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第1行： − '''Robustness''', the ability to withstand failures and [[wikt:perturbation|perturbation]]s, is a critical attribute of many [[complex system]]s including [[complex network]]s. − 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）是许多复杂系统（包括复杂网络）的关键属性。 − − − The study of robustness in complex networks is important for many fields. In [[ecology]], robustness is an important attribute of ecosystems, and can give insight into the reaction to [[ecological disturbance|disturbances]] such as the extinction of species. For [[biology|biologists]], network robustness can help the study of [[disease]]s and [[mutation]]s, and how to recover from some mutations. In [[economics]], network robustness principles can help understanding of the stability and risks of banking systems. And in [[engineering]], network robustness can help to evaluate the [[resilience (network)|resilience]] of [[infrastructure]] networks such as the [[Internet]] or [[power grid]]s. − − The study of robustness in complex networks is important for many fields. In ecology, robustness is an important attribute of ecosystems, and can give insight into the reaction to disturbances such as the extinction of species. For biologists, network robustness can help the study of diseases and mutations, and how to recover from some mutations. In economics, network robustness principles can help understanding of the stability and risks of banking systems. And in engineering, network robustness can help to evaluate the resilience of infrastructure networks such as the Internet or power grids. 第17行：
第10行： − + − − {{Main article|Percolation theory}} − − The focus of robustness in complex networks is the response of the network to the [[Node deletion|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. − + 第35行：
第22行： − We can see the average cluster size suddenly diverges around the critical probability, indicating the formation of a single large cluster. It is also important to note that the exponent <math>\gamma_p</math> is universal for all lattices, while <math>p_c</math> is not. This is important as it indicates a universal [[phase transition]] behavior, at a point dependent on the topology. The problem of robustness in complex networks can be seen as starting with the percolating cluster, and removing a critical fraction of the pebbles for the cluster to break down. Analogous to the formation of the percolation cluster in percolation theory, the breaking down of a complex network happens abruptly during a phase transition at some critical fraction of nodes removed.+ − − We can see the average cluster size suddenly diverges around the critical probability, indicating the formation of a single large cluster. It is also important to note that the exponent <math>\gamma_p</math> is universal for all lattices, while <math>p_c</math> is not. This is important as it indicates a universal phase transition behavior, at a point dependent on the topology. The problem of robustness in complex networks can be seen as starting with the percolating cluster, and removing a critical fraction of the pebbles for the cluster to break down. Analogous to the formation of the percolation cluster in percolation theory, the breaking down of a complex network happens abruptly during a phase transition at some critical fraction of nodes removed. − 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，复杂网络的崩溃突然发生在相变过程中移除某些关键节点。 + − == 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. 第59行：
第39行： − 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. − + 第73行：
第51行： − An important property of this finding is that the critical threshold is only dependent on the first and second moment of the [[degree distribution]] and is valid for an arbitrary degree distribution.+ − − An important property of this finding is that the critical threshold is only dependent on the first and second moment of the degree distribution and is valid for an arbitrary degree distribution. − − 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。 − + − {{Main article|Erdős–Rényi model}} − Using <math>\langle k^2 \rangle = \langle k \rangle(\langle k \rangle+1)</math> for an [[Erdős–Rényi model|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. 第98行：
第68行： − − As a random network gets denser, the critical threshold increases, meaning a higher fraction of the nodes must be removed to disconnect the giant component. − − As a random network gets denser, the critical threshold increases, meaning a higher fraction of the nodes must be removed to disconnect the giant component. 第107行：
第73行： − + − − {{Main article|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. − 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. − +
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'''<font color="#ff8000"> 鲁棒性Robustness</font>'''（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。
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。
== Percolation theory 渗流理论 ==
== 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''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，例如平均簇大小s。它表示所有有限簇的平均大小，并由以下方程式给出:
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>p_c</math>随机放置在<math>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>\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:
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'''<font color="#ff8000"> Molloy-Reed准则</font>''':
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'''<font color="#ff8000"> Molloy-Reed准则</font>''':
'''<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.
'''<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。
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。
=== Random network 随机网络 ===
===随机网络 ===
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。
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。
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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。
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。
=== Scale-free network 无标度网络 ===
=== [[无标度网络]]===
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''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:
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''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: