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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'''<font color="#FF8000">聚类系数 Clustering Coefficient </font>'''和'''<font color="#FF8000">可传递性 Transitivity </font>'''。

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'''<font color="#FF8000">聚类系数 Clustering Coefficient </font>'''和'''<font color="#FF8000">可传递性 Transitivity </font>'''。

===Clustering coefficient===

==Clustering coefficient==

聚类系数

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:

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)}。

我们可以在三个顶点<math>i</math>，<math>j</math>和<math>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>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 > 。

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，因此我们可以表示 G中的三角形为<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>.

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 > 。

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>。

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。

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 \ 在V中，d_i \ ge 2} c（i）</ math>，其中<math> N_2 </ math>是度数至少为2的节点数。

 

 

===Transitivity===

===Transitivity===