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枢纽是拥有大度节点网络的重要构件。与网络中的其他节点相比，枢纽拥有的链接数量明显更多。在保持网络规模''N''和平均度 ''<k>''不变的情况下，无标度网络中枢纽拥有的链接数（度）远远高于随机网络中链接数最大的节点。枢纽的存在是随机网络和无标度网络的最大区别。在随机网络中，对于每个节点而言，度''k''是相当的，因此不可能出现枢纽节点。而在无标度网络中，少数节点（即枢纽）具有较高的度值 ''k''，而其他节点则只拥有少量的链接。

枢纽是拥有大度节点网络的重要构件。与网络中的其他节点相比，枢纽拥有的链接数量明显更多。在保持网络规模''N''和平均度 ''<k>''不变的情况下，无标度网络中枢纽拥有的链接数（度）远远高于随机网络中链接数最大的节点。枢纽的存在是随机网络和无标度网络的最大区别。在随机网络中，对于每个节点而言，度''k''是相当的，因此不可能出现枢纽节点。而在无标度网络中，少数节点（即枢纽）具有较高的度值 ''k''，而其他节点则只拥有少量的链接。

== Emergence 出现==

== Emergence 涌现==

[[Image:Scale-free network sample.png|thumb|Example of a random network and a scale-free network|400px|right|

[[Image:Scale-free network sample.png|thumb|Example of a random network and a scale-free network|400px|right|

Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks (Barabási–Albert model) are different from random networks (Erdős–Rényi model) in two aspects: (a) growth, (b) preferential attachment.

Emergence of hubs can be explained by the difference between scale-free networks and random networks. Scale-free networks (Barabási–Albert model) are different from random networks (Erdős–Rényi model) in two aspects: (a) growth, (b) preferential attachment.

枢纽的成因可以用无标度网络和随机网络的区别来解释。'''<font color="#ff8000">无标度网络 Scale-Free Networks</font>'''（Barabási-Albert模型）与'''<font color="#ff8000">随机网络 Random Networks</font>'''（Erdős–Rényi model）的区别主要存在于如下两个方面: （a）'''<font color="#ff8000">增长 Growth</font>'''，（b）'''<font color="#ff8000">优先链接 Preferential Attachment</font>'''。

枢纽的涌现可以用无标度网络和随机网络的区别来解释。'''<font color="#ff8000">无标度网络 Scale-Free Networks</font>'''（Barabási-Albert模型）与'''<font color="#ff8000">随机网络 Random Networks</font>'''（Erdős–Rényi model）的区别主要存在于如下两个方面: （a）'''<font color="#ff8000">增长 Growth</font>'''，（b）'''<font color="#ff8000">优先链接 Preferential Attachment</font>'''。

* (a) Scale-free networks assume a continuous growth of the number of nodes ''N'', compared to random networks which assume a fixed number of nodes. In scale-free networks the degree of the largest hub rises polynomially with the size of the network. Therefore, the degree of a hub can be high in a scale-free network. In random networks the degree of the largest node rises logaritmically (or slower) with N, thus the hub number will be small even in a very large network.

* (a) Scale-free networks assume a continuous growth of the number of nodes ''N'', compared to random networks which assume a fixed number of nodes. In scale-free networks the degree of the largest hub rises polynomially with the size of the network. Therefore, the degree of a hub can be high in a scale-free network. In random networks the degree of the largest node rises logaritmically (or slower) with N, thus the hub number will be small even in a very large network.

New nodes are added to the network one at a time. Each new node is connected to <math>m \le m_0</math> existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability <math>p_i</math> that the new node is connected to node <math>i</math> is

New nodes are added to the network one at a time. Each new node is connected to <math>m \le m_0</math> existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability <math>p_i</math> that the new node is connected to node <math>i</math> is

每次向网络添加一个新节点。每个新的节点都被连接到<math>m \le m_0</math>现有的节点，其概率与现有节点已经拥有的链接数量成正比。形式上，新节点与节点<math>p_i</math>相连的概率是

每次向网络添加一个新节点。每个新的节点都被链接到<math>m \le m_0</math>个现有的节点，其概率与现有节点已经拥有的链接数量成正比。形式上，新节点与现有节点<math>p_i</math>相连的概率为

 

 

<math>p_i = \frac{k_i}{\sum_j k_j},</math>

<math>p_i = \frac{k_i}{\sum_j k_j},</math>

<math>p_i = \frac{k_i}{\sum_j k_j},</math>

<math>p_i = \frac{k_i}{\sum_j k_j},</math>

where <math>k_i</math> is the degree of the node <math>i</math> and the sum is taken over all pre-existing nodes <math>j</math> (i.e. the denominator results in twice the current number of edges in the network).

where <math>k_i</math> is the degree of the node <math>i</math> and the sum is taken over all pre-existing nodes <math>j</math> (i.e. the denominator results in twice the current number of edges in the network).

where <math>k_i</math> is the degree of the node <math>i</math> and the sum is taken over all pre-existing nodes <math>j</math> (i.e. the denominator results in twice the current number of edges in the network).

where <math>k_i</math> is the degree of the node <math>i</math> and the sum is taken over all pre-existing nodes <math>j</math> (i.e. the denominator results in twice the current number of edges in the network).

其中<math>k_i</math>是节点<math>i</math>的度数，这个和取自所有预先存在的节点<math>j</math>。（分母的结果是当前网络中边的两倍）。

其中<math>k_i</math>是节点<math>i</math>的度，求和则是针对所有预先存在的节点<math>j</math>进行的。（分母的数值是网络边数的两倍）。

 

 

Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.

Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.

Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.

Emergence of hubs in networks is also related to time. In scale-free networks, nodes which emerged earlier have a higher chance of becoming a hub than latecomers. This phenomenon is called first-mover advantage and it explains why some nodes become hubs and some do not. However, in a real network, the time of emergence is not the only factor that influences the size of the hub. For example, Facebook emerged 8 years later after Google became the largest hub on the World Wide Web and yet in 2011 Facebook became the largest hub of WWW. Therefore, in real networks the growth and the size of a hub depends also on various attributes such as popularity, quality or the aging of a node.

网络中枢纽节点的涌现也与时间有关。在无标度网络中，较早出现的节点比后来者更有可能成为枢纽。这种现象被称为'''<font color="#ff8000">先发优势 First-Mover Advantage</font>'''，它解释了为什么一些节点成为枢纽，而一些没有。然而，在一个真实的网络中，出现的时间并不是影响枢纽规模的唯一因素。例如，在谷歌成为全球最大的互联网中心8年后，Facebook 出现了，然而在2011年，Facebook 成为了全球最大的互联网中心。因此，在实际网络中，枢纽的增长和规模也取决于各种属性，如节点的流行程度、质量或老化程度。

网络中枢纽的涌现也与时间有关。在无标度网络中，较早出现的节点比后来者更有可能成为枢纽。这种现象被称为'''<font color="#ff8000">先发优势 First-Mover Advantage</font>'''，它解释了为什么一些节点成为枢纽，而一些没有。然而，在一个真实的网络中，涌现的时间并不是影响枢纽规模的唯一因素。例如，在谷歌成为全球最大互联网枢纽的8年后，Facebook出现了，并于2011年成为了全球最大的互联网枢纽。因此，在实际网络中，枢纽的增长和规模也取决于各种各样的其他特性，如节点的流行程度、质量或老化程度。

== Attributes 属性==

== Attributes 属性==