkkyPnd5hZBAM6dA65cAbq7AZdr8OVbtwL5+d5/PMYYOjo6oNfrYTAYEBIS4mk/oKNZgtOhQ4eQmJiIwsJCsUvxK729vSgrK4NGo8GyZctogxcRjd+t+I4dA86fH/yzVAoMLbS8fSutp6fHc/qBy+WCTqdDSUkJEhMTvftAxK/Y7XZUVlaiqKhI7FL8TnR0NJ599lnPyu/xxx+nDV9EFH6z4rM63bhUa8L//l4Gh4thzkIXlj4qgTYtAqEK76TewMCAp/2gr68PGo0GOp0OaWlp9OyUAAAuXLiA+vp6bNq0SexS/JbD4cD+/fuhUCiwceNG2gBGJpzPB197vx2na404X9eD3i4pyk/EgQMwf0MnmISHTCrBvKwYLMiOQ1Lk2HfY2e12XLt2DXq9Hq2trZgyZQp0Oh2ys7Pp2SgZ4fXXX8fKlSuRk5Mjdil+jed5HDp0CAMDA9i2bRvtjiUTyqefal1p6sOecw2QSDjEqhWQWxWoDxksOSV2cOekmxdwtrYHZ2q6UTovE9PTo+77cV0uF6qqqmAwGFBbW+tpQs7LywvaQcPk/tra2mCz2WhbvhdIpVI8/fTTOHbsGN58801s374d4eHhYpdFgoTPrviuNPVh95l6JISHoLshFPrTd981GZPowrTlvegccOBbCybdMfwEQfC0H1RWViIlJQU6nQ4FBQX0bJOMyvvvv4+wsDAsWbJE7FICBmMMn3/+Ob766ivs2LEDsbGxYpdEgoBPrvja++3Yc64BCeEhUMqlkEgZQlQCBB5wOQcvP4aoBM/by0MEKOVSJISHYM+5BiRFKpEYoQRjDM3NzdDr9SgvL0dUVBR0Oh1WrlxJzy7JmLhcLhgMBnzve98Tu5SAwnEcHn30UajVas+Ul5SUFLHLIgHOJ4PvdK0REgkHpVwKAEjOciI5y4medhnOfzh4gsGyTb0j3k8pl4LjOBy9WIs0dysMBgNkMhmmTZuG5557jmYGkgdWUVGBtLQ0OkFjnMyaNQuhoaHYs2cPNmzYQPdQybjyueCzOt04X9eDWLXigd7f1tuBdwydeGFePLZu3YrExETakUke2ldffYUFCxaIXUZAy8/Ph0qlwttvv401a9ZAq9WKXRIJUD4XfOUtJrh5ATLpg+2ozMrMgDwyHolTMpCURCs88vCMRiN6enowefJksUsJeJmZmfjGN77h6fWbN2+e2CWRAORz+/U7BuwPHHoAAI6DTCpBx4Dde0WRoHbx4kXMmDEDUqlU7FKCQmJiIp599lmcP38eH3/8MXx0/x3xYz4XfDYnD8lDXpqUchzsLt5LFZFgxvM8rly5gpkzZ4pdSlCJiorCs88+i9raWrz33nsQBOH+70TIKPlc8KkUUggP+QyPZwxKmQTd3d24ePEi9u7di//5n/+hZ45kzCorKxEfH0/b7EWgVqvxzW9+E/39/di/fz9cQ4N5CXlIPnePLzFcCTf/cM/u+vr68Zv//B3kfU0ICwtDSEgIMjIyaJMLGbOLFy9i1qxZYpcRtBQKBUpKSvDuu+/ib3/7G7Zt2waVSiV2WcTP+dyKT5MaAZlU8sDh5+YFREZEYP3SwaNPHA4Hbty4gY6ODhw+fBhXr15FX18frf7IffX19aGlpQUFBQVilxLUpFIpNmzYgJSUFLz55pswmUxil0T8nM+t+EIVMszNisH5uh4kRox9oorR7MSCnFis3/hdzJ01HX/605+gVquxbds2REVFobKyEh9++CEkEgkyMjI8vxISEmg2Jxnm8uXL0Ol0NMbOB3Ach1WrVuGLL77Arl27sH37dsTFxYldFvFTPhd8AFCYHYezNd2wu3hPE/to2F08GGMozBn8hpg/fz4SEhKwe/duzJ49G7GxsZg3bx4YY+jr60NjYyMaGxtx4cIFmEwmpKene4IwNTWVfuAFMUEQcOnSJWzbtk3sUshNHMdh0aJFUKvV2L17N7Zt24bU1FSxyyJ+yC9mdQ6F362TW1Z/o3vY29td/F1ndTLG7nt/z2q1oqmpyROGHR0dSEhI8ARheno61Gq19z5B4tOqq6vxySef4Pnnnxe7FHIHlZWV+Mc//oENGzYgNzdX7HKIn/HZ4AO+Pp2B4zjEhSlg6lKMCD43L8BodoIxNurTGUbD5XKhtbXVE4RNTYMbZW69PBodHU0bZgLU/v37kZubi0ceeUTsUshdNDY2Yv/+/Vi1ahWmTZsmdjnEj/h08AFAh8mO0zVGnKvrgZsXIJVKIOU48IyBvznhZV5WDApz4h7onuBoCYKArq4uTxA2NDRAEIRhQZiUlET3CQOA2WzGa6+9hhdffBEhISFil0PuobOzE2VlZViwYAGNlCOj5vPBN8TqdKO8xYSOAbvn3l9iuBKaVO+dwD5W/f39niBsbGxEX18fUlNTPUGYlpYGheLBZo4S8XzxxRcwGo146qmnxC6FjEJ/fz/+9re/YcqUKVi5ciVdhSH35TfB5w9sNtuw+4Tt7e2Ii4sbdp+QjkPybYwxvPbaa1i3bh3S09PFLoeMktVqxd69exEXF4e1a9fSlRdyTxR848jtdo+4T6hSqYZdHo2NjaVnqD6koaEBhw8fxve//336f/EzTqcT77zzDjiOw6ZNm2hXNrkrCr4JxBgbdp+wqakJDodjWBAmJyfTMGQRHTp0CElJSXS/yE/xPI9//OMf6O3tRUlJCU15IXdEwScyk8nkuTza0NCAnp4epKSkDLtPqFSO36Yd8jW73Y7f/va3+NGPfoTQ0FCxyyEPiDGGjz76CNXV1dixYwciIiLELon4GAo+H2O329Hc3OxZFba2tiImJmbYqpC+kcfHhQsXUF9fj02bNoldCvGC06dP49y5c9i+fTvi4+PFLof4EAo+H8fzPNra2obtHlUoFMOCMD4+nu5HPSTGGF5//XUUFRUhOztb7HKIl1y5cgUfffQRtm7dirS0NLHLIT6Cgs/PMMbQ3d09LAhtNtuwcWspKSmQyXxyGp3Pam1txdtvv40f//jH9CQiwFRXV+PQoUNYv3498vLyxC6H+AAKvgBgNpuHBaHRaERSUtKwNgq6yX9vhw8fRkREBBYvXix2KWQcNDc346233sJjjz2G6dOni10OERkFXwByOp3D7hO2tLQgMjJy2OXRyMhIWtnc5HQ68eqrr+KFF16g+6cBrKurC2VlZZg3bx4KCwvFLoeIiIIvCAiCgPb29mGrQjqW6WuXL19GRUUFSkpKxC6FjLP+/n6UlZUhLy8Pjz32GD35C1IUfEGIMYbe3t5hQTgwMBB0xzL96U9/QnR0NBobG1FcXEwHzgYJm82GvXv3IiYmBmvXrqW+2SBEwUcAABaLZcSxTImJiQF9LNO//uu/4tq1a2hsbMQTTzyBp59+GlqtVuyyyARwuVx45513wBjDpk2baKZukKHgI3fkcrnQ0tLiCcLm5uaAO5bpN7/5DT777DMolUowxvDcc8+hqKhI7LLIBOF5Hu+99x6MRiNKSkpoaEEQoeAjoyIIAjo7O4ddHvX3Y5l27dqFnTt3Ijc3Fz/4wQ9oR2cQYozh+PHjqKysxI4dOxAZGSl2SWQCUPCRB8IYG3EsU39/v18dy7Rz506UlZXhD3/4A2bOnCl2OUREZ86cwdmzZ1FaWoqEhASxyyHjjIKPeM39jmXKyMhAWFjYhNd161mONicPlWLwLEfW2wi4nZg7d+6E10R8z9WrV/HBBx9g69atdCRVgKPgI+NG7GOZ2vvtOF1rxPm6Hrh5ATKpBBKOg8CY5+/zsmKwIDsOSZE0CJwAN27cwMGDB7Fu3TpMnjxZ7HLIOKHgIxPm9mOZGhsb4XQ6x+VYpitNfdhzrgESCYdYtQIy6ch7j25egNHsBGMMpfMyMT096qEfl/i/oSkvK1euxIwZM8Quh4wDCj4iKpPJNCwIbz+WKT09HSEhIWP6mFea+rD7TD0SwkOglA+GqM0iQeN1Jbpb5bCaJWAChxCVgLAoHtGpNnAxJjy7aBKFHwEAGI1GlJWVYfbs2Vi4cKFf714mI1HwEZ/ysMcytffb8cpHlYhWKzyh11KjQMXZMPD84A8viYRBKmNwOb9eBT6yqhvuEBt+VjQFiRF02ZMMPikrKytDTk4OioqKKPwCCAUf8WljPZbp4KVmnK/r8YRXV7McFz8JB2McohNcmDzLiqh4NzgOcDs5mHqlaKsNQWaBHWbOgvnZMVg/k46vIYNsNhv27duHyMhIrFu3jqa8BAgKPuJX7nUsU0JKOo52hCIpSg2ZVAJBAD5/NwpWsxTRCS7MecwEyT1+brl5Ad0WJ375xFSEKuhYJzLI5XLhwIED4Hkemzdv9ukWHTI6FHzE7w0MDKCpqQkfX61HuT0SU9IGT9s2tsrx5fHBy6ILivsQGcff92O19FqxZXYG5mTFjGvNxL8IgoD33nsPnZ2dKCkpCbjxfcGGntYSvxceHo6pU6ei1h2FlhtGz8v7uga/vENUwqhCDwCkUgk6BuzjUifxXxKJBGvXrsWJEyewa9cu7NixA1FRUcPe5m79oprUCLqC4GPof4MEDJuTh+SWDQgO2+DmFZV6dKEHAFazGYf+71nUnrRDoVBAoVAgJCRk2O93+/OdXuZPI9zIvXEchxUrVkCtVmPXrl0oLS1FYmLi/ftFL1G/qK+h4CMBQ6WQQnjIK/ehajWWFK3EspxIOJ1OOBwOOJ3OEX+2WCzo7e296+uH/iyVSkcdnKMJWdpcIb758+dDrVbjr3/9K2YXbcDRajM4joOjPRxtNSqY+2QAx6COFJCWa0dMtg1/e8uN/67rw8aicLz4bbpMKjYKPhIwEsOVcPOC5+8hqsE/2yyjDwteYMiMj0Js7MPf42OMweVy3TMYh3632Wzo6+u76+uH/iyVSh8qOG9/PQXpg9HpdGhzKnDgqhEpMRGoPheJtvrBflOOY5DJAVO3FOXGMPR0yBEVytAvl+JEZSeWN0VTv6jIKPhIwNCkRkB2SeK53BSd4AYweMmz3yi9732+offTpN69T3AsOI7zBI43MMbgdrvvG6JDQdrf33/PEHU6neA4zmshqlAoIJMFx4+U9n47jtfZkRYXgdZKtSf0JhXYkK2zQaFkcDs5NFYqUX1ZBZmcQSaVIEopw55zDUiKVFK/qIiC46uUBIVQhQxzs2I8fXzRiS6EhvGwmqW4/qX6vu0MRrMT87NjfHYjAsdxkMvlkMvlXtlVyBgDz/P3DMahPzscDphMpvuuXofC3psrUjEbx1tbWxEVFTXirL7TtUZIJBzknBS1ehUAIC3Xjvw5Vs/byBQM2TobBAG4cWXw/eUyCTiOw+kaI/WLisg3v8MJeUCF2XE4W9MNu4uHUi5FwVwLLn4Sjt5OOS58FDGygb1HiuZqJVIKzGAyhsKcOLE/hQnDcRxkMhlkMpnXtuffa0V6e0gODAzcd/XKGBt1cI4mTMcapLt27UJbWxtKSkowf/58SKVSWJ1unK/rQaxaAWOzAm7X4AambJ3tjh9jUoEddQaVZ3JQXJgC5+p6sEqT5LNPsgId/auTgJIUqUTpvEzPrM74NEBbaEbF2TD0dspx7lgkpFIGifTrkWVuXoA81YkfrMmky08PaShIvXWa+a0r0vtdtrVYLPddvQqCMKbgrK6uhtPpxMsvv4z09HSUlpZCmjjZc1nc1DN4CUGl5hEaLtzxc5ApGCJj3ejplA/+XTp4Ob68xUT9oiKh4CMBZ3p6FL6FSdhzrgEcxyFxkoCYRDfqrw0OqbZbpBB4Dkq1G0xpR3KaHd99LI02HPggqVSK0NBQrwbpaEJ0KEi7u7shk8lgsVhw5swZNDc345v/z+88p304HYO/D22kupuQ0OGvp35RcVHwkYA0PT0KSZFKnK4x4lxdD9y8HdFTrIjL58AzBv6W8/gKc2ilFyykUilUKhVUKtWo3v7QoUOw2+2YN28eHn/8cWg0Ghy81Pp1v+hQ98wYb0NKOQ521+j7S4l3UfCRgJUYocT6mWlYpUnyTNQYuvdHEzXIaGzduhW5ublIT0/33Bu8tV9UoRxcyQ0NS7gbh3X463nGPKeHkIlH3/Uk4IUqZHQvhTyQ5cuXj3jZrf2iETGDqzabWQqbWQJV2MhLnm4X0N89/EctzwtIDKerDGKheUqEEDIGmtQIzwaV2BQnZPLBsBtqa7hdfcXXOzoB7/eLkrGj4COEkDEY6hfttjghkwPZ2sFNKk3VSlR+FQqnYzDk3E4OtQYlaq6qIFd8vRI0mp2Yl+W7/aLBgP7lCSFkjG7tF52kscHUI0V7QwjqylWor1BCJmdwuzgwxiEl2wEAaK0NAc8EMBZc/aK+iFZ8hBAyRkP9op0DDjh5HtMXm6FdYEZknBtSKcAYEBnrhnaBGdMWmeF2cnDzAsxuB0rn0S5isdGKjxBCHsDt/aJJ2QLS8hwj3s7lFtDSyuBw8djyaAKmp9O9PbFR8BFCyAMa2S8qQCqVQMp93S/a0xSK+JAwZKaqsKZQLnbJBBR8hBDyUBIjlOBr0rApLwlWuQn97sF+UeaSoa9RjSaTGpJkKWbOBMLDxa6WABR8hBDy0G7cAAwGGYAYyOWAVArYb04kkwDIzARWrxazQnIrjrGHPLKaEEKC3JUrQHU10N4OWCyA0wkolUBSEqDVAtOmDYYh8Q0UfIQQQoIKtTMQQggJKhR8hBBCggoFHyGEkKBCwUcIISSoUPARQggJKhR8hBBCggoFHyGEkKBCwUcIISSoUPARQggJKhR8hBBCggoFHyGEkKBCwUcIISSoUPARQggJKhR8hBBCggoFHyGEkKBCwUcIISSoUPARQggJKhR8hBBCggoFHyGEkKDy/wN0GFiziDl42QAAAABJRU5ErkJggg==\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": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-06-14T02:53:29.026829Z",
     "start_time": "2020-06-14T02:53:28.988464Z"
    },
    "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": "2018-05-05T12:45:45.265498Z",
     "start_time": "2018-05-05T12:45:45.261530Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "流量作弊\n",
    "\n",
    "<img src = './img/pagerank_trap.png' width = 800>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-06-14T02:57:37.082611Z",
     "start_time": "2020-06-14T02:57:37.072015Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 0.061818139738485794,\n",
       " 1: 0.11515299000945312,\n",
       " 2: 0.11003408408427637,\n",
       " 3: 0.10714737750289349,\n",
       " 4: 0.10584740866489115,\n",
       " 5: 0.10584740866489115,\n",
       " 6: 0.10714737750289349,\n",
       " 7: 0.11003408408427637,\n",
       " 8: 0.11515299000945312,\n",
       " 9: 0.061818139738485794}"
      ]
     },
     "execution_count": 20,
     "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": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "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",
    "\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": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cycle  环\n",
    "A path with the same start and end node. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "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",
    "\n",
    "> A disconnected graph is made up by two or more connected components.   \n",
    "\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",
    "<img src = './img/block.png' width = 600>"
   ]
  },
  {
   "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！结构“坑”"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "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",
    "> 万维网的蝴蝶结模型🎀 bowtie model\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Clustering coefficient 聚集系数\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "what fraction of your neighbors are connected? Watts & Strogatz, Nature 1998.\n",
    "\n",
    "节点$i$的朋友之间是否也是朋友？\n",
    "\n",
    "Node i with degree $k_i$  节点i有k个朋友\n",
    "\n",
    "> $e_i$ represents the number of links between the $k_i$ neighbors of node i.\n",
    "\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",
    "\n",
    "$C_i = \\frac{2e_i}{k_i(k_i -1)}$\n",
    "\n",
    "$C_i$ in [0,1]\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 节点的**聚集系数**\n",
    "\n",
