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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.
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。
The study of robustness in complex networks is important for many fields.
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.
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.
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 ==
==Percolation theory==
{{Main article|Percolation theory}}
{{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 [[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>.<ref name="Stauffer1994">D. Stauffer and A. Aharony. Introduction to Percolation Theory. Tay-lor and Francis. London, 1994.</ref> 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.
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.