更改

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。<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> 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。<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">三元闭包 Triadic Closure </font>'''是'''<font color="#FF8000">社会网络 Social Network</font>'''理论中的一个概念，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>'''<font color="#FF8000">三元闭包</font>'''指的是由A，B，C三个'''<font color="#FF8000">节点 Nodes</font>'''所组成的'''<font color="#FF8000">三元组</font>'''的一种性质，即如果A-B和A-C之间存在'''<font color="#FF8000">强联系 Strong Tie </font>'''，则B-C之间也仅存在'''<font color="#FF8000">强联系</font>'''。<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> 由于这一性质过于极端，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。<ref name=Easley>Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.</ref>

==History==

==History==

==历史==

==历史==

马克·格兰诺维特 Mark Granovetter在1973年发表的《弱链接的力量》 The Strength of Weak Ties一文使得三元闭包性质变得流行。 <ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> 在那里，他综合了弗里茨·海德 Fritz Heider于1946年提出的'''<font color="#FF8000">认知平衡理论 The Theory Of Cognitive Balance </font>'''以及Georg Simmel对社会网络的理解。 一般而言，认知平衡是指两个个体对同一事物具有产生相同感觉的倾向。 如果三个个体所组成的三元组没有闭合，那么与同一个体联系的其余两个个体均将想要闭合这一三元组，进而在关系网络中形成闭包。

马克·格兰诺维特 Mark Granovetter在1973年发表的《弱链接的力量》 The Strength of Weak Ties一文使得'''<font color="#FF8000">三元闭包</font>'''性质变得流行。 <ref>Granovetter, M. (1973). "[http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf The Strength of Weak Ties] {{webarchive|url=https://web.archive.org/web/20080216103216/http://www.stanford.edu/dept/soc/people/mgranovetter/documents/granstrengthweakties.pdf |date=2008-02-16 }}", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80.</ref> 在那里，他综合了弗里茨·海德 Fritz Heider于1946年提出的'''<font color="#FF8000">认知平衡 Cognitive Balance </font>'''理论以及Georg Simmel对'''<font color="#FF8000">社会网络 Social Network</font>'''的理解。 一般而言，'''<font color="#FF8000">认知平衡 Cognitive Balance </font>'''是指两个个体对同一事物具有产生相同感觉的倾向。 如果三个个体所组成的'''<font color="#FF8000">三元组</font>'''没有闭合，那么与同一个体联系的其余两个个体均将想要闭合这一'''<font color="#FF8000">三元组</font>'''，进而在关系网络中形成'''<font color="#FF8000">闭包 Closure</font>'''。

==Measurements==

==Measurements==

==测量==

==测量==

最常见的测量一张图三元闭包性质的两种方法（排名不分先后）是采用该图的'''<font color="#FF8000">聚集系数 Clustering Coefficient </font>'''和'''<font color="#FF8000">传递性 Transitivity </font>'''。

最常见的测量一张'''<font color="#FF8000">图 Graph</font>''''''<font color="#FF8000">三元闭包</font>'''性质的两种方法（排名不分先后）是采用该'''<font color="#FF8000">图</font>'''的'''<font color="#FF8000">聚集系数 Clustering Coefficient </font>'''和'''<font color="#FF8000">传递性 Transitivity</font>'''。

==Clustering coefficient==

==Clustering coefficient==

==聚类系数==

=='''<font color="#FF8000">聚集系数</font>'''==

One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is [[clustering coefficient]], as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

One measure for the presence of triadic closure is clustering coefficient, as follows:

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。

令<math>G =(V，E)</math>是'''<font color="#FF8000">无向简单图 Undirected Simple Graph</font>'''（即，没有'''<font color="#FF8000">自环 Self-loops</font>'''或'''<font color="#FF8000">多重边 Multiple Edges</font>'''的'''<font color="#FF8000">图</font>'''），其中<math>V</math>为'''<font color="#FF8000">顶点 Vertices</font>'''集，<math>E</math>为'''<font color="#FF8000">边 Edges</font>'''集。 另外，令<math>N = |V|</math>和<math>M = |E|</math>分别表示'''<font color="#FF8000">图</font>'''<math>G</math>中'''<font color="#FF8000">顶点</font>'''和'''<font color="#FF8000">边</font>'''的数量，并令<math>d_i</math> 是'''<font color="#FF8000">顶点</font>'''<math>i</math>的'''<font color="#FF8000">度 Degree</font>'''。

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。

我们可以将由'''<font color="#FF8000">顶点</font>'''<math>i</math>,<math>j</math>和<math>k</math>所组成的'''<font color="#FF8000">三元组</font>'''定义为一个由'''<font color="#FF8000">边 Edges</font>'''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>。

我们也可以将'''<font color="#FF8000">顶点</font>'''<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'''<font color="#FF8000">图</font>''<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>.

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>。

假设'''<font color="#FF8000">三元闭包</font>'''性质成立，则一个'''<font color="#FF8000">三元组</font>'''仅需要两条'''<font color="#FF8000">强联系</font>'''便可形成三角形。 因此，在'''<font color="#FF8000">三元闭包</font>'''性质成立的前提下理论上'''<font color="#FF8000">顶点</font>'''<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。

现在，对于具有<math>d_i \ge 2</math>的'''<font color="#FF8000">顶点</font>'''<math>i</math>，'''<font color="#FF8000">顶点</font>'''<math>i</math>的'''<font color="#FF8000">聚集系数</font>'''<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，'''<font color="#FF8000">图</font>'''<math>G</math>的'''<font color="#FF8000">聚集系数</font>'''<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'''<font color="#FF8000">度</font>'''至少为2的'''<font color="#FF8000">顶点</font>'''数量。

==Transitivity==

==Transitivity==

==传递性==

=='''<font color="#FF8000">传递性 Transitivity</font>'''==

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>.

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>。

测量'''<font color="#FF8000">三元闭包</font>'''是否出现的另一方法是'''<font color="#FF8000">传递性</font>'''，定义为<math>T(G)= \frac{3\delta(G)}{\tau(G)}</math>。

==Causes and effects==

==Causes and effects==

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.

遵循此原理的网络将高度互连且具有极高的聚集系数。 与此相反，不遵循该原理的网络的连通性则较差，且一旦包含负面关系，网络则可能会变得较不稳定。

遵循此原理的网络将高度互连且具有极高的'''<font color="#FF8000">聚集系数</font>'''。 与此相反，不遵循该原理的网络的连通性则较差，且一旦包含负面关系，网络则可能会变得较不稳定。

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.

在社会网络中，三合闭包将促进合作行为，但是当通过从现有联系中推荐人时，合作者的平均全球比例要小于个人从总体人口中随机选择新联系时的比例。 对此的两个可能的影响是结构和信息结构。 结构构造来自于倾向于高度可聚性的倾向。 信息结构来自这样一个假设：与随机的陌生人不同，一个人对朋友的朋友有所了解。

在'''<font color="#FF8000">社会网络</font>'''中，'''<font color="#FF8000">三元闭包</font>'''将促进合作行为，但是当通过从现有联系中推荐人时，合作者的平均全球比例要小于个人从总体人口中随机选择新联系时的比例。 对此的两个可能的影响是结构和信息结构。 结构构造来自于倾向于高度可聚性的倾向。 信息结构来自这样一个假设：与随机的陌生人不同，一个人对朋友的朋友有所了解。

==Strong Triadic Closure Property and local bridges==

==Strong Triadic Closure Property and local bridges==