三元闭包

本词条由信白初步翻译

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.

三元闭包 Triadic Closure 是社会网络 Social Network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。三元闭包指的是由A，B，C三个节点 Nodes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 Strong Tie ，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

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.

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。 在那里，他综合了弗里茨·海德 Fritz Heider于1946年提出的认知平衡 Cognitive Balance 理论以及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 Closure。

Measurements

模板:Unreferenced section 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.

测量

最常见的测量一张图 Graph的三元闭包性质的两种方法（排名不分先后）是采用该图的聚集系数 Clustering Coefficient 和传递性 Transitivity。

聚集系数

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:

测量三元闭包是否出现的方法之一是聚集系数，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]是无向简单图 Undirected Simple Graph（即，没有自环 Self-loops或多重边 Multiple Edges的图），其中[math]\displaystyle{ V }[/math]为顶点 Vertices集，[math]\displaystyle{ E }[/math]为边 Edges集。 另外，令[math]\displaystyle{ N = |V| }[/math]和[math]\displaystyle{ M = |E| }[/math]分别表示图[math]\displaystyle{ G }[/math]中顶点和边的数量，并令[math]\displaystyle{ d_i }[/math] 是顶点[math]\displaystyle{ i }[/math]的度 Degree。

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)}.

我们可以将由顶点[math]\displaystyle{ i }[/math],[math]\displaystyle{ j }[/math]和[math]\displaystyle{ k }[/math]所组成的三元组定义为一个由边 Edges集[math]\displaystyle{ ((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]\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，因此图[math]\displaystyle{ G }[/math]中三角形的个数为[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]\displaystyle{ i }[/math]所涉及的三角形的数量为[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]的聚集系数[math]\displaystyle{ c(i) }[/math] 是其拥有的三角形的占比，即[math]\displaystyle{ \frac{\delta(i)}{\tau(i)} }[/math]。 因此，图[math]\displaystyle{ G }[/math]的聚集系数[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]给出，其中[math]\displaystyle{ N_2 }[/math]是度至少为2的顶点数量。

传递性 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].

测量三元闭包是否出现的另一方法是传递性，定义为[math]\displaystyle{ T(G)= \frac{3\delta(G)}{\tau(G)} }[/math]。

形成及其影响

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.

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.

在一个信任网络中，三元闭包性质的出现往往是由于传递性。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。在社会网络中，强三元闭包 Strong Triadic Closure性质的出现往往是由于节点A与节点C拥有共同邻居节点B，因为他们相遇的机会将会增加进而至少产生一条弱联系 Weak Tie。此外，由于两段分离的关系所带来的潜在压力，节点B也具有将节点A和节点C聚在一起的动机。

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。

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.

三元闭包是分析网络如何随时间演变的一个良好模型。 简单图论 Simple Graph Theory倾向于在某个时点分析网络，而应用三元闭包原理则可以预测网络中联系的形成，以及网络连通性的发展。

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，三元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。

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