Dynamics of the Global Stock Market Networks Generated by DCCA Methodology

Featured Application: Authors are encouraged to provide a concise description of the speciﬁc application or a potential application of the work. This section is not mandatory. Abstract: A group of stock markets can be treated as a complex system. We tried to ﬁnd the ﬁnancial market crisis by constructing a global 24 stock market network while using detrended cross-correlation analysis. The community structures by the Girvan-Newman method are observed and other network properties, such as the average degree, clustering coe ﬃ cient, e ﬃ ciency, and modularity, are quantiﬁed. The criterion of correlation between any two markets on the detrended cross-correlation analysis was considered to be 0.7. We used the return (r t ) and volatility ( | r t | ) time series for the periods of 1, 4, 10, and 20-year of composite stock price indices during 1997–2016. Europe (France, Germany, Netherland, UK), USA (USA1, USA2, USA3, USA4) and Oceania (Australia1, Australia2) have been conﬁrmed to make a solid community. This approach also detected the signal of ﬁnancial crisis, such as Asian liquidity crisis in 1997, world-wide dot-com bubble collapse in 2001, the global ﬁnancial crisis triggered by the USA in 2008, European sovereign debt crisis in 2010, and the Chinese stock price plunge in 2015 by capturing the local maxima of average degree and e ﬃ ciency.


Introduction
Time series analysis and complex networks in the field of statistical physics are the main research areas for decades. In particular, non-stationary time series has been considered as a difficult field to work with and various methods have been developed in an effort to analyze these time series. R/S analysis, detrended fluctuation analysis (DFA) [1,2], and detrended cross correlation analysis (DCCA) [3] have been known to detect long-range correlation of time series by applying them to non-stationary time series. In addition, the complex network enables a new perspective analysis by expressing a system as nodes and edges. These two analytical methods have been applied in a variety of fields, such as physics [4], economics [5], biology [6,7], physiology [8], climate science [9], brain sciences [10], and so on.
Zebende [11] proposed the new correlation coefficient, which is the DCCA correlation coefficient that quantifies non-stationary time series using the DFA and DCCA methods. Piao and Fu [12] compared the correlation coefficients of DCCA and Pearson methods. They used temperature and relative humidity data from stations in Beijing, Jinan, Shijiazhuang, and Zhengzhou to show that the correlation coefficient of DCCA was more accurate than Pearson in the annual time series.
Newman and Girvan [13] calculated modularity to determine community structures in the network: modularity is a quantitative measurement of the network's optimal community structure.
(|r t |), as following [25]. The difference of logarithms for consecutive closing indices is almost identical to the return. Additionally, the absolute value is approximately same as the volatility. r t = log(P t+1 ) − log(P t ) |r t | = log(P t+1 ) − log(P t ) (1) P t is a stock price index at time t and P t+1 is one at time t + 1. Table 1 lists the names of surveyed stock markets. There are 24 markets of world-wide 19 countries. More specifically, 11 in Asia (eight countries), five in Europe (five countries), five in North America (two countries), three in South America (three countries), and two in Oceania (one country) were surveyed. Figure 1 shows the time series of stock price indices in 24 stock markets. There are markets that rise significantly overall, whereas there are markets that repeat rising and falling. It is common that the index plunged at the midpoint of the period (the 2008 global financial crisis). However, intuitively, it is hard to understand the correlations between these market indices. In general, the Pearson correlation coefficient is used to analyze the correlation between the two time series. However, the Pearson correlation coefficient was extraordinary high for the stock price index between countries, the temperature distribution between neighboring regions, and so on. Thus, the two time series appear to be self-evidently correlated. indices is almost identical to the return. Additionally, the absolute value is approximately same as the volatility.
= log( +1 ) − log ( ) is a stock price index at time and +1 is one at time + 1. Table 1 lists the names of surveyed stock markets. There are 24 markets of world-wide 19 countries. More specifically, 11 in Asia (eight countries), five in Europe (five countries), five in North America (two countries), three in South America (three countries), and two in Oceania (one country) were surveyed. Figure 1 shows the time series of stock price indices in 24 stock markets. There are markets that rise significantly overall, whereas there are markets that repeat rising and falling. It is common that the index plunged at the midpoint of the period (the 2008 global financial crisis). However, intuitively, it is hard to understand the correlations between these market indices. In general, the Pearson correlation coefficient is used to analyze the correlation between the two time series. However, the Pearson correlation coefficient was extraordinary high for the stock price index between countries, the temperature distribution between neighboring regions, and so on. Thus, the two time series appear to be self-evidently correlated.  1  Argentina  MERVALS  13  Korea1  KOSPI  2  Australia1  AORD  14  Korea2  KOSDAQ  3  Australia2  S&P ASX200  15  Mexico  IPC  4  Brazil  IBOVESPA  16  Netherland  AEX  5  Chile  IGPA  17  Pakistan  KSE100  6  France  CAC40  18  Singapore  STI  7  Germany  DAX  19  Taiwan  TAIMEX  8  Hong Kong  HANGSENG  20  USA1  DOW  9  Hungary  BUX  21  USA2  S&P500  10  India  BSE SENSEX  22  USA3  NASDAQ  11  Indonesia  IDX  23  USA4  DOW TRANS  12 Japan NIKKEI225 24 UK FTSE100 Thus, we will analyze the correlation between stock markets while using the detrended cross correlation analysis (DCCA) method to more strictly judge the correlation. The derived correlation is used to determine whether to connect the links between two nodes in network analysis.

