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=== 配置模型 ===

=== 配置模型 ===

配置模型以一个度序列<ref>{{Cite journal|last=Bender|first=Edward A|last2=Canfield|first2=E.Rodney|date=May 1978|title=The asymptotic number of labeled graphs with given degree sequences|journal=Journal of Combinatorial Theory, Series A|volume=24|issue=3|pages=296–307|doi=10.1016/0097-3165(78)90059-6|issn=0097-3165}}</ref><ref name=":0">{{Cite journal|last=Molloy|first=Michael|last2=Reed|first2=Bruce|date=March 1995|title=A critical point for random graphs with a given degree sequence|journal=Random Structures & Algorithms|language=en|volume=6|issue=2–3|pages=161–180|doi=10.1002/rsa.3240060204|issn=1042-9832|citeseerx=10.1.1.24.6195}}</ref> 或度分布<ref name=":1">{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|journal=Physical Review E|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|pmid=11497662|arxiv=cond-mat/0007235|bibcode=2001PhRvE..64b6118N}}</ref> （用于生成度序列）作为输入, 生成除了度序列外各方面随机连接的图。这意味着对于给定的度序列，图是一致随机地从符合这个度序列的所有图的集合中选择的。图中任意节点的度<math>k</math>是一个[[独立同分布的随机变量|独立同分布]]，其值为整数。When <math display="inline">\mathbb E [k^2] - 2 \mathbb E [k]>0</math>, the configuration graph contains the [[Giant component|giant connected component]], which has infinite size.<ref name=":0" /> The rest of the components have finite sizes, which can be quantified with the notion of the size distribution. The probability <math>w(n)</math> that a randomly sampled node is connected to a component of size <math>n</math> is given by [[convolution power]]s of the degree distribution:<ref>{{Cite journal|last=Kryven|first=Ivan|date=2017-05-02|title=General expression for the component size distribution in infinite configuration networks|journal=Physical Review E|volume=95|issue=5|pages=052303|doi=10.1103/PhysRevE.95.052303|pmid=28618550|arxiv=1703.05413|bibcode=2017PhRvE..95e2303K}}</ref><math display="block">w(n)=\begin{cases}

配置模型以一个度序列<ref>{{Cite journal|last=Bender|first=Edward A|last2=Canfield|first2=E.Rodney|date=May 1978|title=The asymptotic number of labeled graphs with given degree sequences|journal=Journal of Combinatorial Theory, Series A|volume=24|issue=3|pages=296–307|doi=10.1016/0097-3165(78)90059-6|issn=0097-3165}}</ref><ref name=":0">{{Cite journal|last=Molloy|first=Michael|last2=Reed|first2=Bruce|date=March 1995|title=A critical point for random graphs with a given degree sequence|journal=Random Structures & Algorithms|language=en|volume=6|issue=2–3|pages=161–180|doi=10.1002/rsa.3240060204|issn=1042-9832|citeseerx=10.1.1.24.6195}}</ref> 或度分布<ref name=":1">{{Cite journal|last=Newman|first=M. E. J.|last2=Strogatz|first2=S. H.|last3=Watts|first3=D. J.|date=2001-07-24|title=Random graphs with arbitrary degree distributions and their applications|journal=Physical Review E|volume=64|issue=2|pages=026118|doi=10.1103/PhysRevE.64.026118|pmid=11497662|arxiv=cond-mat/0007235|bibcode=2001PhRvE..64b6118N}}</ref> （用于生成度序列）作为输入, 生成除了度序列外各方面随机连接的图。这意味着对于给定的度序列，图是一致随机地从符合这个度序列的所有图的集合中选择的。图中任意节点的度<math>k</math>是一个[[独立同分布的随机变量|独立同分布]]，其值为整数。当 <math display="inline">\mathbb E [k^2] - 2 \mathbb E [k]>0</math> 时，配置图包含[[Giant component|giant connected component]]，其大小是无穷。<ref name=":0" /> 其他分量大小是有限的，可以用大小分布来定量表示。图中任意取的一个节点和大小为<math>n</math>的分量相连接的概率由度分布的[[卷积幂]]给出：<ref>{{Cite journal|last=Kryven|first=Ivan|date=2017-05-02|title=General expression for the component size distribution in infinite configuration networks|journal=Physical Review E|volume=95|issue=5|pages=052303|doi=10.1103/PhysRevE.95.052303|pmid=28618550|arxiv=1703.05413|bibcode=2017PhRvE..95e2303K}}</ref><math display="block">w(n)=\begin{cases}

