Optimization method for protecting the robustness of first-order nodes in complex networks

The robustness of complex networks remains a significant challenge in network science. This study proposes a method aimed at optimizing network robustness by introducing a protection mechanism for the first-order neighbor nodes of a failed node. The load adjustment parameter α and the protection strength parameter δ in the protection mechanism affect the threshold Tc have been determined through theoretical analysis. In addition, in the experiment of a scale-free network, it was proven that α and Tc exhibit a positive proportional relationship, while δ and Tc exhibit an inverse proportional relationship. Notably, the introduction of the protective mechanism consistently resulted in a lower Tc compared to scenarios without protection, validating its efficacy in preventing cascading failures. Finally, the robustness of empirical networks, which include the American Football network, Wikiquote Edits network, and Euroroads network, is compared before and after adding protection. The results demonstrate that the first-order neighbors of failed node are protected, which is an effective method for improving the robustness of complex networks.