Finding the key nodes to minimize the victims of the malicious information in complex network

Safeguarding crucial nodes provides a direct approach to impede the dissemination of malicious information in complex networks, such as the Internet. However, determining the optimal budget size, represented as k, for protecting nodes is a challenging problem classified as NP-hard. In this study, we investigate the origin of the NP-hard property, known as the influence redundancy mechanism, as a means to address this problem. The influence redundancy characterizes the intricate interactions among key nodes. Subsequently, we introduce an objective function that allows for the optimization of the set of key nodes. Our objective is to minimize the spectral radius of the adjacency matrix after removing these key nodes. We mathematically prove that the objective function exhibits the submodular property, and our proposed method achieves an approximation ratio of (1−1/e) with a time complexity of O(NlogN), where N represents the size of the network. Experimental results show that the identified key nodes outperform classical methods in 20 empirical networks, specifically in the Susceptible-Infected-Recovered (SIR) model and Independent Cascade (IC) model, thus confirming their improved performance quality.