Detrended cross Correlation Analysis (DCCA)
The detrended cross correlation analysis (DCCA) is a method for finding long range cross correlation properties in the non-stationary time series that were proposed by Podobnik [3]. In fact, Pearson correlation coefficient is a sum of product that removes 0th order global trend (i.e., average) for the whole box size. On the other hand, DCCA is a sum of product that removes each local trend (usually 1st order) for given sized boxes. Therefore, DCCA has advantages that it can analyze the noise effect by removing various types of trend for various box sizes when compared to Pearson method [12,26]. Here, we used MATLAB as a simulation tool to apply the DCCA method, and R for network analysis and visualization.
The first step in the DCCA is to accumulate two different time series {x i } and y i of equal lengths N.
Next, the local fluctuations are calculated, as follows.
Equation (3) refers to the following process. We divide total time series by a specific box-length n to create N − n + 1 overlapping boxes and the limit size of the box-lengths is set to 1/5 of the total time series [27]. Here, X k,i and Y k,i are linear least squares in the i-th box and they are called 'local trends'. Subtracting 'local trends' X k,i from the accumulated data X k and averaging the sum of squares make the local fluctuation f 2 DFA (n, i) of i-th box with box-length n + 1. If we want to see the cross fluctuation of the two time series, we can calculate f 2 DCCA (n, i) by replacing it with the mutual product for X k and Y k instead of the square for X k .
Averaging all of the local fluctuations for N − n overlapping boxes, the fluctuations F 2 DFA (n) and F 2 DCCA (n) are induced as in Equation (4).
Equation (4) is, in fact, an extended version of variance and co-variance. 'local trend' can be any functions as well as a linear function. If the 'local trend' is a 0-th order monomial of length n = N, the trend becomes the average value of time series, and F 2 DFA (n) and F 2 DCCA (n) are exactly same as variance and co-variance of time series. Therefore, the correlation coefficient ρ DCCA (n) of box-length n can be made as Equation (5). Unlike the Pearson correlation coefficient, ρ DCCA (n) can be obtained several values with varying box-length n.
The correlation coefficient has a value of −1 ≤ ρ DCCA (n) ≤ 1. If ρ DCCA (n) is close to 1, two time series are strongly correlated. Conversely, if the value is close to -1, they are strongly anti-correlated. In both cases, we can easily infer information regarding another time series from one. If ρ DCCA (n) is close to 0, there is no correlation between the two time series.

