## 6 Comparative Analysis of DNC Performance in Small World, Erdos-Renyi, and Scale-Free Networks  
The Watts-Strogatz small world network (WS) is renowned for its high local clustering and short average path length, making it suitable for simulating social networks. The Erdos-Renyi random graph (ER) provides a simple randomly connected network structure. The Barabasi-Albert scale-free network (BA) better characterizes the power-law distribution \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Network** & **DNC** & **CI** & **KS** & **CC** & **KSGC** & **LGM** & **LGC** & **SC** & **IGC** \\ \hline Dolphins & 0.0008 & 0.0012 & 0.0002 & 0.0016 & 0.0136 & 0.0131 & 0.0733 & 0.0006 & 0.0126 \\ Polbooks & 0.0020 & 0.0039 & 0.0004 & 0.0043 & 0.0419 & 0.0412 & 0.3333 & 0.0034 & 0.0442 \\ Adjnoun & 0.0022 & 0.0054 & 0.0004 & 0.0052 & 0.0479 & 0.0469 & 0.2430 & 0.0017 & 0.0837 \\ Jazz & 0.0175 & 0.0368 & 0.0015 & 0.0359 & 0.3764 & 0.4438 & 1.0518 & 0.0063 & 0.3777 \\ C\_elegans & 0.0110 & 0.0413 & 0.0015 & 0.0457 & 0.4396 & 0.4176 & 2.4965 & 0.0078 & 0.7293 \\ USAir97 & 0.0195 & 0.0520 & 0.0050 & 0.1501 & 1.3578 & 1.1940 & 6.6021 & 0.0084 & 1.1002 \\ Vote & 0.0206 & 0.0964 & 0.0046 & 0.6506 & 5.6010 & 5.5481 & 115.3479 & 0.0186 & 4.4040 \\ Email & 0.0294 & 0.1170 & 0.0045 & 0.6634 & 5.7952 & 5.7732 & 130.6723 & 0.0225 & 6.0688 \\ Yeast & 0.0736 & 0.1971 & 0.0105 & 3.1377 & 27.2292 & 27.7196 & 1832.8288 & 0.4662 & 9.3199 \\ Hamsterster & 0.1881 & 1.0533 & 0.0157 & 6.7333 & 44.7648 & 45.4922 & 3458.4607 & 0.6355 & 27.4336 \\ Kohonen & 0.3810 & 3.7765 & 0.0425 & 23.3436 & 137.8985 & 139.2541 & 138,334.9177 & 0.5790 & 243.8257 \\ Dmela & 0.1596 & 3.0046 & 0.1210 & 141.5445 & 615.0009 & 597.0216 & 342,162.0082 & 0.1133 & 268.1242 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: CPU running time of DNC and the baseline methods across different datasets. Figure 5: Subfigures (**a**), (**b**), (**c**), and (**d**) respectively illustrate the CPU time comparison between DNC and the baseline methods on the Adjnoun, Jazz, Dolphins, and USAir97 networks. From these three figures, it is evident that removing crucial nodes identified by the DNC method leads to the fastest network collapse. The robustness results 'R' compared with eight other algorithms are presented in Table 7. This comparison is presented for academic accuracy. From Table 7, it can be observed that the robustness value of the DNC method is the lowest. Therefore, in the aforementioned three networks, DNC demonstrates higher accuracy compared to the baseline methods. Figure 9 presents the rank distribution of the DNC method and baseline methods in the BA network. Figure 10 displays the rank distribution of the DNC method and baseline methods in the ER graph. Figure 11 illustrates the rank distribution plot on the WS network. From these three rank distribution graphs, it is evident that the DNC method is capable of drawing a nearly flat line at the bottom. Table 8 displays the monotonicity values 'M' of the DNC method and the baseline methods in the WS, ER, and BA networks. While the monotonicity 'M' of the DNC method may not be the highest, it is still relatively close to 1. As mentioned earlier, DNC outperforms the baseline methods in terms of robustness. This is presented for academic accuracy. \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 BA & 0.1973 & 0.2323 & 0.2937 & 0.3323 & 0.2937 & 0.2410 & 0.3209 & 0.3094 & 0.2467 \\ WS & 0.4385 & 0.4395 & 0.4867 & 0.4595 & 0.4831 & 0.4438 & 0.4532 & 0.4538 & 0.4562 \\ ER & 0.3771 & 0.3989 & 0.4305 & 0.4239 & 0.4390 & 0.3914 & 0.4053 & 0.4061 & 0.4157 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Robustness of DNC and the baseline methods on the BA, WS, and ER networks. Figure 10: Rank distribution plots of DNC and the baseline methods on ER networks. Figure 9: Rank distribution plots of DNC and the baseline methods on BA networks.