# A Key Node Mining Method Based on K-Shell and Neighborhood Information
## 1 Introduction  
In our interconnected world, complex systems permeate nearly every facet of human society and the natural world, playing indispensable roles in our lives, economies, sciences, and beyond [1; 2; 3; 4; 5; 6]. Therefore, the understanding, analysis, and prediction of complex systems have attracted the attention of researchers [7]. Every complex system can be abstracted as a complex network, where the objects of the system are viewed as nodes, and the relationships between objects are viewed as edges [8]. The network composed of nodes and edges reveals the interrelationships between different parts of the complex system. For example, the power system that provides energy to the world is composed of generators and transmission lines [9]. The biological network, which is a necessary condition for life, is composed of cells and their interactions [10]. The social network that dictates the dissemination of knowledge and resources in society is made up of relationships between individuals, such as family, friends, and professional connections [11]. Due to the heterogeneity of complex networks, certain \"special nodes\" in the network have significant influence [12]. Operating on these \"special nodes\" often has a significant impact on the network [12], and these \"special nodes\" are referred to as key nodes. Therefore, keynode mining in complex networks has been widely applied in many fields [13], such as the prevention of infectious diseases [14], viral marketing [15], and prevention of social network rumors [16]. As one of the important research directions in complex networks, important node mining has been proposed by many researchers [17]. Among these methods, the centrality-based methods are the main category, and the important node-mining methods based on centrality can be divided into three types: local indicators, global indicators, and semi-local indicators [18]. Alternatively, these methods can be categorized into centrality methods relying on local information and centrality methods based on global information. Within this classification, the centrality methods based on local information encompass both local indicators and semi-local indicators. The distinction between the two lies in their scope: local indicators solely consider the node and its immediate neighbors, whereas focusing solely on the node and its first-order neighbors often yields less precise results and greater limitations. Therefore, semi-local indicators not only consider the node and its first-order neighboring nodes but also consider higher-order neighboring nodes. Degree centrality (DC) stands as the quintessential method grounded in local indicators, which posits that a node's significance increases with its degree [19]; the social capital (SC) method, proposed by [PERSON] et al., suggests that a node's importance correlates with the sum of its degree and that of its immediate neighbors [20]; the semi-local centrality (SLC) method, proposed by [PERSON] et al., mirrors SC by evaluating the importance of a node by the sum of the degrees of the node and its neighbors [21], albeit with the distinction of incorporating information from the node's higher-order neighbors. In addition to the aforementioned methods that assess the importance of nodes based on their degree, there are other centrality methods based on local information, such as the H-index method, which was initially used to evaluate academic output and was applied to the field of complex networks by [PERSON] et al. [22]; and [PERSON] et al. proposed the entropy-based ranking measure (ERM), which introduces information entropy theory and evaluates the importance of nodes by combining different disciplinary knowledge [23]. In comparison, centrality methods based on global information have the capacity to incorporate more extensive data. For instance, closeness centrality (CC) [24] and betweenness centrality (BC) [25] are classic centrality methods based on global information.