# A Complex Network Important Node Identification Based on the KPDN Method
## 2 Materials and Methods
### The Existing Benchmark Methods  
In this paper, we use several well-known metrics to study the performance of the proposed method. These measures include:  
1. Degree centrality (DC)) Degree centrality is a basic ranking algorithm for identifying node importance. The degree of node \(i\) is defined as follows: \[k_{i}=\sum_{j=1}^{N}a_{ij}\] (6)
2. Collective influence (CI) The collective influence of a node is defined as follows: \[CI_{i}=(k_{i}-1)\sum_{j\in set(i,j)}(k_{j}-1)\] (7)
3. Betweenness centrality (BC) Betweenness centrality refers to the number of times a node appears in all shortest paths in a graph, that is, the importance of the node in connecting other nodes in the network. It is defined as follows: \[BC_{i}=\sum_{i\in d(i,j)_{\text{min}}+j}1\] (8)  
Figure 3: Comparison of node deletion in a small-world network.  
The K-shell decomposition method proposed by [PERSON] et al. [19; 20; 21] is a graph-theoretical algorithm used to decompose an undirected graph into multiple K-shell subgraphs. A K-shell subgraph refers to a subgraph composed of nodes with degrees greater than or equal to k in the graph; that is, any node in the subgraph has a degree greater than or equal to k. The core idea of the algorithm is to repeatedly prune the nodes with the smallest degrees and update the degrees of their adjacent nodes until all nodes are decomposed into K-shell subgraphs.  
5. WL algorithm  
The WL algorithm [22] is a ranking method based on node degree and adjacent node degree. It is defined as follows:  
\[WL_{i}=\sum_{j\in N(i)}\bigl{(}k_{i}\times k_{j}\bigr{)} \tag{9}\]  
6. DWT algorithm  
The DWT algorithm is a decomposition method proposed by [PERSON] et al. [23]. It quantifies the strength of links based on local information of network topology and designs a simple and effective method to evaluate the importance of nodes according to the number of connections and overlap of neighbors.  
\[DWT_{i}=\sum_{j\in N(i)}\left(1\Big{/}k_{i}+S_{ij}\Big{/}k_{i}\right)^{2},S_{ ij}=\frac{|N(i)\cap N(j)|}{\sqrt{k_{i}\times k_{j}}} \tag{10}\]  
7. Harmonic centrality  
The harmonic centrality (HC) method is defined as follows:  
\[HC_{i}=\sum_{j\
eq i}\frac{1}{d(i,j)} \tag{11}\]  
8. Closeness centrality  
Closeness centrality (CC) is a node centrality measurement method used to measure the distance from a node to other nodes, that is, its average shortest path length to other nodes. It is defined as follows:  
\[CC_{i}=\frac{1}{n}\sum_{j\
eq i}d(i,j) \tag{12}\]  
9. Pagerank  
PageRank [24] is an algorithm proposed by [PERSON] and [PERSON], the co-founders of Google, to rank web pages. The specific formula is as follows:  
\[PR(i)=\frac{(1-d)}{n}+d*\sum_{j\in\Gamma(i)}\frac{PR(j)}{C(j)}\]  
where \(PR(i)\), \(PR(j)\) represent the PageRank values of nodes \(i\) and \(j\), \(C(j)\) represents the outdegree of the node, \(d\) is the damping factor, usually 0.85, \(n\) is the total number of web pages, and \(\Gamma(i)\) is the set of neighboring nodes of node \(i\).  
Incc  
The INCC method [25] combines the direct and indirect influences of a node's nearest neighbors and next-nearest neighbors. The specific formula is as follows:  
\[INCC_{i}=\sum\limits_{j\in\Gamma(i)}\left(p^{\prime}_{ij}+\sum\limits_{k=1,k\
eq i,j}^{n}p^{\prime}_{ik}p^{\prime}_{kj}\right)^{2}\]  
where \(p^{\prime}_{ij}=\frac{Q_{i}}{LR_{i}}\), \(LR_{i}=\sum\limits_{j\in\Gamma(i)}Q_{i}\), \(Q_{j}=\sum\limits_{w\in\Gamma(j)}N_{w}\), \(N_{w}\) represent the sum of the degrees of the two-step neighbors of node \(w\); \(\Gamma(i)\) and \(\Gamma(j)\) represent the sets of neighbors of nodes \(i\) and \(j\), respectively.  
11 Randomness method  
Rank the importance of network nodes by random scoring.