## 5 Results and Discussion
### SIR Spreading Dynamic Results  
Table 10 demonstrates the efficiency of our method compared with baseline measures through a truncation radius \(R=2\) and infection probability \(\beta=\beta_{c}\). Although DCGM\({}^{++}\), which is another gravity centrality measure, demonstrates the highest accuracy in Dolphins and USAir, the results show that HCGC outperforms the existing approaches on six networks including EEC, Email, Euroroad, Blogs, GDciting, and Celegan. Moreover, our measure achieves the second-best performance on Dolphins and USAir networks. Notably, HCGC shows good performance on transportation networks (USAir and Euroroad). Additionally, the results show that DC\({}^{+}\) demonstrates better efficiency than DC on every network, which is why we chose DC\({}^{+}\) instead of traditional DC as a component in our method. Figure 2 presents the spreading capability of different measures on nine networks via [PERSON]'s correlation coefficient. The figure consists of the accuracy of various infection probabilities (\(\beta\)) ranging from \(0.5\beta_{c}\) to \(1.5\beta_{c}\) and \(R=2\) for each network. HCGC shows better performance in Dolphins, USAir, EEC, Email, Euroroad, Blogs, GDciting, and Celegan when the infection probability is greater than \(\beta_{c}\). Moreover, the accuracy of our measure consistently increases compared to others. In particular, in the USAir, EEC, Email, Blogs, GDciting, and Celegan networks, the accuracy curve shows an upward trend compared to other measures that remain relatively stable. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Network** & \(N\) & \(E\) & \(\langle k\rangle\) & \(\langle d\rangle\) & \(\langle c\rangle\) & \(\beta_{c}\) \\ \hline Province & 77 & 179 & 4.649 & 5.586 & 0.505 & 0.230 \\ District & 928 & 3057 & 6.588 & 16.594 & 0.491 & 0.149 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Basic properties of the province and district networks in the ThaiNet dataset, including number of nodes (\(N\)), edges (\(E\)), average degree (\(\langle k\rangle\)), shortest path length (\(\langle d\rangle\)), clustering coefficient (\(\langle c\rangle\)), and epidemic threshold (\(\beta_{c}\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**(Phra Nakhon, Bangkok)** &
\begin{tabular}{c} **(Dusit,** \\ **Bangkok)** \\ \end{tabular} & \(\cdots\) & **(Cho-airong, Narathiwat)** \\ \hline
**(Phra Nakhon, Bangkok)** & 0 & 1 & \(\cdots\) & 0 \\
**(Dusit, Bangkok)** & 1 & 0 & \(\cdots\) & 0 \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\
**(Cho-airong, Narathiwat)** & 0 & 0 & \(\cdots\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Adjacency matrix for districts in Thailand, where a 1 indicates direct adjacency between districts and a 0 denotes no adjacency. Table 11 presents the efficiency of our method compared with baseline measures through a truncation radius \(R=3\) and infection probability \(\beta=\beta_{c}\). Similar to the previous setting with a truncation radius \(R=2\), the HCGC outperforms existing works in most networks. However, DC\({}^{+}\) shows better performance in the EEC network. Comparing the results between \(R=2\) and \(R=3\), \(R=3\) shows better accuracy on USAir, Euroroad, Karate, and Celegan networks. This demonstrates that HCGC performs effectively on transportation networks.