The robustness of complex networks responding to attacks has long been the focus of network science researching. Nonetheless, the precious studies mostly focus on network performance when facing malicious attacks and random failures while rarely pay attention to the influences of scales of attacking. It is wondering if it is an actual fact that the network is more fragile when attacking scale is exacerbated. In this paper, we are committed to exploring the influences related to the very factor of attacking scale from the perspective of cascading failure problem of dynamic network theory. We construct the model with a regular ranking edge deletion method by simulating attacking scale with Na and a is denoted as attacked edge number. To be specific, we rank the edges according to initial distributed loads and delete edges in the ranked list, and subsequently observe the changes of robustness in the networks, including BA scale-free network, WS small-world network and several real traffic networks. During the process, an unusual counterintuitive phenomenon captures our attention that the network damages caused by attacks do not always grow with the increase of attacked edges number. We specifically demonstrate and analyze this abnormal cascading propagation phenomenon, ascribing this paradox to the dynamics of the load and the connections of the network structure. Our work may offer a new angle on better controlling the spread of cascading failure and remind the importance of effectively protecting networks from underlying dangers.

中文翻译：

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级联故障引发的攻击行为的鲁棒性悖论：攻击越大，网络越脆弱？

复杂网络响应攻击的鲁棒性一直是网络科学研究的重点。尽管如此，珍贵的研究大多集中在面对恶意攻击和随机故障时的网络性能，而很少关注攻击规模的影响。它想知道当攻击规模加剧时网络更加脆弱是否是一个实际的事实。在本文中，我们致力于从动态网络理论的级联故障问题的角度探索与攻击规模因素相关的影响。我们通过模拟攻击规模来构建具有规则排名边缘删除方法的模型ñ一种和一种表示为攻击边数。具体来说，我们根据初始分布负载对边进行排序，并在排序列表中删除边，然后观察网络中鲁棒性的变化，包括BA无标度网络、WS小世界网络和几个真实流量网络。在此过程中，一个不寻常的违反直觉的现象引起了我们的注意，即攻击造成的网络损害并不总是随着被攻击边数的增加而增加。我们具体演示和分析了这种异常级联传播现象，将这种悖论归因于负载的动态和网络结构的连接。我们的工作可能为更好地控制级联故障的传播提供一个新的角度，并提醒有效保护网络免受潜在危险的重要性。