## 4 Numerical Analysis

Table 2: Community list of global 24 stock markets of return and volatility Figure 8 describes macroscopic network properties, i.e., (a) average degree \(\langle k\rangle\), (b) characteristic path length \(L\), (c) global efficiency \(E^{\text{global}}\), (d) local efficiency \(E^{\text{local}}\), (e) clustering coefficient \(C\), (f) modularity \(Q\) for return (\(r_{t}\)), and volatility (\(|r_{t}|\)) of global 24 stock market networks. The time series are partitioned by one-year length. Table A1 of Appendix A shows the specific values. Except for modularity \(Q\) that is shown in Figure 8f, the remaining five properties have similar values of return and volatility. In fact, modularity \(Q\) is important for macroscopic values, but it is also important to see the mesoscopic community structures, as mentioned in Figures 5-7 in detail. As mentioned earlier, community structures showed similarities in return and volatility. Thus, the network properties of global stock markets have little difference between return and volatility when we put together Figures 5-8. Therefore, it can be inferred that the results will not change much, even if we analyze with any terms of return and volatility. However, the network properties were slightly larger in the return (\(r_{t}\)), dealing with the magnitude and direction than the volatility (\(|r_{t}|\)) only dealing with the magnitude. Looking more closely at each graph, we can see that Figure 8a,c,d have very similar shapes. There are five local maxima (1997, 2001, 2008, 2010, and 2015) and one global maximum (2008) in common years. The network properties (\(\langle k\rangle\), \(E^{\text{global}}\), \(E^{\text{local}}\)) in 2008 are (8.583, 0.590, 0.768) for return and (7.500, 0.547, 0.759) for volatility. That is the year of the global financial crisis that was triggered by the subprime mortgage crisis from the United States. This suggests that the global stock markets are more closely aligned with each other in crisis situations. In addition, there was an Asian liquidity crisis in 1997, the dot-com bubble collapse in 2001, European sovereign debt crisis in 2010, and Chinese stock price plunge in 2015. Thus, in our results, average degree \(\langle k\rangle\), global efficiency \(E^{\text{global}}\), and local \begin{table}
\begin{tabular}{c c c} \hline
**Figure No.** & **Community** & **Stock Market** \\ \hline Figure 6a & (G1) Europe & France, Germany, Netherland, UK \\ Return Network & USA & USA1, USA2, USA3 \\ (1997–2006) & (G2) C./S. America & Brazil, Mexico \\  & (G3) Asia & Korea1, Korea2 \\ \hline Figure 6b & (G1) Europe & France, Germany, Netherland, UK \\ Volatility Network & USA & USA1, USA2, USA4 \\ (1997–2006) & (G2) Oceania & Australia1, Australia2 \\ \hline Figure 6c & (G1) Europe & France, Germany, Hungary, \\ Return Network & USA & USA1, USA2, USA3, USA4 \\ (2007–2016) & Oceania & Australia1, Australia2 \\  & C. America & Mexico \\  & Asia & Hong Kong, Japan, Korea1, Taiwan \\ \hline Figure 6d & (G1) Europe & France, Germany, Hungary, \\ Volatility Network & USA & USA1, USA2, USA3, USA4 \\  & Oceania & Australia1, Australia2 \\  & C./S. America & Argentina, Brazil, Chile, Mexico \\  & Asia & Korea1, Korea2, Taiwan \\ \hline Figure 7a & (G1) Europe & France, Germany, Netherland, UK \\ Return Network & USA & USA1, USA2, USA3 \\ (1997–2016) & Oceania & Australia1, Australia2 \\  & (G2) Asia & Korea1, Korea2 \\ \hline Figure 7b & (G1) Europe & France, Germany, Netherland, UK \\ Volatility Network & USA & USA1, USA2, USA4 \\ (1997–2016) & Oceania & Australia1, Australia2 \\ \hline \end{tabular}
\end{table} 
Table 2: Cont. efficiency \(E^{\text{local}}\) are seen as important indicators for analyzing or forecasting major events in the stock market. Looking at Figure 8b as an auxiliary indicator, it can be confirmed that characteristic path length \(L\) also makes local maxima in 1997, 2000-2001, 2007, 2010-2011, and 2015-2016. On the other hand, Figure 8e is fluctuating without prominent maxima.