# A Key Node Mining Method Based on K-Shell and Neighborhood Information
## 4 Experiments and Results
### Performance of KCH
#### 4.3.3 Correlation Analysis  
After discussing the connectivity and monotonicity of each algorithm, we are ready to use the Kendall coefficient to measure the correlation between different algorithms. The correlation metric, although often overlooked, holds significant importance, as it has the potential to provide valuable insights. Analyzing the correlation metrics between the proposed method and the baseline method can indicate whether the proposed method offers deeper insights than the baseline method [51]. Hence, this paper adopts the Kendall coefficient as the correlation metric to study whether the KCH algorithm uses more information to mine the importance of nodes. Figure 4 illustrates the correlation matrices between different centrality metrics in the Dolphins, USpowerGrid, Tribes, and HepPh networks, with the correlation matrices for all networks displayed in Appendix C. It is evident from Figure 4 and Appendix C that the Kendall coefficients between the KCH and CI methods for all networks are greater than 0.7, with most networks having \(\tau\) values greater than 0.8 and some even exceeding 0.9. This indicates a high degree of similarity between the node importance rankings generated by the KCH and CI methods. Additionally, in most networks, the Kendall coefficients between the KCH and SC methods are above 0.7, and in some networks, even greater than 0.8, suggesting that the node importance rankings of KCH and SC are also quite similar. In some networks, KCH also shows a high correlation with CNC+, and the correlation among CI, SC, and CNC+ is consistently high, with values above 0.75 across the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Networks** & **KCH** & **CI** & **SC** & **HKS** & **KS** & **CC** & **CNC+** \\ \hline Windsurfer & **1.0000** & 0.9824 & 0.9956 & **1.0000** & 0.4269 & 0.9454 & **1.0000** \\ WikiVote & 0.9958 & 0.9100 & 0.9887 & **0.9997** & 0.7265 & 0.9988 & 0.9976 \\ USpowerGrid & 0.8417 & 0.8646 & 0.9048 & **0.9963** & 0.2460 & 0.9998 & 0.9419 \\ Tribes & **1.0000** & 0.8867 & 0.9834 & **1.0000** & 0.0156 & 0.7951 & **1.0000** \\ Seventh & 0.9951 & 0.8622 & 0.9902 & **1.0000** & 0.1365 & 0.8622 & 0.9951 \\ Rhesus & **1.0000** & 0.7656 & 0.9669 & **1.0000** & 0.3501 & 0.7656 & **1.0000** \\ HepPh & 0.9994 & 0.9798 & 0.9941 & **0.9998** & 0.8350 & 0.9995 & 0.9991 \\ Health & 0.9997 & 0.9982 & 0.9894 & **0.9999** & 0.5245 & 0.9994 & 0.9979 \\ Dolphins & 0.9958 & 0.9613 & 0.9675 & **0.9968** & 0.3769 & 0.9737 & 0.9873 \\ C\_ElegansNeural & **0.9977** & 0.9949 & 0.9955 & **0.9977** & 0.6094 & 0.9893 & 0.9975 \\ Average & 0.9825 & 0.9206 & 0.9776 & **0.9990** & 0.4247 & 0.9329 & 0.9917 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of monotonicity M-values of various centrality metrics across different datasets, with the method having the highest M-value highlighted in bold. The last row in red font shows the average monotonicity results, and the leftmost column in blue font presents the monotonicity results for KCH. 10 networks. Therefore, the node importance rankings generated by the KCH algorithm do not significantly differ from those of CI, SC, and CNC+, suggesting that the KCH algorithm does not utilize additional information to determine node importance. In most networks, the HKS and CC methods demonstrate a very high correlation, while the association between KCH and these two metrics is notably weaker. The Kendall coefficient between KCH and the CC method often ranges from 0.5 to 0.6, and in some networks, it even falls below 0.2, indicating significant disparities in the node importance ranking generated by KCH compared to those of the other two methods. In contrast, the ranking results obtained by the HKS and CC methods are closer, implying that KCH utilizes more information than HKS and CC to mine node importance. In the process of measuring the correlations between various algorithms, particular interest lies in comparing the correlation between the KCH and KS methods, as the KCH method was refined based on the KS method.