更改

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==

==Critical threshold for random failures==