\frac{\mathbb E [k]}{n-1} u_1^{*n}(n-2),& n>1, \\

\frac{\mathbb E [k]}{n-1} u_1^{*n}(n-2),& n>1, \\

u(0) & n=1,

u(0) & n=1,

\end{cases}</math>where <math>u(k)</math> denotes the degree distribution and <math>u_1(k)=\frac{(k+1)u(k+1)}{\mathbb E[k]}</math>. The giant component can be destroyed by randomly removing the critical fraction <math>p_c</math> of all edges. This process is called [[Percolation theory|percolation on random networks]]. When the second moment of the degree distribution is finite, <math display="inline">\mathbb E [k^2]<\infty</math>, this critical edge fraction is given by<ref>{{Cite journal|last=Kryven|first=Ivan|date=2018-01-01|title=Analytic results on the polymerisation random graph model|journal=Journal of Mathematical Chemistry|language=en|volume=56|issue=1|pages=140–157|doi=10.1007/s10910-017-0785-1|issn=0259-9791|doi-access=free}}</ref> <math>p_c=1-\frac{\mathbb E[k]}{ \mathbb E [k^2] - \mathbb E[k]}</math>, and the [[Average path length|average vertex-vertex distance]] <math>l</math> in the giant component scales logarithmically with the total size of the network, <math>l = O(\log N) </math>.<ref name=":1" />

\end{cases}</math>其中 <math>u(k)</math> 表示度分布且 <math>u_1(k)=\frac{(k+1)u(k+1)}{\mathbb E[k]}</math>。 The giant component can be destroyed by randomly removing the critical fraction <math>p_c</math> of all edges. This process is called [[Percolation theory|percolation on random networks]]. When the second moment of the degree distribution is finite, <math display="inline">\mathbb E [k^2]<\infty</math>, this critical edge fraction is given by<ref>{{Cite journal|last=Kryven|first=Ivan|date=2018-01-01|title=Analytic results on the polymerisation random graph model|journal=Journal of Mathematical Chemistry|language=en|volume=56|issue=1|pages=140–157|doi=10.1007/s10910-017-0785-1|issn=0259-9791|doi-access=free}}</ref> <math>p_c=1-\frac{\mathbb E[k]}{ \mathbb E [k^2] - \mathbb E[k]}</math>, and the [[Average path length|average vertex-vertex distance]] <math>l</math> in the giant component scales logarithmically with the total size of the network, <math>l = O(\log N) </math>.<ref name=":1" />

In the directed configuration model, the degree of a node is given by two numbers, in-degree <math>k_\text{in}</math> and out-degree <math>k_\text{out}</math>, and consequently, the degree distribution is two-variate. The expected number of in-edges and out-edges coincides, so that <math display="inline">\mathbb E [k_\text{in}]=\mathbb E [k_\text{out}]</math>. The directed configuration model contains the [[giant component]] iff<ref>{{Cite journal|last=Kryven|first=Ivan|date=2016-07-27|title=Emergence of the giant weak component in directed random graphs with arbitrary degree distributions|journal=Physical Review E|volume=94|issue=1|pages=012315|doi=10.1103/PhysRevE.94.012315|pmid=27575156|arxiv=1607.03793|bibcode=2016PhRvE..94a2315K}}</ref><math display="block">2\mathbb{E}[k_\text{in}]  

In the directed configuration model, the degree of a node is given by two numbers, in-degree <math>k_\text{in}</math> and out-degree <math>k_\text{out}</math>, and consequently, the degree distribution is two-variate. The expected number of in-edges and out-edges coincides, so that <math display="inline">\mathbb E [k_\text{in}]=\mathbb E [k_\text{out}]</math>. The directed configuration model contains the [[giant component]] iff<ref>{{Cite journal|last=Kryven|first=Ivan|date=2016-07-27|title=Emergence of the giant weak component in directed random graphs with arbitrary degree distributions|journal=Physical Review E|volume=94|issue=1|pages=012315|doi=10.1103/PhysRevE.94.012315|pmid=27575156|arxiv=1607.03793|bibcode=2016PhRvE..94a2315K}}</ref><math display="block">2\mathbb{E}[k_\text{in}]