    "<img src = './img/cc.png' width = 500>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Global Clustering Coefficient 全局聚集系数（i.e., Transtivity 传递性)\n",
    "\n",
    "> triangles 三角形\n",
    "> triplets 三元组\n",
    "\n",
    "- A triplet consists of three connected nodes. \n",
    "- A triangle therefore includes three closed triplets\n",
    "- A triangle forms three **connected triplets** \n",
    "- **A connected triplet** is defined to be a connected subgraph consisting of three vertices and **two edges**. \n",
    "  \n",
    "$C  = \\frac{\\mbox{number of closed triplets}}{\\mbox{number of connected triplets of vertices}}$\n",
    "\n",
    "$C = \\frac{3 \\times \\mbox{number of triangles}}{\\mbox{number of connected triplets of vertices}}$\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\nf/11OnfuzIwZM8jPzzcdSxyYiliMy8rKYt68edSrV4/IyEh27drFihUr8PLyMh3NLjnyiPhCPXv2\nJCoqio0bN9KxY0eOHTtmOpI4KBWxGJOdnc2iRYvw8vJi586dbNmyhXXr1uHt7W06ml1TEf9PrVq1\niIiIoEWLFvj5+bF582bTkcQBqYilxOXm5rJs2TIaNmzI+vXr+eijj9i0aRO+vr6mozkER5+avpST\nkxMTJ07kww8/ZMiQIQwdOpSsrCzTscSBqIilxOTn57NmzRq8vb1ZsmQJ4eHh7Nixg4CAANPRHIpG\nxAVr2bIl8fHxHDt2jGbNmnHgwAHTkcRBqIil2FmtVjZs2ECTJk2YM2cOb7/9NhEREbRq1cp0NIek\nIr6yihUr8uGHHzJs2DBat27NO++8o80jpNhpP2IpNlarla1btxISEkJeXh5TpkyhS5cuWCwW09Ec\n2p133snevXu58847TUcp1Q4dOkTv3r2pXbs2ixcv1gIyUmw0IpZisWvXLlq0aMGoUaMYO3YsMTEx\ndO3aVSVcCtj7pg9FpWHDhuzdu5c6derg6+vLF198YTqS2CmNiKVI7dmzhwkTJvDTTz8xceJEevXq\nRdmyZU3Hkr9YrVacnJw4d+6c9mm+Dtu2bWPgwIEMGDCAiRMnUq5cOdORxI5oRCxFIjo6mk6dOtGn\nTx+Cg4M5ePAgffr0UQmXMllZWTg7O6uEr9Ojjz5KXFwcsbGxtGrVih9++MF0JLEjKmK5Kfv27SMw\nMJDAwEC6d+9OUlISAwcO1Bd9KaVp6RtXvXp1Pv30U3r27ElAQACrVq0yHUnshIpYbsjBgwd56qmn\n6NChA61bt+bw4cMMHjwYZ2dn09HkKnTG9M0pU6YMw4cPZ8eOHUyePJl+/fpp8wi5aSpiuS5Hjhyh\nX79+tGrVCl9fX5KTkxk+fDiurq6mo0khqIiLhq+vLzExMTg7O+Pr60tUVJTpSGLDVMRSKEePHmXQ\noEEEBARQt25dkpOTGTNmjL7UbYxW1So6t9xyC++99x7Tpk2jS5cuvPHGG9o8Qm6Iiliu6tixY7z0\n0kv4+vpSpUoVkpKSCA0NpUKFCqajyQ3QiLjoPfnkk0RHR7N582Y6dOjAr7/+ajqS2BgVsRQoNTWV\nUaNG4e3tTbly5Th48CBTp06lUqVKpqPJTVARF4+7776bzz//nFatWuHn58emTZtMRxIboiKWi6Sl\npTF+/HgaNmxIRkYGCQkJzJo1i2rVqpmOJkVAU9PFx8nJiZCQENavX89LL73Eiy++SGZmpulYYgNU\nxALA6dOnmTJlCl5eXvz222/ExMQwf/58atasaTqaFCGNiItfixYtiI+PJzU1lWbNmvH999+bjiSl\nnIrYwWVkZDBjxgy8vLxITEzkm2++YfHixdSuXdt0NCkGKuKScdttt7FmzRpeeeUV2rRpw8KFC7V5\nhFyRithBZWVlMW/ePOrVq0dkZCS7du1ixYoVeHl5mY4mxUhT0yXHYrEwYMAA9uzZw+LFi3n88cc5\nceKE6VhSCqmIHUx2djaLFi3Cy8uLzz77jK1bt7Ju3Tq8vb1NR5MSoBFxyatfvz5ff/01np6e+Pr6\n8vnnn5uOJKWMithB5ObmsmzZMho2bMj69ev56KOP+OSTT/Dx8TEdTUqQitiM8uXLM3PmTJYsWUJw\ncDDjxo0jJyfHdCwpJVTEdi4/P581a9bg7e3NkiVLCA8PZ8eOHQQEBJiOJgakp6dratqgDh06EBcX\nx3fffcdDDz3EkSNHTEeSUkBFbKesVisbNmygSZMmzJkzh7fffpuIiAhatWplOpoYpE0fzKtWrRqb\nN28mKCiIBx54gBUrVpiOJIZpixw7Y7Va2bp1KyEhIeTl5REWFkaXLl2wWCymo0kpoKnp0sFisTBs\n2DDatGlDr1692L59O/Pnz+fWW281HU0M0IjYjuzatYsWLVowatQoxo4dS0xMDF27dlUJy99UxKVL\nkyZNiImJwc3NDV9fX7799lvTkcQAFbEd2LNnDw8//DDPPfccQ4YMYd++fTzxxBOUKaP/vHIxXb5U\n+ri5ubFo0SKmT59O9+7dCQsLIy8vz3QsKUEWq64yNyIlJYXl4eEk7dtH+qlTuFeoQP3Gjek3YABV\nq1Yt1GtER0czYcIEDh48SEhICH379sXJSUcb5MruuOMOYmJiqFGjhukoUoCffvqJ4OBgypYtywcf\nfKCV7RyEhkwlLCoqiuAePWhQqxYHQ0PxW7mSLps347dyJQcmTqT+3XcT3KPHVfc33bdvH4GBgQQG\nBtK9e3eSkpIYOHCgSliuSVPTpdtdd93Frl27aNu2LX5+fnz88cemI0lJsEqJWbRggfV2NzfrbIvF\nehKs1gJ+ToJ1lsVivd3NzbpowYKLnn/gwAFrz549rdWrV7fOnj3bmpGRYeiTiC3Ky8uzWiwWa25u\nrukoUgh79uyx1q5d2zp48GD9XbdzGhGXkHcXLuSNkSPZnZHBy1YrFa/wuIrAcKuV3RkZvDFyJO8u\nXMiRI0fo27cvrVq1wtfXl+TkZF5++WVcXV1L8iOIjcvIyMDV1ZWyZcuajiKF8OCDDxIfH09aWhr+\n/v4kJCSYjiTFRMeIS0BUVBTd27Rhd0YGnhfcPh8IBxKAIOD9S56XDAQ4OZF3yy0MHz6cl19+mQoV\nKpRMaLE7v/32G02aNOH48eOmo8h1sFqtLF++nJEjRxIaGsqQIUN0JYSdURGXgOAePWi6cSMvX/Jb\nvZE/D9JvBzK5vIgB3gQiu3ZlrTYal5t0+PBhOnXqRHJysukocgMOHz5M7969qVGjBu+//z5VqlQx\nHUmKiKami1lKSgqfbt1KvwL+vRMIdAcqXeX5A4Adn31GampqMSUUR6ETtWybl5cXX3/9NQ0bNsTH\nx4d///vfpiNJEVERF7Pl4eE8Dlc8JnwtlYDHLRaWh4cXXShxSCpi2+fs7Mz06dNZunQpffv2ZcyY\nMWRnZ5uOJTdJRVzMkvbto1lW1k29hn9mJt/Hxmpjcbkp2vDBfrRv3574+Hj2799PixYtdLjBxunC\n02KWfuoUN/vV5wGs//BD1nz8MdWqVbvop2rVqpfddv52Z2fnovgIYie04YN9qVq1Kps2beLtt9+m\nefPmvPnmmzz99NM6kcsGqYiLmXuFCpy5ydc4AzwVFMSshQtJSUm56Cc1NZWff/6Z2NjYi24/ceIE\n7u7uVy3rC38qVaqkJTHtnKam7Y/FYuGll16idevWf28esXDhQm0eYWNUxMWsfuPGRK5fz/MFTE/n\nATl//W8ucI4//4NcepVnlKsr3o0b4+7ujru7O3Xr1r3m++bn55OWlvZ3WV9Y0gcOHOCLL764qMxP\nnTpF5cqVr1rWFxa6h4eH/uVtYzQ1bb8aN25MdHQ0I0aMwNfXl5UrV/LAAw+YjiWFpMuXillKSgoN\natXih6ysy07YmvTXz4V1FgqEXPDrk0A9FxeSjh4t9BrUNyInJ4cTJ05cNuK+sKwv/HVubu41y/rC\n21xcXIotuxTO66+/ztmzZ5k6darpKFKMNmzYwPPPP8/QoUMZM2aMFnCxASriEnCl64gL403gm06d\n+GjLlqIPdhPOnj17WTlf+usLb3d1db3mce3zP5UrV9aXRzEYO3YsHh4ejBs3znQUKWY///wzwcHB\nWCwWPvjgA+68807TkeQqVMQl4Eora11LMvBAuXJYPTx4//33eeyxx4orYrGyWq388ccfVyzqS0s7\nLS2NihUrXvO49vlCr1ChgqbJC+Gll17Cy8uLoUOHmo4iJSAvL4833niDuXPnsmjRIgIDA01HkitQ\nEZeQ82tNby9kGScDHd3cGD1zJvc1aUKfPn3o3LkzM2fOtPs1pnNzc/n9998LPU1+7ty5q549fult\n9v77dyX9+/endevWDBgwwHQUKUF79+4lKCiIDh06MGvWLNzc3ExHkkvoZK0SMmjwYABajhzJ/2Vm\n0v8KGz+cBMItFma4ujJp5sy/nxcXF8fzzz+Pv78/q1evplGjRiUXvoQ5OTlRvXp1qlevXqjHZ2Zm\nXlTOF/7/77///rIid3Z2LtQlYNWqVaNKlSp2s72kzpp2TA888ABxcXG88MILf39/NG7c2HQsuYBG\nxCUsOjqauWFhbN6yhcctFvwzM/Hgz0uUolxd2WC10rVzZ4aNHUvTpk0veq7VamXZsmWMHDmSiRMn\navH3G2C1Wjl9+nShp8l///13brvttkJdAla1alUqVqxYav+bPProowwbNoxOnTqZjiIGWK1WPvjg\nA0aMGEFISAgvvvhiqf2z6mhUxIakpqayPDycpIQEzqSl4VGxIvUbNaJv//7XPDs6KSmJoKAgLf5e\nAvLy8jh58mShpshTUlI4e/bsVUv70vtuueWWEvssLVq04I033uChhx4qsfeU0ic5OZmgoCCqVavG\n0qVLi/VqDCkcFbGNys7OZvz48axatYrw8HDatWtnOpIA586du+I0eUE/ZcqUKfRqaVWqVLmp1dKa\nNGnCsmXL8PHxKcJPLLYoOzubkJAQPvjgA8LDw2nfvr3pSA5NRWzjdu7cSf/+/QkODmbKlCla1tKG\nWK1W0tPTr1nW5+8/ceIEHh4ehZ4mv3S1tHr16rF9+3Y8Pa/n3H2xZ//+97/p168fQUFBvPbaa/r+\nMERFbAdSU1MZMGAAx48fZ9WqVXh5eZmOJMXgwtXSCjNNfubMmYtWS/vqq694+umnqVOnToEjb3d3\ndx0zdEAnTpxg4MCB/Prrr6xevVrfHwaoiO2E1Wrl7bffZvLkycycOZO+ffvqS9XBZWdnX7RaWrdu\n3Zg0aRKnTp0qsMzz8/Ova1OR8uXLm/6IUkSsVisLFixg4sSJzJgxg379+un7owSpiO3Mvn376N27\nN40bN+add96hQoUKpiNJKZCXl4ezszO5ublX/IItaLW0K428U1NTcXNzu65NRbRaWumXkJBA7969\nadSokb4/SpCK2A5lZGQwYsQItm3bxqpVq2jevLnpSGLY6dOnufPOOzl9+nSRvN751dIKO03+xx9/\nUKlSpUJvKnLrrbdqRGZIZmYmI0eOZMuWLfr+KCEqYju2ceNGnnvuOV588UXGjRunEYkD++WXX/D3\n9+fXX3818v65ubkXTZNfa+SdnZ19XZuKOOpqacXp448/5rnnnmPIkCH6/ihmKmI79/PPP/P000+T\nn5/PihUruOuuu0xHEgMSExPp1q0bSUlJpqMUyqWrpV1r5F2+fPnr2lTEXlZLK26//PILTz/9NHl5\nedf8/khJSflzbYR9+0g/dQr3ChWo37gx/QYM0LXK16AidgB5eXlMnz6dOXPmsHDhQnr06GE6kpSw\nmJgYnn32WWJjY01HKXLnV0sr7DT