Complex Network Analysis and Girvan-Newman Method
We will figure out the network properties of 24 stock markets while using the analytical methodology of complex networks. The network consists of nodes and edges: nodes are the stock markets that we want to analyze, and the edges will be connected if the correlation coefficient of Equation (5) is significantly high. That is, ρ DCCA (n) of the time series is higher than that of shuffled time series for all box-length n. We analyzed 24 global stock markets networks by obtaining values, such as degree, characteristic path length, efficiency, clustering coefficient, betweenness centrality [23,28], and modularity [13].

Degree
Consider an adjacency matrix A of a network. A component a ij = 1 if two nodes i and j are connected. Otherwise, a i = 0. In this paper, a ij = 1 if two markets are correlated. Subsequently, the degree k i of node i is sum of a ij 's.

Characteristic Path Length
Define d i→j be the shortest path length from node i to node j. Then, the characteristic path length L is defined as the average value of the shortest path length for all node pairs (i, j).

Efficiency
The efficiency of a network represents a measure of information exchange in the network. Efficiency uses the inverse of the shortest path. That is, shortening the path length increases the efficiency. There are two types of network efficiency: global efficiency and local efficiency. The global efficiency is the efficiency of the entire network.
is the global efficiency of the subgraph of nodes that are directly connected to node i and E local is the average of them.

Clustering Coefficient
Clustering is a measure of how well a network is aggregated. The clustering coefficient is calculated while using the local clustering coefficient of the specific node i. The local clustering coefficient is a measure of how well the neighboring nodes are clustered around the specific node i.
where ∆ i is the number of triangles for the sub-network of node i and its neighborhood and k i is the number of neighboring nodes. C i denotes the local clustering coefficient of node i, and their average value is the clustering coefficient of the network.

Betweenness Centrality
The betweenness centrality of a node indicates how much the node is playing a role as an intermediate bridge in the network.
n i→ j is the number of shortest paths from node i to j and n i→k→ j is the number of shortest paths from node i to j through node k. Thus, the betweenness centrality c between k of a node k means the average of ratio n i→k→ j /n i→ j for all node pairs (i, j).

Modularity
Modularity can be considered to be one of the ways to identify community structures in a network. Suppose that a network is divided into two groups; the nodes in the group are fully connected and those of intergroup are not connected at all. Subsequently, it is obvious to separate the network into two groups, and modularity should have the maximum value. Otherwise, if all of the node pairs are randomly connected of a given probability, it would be difficult to separate the network into sub-groups. In this case, the modularity should have a minimum value. The modularity Q is defined, as following.
where NC is the number of communities, e uv is the fraction of edges with one node is in community u and the other in community v, and a u is the fraction of edges that are attached to the vertices in community u.

Girvan-Newman Method
The Girvan-Newman method uses betweenness centrality, which is defined in Section 3.2.5. The algorithm first calculates the betweenness centrality for the all edges. Then, remove an edge with the highest betweenness centrality value and calculate the modularity Q about the new network. Repeatedly perform this process in descending order of betweenness centrality. The structure of the network with the highest modularity Q means the most divided network.

Create Network Structure
The correlation coefficient varies between -1 and 1, according to correlation strength between two time series. Generally, the criteria of strong correlation are known to be greater than 0.6 and very strong correlation is greater than 0.8. The threshold of a correlation coefficient for constructing a network was set to 0.7 by Wu et al. [15] and 0.66 by Pereira et al. [18]. In our results, a fully connected network was appeared at a threshold of 0.6 and a sparse network at 0.8. Therefore, we set the threshold 0.7, which is the median value of strong and very strong criteria.

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 θ (threshold) for the stock market by performing DCCA. The network was defined by connecting the markets where the average of the ρ DCCA for the two stock markets was 0.6 (strong) and 0.8 (very strong), which is over θ = 0.7. Subsequently, network properties, such as average degree k , characteristic path length L, global efficiency E global , local efficiency E local , clustering coefficient C, betweenness centrality c between k , and modularity Q, were calculated.
We used the newly constructed time series to see how the network changes over time. Figure

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We used the newly constructed time series to see how the network changes over time.

