# A Complex Network Important Node Identification Based on the KPDN Method
## 1 Introduction  
Many complex systems in the real world can be abstracted into models of complex networks [1; 2]. In these complex systems, only a very small number of nodes are critical to the operation of the entire system. These nodes can be referred to as critical nodes because their activation or removal will significantly improve or reduce some functionality of the system. Finding the optimal set of key nodes in complex networks has been a long-standing problem in network science with many real-world applications. This is essential for apprehending suspects to stop terrorist activities [3], suppressing the spread of epidemics [4], controlling the spread of rumors [5], maintaining the stable operation of the network [6], and detecting and evaluating the vulnerability nodes of the network [7; 8]. In recent years, many methods have been proposed to identify important nodes in networks, such as degree centrality (DC) [9], closeness centrality (CC), betweenness centrality (BC) [10], harmonic centrality (HC) [11], K-shell decomposition method (K-shell) [12], subgraph centrality (SC) [13], etc. In summary: (1) Existing methods face trade-offs between computational efficiency and identification accuracy. (2) Degree centrality and K-shell algorithms can roughly determine important nodes in networks, but are difficult to be accurate. (3) Betweenness centrality and harmonic centrality can judge node importance more accurately, but are difficult to apply to large-scale networks, and also face trade-offs. (4) Existing methods are difficult to achieve high efficiency and high precision at the same time. At the same time, with the explosive growth of data in the information age today, the network connections in the real world are becoming more complex, and more methods for evaluating node importance are emerging. [PERSON] and [PERSON] [14] proposed a scalable theoretical framework called collective influence (CI) to identify a minimum set of key nodes. It is worth mentioning that CI has significant advantages over many previous algorithmsin identifying node importance in maintaining network connectivity [15]. Meanwhile, the most common evaluation methods based on network position are the K-shell decomposition method [12] and its improved methods [16; 17; 18; 19; 20; 21], which evaluate the importance of nodes by determining whether the nodes are in the central position of the network. The WL algorithm [22] is a sorting method based on node degree and neighbor node degree. This method relies heavily on local properties. The DWT algorithm [23] identifies isolated nodes with bridging functions based on local weak connections in the network and can determine the local properties of nodes more accurately. PageRank [24] is a node importance algorithm based on network topology. It calculates the global importance of nodes by simulating random walk processes and is significant for recommendation systems, web page ranking, etc. However, it relies solely on network topology without considering other attributes and cannot identify locally important nodes. INCC [25] is an improved method for detecting structural holes. Its advantage lies in its ability to identify structural holes in networks by relying on local connection patterns. Of course, this method has a strong dependence on the network topology. Existing literature indicates that INCC achieves suitable performance in analyzing propagation characteristics. In recent years, some teams have also explored important node ranking methods using machine learning approaches [26]. These methods require constructing large datasets beforehand to train models for identifying important nodes in complex networks. It should be noted that the workload of data preparation and manual labeling is substantial in the early stages when using such methods. In addition, exploring important node identification in complex networks using machine learning approaches also has certain hardware requirements for the universal applicability of the models.