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Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. 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. 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.

Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie [Sociology: Investigations on the Forms of Sociation]. 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. 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.

'''<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">三元闭包 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==

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>.

假设三元闭包成立，则一个三元组的形成仅需要两个牢固的边缘。 因此，在顶点<math>i</math>的三元组关闭假设下应该出现的理论三元组的数量为<math>\tau（i）= \binom{d_i}{2}</math>， 假设<math>d_i \ge 2</math>。 我们可以表示<math>\tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i）</math>。

假设三元闭包性质成立，则一个三元组仅需要两条强联系便可形成三角形。 因此，在三元闭包性质成立的前提下理论上顶点<math>i</math>所涉及的三角形的数量为<math>\tau(i) = \binom{d_i}{2}</math>， 假设<math>d_i \ge 2</math>。 我们可以表示<math>\tau(G) = \frac{1}{3} \sum_{i\in V} \ \tau(i)</math>。

 

 

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the [[clustering coefficient]] <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the [[clustering coefficient]] <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

Now, for a vertex <math>i</math> with <math>d_i \ge 2</math>, the clustering coefficient <math>c(i)</math> of vertex <math>i</math> is the fraction of triples for vertex <math>i</math> that are closed, and can be measured as <math>\frac{\delta (i)}{\tau (i)}</math>. Thus, the clustering coefficient <math>C(G)</math> of graph <math>G</math> is given by <math>C(G) = \frac {1}{N_2} \sum_{i \in V, d_i \ge 2} c(i)</math>, where <math>N_2</math> is the number of nodes with degree at least 2.

现在，对于具有<math>d_i \ge 2</math>的顶点<math>i</math>，顶点<math>i</math>的聚类系数<math>c（i)</math> 是封闭的顶点<math>i</math>的三元组分数，可以测量为<math>\frac{\delta（i）}{\tau（i）}</math>。 因此，图<math> G </math>的聚类系数<math> C（G）</math>由<math> C（G）=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c（i）</math>，其中<math>N_2</math>是度数至少为2的节点数。

现在，对于具有<math>d_i \ge 2</math>的顶点<math>i</math>，顶点<math>i</math>的聚集系数<math>c(i)</math> 是其拥有的三角形的占比，即<math>\frac{\delta(i)}{\tau(i)}</math>。 因此，图<math>G</math>的聚集系数<math>C(G)</math>由<math> C(G)=\frac {1}{N_2} \sum_{i \in V,d_i \ge 2}c(i)</math>给出，其中<math>N_2</math>是度至少为2的顶点数。

==Transitivity==

==Transitivity==

传递性<br>

==传递性==

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

Another measure for the presence of triadic closure is transitivity, defined as <math>T(G) = \frac{3\delta (G)}{\tau (G)}</math>.

关于三元闭包的另一种度量是可传递性，定义为<math>T（G）= \frac{3\delta（G）}{\tau（G）}</math>。

测度三元闭包是否出现的另一方法是传递性，定义为<math>T(G)= \frac{3\delta(G)}{\tau(G)}</math>。

==Causes and effects==

==Causes and effects==

'''<font color="#FF8000">因果 Causes And Effects </font>'''<br>

==前因及影响==

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.<ref name=Easley/>

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.

Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.