三元闭包 Triadic closure

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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.

三合闭包 Triadic Closure 是社会网络理论 Social Network Theory 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》中提出。 三元闭合是三个节点A，B和C之间的属性，因此，如果A-B和A-C之间存在牢固的联系，则B-C之间仅存在牢固的联系。 这个属性太极端了，无法在非常大的复杂网络 Complex Network 中实现，但是它是对现实的有用简化，可以用来理解和预测网络。

History

历史

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.

马克·格兰诺维特（Mark Granovetter）在其1973年的文章《弱链接的力量》（The Strength of Weak Ties）中使三元闭包变得流行。 在那里，他综合了弗里茨·海德（Fritz Heider）于1946年提出的认知平衡理论 The Theory Of Cognitive Balance ，该理论对Simmelian的社交网络有所了解。 一般而言，认知平衡是指两个人想对一个物体感觉相同的倾向。 如果没有关闭三个人的闭合，那么连接到两个人的人都将要关闭，以便在关系网络中实现闭合。

Measurements

测量

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.

一张图的三元闭合的两个最常见的度量是（不按特定顺序）该图的聚类系数 Clustering Coefficient 和可传递性 Transitivity 。

Clustering coefficient

聚类系数

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:

衡量三元闭包的一种方法是聚类系数，如下所示：

Let [math]\displaystyle{ 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]\displaystyle{ N = |V| }[/math] and [math]\displaystyle{ M = |E| }[/math] denote the number of vertices and edges in G, respectively, and let [math]\displaystyle{ d_i }[/math] be the degree of vertex i.

Let [math]\displaystyle{ 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]\displaystyle{ N = |V| }[/math] and [math]\displaystyle{ M = |E| }[/math] denote the number of vertices and edges in G, respectively, and let [math]\displaystyle{ d_i }[/math] be the degree of vertex i.

令[math]\displaystyle{ G =（V，E） }[/math]是无向的简单图 Simple Graph （即没有自环或多个边的图），其中V为顶点集，E为边集。 另外，令[math]\displaystyle{ N = |V| }[/math]和[math]\displaystyle{ M = |E| }[/math]分别表示G中顶点和边的数量，并令[math]\displaystyle{ d_i }[/math] 是顶点的度i。

We can define a triangle among the triple of vertices [math]\displaystyle{ i }[/math], [math]\displaystyle{ j }[/math], and [math]\displaystyle{ 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]\displaystyle{ i }[/math], [math]\displaystyle{ j }[/math], and [math]\displaystyle{ 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]\displaystyle{ i }[/math]，[math]\displaystyle{ j }[/math]和[math]\displaystyle{ 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）}。

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

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

我们也可以将顶点[math]\displaystyle{ i }[/math]所涉及的三角形的数量定义为[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ i }[/math] is [math]\displaystyle{ \tau (i) = \binom{d_i}{2} }[/math], assuming [math]\displaystyle{ d_i \ge 2 }[/math]. We can express [math]\displaystyle{ \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]\displaystyle{ i }[/math] is [math]\displaystyle{ \tau (i) = \binom{d_i}{2} }[/math], assuming [math]\displaystyle{ d_i \ge 2 }[/math]. We can express [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \tau（i）= \binom{d_i}{2} }[/math]， 假设[math]\displaystyle{ d_i \ge 2 }[/math]。 我们可以表示[math]\displaystyle{ \tau（G）= \frac{1}{3} \sum_{i\in V} \ \tau（i） }[/math]。

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

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

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

Transitivity

传递性

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

Another measure for the presence of triadic closure is transitivity, defined as [math]\displaystyle{ 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]\displaystyle{ T（G）= \frac{3\delta（G）}{\tau（G）} }[/math]。

Causes and effects

因果 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.

在信任网络中，由于传递属性，可能会发生三元闭包。 如果节点A信任节点B，并且节点B信任节点C，则节点A将具有信任节点C的基础。在社交网络中，会发生强烈的三元闭包，因为具有共同邻居B的节点A和C拥有更多的机会来满足并因此建立起相对薄弱的纽带。 节点B还具有将A和C聚在一起以减少两个单独关系中的潜在压力的动机。

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，并且具有很高的聚类系数。 但是，不遵循该原理的网络将连接不良，一旦包含负面关系，网络可能会变得不稳定。

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.

三合闭包是网络如何随着时间演变的良好模型。 尽管简单图论倾向于在某个时间点分析网络，但应用三元闭包原理可以预测网络内联系的发展并显示连通性的发展。

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

In social networks, triadic closure facilitates cooperative behavior, but when new connections are made via

在社交网络中，三合闭包有利于合作行为，但是当通过 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.

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.

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

强三元闭包性 Strong Triadic Closure Property 和本地网桥 Local Bridges

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.

强三元闭包性是，如果一个节点与两个邻居有牢固的联系，则这些邻居之间必须至少有一条弱联系。 另一方面，当节点充当两个其他未连接的相邻节点之间的网守时，则发生本地网桥。 在遵循“强三元闭合性”的网络中，本地网桥中涉及的节点之间的联系之一必须是弱联系。

Proof by contradiction

矛盾证明 Proof By Contradiction

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore, B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.

假设节点B是节点A和C之间的本地桥梁，以使所涉及的节点之间没有弱联系。 因此，B与A和C都具有牢固的联系。根据“强三元闭包性”的定义，节点A和C之间将形成弱联系。但是，这与B是本地网守的事实相矛盾。 因此，本地网桥中涉及的至少一个节点需要是弱连接，以防止三元闭包的发生。

References

Category:Social systems

类别: 社会系统

Category:Sociological terminology

类别: 社会学术语

This page was moved from wikipedia:en:Triadic closure. Its edit history can be viewed at 三元闭包/edithistory