# Predicting Critical Nodes in Temporal Networks by Dynamic Graph Convolutional Networks
## 4 Method
### Dynamic Graph Convolutional Network
In this paper, the replaced methods are called N2V-LSTM and S2V-LSTM. These two methods embed nodes of each snapshot and input them into LSTM. ### Results

As we know, for the problem of predicting the importance of nodes in temporal networks, the closer the predicted ranking is to the real ranking (simulated value of the SIR Model), the better the performance of the method. Here, we use the kendall \(\tau\) correlation coefficients [57] to measure the performance of all methods. Firstly, we perform a sensitivity analysis of the number of input snapshots \(s\) on DGCN. In Figure 4, we show the impact of \(s\) on final kendall \(\tau\) between the real ranking and predicted ranking with the infection rate \(\beta=0.05\). From Figure 4, in the email network, the time interval is 168 h (a week), when \(s\) is less than 5, the performance of DGCN improves \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Networks** & \(\mathbf{N}\) & \(\mathbf{M}\) & \(\delta\) & \(\mathbf{L}\) \\ \hline Email & 986 & 332,334 & 168 & 75 \\ Contact & 274 & 28,244 & 1 & 69 \\ DNC & 1865 & 39,623 & 12 & 56 \\ UCI & 1899 & 59,835 & 24 & 58 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical properties of four real-world networks. \(N\) and \(M\) are the total number of nodes and edges, respectively. \(\delta\) is the time interval (hours) used to divide networks and \(L\) is the number of snapshots. as \(s\) increases, and then it stays in a steady state. This means that the importance of a user in the next 10 weeks can be predicted only by using the data in the past 5 weeks and more data is not necessary for improving the performance. In the contact network, the time interval is 1 h, DGCN is almost unaffected by \(s\) while performing extremely well. This also means that only the data of the past 1 h are needed to predict the importance of the user in the next 10 h very accurately. In the DNC network, the time interval is 12 h; it can be seen that the performance of DGCN reaches its peak at \(s=4\) and then shows a downward trend. This indicates that the data of the past 48 h are sufficient to predict the importance of the user in the next 120 h. However, very old historical data will bring a lot of noise, which will seriously affect the performance of DGCN. Similar to the DNC network, in the UCI network, the time interval is 24 h (1 day), the performance of DGCN reaches its peak at \(s=2\) and then shows a downward trend. These results show that in different temporal networks, the historical data needed to predict the importance of nodes are not as much as possible. Very old data may even have a negative impact on the prediction results. So far, the optimal value of \(s\) for different networks can be selected by an exhaustive search. After determining \(s\), from Figure 5, compared with other methods, no matter how the infection rate \(\beta\) changes, DGCN has the highest kendall \(\tau\) correlation coefficients. This means that among all the methods, the ranking result of DGCN is the closest to the real ranking result. In addition, we can see that the performances of N2V-LSTM and S2V-LSTM are very poor in all networks, especially in the contact network. This shows that a simple combination of deep learning models cannot solve some professional problems in the field of complex networks. An effective and feasible learning framework needs to be designed with more knowledge in related fields. In DGCN, we not only consider the local characteristics of the nodes but also the multiple interactions of the nodes in the same snapshot. These features will significantly affect the information transmission ability of the nodes. Figure 4: The impact of the number of input snapshots \(s\) on final kendall \(\tau\). In real life, people usually care more about top \(k\) nodes than the ranking of nodes. So we compared the hit rate of top \(k\) nodes of all methods.