# A Key Node Mining Method Based on K-Shell and Neighborhood Information
## 4 Experiments and Results
### Performance of KCH
#### 4.3.4 Statistical Analysis  
In complex networks, each algorithm used for identifying important nodes employs unique computational methods to measure node importance, resulting in dimensionless absolute values [52]. Additionally, due to the substantial differences in scale and structure among various networks, statistical analysis methods are essential for describing node or network characteristics based on relative results. To address this, this paper constructs several random networks that share similar properties with real networks, namely, the null models in statistics [53].  
Typically, two methods are employed to establish null models of complex networks. The first method entails calculating specific network-specific statistical properties and using these properties to generate a random network, thereby constructing a null model. While this method is straightforward and intuitive, constructing higher-order null models can be challenging. The second method involves randomizing an existing network by randomly breaking and reconnecting edges without altering its network properties. However, this approach is more cumbersome to implement, has long computational times, and is difficult to apply to large networks.  
In this paper, the first method is utilized to construct two types of null models: the \"0 th-order null model\" and the \"1 st-order null model\". The 0 th-order null model maintains  
Figure 4: Subfigures (**a**–**d**) represent the centrality metric correlation matrix for the Dolphins, US-powerGrid, WikiVote, and HepPh networks. The metric names in the graph correspond to the centrality metrics discussed in Section 3. Each element in the graph represents the Kendall coefficient between two metrics. For example, in the USpowerGrid network, the Kendall coefficient between the important node ranking lists generated by CI and CC is 0.25.  
an average degree identical to that of the real network, while the 1 st-order null model matches the degree distribution of the real network, with all other network properties completely randomized.  
Furthermore, 500 instances of both the 0 th-order null models and 1 st-order null models are constructed. By analyzing the distribution of the KCH scores in the null models, this paper obtains the 95% confidence interval for the KCH scores. This information is then used to investigate the impact of certain non-random factors in real networks on the KCH algorithm.  
Figure 5 shows the distribution of KCH scores on four real networks: Dolphins, USpowerGrid, WikiVote, and HepPh (with distribution graphs for all networks shown in Appendix D).  
Each point in the graph represents a node in the network; if the node's KCH score is within the confidence interval obtained from the 0 th-order null model, the node is colored green. If the KCH score is greater than the confidence interval of the 1 st-order null model, it is colored red. Nodes with KCH scores lying between the two confidence intervals are colored blue.  
The figure illustrates that in the Dolphins, USpowerGrid, WikiVote, and HepPh networks, all nodes are colored green, indicating that their KCH scores are concentrated within the confidence interval of the 0 th-order null model. This observation suggests that the KCH method exhibits a high degree of universality and performs well in both real and random networks.  
Even in networks that have the same average degree or degree distribution as real networks but with all other properties randomized, the KCH method's scores still fall within this range. This implies that KCH achieves high accuracy using only structural  
Figure 5: Subfigures (**a**–**d**) represent zero model experiment—distribution of KCH scores on real networks.  
holes, clustering coefficients, and the degree. Essentially, the KCH method attains high accuracy with a smaller amount of information.