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We apply the Girvan-Newman method to find the optimal community in the network.

Girvan-Newman method 
Compute between for every edge in the network  is calculated by removing the edge with the largest between  When has the largest value, find the optimal community in the network  Do this volatility   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 ρ DCCA in both 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.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 18 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.   and between for all edges of the initial network. Subsequently, remove the edge having largest between and the procedure is iterated. If reaches the maximum value, then the network becomes the optimal community structure. In this example, the optimal community was divided in three steps. 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.   and between for all edges of the initial network. Subsequently, remove the edge having largest between and the procedure is iterated. If reaches the maximum value, then the network becomes the optimal community structure. In this example, the optimal community was divided in three steps. for all edges of the initial network. Subsequently, remove the edge having largest c between k 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 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. In other words, the size of community is similar, and the countries that make up the community are nearly common. Second, the size of community grows and becomes smaller during the five periods. Third, community is formed the most in the period around 2008 when the global financial crisis was triggered. Therefore, we can infer that the global stock market cooperates in the event of a financial crisis and it forms a huge community according to the Girvan-Newman method. Figure 6 shows the dynamics of community structures from 1997 to 2016 that were partitioned by 10 years. The context does not change in Figure 6. It is clear that the communities are enlarged for the post decade (2007-2016) when compared to the prior decade (1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006). However, unlike Figures 5 and 7, it can be confirmed that the volatility forms a larger community than return for   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. In other words, the size of community is similar, and the countries that make up the community are nearly common. Second, the size of community grows and becomes smaller during the five periods. Third, community is formed the most in the period around 2008 when the global financial crisis was triggered. Therefore, we can infer that the global stock market cooperates in the event of a financial crisis and it forms a huge community according to the Girvan-Newman method. Figure 6 shows the dynamics of community structures from 1997 to 2016 that were partitioned by 10 years. The context does not change in Figure 6. It is clear that the communities are enlarged for the post decade (2007-2016) when compared to the prior decade (1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006). However, unlike Figures 5  and 7, it can be confirmed that the volatility forms a larger community than return for 10-year time series. Except for Pakistan and Singapore, all stock markets formed a huge community in volatility network for 2007-2016. Figure 7 shows the community structures for 20-year time series, the whole period of this study. The results of analyzing the time series for the whole period show that the similarity between return and volatility is the highest as compared to Figures 5 and 6. Only developed countries, such as Europe, the United States, and Australia, form a community, and the rest of the countries are shown to maintain independence. To make it more specific in Figure 5, Figure 6, Figure 7, the community structures are listed in Table 2. Table 2. Community list of global 24 stock markets of return and volatility     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 ( k , E global , E 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 k , global efficiency E global , and local efficiency E 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. Notable is that, since the 2000s, the values have oscillated between 0.6 and 1.0. These are very large clustering coefficients when compared to general networks, which can be inferred that the global stock market network has small-worldness. Lastly, Figure 8f shows a major difference between return and volatility, as mentioned above. In terms of return, local maxima occur in 1998, 2005, and 2013, and local minima occur in 2001, 2010 and 2014. This is similar to the reversed shapes of (a), (c), and (d).  Table A for detailed values. Table 3 shows average degree < >, characteristic path length , global efficiency global , local efficiency local , clustering coefficient , betweenness centrality between for return ( ), and volatility (| |) of global 24 stock networks. The unit lengths of the time series are four, 10, and 20 years. As the results of the one-year length analysis in Figure 8 showed that < >, global , and  Table A1 for detailed values. Table 3 shows average degree k , characteristic path length L, global efficiency E global , local efficiency E local , clustering coefficient C, betweenness centrality c between k for return (r t ), and volatility (|r t |) of global 24 stock networks. The unit lengths of the time series are four, 10, and 20 years. As the results of the one-year length analysis in Figure 8 showed

