# A Complex Network Important Node Identification Based on the KPDN Method
## 2 Materials and Methods
### The Proposed Method
We incorporate this indicator into the KPD method:  
\[KPDN_{i}=\sum_{j\in N(i)}\left(\frac{S_{ij}+k_{i}}{KPD_{j}}\right)^{2} \tag{5}\]In summary, the above operations can form an initial dictionary during the first traversal of K-shell for storage and constantly call to save calculation time. Therefore, from a theoretical analysis, the computational complexity of the proposed method in this paper should be close to the K-shell method. Therefore, it is determined that this method is more suitable for large-scale networks. The KPDN method is shown in Figure 1.  
The traditional K-shell method divides the levels of complex networks roughly. As shown in Figure 2a-c, nodes A, B, C, D, E, F, G, and H are at the same 1-shell level, and the importance of nodes is not further distinguished. However, in the KPD method, we record the order of peeling off at the same 1-shell level synchronously during the decomposition process. For example, nodes A, B, C, D, E, and H are the first nodes peeled off in the 1-shell, marked as \(l_{A}=l_{B}=l_{C}=l_{D}=l_{E}=l_{H}=1\). Nodes G and F are the second nodes peeled off in the 1-shell layer, marked as \(l_{G}=l_{F}=2\). Because the 1-shell layer is peeled off twice in total, \(l_{\text{max}}=2\). At the same time, because the neighboring nodes of each node are different, as shown in Figure 2e,f, the neighboring nodes of nodes A, B, and C only have F, while the neighboring node of node H is I, and the neighboring node of node D is J. Therefore, the maximum K-shell values of the neighboring nodes of each point are different, with values of \(ks_{A|next}=ks_{B|next}=ks_{C|next}=1\), \(ks_{D|next}=3\), \(ks_{H|next}=3\), \(ks_{G|next}=ks_{F|next}=3\), respectively. Similarly, the maximum degree values of the neighboring nodes of each point can be obtained: \(k_{N(D)\text{max}}=5\), \(k_{N(H)\text{max}}=6\), \(k_{N(A)\text{max}}=k_{N(B)\text{max}}=k_{N(C)\text{max}}=k_{N(E)\text{max}}=2\), \(k_{N(C)\text{max}}=5\), \(k_{N(F)\text{max}}=4\). Figure 1: Flow chart of the KPD method and KPDN method (as shown in the figure, the white box indicates the flow chart of the KPD method, and the gray box indicates the flow chart of the KPDN method). Plugging the above parameters into Formula (2) yields the following values from A to H are as follows:  
\[KPD_{H}\approx 2.0952,KPD_{A}=KPD_{B}=KPD_{C}=KPD_{E}\approx 2.1111,KPD_{D} \approx 2.1666\]  
\[KPD_{G}\approx 3.4444,KPD_{F}\approx 5.5333.\]  
Through the KPD method, we can further distinguish the importance of nodes within the same K-shell layer. The importance of nodes F and G is higher than that of D and H, and also, the importance of nodes D and H is higher than that of A, B, C, and E. This shows that the KPD method can more accurately identify the topological structure of complex networks and effectively correct the deficiencies of the traditional K-shell method. Then we demonstrate calculating the KPDN value of node K. First, traversing the neighbor nodes of node k forms the \(\Gamma(K)=\{F,J,M,L\}\). Since there are 4 nodes adjacent to k, we need to calculate \(S_{KF},S_{KI},S_{KM},S_{KL}\), a total of 4 Solton index values. The results are as follows: First, calculate \(S_{KF}\). To compute \(S_{KF}\), we first need to traverse the neighbor sets of nodes K and F. The neighbor set of K is: \(\Gamma(K)=\{F,J,M,L\}\), and the neighbor set of F is: \(\Gamma(F)=\{A,B,C,K\}\). Then, we perform the \"and\" operation on the two sets and plug them into Formula (4) to obtain \(S_{KF}=0\). Similarly, to compute the value of S2, the neighbor sets of nodes K and J are \(\Gamma(K)=\{F,J,M,L\}\) and \(\Gamma(J)=\{D,K,M,N,G\}\) respectively. Subsequently, plugging them into Formula (4) yields \(S_{KJ}=\frac{1}{\sqrt{4}*5}=\frac{\sqrt{5}}{10}\). And also, calculating \(S_{KM}=\frac{\sqrt{5}}{10},S_{KL}=0\). The degree of node K is: \(K_{K}=4\),  
The KPD value of node K is: \(KPD_{K}\approx 7.222\),  
Plugging the above parameters into Formula (5) yields: \(KPDN_{K}\approx 0.1441655\).