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→Scale-free network
By re-expressing the critical threshold as a function of the gamma exponent for a [[scale-free network]], we can draw a couple of important conclusions regarding scale-free network robustness.<ref name="NetworkBook"/>
By re-expressing the critical threshold as a function of the gamma exponent for a [[scale-free network]], we can draw a couple of important conclusions regarding scale-free network robustness.
By re-expressing the critical threshold as a function of the gamma exponent for a scale-free network, we can draw a couple of important conclusions regarding scale-free network robustness.
By re-expressing the critical threshold as a function of the gamma exponent for a scale-free network, we can draw a couple of important conclusions regarding scale-free network robustness.
通过将临界阈值重新表达为无标度网络的伽马指数函数，我们可以得出有关无标度网络鲁棒性的两个重要结论。
<math>
<math>
f_c &=1-\frac{1}{\kappa-1}\\
f_c &=1-\frac{1}{\kappa-1}\\
\kappa &=\frac{\langle k^2\rangle}{\langle k \rangle}=\left|\frac{2-\gamma}{3-\gamma}\right|A \\
\kappa &=\frac{\langle k^2\rangle}{\langle k \rangle}=\left|\frac{2-\gamma}{3-\gamma}\right|A \\
A &=K_{min},~\gamma > 3 \\
A &=K_{min},~\gamma > 3 \\
A &=K_{max}^{3-\gamma}K_{min}^{\gamma-2},~3 > \gamma > 2 \\
A &=K_{max}^{3-\gamma}K_{min}^{\gamma-2},~3 > \gamma > 2 \\
A &=K_{max},~2 > \gamma > 1 \\
A &=K_{max},~2 > \gamma > 1 \\
&where~K_{max}=K_{min}N^{\frac{1}{\gamma - 1}}
&where~K_{max}=K_{min}N^{\frac{1}{\gamma - 1}}
</math>
</math>
For gamma greater than 3, the critical threshold only depends on gamma and the minimum degree, and in this regime the network acts like a random network breaking when a finite fraction of its nodes are removed. For gamma less than 3, <math>\kappa</math> diverges in the limit as N trends toward infinity. In this case, for large scale-free networks, the critical threshold approaches 1. This essentially means almost all nodes must be removed in order to destroy the giant component, and large scale-free networks are very robust with regard to random failures. One can make intuitive sense of this conclusion by thinking about the heterogeneity of scale-free networks and of the hubs in particular. Because there are relatively few hubs, they are less likely to be removed through random failures while small low-degree nodes are more likely to be removed. Because the low-degree nodes are of little importance in connecting the giant component, their removal has little impact.
For gamma greater than 3, the critical threshold only depends on gamma and the minimum degree, and in this regime the network acts like a random network breaking when a finite fraction of its nodes are removed. For gamma less than 3, <math>\kappa</math> diverges in the limit as N trends toward infinity. In this case, for large scale-free networks, the critical threshold approaches 1. This essentially means almost all nodes must be removed in order to destroy the giant component, and large scale-free networks are very robust with regard to random failures. One can make intuitive sense of this conclusion by thinking about the heterogeneity of scale-free networks and of the hubs in particular. Because there are relatively few hubs, they are less likely to be removed through random failures while small low-degree nodes are more likely to be removed. Because the low-degree nodes are of little importance in connecting the giant component, their removal has little impact.
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，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==