Conclusions
In this study, we looked for community structures and other network properties of return and volatility of global 24 stock market networks. First, we performing the detrended cross-correlation analysis (DCCA) for the return and volatility of the global 24 stock market composite price index time series for 1997-2016. We analyzed the time series divided into one year, four years, 10 years, and 20 years to see the dynamics of the network. Second, two markets are linked by considering that there is a strong correlation if the correlation coefficient of any two markets through DCCA exceeds the threshold value 0.7. Here, the threshold value 0.7 is the median of the general classification principle, strong correlation (0.6), and very strong correlation (0.8), which illustrates our analysis results well. Third, we applied the Girvan-Newman method in the stock market networks to detect the community structure. Finally, we calculated macroscopic network properties and inferred the association with international economic events. Figures 5-7 show the dynamics of optimal community structures for the global 24 stock market networks using the Girvan-Newman method for four-year, 10-year, and 20-year length time series. We can find important properties here. The community structure of return and volatility is similarly formed at each period, and the size of community grows to 2008-2012 (right after the global financial crisis triggered by USA and European fiscal crisis) and becomes smaller during the five periods. Therefore, we can infer that the global stock market cooperates in the event of a financial crisis and forms a huge community according to the Girvan-Newman method. The structures of the return and volatility are similarly formed. The stock market networks have a huge community that consists  (1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006) and the size of communities of volatility is larger than that of return ( Figure 6).
Pereira et al. [18] analyzed 20 stock markets for subprime crisis. In these results, three groups were found: (Europe), (America), and (Asia, Australia) in the pre-subprime crisis period. In the post-subprime crisis period, two groups of (Europe, America) and (Asia, Australia) are detected.
They found the characteristics of the formation of large-scaled intercontinental solidarity after the financial crisis. We found a similar property in this study. Figure 6 shows the solidarities of pre-subprime crisis. Figure 6a shows three groups: (Europe, America), (South America), and (Asia), In Figure 6b, two groups are detected: (Europe, USA) and (Australia). On the other hand, during the post-subprime crisis period, most of the stock markets formed a huge network. We analyzed 24 stock markets instead of 20 in Pereira et al. [18].
Comparing and analyzing the global stock market structure from the community perspective with their research, we can see the similar result that the structure is different before the financial crisis, but it has a huge community after the financial crisis. On the other hand, in the 20-year length time series, the community structure for the whole period can be seen as one picture, it is commonly making a community that consists of Europe, USA, and Oceania in return and volatility (Figure 7). Table 2 shows the detailed list of community.
Additionally, we inferred macroscopic properties of the global stock market networks by calculating average degree k , characteristic path length L, global efficiency E global , local efficiency E local , clustering coefficient C, and modularity Q in Equations (6)-(13) (Figure 8). The remaining five properties have similar values of return and volatility, except for modularity Q. In conclusion, we found that there was no superiority between return and volatility in network analysis. We were able to catch the events that Asian liquidity crisis in 1997, world-wide dot-com bubble collapse in 2001, global financial crisis triggered by USA in 2008, European sovereign debt crisis in 2010, and Chinese stock price plunge in 2015. Silva et al. [16] confirmed that the modularity value temporarily plummeted during the black Monday crisis. On the other hand, Yan et al. [17] showed that the E global value increased before the subprime crisis and rapidly decreased during the crisis in the SSE (Shanghai Stock Exchange) network. Both studies show that the network property value increases or decreases rapidly and sensitively responds to the financial crisis. In this paper, we confirmed that k , E global and E local are related to the financial crisis. Thus, the network properties are candidate indicators for predicting global financial market trends. However, such singularities have not been obtained for other network properties. In addition, no scientific reason has been found as to whether the three properties k , E global , and E local are related to the financial crisis.
In future studies, stock market data from a wider variety of countries will be needed to derive results that reflect reality. Moreover, we would like to construct a weighted network according to the magnitude of correlation coefficient of DCCA. Additionally, one can use another method, e.g., the Louvain algorithm, instead of Girvan-Newman method to detect optimal community in the network. Also, it is necessary to prove scientific evidence by rigorous statistical test that these network properties, such as k , E global , and E local are crucial indicators of financial crisis.

Conflicts of Interest:
The authors declare no conflicts of interest.