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第5行：
第5行： − 鲁棒性（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。+ 第25行：
第25行： − 复杂网络的鲁棒性主要是关注因删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p随机放置在n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，例如平均聚类s。它表示所有有限簇的平均大小，并由以下方程式给出。+ 第41行：
第41行： − == Critical threshold for random failures 随机故障的临界阈值 == + 第49行：
第49行： − 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。+ 第63行：
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第115行： − 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。+ 第144行：
第146行： − 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。+ + + 第170行：
第174行： − 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，当伽马值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。+ 第178行：
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无编辑摘要
Robustness, the ability to withstand failures and perturbations, is a critical attribute of many complex systems including complex networks.
Robustness, the ability to withstand failures and perturbations, is a critical attribute of many complex systems including complex networks.
'''<font color="#ff8000"> 鲁棒性Robustness</font>'''（承受故障和干扰的能力）是许多'''<font color="#ff8000"> 复杂系统complex network</font>'''（包括复杂网络）的关键属性。
The focus of robustness in complex networks is the response of the network to the removal of nodes or links. The mathematical model of such a process can be thought of as an inverse percolation process. Percolation theory models the process of randomly placing pebbles on an n-dimensional lattice with probability p, and predicts the sudden formation of a single large cluster at a critical probability <math>p_c</math>. In percolation theory this cluster is named the percolating cluster. This phenomenon is quantified in percolation theory by a number of quantities, for example the average cluster size <math>\langle s \rangle</math>. This quantity represents the average size of all finite clusters and is given by the following equation.
The focus of robustness in complex networks is the response of the network to the removal of nodes or links. The mathematical model of such a process can be thought of as an inverse percolation process. Percolation theory models the process of randomly placing pebbles on an n-dimensional lattice with probability p, and predicts the sudden formation of a single large cluster at a critical probability <math>p_c</math>. In percolation theory this cluster is named the percolating cluster. This phenomenon is quantified in percolation theory by a number of quantities, for example the average cluster size <math>\langle s \rangle</math>. This quantity represents the average size of all finite clusters and is given by the following equation.
复杂网络的鲁棒性主要是关注因删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''p''随机放置在n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，例如平均聚类s。它表示所有有限簇的平均大小，并由以下方程式给出。
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>\gamma_p</math> 对于所有晶格都是通用的，而 <math>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，复杂网络的崩溃在相变过程中突然发生，且发生在某些节点的关键部分。
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>\gamma_p</math> 对于所有晶格都是通用的，而 <math>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，复杂网络的崩溃在相变过程中突然发生，且发生在某些节点的关键部分。
== Critical threshold for random failures 随机故障的临界阈值 ==
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.
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'''<font color="#ff8000"> Molloy-Reed准则</font>'''。
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。
'''<font color="#ff8000"> Molloy-Reed准则</font>'''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。
Using <math>\langle k^2 \rangle = \langle k \rangle(\langle k \rangle+1)</math> for an Erdős–Rényi (ER) random graph, one can re-express the critical point for a random network.
Using <math>\langle k^2 \rangle = \langle k \rangle(\langle k \rangle+1)</math> for an Erdős–Rényi (ER) random graph, one can re-express the critical point for a random network.
使用 <math>\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。
使用 <math>\langle k^2 \rangle = \langle k \rangle(\langle k \rangle+1)</math> 表示'''<font color="#ff8000"> Erdős-Rényi（ER）随机图</font>'''，可以重新表达随机网络的临界点。
随着随机网络变得越来越密集，临界阈值会增加，这意味着必须删除更高比例的节点才能断开巨型组件的连接。
随着随机网络变得越来越密集，临界阈值会增加，这意味着必须删除更高比例的节点才能断开巨型组件的连接。
=== Scale-free network 无标度网络 ===
=== Scale-free network 无标度网络 ===
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'''<font color="#ff8000"> 无标度网络Scale-free network</font>'''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。
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''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 无标度网络的针对性攻击 ==
== Targeted attacks on scale-free networks 无标度网络的针对性攻击 ==
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，当伽马值''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。
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。
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。
== Cascading failures 连锁故障 ==
== Cascading failures 连锁故障 ==
有关建模连锁故障的更多详细信息，请参阅全局连锁模型页面。
有关建模连锁故障的更多详细信息，请参阅全局连锁模型页面。
==References==
==References==