## 5 Experiment Conclusions
### Experimental Results and Analysis
#### 5.3.3 Monotonicity  
We assessed the monotonicity of DNC compared to the other comparative methods using the \(M\) and rank distribution charts. Table 5 presents the specific results of \(M\) for DNC and the baseline methods. Figure 4 depicts the rank distribution charts for various methods on the Dmela, Email, Yeast, and Hamsterster networks. As observed in Figure 4 and Appendix C, in all rank distribution plots, the DNC method consistently forms a straight line towards the lower end, indicating the excellent monotonicity of the DNC method. From the above monotonicity index \(M\) values and rank distribution plots, it can be observed that LGC exhibits the best monotonicity, with an average value closest to 1. However, LGC has a high time complexity. The next best in terms of monotonicity are KSGC, LGM, and DNC, while K-Shell demonstrates the weakest monotonicity. LGC considers the cumulative total of the Laplacian centrality scores for nodes within half the distance of the node's truncation radius as an attraction model's mass, leading to its high time complexity. Similarly, the time complexity of KSGC and LGM based on the gravity model is also high. DNC, on the other hand, takes into account only a node's degree and its local clustering coefficient within the first-order neighbors. Unless a node's values for these two factors are identical, DNC can effectively differentiate the importance of different nodes. Furthermore, DNC has a time complexity of \(O(|E|)\) and exhibits strong monotonicity. Consequently, when considering all factors, DNC's performance is relatively superior. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Network** & **DNC** & **CI** & **KS** & **CC** & **KSGC** & **LGM** & **LGC** & **SC** & **IGC** \\ \hline Dolphins & **0.2862** & 0.2882 & 0.2947 & 0.3548 & 0.3005 & 0.3033 & 0.3153 & 0.3122 & 0.3124 \\ Polbooks & **0.2604** & 0.2669 & 0.3481 & 0.3341 & 0.2787 & 0.2796 & 0.3042 & 0.3146 & 0.2994 \\ Adjnoun & **0.2913** & 0.3025 & 0.3308 & 0.3260 & 0.3096 & 0.3050 & 0.3252 & 0.3276 & 0.3166 \\ Jazz & 0.4399 & 0.4384 & 0.4559 & **0.4201** & 0.4459 & 0.4438 & 0.4470 & 0.4463 & 0.4488 \\ C\_elegans & **0.3311** & 0.3480 & 0.3689 & 0.3960 & 0.3534 & 0.3563 & 0.3849 & 0.3849 & 0.3646 \\ USAir97 & **0.1230** & 0.1300 & 0.1546 & 0.1367 & 0.1379 & 0.1414 & 0.1508 & 0.1525 & 0.1562 \\ Vote & **0.1781** & 0.2188 & 0.2200 & 0.2954 & 0.3000 & 0.2169 & 0.2623 & 0.2418 & 0.2641 \\ Email & **0.2573** & 0.2702 & 0.2935 & 0.2893 & 0.2699 & 0.2710 & 0.2846 & 0.2828 & 0.2890 \\ Yeast & **0.2203** & 0.2368 & 0.2833 & 0.2500 & 0.2414 & 0.2452 & 0.2772 & 0.2704 & 0.2953 \\ Hamsterster & 0.1487 & **0.1422** & 0.1760 & 0.1605 & 0.1997 & 0.1461 & 0.1621 & 0.1604 & 0.1624 \\ Kohonen & **0.1085** & 0.1424 & 0.1708 & 0.2683 & 0.3908 & 0.1676 & 0.2722 & 0.2661 & 0.1674 \\ Dmela & **0.1293** & 0.1423 & 0.1681 & 0.1746 & 0.1463 & 0.1486 & 0.1789 & 0.1774 & 0.1673 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The robustness values (R) of DNC and the baseline methods across the different datasets show that, in the majority of networks, DNC exhibits the least robustness. The smallest R values in the different networks are shown in bold.