# Predicting Critical Nodes in Temporal Networks by Dynamic Graph Convolutional Networks
## 4 Method
### Dynamic Graph Convolutional Network
For details, the top \(k\) hit rate can be defined as

\[HR=\frac{|P\bigcap R|}{|R|} \tag{9}\]

where \(P\) is the set of top \(k\) nodes in predicted ranking and \(R\) is the set of top \(k\) nodes in real ranking. Obviously, the best method should have the largest \(HR\). From Figure 6, it can be seen that DGCN outperforms other methods in most cases under a different infection rate \(\beta\). These results further prove the effectiveness of DGCN. Figure 5: The impact of infection rate \(\beta\) on final kendall \(\tau\). Figure 6: The top 5% \(HR\) under different infection rate \(\beta\). Finally, The time costs of the above five methods are compared. Table 2 shows the training time of DGCN on four temporal networks with different \(s\). It can be seen that DGCN spends very little training time. Table 3 shows the time cost of ranking nodes by all methods. From Figure 6 and Table 3, although DGCN is not the fastest algorithm, it achieves the best results in an acceptable time and is scalable in linear time complexity. ## 6 Conclusions

In this work, we introduce a new learning framework, DGCN, which can predict the importance of nodes in temporal networks. The model consists of a special graph convolutional network and a long-short term memory network. We have assessed the performance of DGCN on four real-world networks against some benchmark methods. The results show that the ranking by DGCN is the closest to the real ranking (no matter how the infection rate \(\beta\) changes, DGCN has the highest kendall \(\tau\) correlation coefficients) and has the highest top \(k\) hit rate. Furthermore, the training time of DGCN is linearly related to the number of nodes and can be used for large-scale networks. Like most current models, DGCN relies on snapshots, which are relatively crude temporal representations. Continuous time methods can often capture more in-depth and detailed information. Hence, how to extend DGCN to continuous-time temporal networks is one of the main tasks in the future, and another interesting extension of our work is critical edges prediction in temporal networks. Conceptualization, E.Y. and D.C.; methodology, E.Y. ; validation, E.Y., Y.F. and J.Z. ; formal analysis, E.Y., H.S. and D.C.; investigation, E.Y. ; data curation, E.Y. ; writing--original draft preparation, E.Y. ; writing--review and editing, D.C., Y.F., J.Z. and H.S. ; supervision, Y.F. ; funding acquisition, D.C. and H.S. All authors have read and agreed to the published version of the manuscript. This work is jointly supported by the National Natural Science Foundation of China under Grant Nos. 61673085 and 71901115, by the Science Strength Promotion Programme of UESTC under Grant No. Y03111023901014006, by the International Cooperation Programme of JiangSu Province under Grant No BZ2020008, by the Young Scholar Programme from NUFE under Grant No SHLXW19001 and by the National Key R&D Program of China under Grant No. 2017 YFC1601005. **Institutional Review Board Statement**: Not applicable. **Informed Consent Statement**: Not applicable. **Data Availability Statement**: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the original URL of the data can't achieve now. **Conflicts of Interest**: The authors declare no conflict of interest. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Networks** & \(\mathbf{s=1}\) & \(\mathbf{s=2}\) & \(\mathbf{s=4}\) & \(\mathbf{s=8}\) & \(\mathbf{s=15}\) & \(\mathbf{s=30}\) \\ \hline Email & 69.17 & 77.45 & 94.46 & 132.30 & 195.10 & 341.90 \\ Contact & 21.54 & 22.73 & 28.06 & 38.87 & 58.55 & 106.61 \\ DNC & 128.01 & 147.25 & 175.50 & 247.95 & 362.78 & 643.84 \\ UCI & 126.32 & 142.78 & 176.30 & 261.63 & 385.58 & 752.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The training time (seconds) of four temporal networks with different \(s\).