## 3 Materials and Methods
### Method
#### 3.2.1 Network Structure and Performance Measurements
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Metric** & **Abbreviation** & **Means** & **Formula** \\ \hline \multicolumn{4}{c}{**Network Structure on Global Level _f(G)_**} \\ \hline Density & \(DE\) & The density of edges between nodes in a network. & \(DE=\frac{L}{2N(N-1)}\) \\ Clustering Coefficient & \(CLC\) & The degree of clustering between vertices. & \(CLC=\frac{\text{number of closed triplet}}{\text{number of triplet}(\text{ closed + open})}\) \\ The Average Path & \(APL\) & The average shortest distance between all pairs of nodes. & \(APL=\frac{2}{N(N+1)}\sum\limits_{i\
eq j}d_{ij}\) \\ Small-Worldness & \(SW\) & The degree of small-world nature. & \(SW=\frac{CLC/CLC_{random}}{APL/APL_{random}}\) \\ \hline \multicolumn{4}{c}{**Network Structure on Individual Level**} \\ \hline Degree Centrality & \(DC\) & The number of other nodes in the network directly tie with a referred one. & \(DC_{i}=\frac{\sum(w_{i}+w_{j})}{2(N-1)}\) \\ Closeness Centrality & \(CC\) & The degree of a node is close to all others. & \(CC_{i}=\frac{1}{\sum_{i=1}^{n}d_{ij}}\) \\ Betweenness Centrality & \(BC\) & The frequency of a node which is just in the path of other pairs in the network. & \(BC_{i}=\sum\limits_{j,k\in I}\frac{\sigma(j,k)}{\sigma(j,k)}\) \\ Eigenvector Centrality & \(EC\) & Based on connections to high-scoring nodes contribute more to the score of the node to measure a node’s influence. & \(EC_{i}=\lambda^{-1}\sum_{j=1}^{N}A_{ij}e_{j}\) \\ The Local Clustering & \(LCLC\) & The extent of the neighbors of a referred point, which are also interconnected. & \(LCLC_{i}=\frac{E_{i}}{m_{i}(m_{i}-1)}\) \\ \hline \multicolumn{4}{c}{**International Tourism Performance**} \\ \hline The Amount of Average & \multirow{2}{*}{_AITA_} & The average number of international tourism arrivals in the world divided by the number of countries \\ International Tourism & & \multicolumn{1}{c}{} \\ Arrivals & & & \\ The Number of & & & \\ \multicolumn{4}{c}{} \\ International Tourism & & \multicolumn{1}{c}{} \\ Arrivals From a Country & & & \\ \hline \hline \end{tabular} Note. In the above formula, \(G\) means a definite network, which has \(N\) members and \(L\) edges. \(i\) and \(j\) represent the two nodes \(i\) and \(j\), respectively, if there is a direct tie from \(i\) and \(j\), and the edge is shown as \(e_{ij}\). Notice that the network is directed, the edge from \(i\) to \(j\) is different from the edge from \(j\) to \(i\). \(L(i)\) is the set of nodes directly connected to vertex: \(d_{ij}\) represents the path length of a pair node \(i\) and \(j\). \(A_{ij}\) is the adjacent matrix of the network, where \(A_{ij}\) is 1 if it directly contacts with \(j\); otherwise, \(A_{ij}\) is 0. \(w_{j}\) is the weight of edge from \(i\) to \(j\). \(\sigma(j,k)\) is the sum of the shortest path between a given pair of nodes \(i\) and \(j\), which can be counted by the distance \(d_{ij}\). \(\sigma(j,k\mid i)\) denotes the weighted shortest paths from node \(j\) to \(k\) that pass-through vertex \(i\). \(\lambda\) and \(e_{j}\) are the largest eigenvalues of the adjacent matrix and its eigenvector. \(E_{i}\) is the number of edges in a subgraph, which only contains all the nodes that directly contact with the node \(i\).  
\end{table} 
Table 1: The measurements of network structures and international tourism performance. **Hypothesis 1. **_The global tourism network structure will affect the overall international tourism performance._  
Furthermore, from the measurements above, this paper lists the following detailed hypotheses:  
**Hypothesis 1a (H1a). **_The density (DE) of GNT can positively affect the overall tourism performance._  
**Hypothesis 1b (H1b). **_The degree of clustering between regions (CLC) of GNT can positively affect the overall tourism performance._  
**Hypothesis 1c (H1c). **_The average shortest distance between all pairs of regions (APL) of GNT can negatively affect the overall tourism performance._  
**Hypothesis 1d (H1d).