5yZMnue222wq9qchtt93m0NPkeXl5zJgxg9mzZxf4/REVFcXc\nsDA+3bqVHoB/VtbfqwVG/rVaYJdOnRg2diz+/v4mPkKppyJ2IN9++y1BQUE88sgjzJ49u0RXdRKz\nvvjiC0JDQ4mIiDAdxbi8vLy/NxUpzFKnmZmZ17WpiL1uqhAZGUnv3r155JFHmDNnDm5ubry7cCGh\nI0cyOjOTfldYPz+NP9fPn37J+vnyPypiB3P69GmGDBlCdHQ0q1ev1ipLDmLz5s0sXLiQTz/91HQU\nm3PpamlXG3kfP34cJyen69pUpFy5cqY/YqFd+P3xZI8erJoz54Z2lFMZX0xF7KBWrFjB8OHDefXV\nVxk2bJhDT705gjVr1rBhwwbWrl1rOopdO79aWmGmyFNSUvj999+59dZbr2tTkQtXSzNl8uTJzAoN\nJRouKuE0YCCwE6gKTAV6X3B/MtDSzY1NERGXbWrjyHTGgoMKDg6mefPmBAUFsWPHDpYuXVrobQfF\n9pw5cwYPDw/TMeyexWLBw8MDDw8P6tWrd83H5+fn/72pyKVlnZCQcFlxp6enU6VKlevaVKQ4/pGd\nFB9PqMWC5yXjuBcAFyAViAW6AD7APX/d7wn8X2Ymc8PC+GD9+iLPZas0InZwOTk5TJw4kaVLl7J0\n6VI6duxoOpIUg9mzZ/Pf//6XOXPmmI4iN+HS1dKuNvI+fvw4QKGnyatWrVqotaZTUlJoUKsWP2Rl\nXXRMOANMRSIpAAAXfklEQVSoCBwAzv8TpB9Qkz9HxuedBOq5uJB09KjOpv6LRsQOrly5crz++uu0\na9eOvn370rNnT6ZOnarlC+1Meno67u7upmPITXJ2dqZGjRrUqFGjUI8/e/ZsgWX9yy+/EBcXd9nt\n7u7u15wm/3TTJgLhshOzkoBy/K+EAZoAl54eWAl43GJheXg4I0aNusHfCfuiIhYAHn74YeLj4/nn\nP/9J8+bNWb16NQ0aNDAdS4rI+Q0gxLHccsst1KlThzp16lzzsfn5+fzxxx8FHs8+ePAgERERpKSk\nkBgfz8SsrMuenw7cesltt/LnZUyX8s/MJD4h4UY+kl1SEcvfKleuzL/+9S8WLVpEixYtmDZtGs88\n84xO5LID6enp1KpVy3QMKcXKlClDpUqVqFSp0lX/ER7UrRsemzdfdrs7cOkCqqeAgs5M8ADOpKXd\nRFr7Yv70OylVLBYLzz//PF9++SXz5s2jZ8+epOkvjM3T1LQUFfcKFQoc5dYHcoEjF9z2HeBdwGPP\nAB4VC7rq2DGpiKVA9957L5GRkdSoUQMfHx92795tOpLchPT0dJ01LUWifuPGRLq4XHa7G9ADCOHP\nE7e+AjYBTxfwGlGurtRv1Kg4Y9oUFbFckYuLC3PnzmXBggX07NmT0NBQcnNzTceSG3DmzBmNiKVI\n3HvffazNzqagebL5/FnC1YBg4B3+d+nSeSeBDVYrffv3L9actkRFLNfUpUsXYmNj+frrr2ndujU/\n/vij6UhynTQ1LTcrNjaWrl27MnjwYHwbNya8gHNHKgIb+PPErR+Bpwp4nWUWC107d9alSxdQEUuh\n3HHHHWzfvp3AwECaNWumFZpsjKam5Ubt37+fJ554gm7duvHoo4+SlJTE7HffZbqrK8nX+VrJwHRX\nV4aNHVscUW2WilgKrUyZMowaNYotW7YwYcIEBgwYQHp6uulYUgiampbrlZSURFBQEO3atePBBx/k\n8OHDvPjii5QvXx5/f38mzZxJRze3Qpfx+bWmJ82cqeUtL6EiluvWtGlTYmNjsVgs+Pn5ER0dbTqS\nXIOmpqWw/vOf/zBgwABatGjBfffdx+HDhxkxYsRlu0oNGjyY0TNn0tLNjdkWS4HHjOHPY8KzLBZa\nasOHK1IRyw1xd3fn/fffZ/LkyXTu3JkZM2aQn59vOpZcgYpYruXnn3/m+eefx9/fn7vvvpvDhw8z\nbty4qx7SGDR4MJsiIoh9/HHqurgw0NWVhcAKYCEw0NWVei4uxD3+OJsiIlTCV6C1puWm/fjjj/Tp\n0wc3NzeWL1/OHXfcYTqSXCAnJwdXV1dycnK0OItc5rfffiMsLIwVK1bw7LPPMmrUqBtahS01NZXl\n4eEkJSRwJi0Nj4oVqd+oEX3799eJWdegIpYikZuby5QpU1i0aBGLFy+ma9eupiPJX9LS0qhTpw5/\n/PGH6ShSipw4cYLp06ezZMkS+vbty5gxY7QDmyGampYi4eTkxKRJk1i3bh1Dhgxh6NChZBWwHq2U\nPE1Ly4X++OMPJkyYQIMGDUhPT2ffvn3Mnj1bJWyQiliKVMuWLYmPj+fYsWM0a9aM77//3nQkh6e9\niAX+/HPw2muv4eXlxa+//kpMTAwLFiygZs2apqM5PBWxFLmKFSvy4YcfMmzYMFq3bs3ChQvRERBz\nNCJ2bBkZGcyYMQNPT08OHTrE119/zZIlS6hdu7bpaPIXFbEUC4vFwjPPPMNXX33Fu+++S48ePfj9\n999Nx3JIKmLHlJWVxbx58/D09CQyMpJdu3axYsUKvLy8TEeTS6iIpVg1bNiQvXv3UrduXXx8fPj8\n889NR3I4mpp2LDk5Obz77rvUr1+fzz77jC1btrBu3Tq8vQvaB0lKAxWxFLvy5cvz5ptv8t5779Gn\nTx9effVVcnJyTMdyGBoRO4bc3FzCw8Np0KAB69evZ926dXzyySf4+PiYjibXoCKWEvPoo48SFxdH\nXFwcLVu25IcffjAdySGoiO1bfn4+q1evxtvbm6VLl7Js2TK2b99OQECA6WhSSCpiKVHVq1dn8+bN\n9OrVi4CAAFasWGE6kt3Thg/2yWq1smHDBpo0acK8efOYP38+X3zxBS1btjQdTa6Tk+kA4njKlCnD\nyy+/TJs2bejduzfbt29n/vz53Hrrraaj2SVt+GBfrFYrW7ZsISQkBKvVyrRp0+jcubNWTbNhGhGL\nMT4+PkRHR+Pm5oavry/ffvut6Uh2SVPT9sFqtfLZZ5/x4IMPMnr0aF599VViYmLo0qWLStjGqYjF\nqFtuuYVFixYxffp0unfvTlhYGHl5eaZj2RVNTdu+3bt38