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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，复杂网络的崩溃突然发生在相变过程中移除某些关键节点。

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>'''。

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>''':

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.

'''<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。

'''<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.

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.

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。

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: