Recent theoretical studies on network robustness have focused primarily on attacks by random selection and global vision, but numerous real-life networks suffer from proximity-based breakdown. Here we introduce the multi-hop generalized core percolation on complex networks, where nodes with degree less than [Formula: see text] and their neighbors within [Formula: see text]-hop distance are removed progressively from the network. The resulting subgraph is referred to as [Formula: see text]-core, extending the recently proposed [Formula: see text]-core and classical core of a network. We develop analytical frameworks based upon generating function formalism and rate equation method, showing for instance continuous phase transition for [Formula: see text]-core and discontinuous phase transition for [Formula: see text]-core with any other combination of [Formula: see text] and [Formula: see text]. We test our theoretical results on synthetic homogeneous and heterogeneous networks, as well as on a selection of large-scale real-world networks. This unravels, e.g., a unique crossover phenomenon rooted in heterogeneous networks, which raises a caution that endeavor to promote network-level robustness could backfire when multi-hop tracing is involved.

中文翻译：

---

复杂网络上的多跳广义核心渗透

最近关于网络鲁棒性的理论研究主要集中在随机选择和全局视觉的攻击上，但许多现实生活中的网络都遭受了基于邻近性的故障。这里我们介绍了复杂网络上的多跳广义核心渗透，其中度数小于[公式：见文本]的节点及其在[公式：见文本]-跳距离内的邻居逐渐从网络中移除。生成的子图称为 [Formula: see text]-core，扩展了最近提出的 [Formula: see text]-core 和网络的经典核心。我们开发了基于生成函数形式和速率方程方法的分析框架，例如显示 [Formula: see text] 核心的连续相变和 [Formula: 参见文本]-核心与 [公式：参见文本] 和 [公式：参见文本] 的任何其他组合。我们在合成同构和异构网络以及选择的大型现实世界网络上测试了我们的理论结果。这揭示了，例如，一种植根于异构网络的独特交叉现象，这引发了一个警告，即当涉及多跳跟踪时，努力提高网络级别的鲁棒性可能会适得其反。