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本词条由信白初步翻译<br>

本词条由信白初步翻译

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

'''Triadic closure''' is a concept in [[social network]] theory, first suggested by [[Germany|German]] [[sociology|sociologist]] [[Georg Simmel]] in his 1908 book ''Soziologie'' [''Sociology: Investigations on the Forms of Sociation''].<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref> Triadic closure is the property among three nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is only a strong tie between B-C.<ref>[https://serendipstudio.org/complexity/course/emergence06/bookreviews/kmaffei.html Working concept] of triadic closure: book review of [[Duncan Watts]]' "[[Six Degrees: The Science of a Connected Age]]" at the ''Serendip'' ([[Bryn Mawr College]]) website. Retrieved on December 21, 2007.</ref> This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

'''<font color="#FF8000">三合闭包 Triadic Closure </font>'''是'''<font color="#FF8000">社会网络理论 Social Network Theory </font>'''中的一个概念，最早由德国社会学家格奥尔格·西梅尔 Georg Simmel在其1908年的著作《社会学：社会形式的调查》 Sociology: Investigations on the Forms of Sociation中提出。<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref>三元闭包指的是由A，B，C三个节点所组成的三元组的一种性质，即如果A-B和A-C之间存在强联系，则B-C之间也仅存在强联系。 这一性质过于极端，以至于它难以在规模较大、结构复杂的网络中被满足，然而在理解网络与网络预测等方面，它却是一种十分有用的对现实的简化。

'''<font color="#FF8000">三合闭包 Triadic Closure </font>'''是'''<font color="#FF8000">社会网络理论 Social Network Theory </font>'''中的一个概念，最早由德国社会学家格奥尔格·西梅尔 Georg Simmel在其1908年的著作《社会学：社会形式的调查》 Sociology: Investigations on the Forms of Sociation中提出。<ref>[https://www.nytimes.com/2007/12/17/style/17facebook.html?pagewanted=print Georg Simmel], originator of the concept: "Facebook" article at [[the New York Times]] website. Retrieved on December 21, 2007.</ref>三元闭包指的是由A，B，C三个节点所组成的三元组的一种性质，即如果A-B和A-C之间存在强联系，则B-C之间也仅存在强联系。 这一性质过于极端，以至于它难以在规模较大、结构复杂的网络中被满足，然而在理解网络与网络预测等方面，它却是一种十分有用的对现实的简化。

==History==

==History==

测度三元闭包是否出现的方法之一是聚集系数，如下所示：

测度三元闭包是否出现的方法之一是聚集系数，如下所示：

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the [[degree (graph theory)|degree]] of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

Let <math>G = (V,E)</math> be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let <math>N = |V|</math> and <math>M = |E|</math> denote the number of vertices and edges in G, respectively, and let <math>d_i</math> be the degree of vertex i.

令<math>G =(V，E)</math>是无向简单图（即，没有自环或多重边的图），其中<math>V</math>为节点集，<math>E</math>为边集。 另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示图<math>G</math>中节点和边的数量，并令<math>d_i</math> 是节点<math>i</math>的度。

令<math>G =(V，E)</math>是无向简单图（即，没有自环或多重边的图），其中<math>V</math>为顶点集，<math>E</math>为边集。 另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示图<math>G</math>中顶点和边的数量，并令<math>d_i</math> 是顶点<math>i</math>的度。

 

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.  

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.  

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.  

We can define a triangle among the triple of vertices <math>i</math>, <math>j</math>, and <math>k</math> to be a set with the following three edges: {(i,j), (j,k), (i,k)}.  

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>k</math>中定义一个三角形，以使其具有以下三个边的集合：{（i ，j），（j，k），（i，k）}。

我们可以将由顶点<math>i</math>,<math>j</math>和<math>k</math>所组成的三元组定义为一个由边集<math>((i,j),(j,k),(i,k))</math>所组成的三角形。

 

 

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.  

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.  

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.  

We can also define the number of triangles that vertex <math>i</math> is involved in as <math>\delta (i)</math> and, as each triangle is counted three times, we can express the number of triangles in G as <math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>.  

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta（i）</math>，并且，由于每个三角形都被计数了三次，因此我们可以表示 G中的三角形为<math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>。

我们也可以将顶点<math>i</math>所涉及的三角形的数量定义为<math>\delta(i)</math>，并且由于每个三角形都被计数了三次，因此图<math>G</math>中三角形的个数为<math>\delta (G) = \frac{1}{3} \sum_{i\in V} \ \delta (i)</math>。

 

 

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.

Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex <math>i</math> is <math>\tau (i) = \binom{d_i}{2}</math>, assuming <math>d_i \ge 2</math>. We can express <math>\tau (G) = \frac{1}{3} \sum_{i\in V} \ \tau (i)</math>.