# 鲁棒性

（重定向自网络的鲁棒性

## 渗流理论

$\displaystyle{ \langle s \rangle \sim \left|p - p_c\right|^{\gamma_p} }$

## 随机故障的临界阈值

$\displaystyle{ \kappa \equiv \frac{\langle k^2 \rangle}{\langle k \rangle} \gt 2 }$

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连接。这类似于每个人握住另外两个人的手以形成一条链。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，可以得到一个需要移除的部分节点的临界阈值[7]

$\displaystyle{ f_c=1-\frac{1}{\frac{\langle k^2 \rangle}{\langle k \rangle}-1} }$

### 随机网络

$\displaystyle{ f_c^{ER}=1-\frac{1}{\langle k \rangle} }$

### 无标度网络

\displaystyle{ \begin{align} f_c &=1-\frac{1}{\kappa-1}\\ \kappa &=\frac{\langle k^2\rangle}{\langle k \rangle}=\left|\frac{2-\gamma}{3-\gamma}\right|A \\ A &=K_{min},~\gamma \gt 3 \\ A &=K_{max}^{3-\gamma}K_{min}^{\gamma-2},~3 \gt \gamma \gt 2 \\ A &=K_{max},~2 \gt \gamma \gt 1 \\ &where~K_{max}=K_{min}N^{\frac{1}{\gamma - 1}} \end{align} }

## 无标度网络的针对性攻击

\displaystyle{ \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) \end{align} }

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