# 枢纽节点

In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the network topology. Hubs can be found in many real networks, such as Brain Network or Internet.

In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the network topology. Hubs can be found in many real networks, such as Brain Network or Internet.

A hub is a component of a network with a high-degree node. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links (degrees) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size N of the network and average degree <k> constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree k is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree k while the other nodes have a small number of links.

A hub is a component of a network with a high-degree node. Hubs have a significantly larger number of links in comparison with other nodes in the network. The number of links (degrees) for a hub in a scale-free network is much higher than for the biggest node in a random network, keeping the size N of the network and average degree <k> constant. The existence of hubs is the biggest difference between random networks and scale-free networks. In random networks, the degree k is comparable for every node; it is therefore not possible for hubs to emerge. In scale-free networks, a few nodes (hubs) have a high degree k while the other nodes have a small number of links.

## Emergence 涌现 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.

• (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）无标度网络假设节点数量N保持持续的增长，而随机网络则假设节点数量是固定的。在无标度网络中，最大枢纽的度随着网络规模的增大，呈多项式地上升。因此，在无标度网络中，枢纽的度可以很高。而在随机网络中，最大节点的度随N的增大而呈对数式（或更慢）的上升。因此即使在一个非常大的随机网络中，枢纽的数量也会很小。
• (b) A new node in a scale-free network has a tendency to link to a node with a higher degree, compared to a new node in a random network which links itself to a random node. This process is called preferential attachment. The tendency of a new node to link to a node with a high degree k is characterized by power-law distribution (also known as rich-gets-richer process). This idea was introduced by Vilfredo Pareto and it explained why a small percentage of the population earns most of the money. This process is present in networks as well, for example 80 percent of web links point to 15 percent of webpages. The emergence of scale-free networks is not typical only of networks created by human action, but also of such networks as metabolic networks or illness networks. This phenomenon may be explained by the example of hubs on the World Wide Web such as Facebook or Google. These webpages are very well known and therefore the tendency of other webpages pointing to them is much higher than linking to random small webpages.

The mathematical explanation for Barabási–Albert model:

The mathematical explanation for Barabási–Albert model:

Barabási-Albert模型的数学解释:

The network begins with an initial connected network of $\displaystyle{ m_0 }$ nodes.

The network begins with an initial connected network of $\displaystyle{ m_0 }$ nodes.

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

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

$\displaystyle{ p_i = \frac{k_i}{\sum_j k_j}, }$

$\displaystyle{ p_i = \frac{k_i}{\sum_j k_j}, }$

where $\displaystyle{ k_i }$ is the degree of the node $\displaystyle{ i }$ and the sum is taken over all pre-existing nodes $\displaystyle{ j }$ (i.e. the denominator results in twice the current number of edges in the network).

where $\displaystyle{ k_i }$ is the degree of the node $\displaystyle{ i }$ and the sum is taken over all pre-existing nodes $\displaystyle{ j }$ (i.e. the denominator results in twice the current number of edges in the network).

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.

## Attributes 属性

There are several attributes of Hubs in a Scale-Free Network

There are several attributes of Hubs in a Scale-Free Network

### Shortening the path lengths in a network 缩短网络路径长度

The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes. Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as "small" or "ultra small". While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.

The more observable hubs are in a network, the more they shrink distances between nodes. In a scale-free network, hubs serve as bridges between the small degree nodes. Since the distance of two random nodes in a scale-free network is small, we refer to scale-free networks as "small" or "ultra small". While the difference between path distance in a very small network may not be noticeable, the difference in the path distance between a large random network and a scale-free network is remarkable.

Average path length in scale-free networks:

Average path length in scale-free networks:

$\displaystyle{ \ell\sim\frac{\ln N}{\ln \ln N}. }$

$\displaystyle{ \ell\sim\frac{\ln N}{\ln \ln N}. }$

### Aging of hubs (nodes) 枢纽（节点）的老化

The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks. The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).

The phenomenon present in real networks, when older hubs are shadowed in a network. This phenomenon is responsible for changes in evolution and topology of networks. The example of aging phenomenon may be the case of Facebook overtaking the position of the largest hub on the Web, Google(which was the largest node since 2000).

### Robustness and Attack Tolerance 鲁棒性和抗毁性

During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.

During the random failure of nodes or targeted attack hubs are key components of the network. During the random failure of nodes in network hubs are responsible for exceptional robustness of network.

The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.

The chance that a random failure would delete the hub is very small, because hubs coexists with a large number of small degree nodes. The removal of small degree nodes does not have a large effect on integrity of network. Even though the random removal would hit the hub, the chance of fragmantation of network is very small because the remaining hubs would hold the network together. In this case, hubs are the strength of a scale-free networks.

During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system. On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.

During a targeted attack on hubs, the integrity of a network will fall apart relatively fast. Since small nodes are predominantly linked to hubs, the targeted attack on the largest hubs results in destroys the network in a short period of time. The financial market meltdown in 2008 is an example of such a network failure, when bankruptcy of the largest players (hubs) led to a continuous breakdown of the whole system. On the other hand, it may have a positive effect when removing hubs in a terrorist network; targeted node deletion may destroy the whole terrorist group. The attack tolerance of a network may be increased by connecting its peripheral nodes, however it requires to double the number of links.

### Degree correlation 度相关

The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.

The perfect degree correlation means that each degree-k node is connected only to the same degree-k nodes. Such connectivity of nodes decide the topology of networks, which has an effect on robustness of network, the attribute discussed above. If the number of links between the hubs is the same as would be expected by chance, we refer to this network as Neutral Network. If hubs tend to connected to each other while avoiding linking to small-degree nodes we refer to this network as Assortative Network. This network is relatively resistant against attacks, because hubs form a core group, which is more reduntant against hub removal. If hubs avoid connecting to each other while linking to small-degree nodes, we refer to this network as Disassortative Network. This network has a hub-and-spoke character. Therefore, if we remove the hub in this type of network, it may damage or destroy the whole network.