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Enhancing the robustness of complex networks is of great practical significance as it ensures the stable operation of infrastructure systems. We measure its robustness by examining the size of the largest connected component of the network after initial attacks. However, traditional research on network robustness enhancement has mainly focused on low-order networks, with little attention given to higher-order networks, particularly higher-low order coupling networks(the largest connected component of the network must exist in both higher-order and low-order networks). To address this issue, this paper proposes robust optimization methods for higher-low order coupled networks based on the greedy algorithm and the simulated annealing algorithm. By comparison, we found that the simulated annealing algorithm performs better. The proposed method optimizes the topology of the low-order network and the higher-order network by randomly reconnecting the edges, thereby enhancing the robustness of the higher-order and low-order coupled network. The experiments were conducted on multiple real networks to evaluate the change in the robustness coefficient before and after network optimization. The results demonstrate that the proposed method can effectively improve the robustness of both low-order and higher-order networks, ultimately enhancing the robustness of higher-low order coupled networks.

提升复杂网络（complex networks）的鲁棒性具有重要的现实意义，因其可保障基础设施系统的稳定运行。本文通过考察初始攻击后网络最大连通分量（largest connected component）的规模来衡量其鲁棒性。然而，传统的网络鲁棒性增强研究主要聚焦于低阶网络（low-order networks），对高阶网络（higher-order networks）尤其是高低阶耦合网络（higher-low order coupling networks）的关注较少——此类网络的最大连通分量需同时存在于高阶与低阶子网络中。针对这一研究空白，本文提出基于贪心算法（greedy algorithm）与模拟退火算法（simulated annealing algorithm）的高低阶耦合网络鲁棒优化方法。经对比实验发现，模拟退火算法的优化效果更优。所提方法通过随机重连边的方式优化低阶网络与高阶网络的拓扑结构，进而提升高低阶耦合网络的整体鲁棒性。本文在多个真实网络数据集上开展实验，以评估网络优化前后鲁棒性系数（robustness coefficient）的变化情况。实验结果表明，所提方法可有效提升低阶网络与高阶网络各自的鲁棒性，最终实现高低阶耦合网络整体鲁棒性的增强。