Complex network analysis of global stock market co-movement during the COVID-19 pandemic based on intraday open-high-low-close data

Abstract

This study uses complex network analysis to investigate global stock market co-movement during the black swan event of the Coronavirus Disease 2019 (COVID-19) pandemic. We propose a novel method for calculating stock price index correlations based on open-high-low-close (OHLC) data. More intraday information can be utilized compared with the widely used return-based method. Hypothesis testing was used to select the edges incorporated in the network to avoid a rigid setting of the artificial threshold. The topologies of the global stock market complex network constructed using 70 important global stock price indices before (2017–2019) and after (2020–2022) the COVID-19 outbreak were examined. The evidence shows that the degree centrality of the OHLC data-based global stock price index complex network has better power-law distribution characteristics than a return-based network. The global stock market co-movement characteristics are revealed, and the financial centers of the developed, emerging, and frontier markets are identified. Using centrality indicators, we also illustrate changes in the importance of individual stock price indices during the COVID-19 pandemic. Based on these findings, we provide suggestions for investors and policy regulators to improve their international portfolios and strengthen their national financial risk preparedness.

Introduction

Stock market co-movement refers to a phenomenon in which multiple national stock markets experience the same trend of rising and falling under the deepening economic globalization and financial market integration (Forbes and Rigobon 2002). The classical theory holds that the co-movement of international stock markets stems primarily from two mechanisms. On the one hand, the economic fundamentals of various stock markets are interconnected. The core proposition of this point is that the stock market is a ‘barometer’ of the macroeconomy, and the macro fundamentals of various countries are interconnected, thus triggering the transnational co-movement of different stock markets (McQueen and Roley 1993). On the other hand, the market contagion mechanism leads to up-and-down linkages. This view holds that, in the event of a black swan event, such as a financial crisis, the herding effect in the financial market amplifies the speculative behavior of investors, aggravates the price volatility of the stock market, and enhances the co-movement between stock markets (King and Wadhwani 1990).

The global stock market is a complex economic system that comprises the stock markets of many countries and regions. This is an ideal testing ground for complex network analysis techniques to explore the complex co-movements of international stock markets (Wen et al. 2019). With their unique topology, complex networks can effectively capture the behavioral characteristics of various stock markets, intuitively represent their interdependence, and identify influential stock markets (Roy and Sarkar 2011). In recent years, a popular research topic in complex networks has explored the change in the topological characteristics of complex networks in black swan events (Jin et al. 2012). The co-movement of global stock markets has been investigated using the complex network during the subprime crisis and global financial crisis (Liu and Tse 2012; Li and Pi 2018), the collapse of Lehman Brothers (Roy and Saker 2011, 2013), the European debt crisis (Nobi et al. 2014; Gong et al. 2019), and Coronavirus Disease 2019 (COVID-19) (Aslam et al. 2020; Samitas et al. 2022).

In 2020, acute pneumonia COVID-19 swept the world, spread to more than 200 countries, and quickly developed into a global public health and economic disaster. Three months after the outbreak, more than 500,000 people were diagnosed with COVID-19 (Zhang et al. 2020). As of December 31, 2022, COVID-19 has caused 660.4 million infections and 6.6902 million deaths in 289 countries or territories (Dong et al. 2020). The COVID-19 pandemic has significantly increased uncertainty and volatility in global financial markets (Haroon and Rizvi 2020; Okorie and Lin 2021). In the context of uncertainty, investors become more cautious and seek safe havens to avoid possible financial losses, significantly weakening the liquidity of financial markets (Omay and Iren 2019). Global stock markets reacted quickly to the COVID-19 outbreak, and stock price indices in various countries experienced significant declines (Aslam et al. 2020). In March 2020, the United States stock market hit the circuit breaker mechanism four times in 10 days, while the last meltdown returned from 2007 to 2008 during the global financial crisis (Zhang et al. 2020). The European STOXX 600 index fell by more than 20% compared with its high index at the beginning of 2020. The day after the World Health Organization declared COVID-19 a global pandemic, March 12, was the worst day in global stock markets. The Nikkei 225 index of the Tokyo Stock Exchange plunged by more than 20% compared with its high value in December 2019. European stock markets fell by 11%, the United’s stock price index fell by more than 10%, and the S&P 500 index fell by 9.5% (Samitas et al. 2022). These declines forced a halt in trading on the Asia–Pacific and New York stock exchanges. the immense black swan event of COVID-19 had a huge impact on global stock markets, complicating the international economic and financial situation.

This study models the topology of global stock market networks before and after the COVID-19 outbreak using complex network theory to reveal the hidden information and relationships of global stock market co-movements. In the context of the COVID-19 outbreak, studying the co-movements among the world’s stock markets will help policymakers take appropriate measures to resist international shocks, prevent financial risks, and maintain macroeconomic security while opening domestic capital markets (Roy and Sarkar 2013). Investors also need to clearly understand the co-movement changes in the international market to improve their investment judgment abilities and make adjustments to an internationally diversified portfolio (Samitas et al. 2022). This study provides a dynamic and visual paradigm for complex network research, which will provide policymakers and investors with a better understanding of global stock markets in the event of a black swan event. The contributions of this study to the literature on stock market co-movement are fourfold.

• (1) A novel method for calculating the similarity between a pair of stock price indices was proposed. Most of the existing literature calculates stock price index similarity based solely on the return on the close price (e.g., Liu and Tse 2012; Roy and Sarker 2013; Li and Pi 2018; Zhang et al. 2020; Aslam et al. 2020). This practice may lead to the loss of important trading information, including open, high, and low prices (Huang et al. 2022a). In addition, the return-based method cannot provide a reliable measure of the similarity between two stock price indices in some cases (see Fig. 3) because the close price fails to fully reflect the intraday gaming dynamics of market buyers and sellers. By contrast, the proposed open-high-low-close (OHLC) data-based method can take full advantage of intraday trading information and guarantee a reliable similarity measure by additionally considering intraday volatility and the relative positions of open and close prices.

• (2) The proposed hypothesis testing-based edge-selection approach provides new insights for building complex networks. Most existing stock price index complex networks in the literature are threshold networks, that is, when the similarity of two stock price indices is higher than the threshold. A connected edge between two corresponding nodes is revealed in the network. For example, the threshold values used by Roy and Sarkar (2011, 2013), Nobi et al. (2014), and Li and Pi (2018) are 0.6, 0.6, 0.3, and 0.9. Differences in threshold values can significantly affect the topology of the network structure. When the threshold value is significant, the network is sparse; when the threshold value is small, the network is dense. However, threshold values are often set artificially. This study examines the degree of similarity between each pair of stock price indices using t-statistics for hypothesis testing. An edge between a pair of nodes with a significant similarity coefficient was incorporated into the complex network to avoid the rigid setting of artificial thresholds.

• (3) The degree centricity of the OHLC data-based network exhibited better power-law distribution characteristics than the widely used return-based network. The degree distribution of complex networks in the financial domain should follow a power-law distribution (Aiello et al. 2001; Boginski et al. 2006), which can be used as a criterion to measure whether the constructed financial complex network is reasonable. The maximum likelihood estimation for the degree of centricity of the constructed network indicates that the goodness-of-fit of the OHLC data-based network is 0.5939, which is higher than that of the return-based network (0.5369). The Kolmogorov–Smirnov statistics based on bootstrapping further prove that the degree distribution of the OHLC data-based network has a 78.6% probability of obeying a power law distribution. By comparison, that of the return-based network was only 10.4%.