/DDD/+9at13331Hjx49VMB2QkUspcIT\nTzxBdHQ027Zto3379vzyyy+mI9kNTU3brsjISDp27Ei/fv3o378/33//Pb1796Zs2bKmo0kRUhFL\nqXHXXXexa9cu2rZti5+fHxs3bjQdyS5oatr2xMfH0717d5544gl69OjBoUOH6N+/P05OOq3HHqmI\npVQpW7Ys48ePZ+PGjbzyyisMHjyYjIwM07FsmorYdhw4cIB//OMfdO7cmXbt2nH48GGee+45nJ2d\nTUeTYqQillKpefPmxMXFcerUKfz9/dm3b5/pSDZLK2uVfocPHyY4OJiHH36YZs2akZyczNChQ3Fx\ncTEdTUqAilhKrQoVKrBy5UpGjx7NI488wltvvaXNI26ARsSl148//sgzzzxD8+bNadiwIcnJyYwa\nNQo3NzfT0aQEqYilVLNYLPTt25dvvvmG5cuX061bN1JTU03Hsikq4tLnl19+4YUXXuD++++nRo0a\nHD58mPHjx2vmwkGpiMUmeHp6smfPHu677z58fHzYuXOn6Ug2Q1PTpcfx48cZPnw4jRs3xt3dncTE\nRKZMmULFihVNRxODVMRiM5ydnZk2bRrLly9nwIABjBo1iuzsbNOxSjWr1aoRcSnw+++/M2bMGO69\n917y8/P5/vvvmT59OlWqVDEdTUoBFbHYnEceeYS4uDgSExN58MEHSUpKMh2p1MrOzsZiseisW0NO\nnTpFaGgoDRo04I8//iA+Pp65c+dy++23m44mpYiKWGxS1apV+fjjjxkwYAAPPvggS5cu1YlcBdCq\nWmakp6czdepUPD09OXr0KJGRkbzzzjvcddddpqNJKaQiFptlsVgYMmQIn3/+OW+++Sa9e/fmjz/+\nMB2rVNGqWiUrMzOTN998E09PT/bv389XX33F0qVLqVu3ruloUoqpiMXmNWrUiKioKCpXroyPjw97\n9uwxHanU0PHhknHu3DnefvttPD09+frrr/nss89YtWoVDRo0MB1NbICKWOyCq6sr8+fPZ+7cufTo\n0YPJkyeTm5trOpZxmpouXjk5OSxevJj69euzbds2Nm3axPr167nvvvtMRxMboiIWu/LYY48RGxtL\nREQEDz/8MEePHjUdyShNTRePvLw8li9fzj333MPatWtZs2YNmzdvxs/Pz3Q0sUEqYrE7NWvWZMeO\nHXTp0oWmTZvy0UcfmY5kjKami1Z+fj5r167lvvvu47333mPx4sXs3LmT5s2bm44mNkxbeYhdKlu2\nLGPGjKFt27YEBQWxbds25s6dyy233GI6WolSERcNq9XKJ598QkhICC4uLsydO5f27dtrP2ApEhoR\ni11r1qwZcXFx5OTkcP/99xMXF2c6UonSqlo3x2q1snXrVpo1a0ZoaCivvfYae/fupUOHDiphKTIq\nYrF7Hh4eLFu2jJCQEDp06MDs2bPJz883HatEaER84z7//HMeeughRowYwejRo4mNjaVbt24qYCly\nKmJxGEFBQXz77besXbuWLl26cPz4cdORip2K+Prt2bOHtm3bMmjQIF544QUSEhJ48sknKVNGX5dS\nPPQnSxxK3bp12b17N/fffz++vr5s27bNdKRipanpwouOjqZTp0706dOH4OBgDh48SJ8+fShbtqzp\naGLnVMTicMqVK8drr73G6tWrGTRoEMOHD+fcuXOmYxULjYivbd++fQQGBhIYGEj37t1JSkpi4MCB\nODnpXFYpGSpicVitW7cmPj6e//73vwQEBHDw4EHTkYqcivjKDh48yFNPPUWHDh1o06YNhw8fZvDg\nwdogQ0qcilgcWqVKlVi/fj0vvPACrVq14r333rOrzSM0NX25I0eO0LdvX1q3bo2fnx9Hjhzh5Zdf\nxtXV1XQ0cVAqYnF4FouFQYMGERERwdtvv80//vEPTp48aTpWkdCI+H+OHj3Ks88+S0BAAJ6eniQn\nJzN69GiHu7ZcSh8Vschf7r33Xr799lvuvPNOfHx8+PLLL01HumkqYvj111958cUX8fX1pVq1aiQl\nJRESEsKtt95qOpoIoCIWuYiLiwtz5szhnXfe4amnnmLChAk2vXmEI2/6kJKSwogRI2jUqBEuLi4c\nPHiQ119/nUqVKpmOJnIRFbFIATp37kxcXByRkZG0atWK//znP6Yj3RBH3PTh5MmTjBs3jnvuuYfs\n7GwSEhKYOXMm1apVMx1NpEAqYpEruP3229m6dStPPvkkAQEBrF692nSk6+ZIU9OnT59m0qRJ1K9f\nnxMnThAXF8dbb71FjRo1TEcTuSqL1Z5OERUpJrGxsfTu3ZvmzZvz1ltv2cR0r9VqxcnJiXPnztn1\nNbFnz57lrbfeYtasWXTq1ImQkBDq1atnOpZIoWlELFIIfn5+xMTE4OTkhJ+fH1FRUaYjXVNmZibO\nzs52W8KZmZnMnj2bevXqER8fz5dffsmyZctUwmJzVMQiheTu7s7ixYt5/fXX6dKlC9OnTy/Vm0fY\n67R0dnY2CxYswMvLi4iICHbs2MGaNWto2LCh6WgiN0RFLHKdevbsSVRUFJ988gkdOnTg119/NR2p\nQPZWxDk5OSxZsoT69euzadMmNmzYwMaNG2ncuLHpaCI3RUUscgNq1arFF198QcuWLfHz82PTpk2m\nI13GXlbVysvLY8WKFdx7772sXLmSlStXsnXrVvz9/U1HEykS9nnwSKQEODk5ERoayiOPPEJwcDDb\nt29nxowZpWapRFsfEefn5/Ovf/2LkJAQKlasyKJFi2jbtq3pWCJFTiNikZv00EMPERcXR0pKCs2a\nNWP//v2mIwG2W8RWq5VPPvkEPz8/pk2bxqxZs/jqq69UwmK3VMQiRaBixYqsXbuW4cOH06ZNGxYs\nWGB88whbm5q2Wq1s376dgIAAxo8fz6RJk4iKiuLRRx/FYrGYjidSbFTEIkXEYrEwcOBA9uzZw5Il\nSwgMDOTEiRPG8tjSiDgiIoJWrVrx8ssvM3LkSOLj43nsscdUwOIQVMQiRaxBgwZ8/fXXeHl54ePj\nw65du4zksIUi/uabb2jXrh0DBw5k0KBB7N+/n549e1KmjL6axHHoT7tIMShfvjwzZ87k/fffJzg4\nmDFjxpCTk1OiGUrz1HRMTAxdunShV69e9OrVi0OHDvH0009TtmxZ09FESpyKWKQYdejQgfj4eBIS\nEmjRogXJyckl9t6lcUSckJBAjx496N69O507dyYpKYl//vOflCtXznQ0EWNUxCLFrFq1amzevJng\n4GCaN2/O8uXLS+RErtJUxImJifTu3Zv27dvz0EMPkZyczJAhQyhfvrzpaCLGqYhFSoDFYmHo0KF8\n9tlnTJs2jeDgYE6fPl2s71ka9iL+4Ycf6N+/Pw899BCNGjUiOTmZV155pdRcay1SGqiIRUpQkyZN\niI6OxsPDA19fX/bu3Vts72VyL+KffvqJ5557jmbNmlGrVi0OHz7MuHHjSs0IXaQ0URGLlDA3Nzfe\neecdZsyYwWOPPcbrr79OXl5ekb+PianpY8eOMXToUHx8fKhUqRKJiYlMmjSJ2267rURziNgSFbGI\nIT169CAmJoadO3fSrl07fv755yJ9/ZIs4tTUVEaNGoW3tzdly5blwIEDhIWFUbly5RJ5fxFbpiIW\nMejOO+/k3//+N+3bt+f+++9nw4YNRfbaJXH5UlpaGuPHj6dhw4acPXuWhIQEZs+eTfXq1Yv1fUXs\niYpYxLCyZcsybtw4Pv74Y0aMGMHzzz9PRkbGTb9ucY6IT58+zZQpU/Dy8uK3334jJiaGBQsWULNm\nzWJ5PxF7piIWKSUeeOAB4uLiOHPmDE2bNuW77767qdcrjiI+e/Ys06dPx8vLi8TERL755hsWL15M\n7dq1i/R9RByJilikFKlQoQIrV65k7NixtGvXjnnz5t3wNcdFOTWdlZXF3Llz8fT0JCoqil27drFi\nxQq8vLyK5PVFHJnFanqLGBEp0JEjRwgKCqJKlSosXbqUatWqFfq5+fn5ODk5kZOTc1PLRmZnZ/P+\n++/z+uuv4+vry+TJk/Hx8bnh1xORy2lELFJK1atXj6+++orGjRvj4+PDjh07Cv3cjIwMXF1db7iE\nc3NzWbp0KQ0aNGDDhg2sX7+eTz75RCUsUgw0IhaxAbt27aJfv3489dRTTJ06FWdn5wIfl5KSwvLw\ncL779lu2bt5Mj3/8g/qNG9NvwACqVq16zffJy8tj7dq1TJw4kTvuuIPXXnuNli1bFvXHEZELqIhF\nbMSJEyd45pln+Pnnn1m1ahUNGjT4+76oqCjmhoXx6dat9AD8s7LwAM4Aka6ubLBa6dKpE8PGjsXf\n3/+y17ZarWzYsIGQkBA8PDyYMmUKjzzyiPYDFikJVhGxGfn5+dYFCxZYq1SpYl2yZIk1Pz/fumjB\nAuvtbm7W2RaL9SRYrQX8nATrLIvFerubm3XRggUXvd6mTZusvr6+Vl9fX+unn35qzc/PN/gJRRyP\nRsQiNmj//v307t0bF2dnTh48yPbMTDwL8bxkoKObG/83cyZ1PT2ZMGEC6enpTJkyhcDAQI2ARQxQ\nEYvYqN27dxPYti3f5ub+XcLZwAvAZ0AaUA+YCjx6wfOSAf8yZbj1zjuZNm0aPXv2vKkzq0Xk5jiZ\nDiAiN2bR7NlMyMu7aCScC9wN7AbuAj4FegL7/7odwBOYYLUS4+dH7969SzKyiBRAI2IRG5SSkkKD\nWrX4ISuLitd4bBNgIvD4BbedBOq5uJB09GihzqYWkeKj64hFbNDy8HAeh2uW8HHgMOB9ye2VgMct\nFpaHhxdDOhG5HipiERuUtG8fzbKyrvqYXCAY6A/UL+B+/8xMkhISij6ciFwXFbGIDUo/dYqrrSJt\n5c8SLg+8dYXHeABn0tKKOpqIXCedrCVig9wrVODMVe5/BjgBbAGudD70GcCj4rUmt0WkuGlELGKD\n6jduTKSLS4H3PQ8cAj4BCl4I809Rrq7Ub9SoGNKJyPXQWdMiNuhKZ00fBWoDLvxvJGwBFgEXXqik\ns6ZFSg+NiEVsULVq1ejSqRPLLlkJ624gH8jgz6nnM8BpLi5hgGUWC107d1YJi5QCGhGL2KioqCi6\nt2nD7oyMQi1veV4y0NLNjU0RETRt2rS44olIIWlELGKj/P39mTRzJh3d3Egu5HPOrzU9aeZMlbBI\nKaEiFrFhgwYPZvTMmbR0c2O2xcKVLkY6CcyyWGjp5sbomTMZNHhwScYUkavQ1LSIHYiOjmZuWBib\nt2zhcYsF/8zMv/cjjvprP+KunTszbOxYjYRFShkVsYgdSU1NZXl4OEkJCZxJS8OjYkXqN2pE3/79\ndWKWSCmlIhYRETFIx4hFREQMUhGLiIgYpCIWERExSEUsIiJikIpYRETEIBWxiIiIQSpiERERg1TE\nIiIiBqmIRUREDFIRi4iIGKQiFhERMUhFLCIiYpCKWERExCAVsYiIiEEqYhEREYNUxCIiIgapiEVE\nRAxSEYuIiBikIhYRETFIRSwiImKQilhERMQgFbGIiIhBKmIRERGDVMQiIiIGqYhFREQMUhGLiIgY\npCIWERExSEUsIiJikIpYRETEIBWxiIiIQSpiERERg1TEIiIiBqmIRUREDFIRi4iIGKQiFhERMUhF\nLCIiYpCKWERExCAVsYiIiEEqYhEREYNUxCIiIgapiEVERAxSEYuIiBikIhYRETFIRSwiImKQilhE\nRMQgFbGIiIhBKmIRERGDVMQiIiIGqYhFREQMUhGLiIgYpCIWEREx6P8BQKb7cL+RpaYAAAAASUVO\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": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "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": {
    "collapsed": true,
    "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."
   ]
  },
  {
   "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"
   ]
  }
 ],
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