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第45行： − + − 复杂网络失去巨分量阈值的数学推导是基于莫莱-里德准则。+ − − − <math> − − 《数学》 − − \begin{align} − − \begin{align} − − 开始{ align } − − \kappa \equiv \frac{\langle k^2 \rangle}{\langle k \rangle} > 2 − − \kappa \equiv \frac{\langle k^2 \rangle}{\langle k \rangle} > 2 − − \end{align} − − \end{align} − − 结束{ align } − </math> − 数学+ − − − − − 莫洛伊-里德准则的基本原理是，为了使一个巨型部件存在，网络中的每个节点平均必须至少有两个链路。这类似于每个人牵着另外两个人的手形成一个链条。使用这个标准和一个复杂的数学证明，我们可以推导出一个临界阈值的节点的部分需要删除的分解复杂网络的巨大组成部分。+ − − − <math> − − 《数学》 − − \begin{align} − − \begin{align} − − 开始{ align } − − f_c=1-\frac{1}{\frac{\langle k^2 \rangle}{\langle k \rangle}-1} − − 1-frac { frac { langle k ^ 2 rangle }{ langle k rangle }-1} − − \end{align} − − \end{align} − − 结束{ align } − − </math> − − 数学 − 第268行：
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→Critical threshold for random failures
The mathematical derivation for the threshold at which a complex network will lose its [[giant component]] is based on the [[Molloy–Reed criterion]].<ref name="Molloy1995">Molloy, M. and Reed, B. (1995) ''Random Structures and Algorithms 6'', 161–180.</ref>
The mathematical derivation for the threshold at which a complex network will lose its [[giant component]] is based on the [[Molloy–Reed criterion]].
The mathematical derivation for the threshold at which a complex network will lose its giant component is based on the Molloy–Reed criterion.
The mathematical derivation for the threshold at which a complex network will lose its giant component is based on the Molloy–Reed criterion.
关于复杂网络失去其庞大组成部分的触发阈值，其数学推导遵循Molloy-Reed准则。
<math>
<math>
\kappa \equiv \frac{\langle k^2 \rangle}{\langle k \rangle} > 2
\kappa \equiv \frac{\langle k^2 \rangle}{\langle k \rangle} > 2
</math>
</math>
The Molloy–Reed criterion is derived from the basic principle that in order for a giant component to exist, on average each node in the network must have at least two links. This is analogous to each person holding two others' hands in order to form a chain. Using this criterion and an involved mathematical proof, one can derive a critical threshold for the fraction of nodes needed to be removed for the breakdown of the giant component of a complex network.
The Molloy–Reed criterion is derived from the basic principle that in order for a giant component to exist, on average each node in the network must have at least two links. This is analogous to each person holding two others' hands in order to form a chain. Using this criterion and an involved mathematical proof, one can derive a critical threshold for the fraction of nodes needed to be removed for the breakdown of the giant component of a complex network.<ref name="Cohen2000">{{cite journal |last1=Cohen |first1=R. |last2=Erez |first2=K. |last3=Havlin |first3=S. |title=Resilience of the Internet to random breakdowns |url=|journal=Phys. Rev. Lett. |volume=85 |issue=21 |page=4626|year=2000 |doi=10.1103/physrevlett.85.4626 |bibcode=2000PhRvL..85.4626C|arxiv=cond-mat/0007048 |pmid=11082612}}</ref>
The Molloy–Reed criterion is derived from the basic principle that in order for a giant component to exist, on average each node in the network must have at least two links. This is analogous to each person holding two others' hands in order to form a chain. Using this criterion and an involved mathematical proof, one can derive a critical threshold for the fraction of nodes needed to be removed for the breakdown of the giant component of a complex network.
The Molloy–Reed criterion is derived from the basic principle that in order for a giant component to exist, on average each node in the network must have at least two links. This is analogous to each person holding two others' hands in order to form a chain. Using this criterion and an involved mathematical proof, one can derive a critical threshold for the fraction of nodes needed to be removed for the breakdown of the giant component of a complex network.
Molloy-Reed准则基于以下基本原理：为了形成一个巨大的组件，网络中的每个节点平均必须至少具有两个链接。这类似于每个人握住两只手以形成一条链。依据这一标准和相关的数学证明，对于复杂网络巨型组件的故障，可以通过删除其中部分节点来得出一个临界阈值。该发现具有一个极其重要的性质，其临界阈值仅取决于度分布的第一和第二矩，并且对于任意度分布均有效。
<math>
<math>
f_c=1-\frac{1}{\frac{\langle k^2 \rangle}{\langle k \rangle}-1}
f_c=1-\frac{1}{\frac{\langle k^2 \rangle}{\langle k \rangle}-1}
</math>
</math>
对于 γ 大于3的情况，临界阈值只取决于 γ 和最小度，在这种情况下，当有限部分的节点被移除时，网络就像一个随机的网络断裂。对于小于3的 γ，k 在极限处发散，n 趋向于无穷大。在这种情况下，对于大规模无标度网络，临界阈值接近1。这基本上意味着几乎所有的节点必须被删除，以摧毁巨大的组成部分，大规模无尺度网络是非常健壮的随机故障。通过考虑无标度网络的异质性，特别是枢纽的异质性，人们可以对这一结论有直观的理解。由于集线器相对较少，它们不太可能通过随机故障被移除，而小的低度节点更有可能被移除。由于低度节点对于连接巨型构件的重要性不大，因此去除这些节点的影响不大。
对于 γ 大于3的情况，临界阈值只取决于 γ 和最小度，在这种情况下，当有限部分的节点被移除时，网络就像一个随机的网络断裂。对于小于3的 γ，k 在极限处发散，n 趋向于无穷大。在这种情况下，对于大规模无标度网络，临界阈值接近1。这基本上意味着几乎所有的节点必须被删除，以摧毁巨大的组成部分，大规模无尺度网络是非常健壮的随机故障。通过考虑无标度网络的异质性，特别是枢纽的异质性，人们可以对这一结论有直观的理解。由于集线器相对较少，它们不太可能通过随机故障被移除，而小的低度节点更有可能被移除。由于低度节点对于连接巨型构件的重要性不大，因此去除这些节点的影响不大。
==Targeted attacks on scale-free networks==
==Targeted attacks on scale-free networks==