## 4 Numerical Analysis  
The detrended cross correlation analysis (DCCA) was conducted in world-wide 24 stock markets to obtain correlation coefficients between stock markets. When DCCA was applied, the box-length was shifted by 10 days. When we formed each network, we obtained \(\theta\) (threshold) for the stock market by performing DCCA. The network was defined by connecting the markets where the average of the \(\rho_{DCCA}\) for the two stock markets was 0.6 (strong) and 0.8 (very strong), which is over \(\theta\) = 0.7. Subsequently, network properties, such as average degree \(\langle k\rangle\), characteristic path length \(L\), global efficiency \(E^{\text{global}}\), local efficiency \(E^{\text{local}}\), clustering coefficient \(C\), betweenness centrality \(c_{k}^{\text{between}}\), and modularity \(Q\), were calculated. We used the newly constructed time series to see how the network changes over time. Figure 2 shows the algorithm of detrended cross correlation analysis and network analysis of this research. Figure 3 shows the results of DCCA on time series reconstructed into four, 10, and 20 year periods. (a) return and (b) volatility in Figure 3 are all examples of stock markets that have strongly correlated. This means that the trends of the two stock markets are almost coupled. If the average of \(\rho_{DCCA}\) in both  
Figure 2: Schematic diagram of detrended cross correlation analysis and network analysis. markets is greater than 0.7, they are connected in the network analysis, as mentioned above. In this way, the network properties for the 24 nodes were calculated for one, four, 10 and 20-year periods. We apply the _Girvan-Newman method_ to find the optimal community in the network. Figure 4 shows the network analysis of return for 2013-2016, which shows that the network was divided in step 3 to find the best community. Here, decreasing \(Q\) value gradually after step 3 means that the network is not divided into communities. We followed the process, as shown in Figure 4, to find the best community. Figures 5-7 show the dynamics of community structures for global 24 stock market networks of return and volatility generated by DCCA partitioned by four, 10, and 20 years, respectively. We have colored nodes to identify communities, and our results show that some stock markets are scattered in small communities, or they form one large community. The left figures are the community structures for return and the rights are for volatility. As can be easily compared, the community structures are similarly formed both return and volatility. The community structure analysis using DCCA and the _Girvan-Newman method_ showed that more than one huge community was not formed. Figure 4: Modularity \(Q\) according to step number using the Girvan-Newman method. At first, calculate the Newman modularity \(Q\) and \(c_{k}^{\text{between}}\) for all edges of the initial network. Subsequently, remove the edge having largest \(c_{k}^{\text{between}}\) and the procedure is iterated. If \(Q\) reaches the maximum value, then the network becomes the optimal community structure. In this example, the optimal community was divided in three steps. Figure 3: Examples of pairs with strong correlation as a result of detrended cross-correlation analysis (DCCA): we construct a network by linking two stock markets where the average \(\rho_{DCCA}\) is above 0.7 (intermediate value of strong correlation and very strong correlation). Figure 5: _Cont_. Figure 5: Community structures of return and volatility for global 24 stock markets of four-year length time series. The networks are formed by DCCA and the community structure and the communities are structured by the _Girvan-Newman method_. The communities are gradually larger to 2005–2008, and then they are divided into small communities. Figure 6: \(Cont\). Figure 5 shows the dynamics of community structures from 1997 to 2016 partitioned by four years. Here, we can find three interesting properties: First, the community structure of return and volatility is similarly formed at each period.