鲁棒性

此词条暂由Jie翻译

鲁棒性Robustness（承受故障和干扰的能力）是许多复杂系统（包括复杂网络）的关键属性。

复杂网络的鲁棒性研究对许多领域都非常重要。在生态学中，鲁棒性是生态系统的重要属性，可以使人们深入了解对诸如物种灭绝等干扰因素的反应。对于生物学家而言，网络鲁棒性可以帮助研究疾病和突变，以及如何从某些突变中恢复过来。对于经济学，网络鲁棒性原则可以帮助理解银行系统的稳定性和风险。同时在工程中，网络鲁棒性可以帮助评估基础建设网络（如互联网或电网）的恢复能力。

渗流理论

复杂网络的鲁棒性主要是关注移除节点或链接后的网络响应情况。可以将这个过程的数学模型视为逆渗透过程。渗流理论模拟了将小卵石以概率[math]\displaystyle{ p_c }[/math]随机放置在[math]\displaystyle{ n }[/math]维晶格上的过程，并以临界概率来预测突然形成单个簇群的过程。在渗流理论中，该簇群被称为渗流簇。这种现象可以通过许多参数来量化，例如平均簇大小[math]\displaystyle{ \langle s \rangle }[/math]。它表示所有有限簇的平均大小，并由以下方程式给出:

[math]\displaystyle{ \langle s \rangle \sim \left|p - p_c\right|^{\gamma_p} }[/math]

我们可以看到平均聚类大小的簇群突然在临界概率附近发散，表明形成了单个大簇群。需要注意的是指数 [math]\displaystyle{ \gamma_p }[/math] 对于所有晶格都是通用的，而 [math]\displaystyle{ p_c }[/math] 却不是的。这点非常重要，因为它表明在基于拓扑学观点上存在通用相变行为。可以将复杂网络中的鲁棒性问题视为渗透一个簇群开始，紧接着除去该簇群中卵石的关键部分最终以使该簇群瓦解。类似于渗流理论中渗流簇的形成，复杂网络的崩溃突然发生在相变过程中移除某些关键节点。

随机故障的临界阈值

关于复杂网络失去其庞大组成部分的阈值，其数学推导遵循 Molloy-Reed准则:

[math]\displaystyle{ \kappa \equiv \frac{\langle k^2 \rangle}{\langle k \rangle} \gt 2 }[/math]

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。

[math]\displaystyle{ f_c=1-\frac{1}{\frac{\langle k^2 \rangle}{\langle k \rangle}-1} }[/math]

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一阶矩和二阶矩，并且对于任意度分布均有效。

随机网络

使用 [math]\displaystyle{ \langle k^2 \rangle = \langle k \rangle(\langle k \rangle+1) }[/math] 表示 Erdős-Rényi（ER）随机图，可以重新表达随机网络的临界点。

[math]\displaystyle{ f_c^{ER}=1-\frac{1}{\langle k \rangle} }[/math]

随着随机网络变得越来越密集，临界阈值会增加，这意味着必须移除更高比例的节点才能断开巨型组件的连接。

无标度网络

通过将临界阈值重新表达为 无标度网络 Scale-free network的γ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]\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} }[/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]\displaystyle{ \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]\displaystyle{ \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γ，临界阈值仅取决于γ和最小度。这种情况下，网络的部分节点被移除，之后该网络会像随机网络瓦解一般。对于小于3的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，这意味着几乎要移除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移除，而较小的低度节点则更可能被移除。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 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.

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]\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} }[/math]

This equation cannot be solved analytically, but can be graphed numerically. To summarize the important points, when gamma is large, the network acts as a random network, and attack robustness become similar to random failure robustness of a random network. However, when gamma is smaller, the critical threshold for attacks on scale-free networks becomes relatively small, indicating a weakness to targeted attacks.

This equation cannot be solved analytically, but can be graphed numerically. To summarize the important points, when gamma is large, the network acts as a random network, and attack robustness become similar to random failure robustness of a random network. However, when gamma is smaller, the critical threshold for attacks on scale-free networks becomes relatively small, indicating a weakness to targeted attacks.

该方程无法解析求解，但可以用数字表示。从中得出的结论是，当γ很大时，该网络可近似看作随机网络，其对抗攻击的鲁棒性变得类似于随机网络的随机故障鲁棒性。但是，当γ较小时，针对无标度网络攻击的临界阈值将变得相对较小，其抵抗定向攻击的脆弱性质逐渐显现出来。

For more detailed information on the attack tolerance of complex networks please see the attack tolerance page.

For more detailed information on the attack tolerance of complex networks please see the attack tolerance page.

有关复杂网络攻击耐受性的更多详细信息，请参阅攻击耐受性页面。

级联失效

许多网络中的故障的一个重要方面是，一个节点中的单个故障可能会导致相邻节点中的故障。当少量故障导致更多故障，导致相对于网络规模的大量故障时，就发生了级联故障。级联故障有很多模型。这些模型在许多细节上都不同，并且对从电源故障到Twitter上的信息流的不同物理传播现象进行建模，但是具有一些共享的原理。每个模型都专注于某种传播或级联，有一些阈值确定节点何时将发生故障或激活，并有助于传播，并且定义了某种机制，通过该机制，当节点发生故障或激活时将定向传播。所有这些模型都预测了某种临界状态，其中潜在级联的大小分布与幂律相匹配，并且指数由基础网络的度指数唯一确定。由于模型之间的差异以及结果的共识，我们认为潜在的现象是普遍的且与模型无关。

有关建级联故障的更多详细信息，请参阅全局级联模型页面。