• (4) This study uses an extensive sample to describe the data accurately, and is therefore able to observe structural changes in global financial networks over the COVID-19 period. Literature on global stock market co-movement in the context of public health crises, especially during COVID-19, is limited. Only a few studies, such as that of Aslam et al. (2020), used a complex network analysis method to study the impact of the COVID-19 outbreak on 56 global stock price indices in the early stage from October 15, 2019, to August 7, 2020, while Samitas et al. (2022) investigated volatility and contagion risk in 51 major global stock markets from January 1, 2018, to June 18, 2020, based on network analysis. However, many uncertainties remain regarding the impact of COVID-19 on global stock market co-movement and the comparison of global stock market networks before and after the outbreak. This study explored a long window between January 1, 2017, and December 31, 2022, spanning the COVID-19 outbreak period. In addition, we estimate a series of stock price indices for 70 major global stock markets. A complex network analysis identifies the dynamic topological characteristics of the global financial market network before and after the COVID-19 outbreak. The findings provide an in-depth and comprehensive understanding of stock market co-movements.

The remainder of this paper is organized as follows: The “Literature review” section discusses the primary literature on complex network analyses of global stock market co-movement. The “Data and method” section provides the data and methods employed for complex networks, and the “Empirical analysis of global stock market complex network” section presents an empirical analysis of the global stock market complex network. Finally, conclusions are presented in the “Conclusions”.

Literature review

Complex network analysis is a powerful tool for exploring topological relationships among actors (Scott 1988). In recent decades, complex network analysis has been widely used in various sociological research fields, such as international trade (Kim and Shin 2002), epidemic spread (Firestone et al. 2011), and smuggling networks (Huang et al. 2020). Integrating complex networks and finance involves studying stock market co-movement (Li and Pi 2018; Aslam et al. 2020). For instance, Roy and Sarkar (2011) use the Pearson correlation coefficient to measure the similarity between the returns of 93 global stock price indices from 2006 to 2010. They used the correlation coefficient as a weight to construct a complex network and a minimum-spanning tree with a correlation threshold of 0.6. The results indicate that SXXP and SXXE from Europe were the most influential stock price indices in the global stock market complex network before and after the collapse of Lehman Brothers. Liu and Tse (2012) employed a complex network analysis to examine the co-movement between the close price returns of the stock price indices of 67 member countries of the World Federation of Exchanges from 2006 to 2010. The results indicate that, before the 2008 financial crisis, the global stock market network exhibited cyclical synchronized behavior, and co-movement became pronounced after the financial crisis. In addition, developed markets are more interconnected than other ones. Roy and Sarkar (2013) conduct a complex network analysis based on 93 global stock price indices from 2006 to 2010. They detected stock market volatility in different periods through changes in the degree centrality ranking. The results indicate that the global stock market network became more interconnected during the financial crisis. Nobi et al. (2014) constructed a complex threshold network of 30 global stock price indices and 145 local Korean stocks from 2000 to 2012 based on the Pearson correlation coefficient with a threshold of 0.3. The results indicate that the average correlation of global stock price indices strengthened over time, whereas the average correlation between local Korean stocks tended to decrease.

Cao et al. (2017) construct a complex network based on the fluctuation correlations of 27 global stock price indices from January 1999 to December 2014. The dynamic evolution of the Chinese and international stock market relationships was analyzed using a sliding window approach. The results show that the connection between the Chinese and foreign stock markets became more vigorous, especially after China joined the WTO. Li and Pi (2018) construct a complex weighted network, minimum spanning tree, and threshold complex network of 38 global stock price indices from 2005 to 2010 based on the Pearson correlation coefficient. The results indicate that the United States, Southeast Asia, and European stock markets formed three clusters. Gong et al. (2019) analyzed stock market network connectivity using the transfer entropy method. The results showed that the overall connectivity of the network increased during the financial crisis. The closer the stock market is to the center of the network, the more likely it is to be affected by a financial crisis. Tang et al. (2019) applied the Granger causality test to construct a Granger causality-oriented network of 33 major global stock price indices. The results show that the United States stock price index dominates the network, with European and Asian indices not far behind. Wen et al. (2019) use tail-dependent networks to capture financial markets characterized by extreme volatility. According to the close price data of stock price indices in 73 countries from 2000 to 2016, the global efficiency of the tail-dependent network is higher than that of the Pearson’s correlation coefficient network. Moreover, the European market is more influential than the Asian and African markets.

