Identifying the vital nodes in networks is of great significance for understanding the function of nodes and the nature of networks. Many centrality indices, such as betweenness centrality (Bc), eccentricity centrality (ec), closeness centricity (cc), structural holes (SH), degree centrality (Dc), pageRank (pR) and eigenvector centrality (VC), have been proposed to identify the influential nodes of networks. However, some of these indices have limited application scopes. ec and cc are generally only applicable to undirected networks, while pR and Vc are generally used for directed networks. to design a more applicable centrality measure, two vital node identification algorithms based on node adjacency information entropy are proposed in this paper. To validate the effectiveness and applicability of the proposed algorithms, contrast experiments are conducted with the BC, EC, CC, SH, DC, PR and VC indices in different kinds of networks. The results show that the index in this paper has a high correlation with the local metric Dc, and it also has a certain correlation with the pR and Vc indices for directed networks. in addition, the experimental results indicate that our algorithms can effectively identify the vital nodes in different networks. The vital nodes in networks are the nodes that have great impacts on the network structure and function 1. Previous studies have described many centralities that can rank the nodes in networks, such as degree centrality 2 , eccentricity 3 , closeness centricity 4 , betweenness centrality 5-7 , eigenvector centrality 8 and PageRank 9. Identifying the influential nodes in networks is not only of theoretical significance but also of practical value. For example, identifying the important junctions in traffic networks can prevent the paralysis of traffic networks caused by traffic congestion. Locking key sources in virus transmission networks can significantly reduce the speed and scope of virus transmission. These examples and others are all related to identifying the vital nodes in networks. The paper of Gino et al. applied the optimal percolation theory to predict the influential nodes in memory networks 10. Considering that the local metrics have lower computational complexity and the global metrics have higher computational accuracy, in recent work, many vital node identification methods that consider both local and global metrics have been proposed. A semi-local metric that balances the accuracy and efficiency was proposed by Chen et al. 11. Another neighbourhood centrality that takes into account the importance of a node and its neighbours' was proposed 12. In the paper by Yu et al. 13 , an improved method called improved structural holes (ISH) that identifies the key nodes in complex networks was proposed; unlike the eccentricity and betweenness centrality, this method can be applied to large-scale and disconnected networks. Zhang et al. 14 presented an effective method named VoteRank to identify a set of dispersive spreaders with the best spreading ability. By consideri...

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Stealth malware is a representative tool of advanced persistent threat (APT) attacks, which poses an increased threat to cyber-physical systems (CPS) today. Due to the use of stealthy and evasive techniques, stealth malwares usually render conventional heavy-weight countermeasures inapplicable. Light-weight countermeasures, on the other hand, can help retard the spread of stealth malwares, but the ensuing side effects might violate the primary safety requirement of CPS. Hence, defenders need to find a balance between the gain and loss of deploying light-weight countermeasures, which normally is a challenging task. To address this challenge, we model the persistent anti-malware process as a shortest-path tree interdiction (SPTI) Stackelberg game with both static version (SSPTI) and multi-stage dynamic version (DSPTI), and safety requirements of CPS are introduced as constraints in the defender’s decision model. The attacker aims to stealthily penetrate the CPS at the lowest cost (e.g., time, effort) by selecting optimal network links to spread, while the defender aims to retard the malware epidemic as much as possible. Both games are modeled as bi-level integer programs and proved to be NP-hard. We then develop a Benders decomposition algorithm to achieve the Stackelberg equilibrium of SSPTI, and design a Model Predictive Control strategy to solve DSPTI approximately by sequentially solving an 1+δ approximation of SSPTI. Extensive experiments have been conducted by comparing proposed algorithms and strategies with existing ones on both static and dynamic performance metrics. The evaluation results demonstrate the efficiency of proposed algorithms and strategies on both simulated and real-case-based CPS networks. Furthermore, the proposed dynamic defense framework shows its advantage of achieving a balance between fail-secure ability and fail-safe ability while retarding the stealth malware propagation in CPS.