## 5 Results and Discussion
### Centrality Measurement in the ThaiNet Dataset
#### 5.3.1 Centrality Measurement in Province Network  
For the province network, measures across 77 nodes are applied to identify potential hubs. Each node represents a Thai province across six regions, including northern, northeastern (Isan), western, central, eastern, and southern. Our analysis specifically focuses on degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), and eigenvector centrality (EC) as baselines for comparison with HCGC with a truncation radius of \(R=3\). Each method offers valuable insights into the relative importance of provinces within the network.  
Table 15 demonstrates the top 10 nodes with their scores for each measure. The results from HCGC show that Loppuri province has the highest score, followed by Khon Kaen, Nakhon Ratchasima, Nakhon Sawan, and Phetchabun. These results include nodes from the top-ranked nodes of baseline measures, demonstrating that the influential nodes identified by our approach consider not only local information (like DC and EC) but also global information (like CC and BC). Moreover, our proposed method produces scores with high variance, which helps identify vital hubs more easily compared to traditional DC or CC, which often produce similar scores that are difficult to distinguish and rank. Additionally, the top 10 provinces come from diverse regions of Thailand, including central, northeastern, and northern.  
Figure 5: Comparison of imprecision function of different measures on public real networks dataset with \(\beta=\beta_{c}\) and \(R=3\).  
Figure 6 presents nodes in the province network with their province labels and communities categorized by the Louvain algorithm. Larger node sizes represent higher HCGC scores. There are six communities in the network, which corresponds to the number of regions in Thailand.  
\begin{table}
\begin{tabular}{l l l l l l l l l l l l l} \hline \hline \multirow{2}{*}{**Rank**} & \multicolumn{2}{c}{**DC**} & \multicolumn{2}{c}{**CC**} & \multicolumn{2}{c}{**BC**} & \multicolumn{2}{c}{**EC**} & \multicolumn{2}{c}{**HCGC**} & \multirow{2}{*}{**Region**} & \multirow{2}{*}{**Score**} \\  & **Node** & **Score** & **Node** & **Score** & **Node** & **Score** & **Node** & **Score** & **Node** & **Province** & **Region** & **Score** \\ \hline
1 & 14 & 0.1184 & 6 & 0.2533 & 61 & 0.3211 & 6 & 0.2734 & 6 & Lopburi & Central & 217,858.6 \\
2 & 28 & 0.1184 & 58 & 0.2476 & 55 & 0.3062 & 18 & 0.2454 & 28 & Khon Kaen & North & 216,276.7 \\  & & & & & & & & & & & eastern & \\
3 & 50 & 0.1184 & 47 & 0.2460 & 62 & 0.3046 & 47 & 0.2357 & 18 & Nakhon & North & 200,406.2 \\  & & & & & & & & & & & & & \\
4 & 18 & 0.1053 & 4 & 0.2452 & 69 & 0.2874 & 28 & 0.2286 & 47 & Nakhon Sawan & Central & 200,112.5 \\
5 & 6 & 0.1053 & 50 & 0.2452 & 67 & 0.2293 & 54 & 0.2258 & 54 & Phetchabun & Northern & 191,681.5 \\
6 & 47 & 0.1053 & 56 & 0.2428 & 50 & 0.2164 & 4 & 0.2182 & 50 & Tak & Northern & 172,389.4 \\
7 & 40 & 0.0921 & 9 & 0.2397 & 28 & 0.1930 & 9 & 0.1897 & 4 & Si Ayuthaya & Central & 152,883.1 \\
8 & 57 & 0.0921 & 48 & 0.2397 & 56 & 0.1929 & 57 & 0.1883 & 16 & Nakhon Sayok & Central & 133,072.0 \\
9 & 54 & 0.0921 & 18 & 0.2397 & 58 & 0.1702 & 16 & 0.1863 & 58 & Nakhon Puthom & Central & 130,613.8 \\
10 & 4 & 0.0921 & 54 & 0.2390 & 18 & 0.1637 & 3 & 0.1822 & 9 & Sarabur & Central & 127,022.9 \\ \hline \hline \end{tabular}
\end{table}
Table 15: Top 10 provinces in the province network ranked by different measures.  
Figure 6: The nodes in province network with their province labels and communities.  
Figure 7 illustrates the top 20% of nodes representing the vital hubs in the province network with their province labels and communities. The vital hubs exist in Communities 1, 3, 4, and 5. The results show that Loppuri province, which achieves the highest score using the HCGC method by connecting nodes from other communities, can be considered the most vital hub in this network.