# A Complex Network Important Node Identification Based on the KPDN Method
## 2 Materials and Methods
### The Proposed Method  
In response to the above problems, inspired by [20; 23], this paper proposes a K-shell method positioned with degrees and neighbors (KPDN) and the KPDN method that combines the characteristics of degree, K-shell, and Solton [28]. Compared with traditional methods, this method comprehensively considers global and local network characteristics. It applies the K-shell method hierarchically on a global scale and records the isolation order in each layering process as the basis for secondary sorting. Finally, at the local level, it conducts a third round of sorting based on the relationship between each node's degree value and adjacent nodes and the Solton index to distinguish node importance. Since the KPDN method does not contain free parameters and each calculation step is recorded along with the K-shell expansion process, it does not require pre-computation of parameter values, reducing computational overhead and complexity. The specific steps are as follows:  
The improved fused degree K-shell method (recording the peeling order of each node under each shell layer while comparing degree values of the same peeling order). First, use K-shell to record which K-shell layer each node belongs to. Second, when nodes in the same K-shell layer are peeled off sequentially, record the order in which the node is peeled off as well as the layer number and sequence number. Finally, in each set of nodes with the same layer and peeling order, sort in descending order according to the proportion of node degree values. The calculation steps are as follows. Step 1: Inspired by the literature [20], we construct a new formula to calculate important nodes, calculate the peeling order of nodes under the same shell layer, and mark the nodes. To first record the peeling order of nodes under the same shell layer, the formula is as follows:  
\[KSL_{ks_{i},l_{i}}=\frac{l_{i}}{l_{\text{max}}+1}MAX\Big{(}1,\ ks_{i|next}-ks_{ i}\Big{)} \tag{1}\]  
where \(ks_{i}\) is the _k-shell_ value of node \(i\), \(l_{i}\) is the peeling order of node \(i\) under the same \(ks_{i}\) shell iteration peeling operation, and \(l_{\text{max}}\) is the maximum peeling order number under the same shell layer; \(ks_{i|next}\) is the _k-shell_ value adjacent to node \(i\), and when node \(i\) is at the maximum _k-shell_ layer, the \(ks_{i|next}-ks_{i}\) value is automatically set to 1. Where KSL is (K-shell methods with layers). Step 2: We summarize the nodes under the same peeling order layer (i.e., the same value) in the same shell layer (i.e., the same value) and sort them in descending order of the current degree values of each node. Where KPD is (K-shell methods positioned with degrees). The formula is as follows:  
\[KPD_{i}=ks_{i}+k_{i}+\frac{k_{i}}{k_{N(i)\text{max}}+1}KSL_{ks_{i},l_{i}} \tag{2}\]  
where \(k_{i}\) is the degree value of node \(i\), \(k_{N(i)\text{max}}\) is the maximum degree value of the neighboring nodes of node \(i\), and \(ks_{i}\) is the _k-shell_ value of node \(i\), the same as above. Step 3: Inspired by the literature [23; 24; 25], we calculate the adjacent nodes of each obtained node. Here, inspired by previous research, we define the edge intensity between two connected nodes based on the local information of network topology, such as the Solton index proposed in [23; 24; 25]: The formula is as follows:  
\[S_{ij}=\frac{|N(i)\cap N(j)|}{|N(i)\cup N(j)|} \tag{3}\]  
Of course, in the actual debugging process, we found that replacing the denominator (4) with this method produced slightly better results. \[S_{ij}=\frac{|N(i)\cap N(j)|}{\sqrt{k_{i}*k_{j}}} \tag{4}\]  
where \(N(i)\) and \(N(j)\) are the set of adjacent nodes of nodes \(i\) and \(j\), respectively. The above operations can form an initial dictionary during the first traversal of the K-shell and store it, and constantly call it to save computation time.