三元闭包

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

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.

三元闭合是社会网络理论中的一个概念，最早由德国社会学家格奥尔格 · 西梅尔在他1908年出版的《社会学》一书中提出。三元闭包是三个节点 a、 b 和 c 之间的性质，如果 A-B 和 A-C 之间存在强联系，则 B-C 之间只存在强联系。这个特性在非常大的复杂网络中过于极端，不适用，但是它是一个有用的现实简化，可以用来理解和预测网络。

History

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.

在1973年发表的文章《弱关系的力量》中，三合一的封闭马克·格兰诺维特广受欢迎。在那里，他综合了弗里茨 · 海德1946年首次提出的认知平衡理论和西梅尔对社会网络的理解。一般来说，认知平衡指的是两个人倾向于对一个事物有相同的感觉。如果三个人组成的三位一体不是封闭的，那么与两个人都有联系的人就会想要封闭这个三位一体，以便在关系网络中实现封闭。

Measurements

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.

一个图的三元闭包的两个最常见的度量是该图的集聚系数和传递性(没有特定的顺序)。

Clustering coefficient

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:

三元闭合的一个衡量标准是集聚系数，具体如下:

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.

设 g = (v，e) </math > 是一个无向简单图(即没有自回路或多条边的图) ，v 是顶点集，e 是边集。同时，让 < math > n = | v | </math > 和 < math > m = | e | </math > 分别表示 g 中的顶点数和边数，让 < math > di </math > 表示顶点的度数。

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)}.

我们可以定义三个顶点之间的三角形 < math > ，< math > j </math > ，< math > k </math > 是一个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].

我们还可以定义顶点 < math > i </math > 所涉及的三角形的个数为 < math > delta (i) </math > ，并且，当每个三角形被计算三次时，我们可以将 g 中三角形的个数表示为 < 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]\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，一个三元组只需要两条强边即可形成。因此，在三元闭包假设下，顶点 i </math > i </math > 的理论三元数应该是 < 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]\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，对于一个顶点 < math > i </math > 与 < math > d _ i ge 2 </math > 相关的集聚系数 </math > c (i) </math > i </math > 是顶点 < math > i </math > 的三元组的分数，它是封闭的，可以被测量为 < math > frac { delta (i)}{ tau (i)} </math > 。因此，图的 c (g) </math > g </math > 的集聚系数由 < math > c (g) = frac {1}{ n _ 2} sum { i in v，d _ i ge2} c (i) </math > 给出，其中 < math > n _ 2 </math > 是度数至少为2的节点数。

Transitivity

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].

存在三元闭包的另一个度量是传递性，定义为 < math > t (g) = frac {3 delta (g)}{ tau (g)} </math > 。

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，由于三元闭包具有传递性，容易产生三元闭包。如果节点 a 信任节点 b，而节点 b 信任节点 c，则节点 a 将有信任节点 c 的基础。在社交网络中，强三元闭包的出现是因为与共同邻居 b 相遇的节点 a 和 c 有更多的机会相遇，从而产生至少个弱关系。节点 b 也有动机将 a 和 c 结合在一起，以减少两个独立关系中的潜在压力。

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.

遵循这一原则的网络变得高度互联，并具有非常高的聚类系数。然而，不遵循这一原则的网络结果是连接不良，一旦包括负面关系，就可能遭受不稳定。

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.

三元闭包是一个很好的模型，用来说明网络将如何随着时间的推移而演化。简单的图论倾向于在一个时间点上分析网络，而应用三并元闭包原理可以预测网络中关系的发展，并显示连通性的进展。

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，三元闭包促进了合作行为，但是当新的连接是通过

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.

根据现有关系的推荐，合作者的全球平均比例比个人从总体人口中随机选择新关系的比例要低。两个可能的影响是通过结构和信息化建设。结构构造起源于向高集聚性的倾向。这种信息结构来自于这样一种假设，即一个人知道一个朋友的朋友的某些事情，而不是一个随机的陌生人。

Strong Triadic Closure Property and 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.

强三元闭包性质是指，如果一个节点与两个邻居之间有强关联，那么这两个邻居之间至少有弱关联。另一方面，当一个节点在两个没有连接的相邻节点之间充当网守时，就会出现本地桥接。在遵循强三元闭包性质的网络中，局部桥梁中涉及的节点之间的联系之一需要是弱联系。

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.

设 b 节点是 a 节点和 c 节点之间的本地桥梁，这样所涉及的节点之间就不存在弱联系。因此，b 与 a 和 c 都有很强的联系。根据强三元闭包的定义，在节点 a 和 c 之间会产生弱联系。因此，局部桥梁中至少有一个节点需要是弱连接，以防止三分闭合发生。

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