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第146行： − + − + − − − 尽管无标度网络对于随机故障具有弹性，但我们可以想象它们对于有针对性的中心移除是非常脆弱的。在这种情况下，我们考虑了无标度网络对于目标攻击的鲁棒性，这些攻击是在完全了解网络拓扑的情况下进行的。通过考虑移除枢纽引起的变化，特别是连通节点的最大度和度的变化，我们可以推导出无标度网络上考虑目标攻击的临界阈值公式。+ − − 第167行：
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→Targeted attacks on scale-free networks
对于大于3的伽玛γ，临界阈值仅取决于伽玛γ和最小度。这种情况下，网络的部分节点被删除，之后该网络会像随机网络瓦解一般。对于小于3的伽玛γ，随着N趋于无穷大，κ的极限会发散。在这种情况下，对于大型无标度网络，关键阈值接近1。从本质上讲，这意味着几乎要除去所有节点才能破坏巨型组件，该大型无标度网络在应对随机故障方面非常强大。通过考虑无标度网络尤其是枢纽的异构性，可以直观地理解这一点。由于相对较少的枢纽节点，因此不太可能通过随机故障将其删除，而较小的低度节点则更可能被删除。同时由于低度节点在连接巨型部件方面不重要，因此将其移除几乎没有多大影响。
对于大于3的伽玛γ，临界阈值仅取决于伽玛γ和最小度。这种情况下，网络的部分节点被删除，之后该网络会像随机网络瓦解一般。对于小于3的伽玛γ，随着N趋于无穷大，κ的极限会发散。在这种情况下，对于大型无标度网络，关键阈值接近1。从本质上讲，这意味着几乎要除去所有节点才能破坏巨型组件，该大型无标度网络在应对随机故障方面非常强大。通过考虑无标度网络尤其是枢纽的异构性，可以直观地理解这一点。由于相对较少的枢纽节点，因此不太可能通过随机故障将其删除，而较小的低度节点则更可能被删除。同时由于低度节点在连接巨型部件方面不重要，因此将其移除几乎没有多大影响。
==Targeted attacks on scale-free networks==
== Targeted attacks on scale-free networks 无标度网络的针对性攻击 ==
Although scale-free networks are resilient to random failures, we might imagine them being quite vulnerable to targeted hub removal. In this case we consider the robustness of scale free networks in response to targeted attacks, performed with thorough prior knowledge of the network topology. By considering the changes induced by the removal of a hub, specifically the change in the maximum degree and the degrees of the connected nodes, we can derive another formula for the critical threshold considering targeted attacks on a scale free network.
Although scale-free networks are resilient to random failures, we might imagine them being quite vulnerable to targeted hub removal. In this case we consider the robustness of scale free networks in response to targeted attacks, performed with thorough prior knowledge of the network topology. By considering the changes induced by the removal of a hub, specifically the change in the maximum degree and the degrees of the connected nodes, we can derive another formula for the critical threshold considering targeted attacks on a scale free network.<ref name="Cohen2001">{{cite journal |last1=Cohen |first1=R. |last2=Erez |first2=K. |last3=ben-Avraham |first3=D. |last4=Havlin |first4=S. |title=Breakdown of the Internet under intentional attack |url=|journal=Phys. Rev. Lett. |volume=86 |issue=16 |page=3682|year=2001 |doi=10.1103/physrevlett.86.3682 |bibcode=2001PhRvL..86.3682C|arxiv=cond-mat/0010251 |pmid=11328053}}</ref>
Although scale-free networks are resilient to random failures, we might imagine them being quite vulnerable to targeted hub removal. In this case we consider the robustness of scale free networks in response to targeted attacks, performed with thorough prior knowledge of the network topology. By considering the changes induced by the removal of a hub, specifically the change in the maximum degree and the degrees of the connected nodes, we can derive another formula for the critical threshold considering targeted attacks on a scale free network.
Although scale-free networks are resilient to random failures, we might imagine them being quite vulnerable to targeted hub removal. In this case we consider the robustness of scale free networks in response to targeted attacks, performed with thorough prior knowledge of the network topology. By considering the changes induced by the removal of a hub, specifically the change in the maximum degree and the degrees of the connected nodes, we can derive another formula for the critical threshold considering targeted attacks on a scale free network.
尽管无标度网络可以抵抗随机故障，但可以想象它对枢纽节点针对性的攻击其实非常脆弱。此时，我们就需要考虑无标度网络对目标攻击的鲁棒性，这需要在充分了解网络拓扑结构的前提下进行。通过研究删除枢纽节点时网络产生的变化，特别是最大程度与所连接节点的程度变化，我们就可以考虑到无标度网络上的针对性攻击，得出临界阈值的另一个公式。
<math>
<math>
\begin{align}
\begin{align}
f_c^{\frac{2-\gamma}{1-\gamma}}=2+\frac{2-\gamma}{3-\gamma}K_{min}(f_c^{\frac{3-\gamma}{1-\gamma}}-1)
f_c^{\frac{2-\gamma}{1-\gamma}}=2+\frac{2-\gamma}{3-\gamma}K_{min}(f_c^{\frac{3-\gamma}{1-\gamma}}-1)
\end{align}
\end{align}
</math>
</math>
</math>
</math>
有关复杂网络的攻击容忍度的更多详细信息，请参阅攻击容忍页面。
有关复杂网络的攻击容忍度的更多详细信息，请参阅攻击容忍页面。
==Cascading failures==
==Cascading failures==