更改

添加4,091字节 、 2020年5月18日 (一) 17:21
第472行: 第472行:     
=== Barabási–Albert (BA) 优先链接模型 ===
 
=== Barabási–Albert (BA) 优先链接模型 ===
 +
The [[Barabási–Albert model]] is a random network model used to demonstrate a preferential attachment or a "rich-get-richer" effect.  In this model, an edge is most likely to attach to nodes with higher degrees.The network begins with an initial network of ''m''<sub>0</sub> nodes.  ''m''<sub>0</sub>&nbsp;≥&nbsp;2 and the degree of each node in the initial network should be at least&nbsp;1, otherwise it will always remain disconnected from the rest of the network.
    
[[BA模型]]是一个随机网络模型,用于说明偏好依附效应(优先链接)preferential attachment或“富人越富”效应。 在这个模型中,边最有可能附着在度数较高的节点上。 这个网络从一个 ''m''<sub>0</sub>节点的初始网络开始。 ''m''<sub>0</sub>&nbsp;≥&nbsp;2,初始网络中每个节点的度至少为&nbsp;1,否则它将始终与网络的其余部分断开。
 
[[BA模型]]是一个随机网络模型,用于说明偏好依附效应(优先链接)preferential attachment或“富人越富”效应。 在这个模型中,边最有可能附着在度数较高的节点上。 这个网络从一个 ''m''<sub>0</sub>节点的初始网络开始。 ''m''<sub>0</sub>&nbsp;≥&nbsp;2,初始网络中每个节点的度至少为&nbsp;1,否则它将始终与网络的其余部分断开。
 +
 +
 +
In the BA model, new nodes are added to the network one at a time. Each new node is connected to <math>m</math> existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability ''p''<sub>''i''</sub> that the new node is connected to node ''i'' is<ref name=RMP>{{Cite journal
 +
|url        = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf
 +
|author1    = R. Albert
 +
|author2    = A.-L. Barabási
 +
|title      = Statistical mechanics of complex networks
 +
|journal    = [[Reviews of Modern Physics]]
 +
|volume      = 74
 +
|issue    = 1
 +
|pages      = 47–97
 +
|year        = 2002
 +
|doi        = 10.1103/RevModPhys.74.47
 +
|bibcode    = 2002RvMP...74...47A
 +
|arxiv      = cond-mat/0106096
 +
|url-status    = dead
 +
|archiveurl  = https://web.archive.org/web/20150824235818/http://www3.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf
 +
|archivedate = 2015-08-24
 +
|citeseerx    = 10.1.1.242.4753
 +
}}</ref>
 +
 +
: <math>p_i = \frac{k_i}{\sum_j k_j},</math>
 +
 +
where ''k''<sub>''i''</sub> is the degree of node ''i''. Heavily linked nodes ("hubs") tend to quickly accumulate even more links, while nodes with only a few links are unlikely to be chosen as the destination for a new link. The new nodes have a "preference" to attach themselves to the already heavily linked nodes.
 +
 +
[[File:Barabasi-albert model degree distribution.svg|thumb|The degree distribution of the BA Model, which follows a power law. In loglog scale the power law function is a straight line.<ref name=Barabasi1999>{{Cite journal
 +
|url        = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/EmergenceRandom_Science%20286,%20509-512%20(1999).pdf
 +
|author      = [[Albert-László Barabási]] & [[Réka Albert]]
 +
|title      = Emergence of scaling in random networks
 +
|journal    = [[Science (journal)|Science]]
 +
|volume      = 286
 +
|pages      = 509&ndash;512
 +
|date        = October 1999
 +
|doi        = 10.1126/science.286.5439.509
 +
|issue      = 5439
 +
|pmid        = 10521342
 +
|arxiv      = cond-mat/9910332
 +
|bibcode    = 1999Sci...286..509B
 +
|url-status    = dead
 +
|archiveurl  = https://web.archive.org/web/20120417112354/http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/EmergenceRandom_Science%20286,%20509-512%20(1999).pdf
 +
|archivedate = 2012-04-17
 +
}}</ref>]]
 +
The degree distribution resulting from the BA model is scale free, in particular, it is a power law of the form:
 +
: <math>P(k)\sim k^{-3} \, </math>
    
在BA模型中,每次向网络中添加一个新节点。每个新节点和已有的<math>m</math>个节点相连接,连接的概率正比于每个已存在节点当前的度。形式上,新节点与节点''i''相连的概率''p''<sub>''i''</sub>为<ref name=RMP>{{Cite journal
 
在BA模型中,每次向网络中添加一个新节点。每个新节点和已有的<math>m</math>个节点相连接,连接的概率正比于每个已存在节点当前的度。形式上,新节点与节点''i''相连的概率''p''<sub>''i''</sub>为<ref name=RMP>{{Cite journal
第517行: 第562行:  
BA模型得到的度分布是无标度的,特别是它的形式是幂律:
 
BA模型得到的度分布是无标度的,特别是它的形式是幂律:
 
: <math>P(k)\sim k^{-3} \, </math>
 
: <math>P(k)\sim k^{-3} \, </math>
 +
 +
 +
Hubs exhibit high betweenness centrality which allows short paths to exist between nodes. As a result, the BA model tends to have very short average path lengths. The clustering coefficient of this model also tends to 0.
 +
While the diameter, D, of many models including the Erdős Rényi random graph model and several small world networks is proportional to log N, the BA model exhibits D~loglogN (ultrasmall world).<ref>{{cite journal|last=Cohen|first=R. |title=Scale-free networks are ultrasmall|journal=Phys. Rev. Lett.|year=2003|volume=90|pages=058701|url=http://havlin.biu.ac.il/Publications.php?keyword=Scale-free+networks+are+ultrasmall&year=*&match=all|doi=10.1103/PhysRevLett.90.058701|pmid=12633404|first2=S.|last2=Havlin|issue=5|bibcode=2003PhRvL..90e8701C |arxiv=cond-mat/0205476}}</ref>
 +
Note that the average path length scales with N as the diameter.
    
中心节点表现出很高的介数中心性,这使得节点之间存在捷径。因此,BA模型的平均路径长度往往很短。该模型的聚类系数也趋于0。
 
中心节点表现出很高的介数中心性,这使得节点之间存在捷径。因此,BA模型的平均路径长度往往很短。该模型的聚类系数也趋于0。
198

个编辑