From the literature review above, the existing literature on the complex networks of global stock markets focuses on comparing network changes before and after a black swan event, such as the mortgage, global financial, and European debt crises. Iwanicz-Drozdowska et al. (2021) investigate the impact of various economic and non-economic events on stock market spillover effects in 16 major developed and emerging countries from 2000 to 2020. The results show that viruses (e.g., the COVID-19 pandemic) were the most widespread sources of market contagion. Hence, COVID-19 can be considered a significant research event affecting global stock markets. According to the Johns Hopkins University Center for Systems Science and Engineering COVID-19 data repository, COVID-19 has caused 660.4 million infections and 6.6902 million deaths in 289 countries or territories as of December 31, 2022 (Dong et al. 2020). Figure 1 shows the cumulative number of COVID-19 cases in different continents from 22/1/2020 to 31/12/2022, which illustrates that the number of infected people maintained a rapid growth trend throughout the study period. COVID-19 poses an unprecedented threat to the economic functioning of countries worldwide (Altig et al. 2020; Deb et al. 2022a). An outstanding issue is severe unemployment (Aslam et al. 2020). For example, according to the Bureau of Labour Statistics, more than 22 million Americans lost their jobs between February and October 2020 (Milovanska-Farrington 2022); the South Asia Report 2020 pointed out that approximately 140 million people in South Asian countries were unemployed owing to lockdown measures (UNDP 2020). The Center for Monitoring the Indian Economy stated that approximately 38 million Indians have lost their jobs due to COVID-19 (Gururaja and Ranjitha 2022). In an economically integrated world where production and trade are closely linked, the impact of COVID-19 has long exceeded the loss of labor due to death from the disease and the inability to work due to illness. COVID-19 has led to a dramatic decline in industrial production, disruptions in global supply chain operations, restrictions on trade shipments between countries, the spread of global panic, massive business bankruptcy, halving of global economic growth, and a plunge in global stock price indices (Ashraf 2020; Gupta et al. 2020; Jackson 2021).

Fig. 1

Cumulative daily new COVID-19 cases by continent

The existing literature based on complex networks to study stock market co-movement still lacks exploration in the global pandemic context. The literature on the economic and financial impacts of public health crisis-type shocks is scarce for two reasons. First, the spread of infectious diseases was limited in the past and the extent and severity of the infected areas were much lower than those of COVID-19. Second, global stock market correlations were weak before the 1990s (Claessens et al. 2011). When financial markets are relatively independent, public health shocks external to the economic system hardly cause significant stock market co-movement. However, studying stock market co-movement responses in the context of public health crises is essential for the development of financial globalization and the gradual increase in financial system correlation. The limited literature on global stock market co-movement in the context of COVID-19 includes Aslam et al. (2020), who use a complex network approach to analyze the impact of COVID-19 on 56 stock price indices worldwide between November 15, 2019, and August 7, 2020. They divided the 56 stock markets into developed, emerging, and frontier markets. The findings show an increase in the number of global stock price indices that are positively correlated during the pandemic. France and Germany were at the center of developed markets, whereas Taiwan and Slovenia were at the center of emerging and frontier markets. Samitas et al. (2022) investigate the impact of the COVID-19 pandemic on 51 major stock markets based on dependence dynamics and network analysis.

The study was conducted from January 1, 2018, to June 18, 2020. Evidence suggests that the lockdown and coronavirus transmission led to an immediate financial contagion. They provide investors and policymakers with important information on the use of financial networks to improve portfolio selection. Yuan et al. (2022) construct a nonlinear financial contagion network for 26 major global stock markets during the COVID-19 pandemic using a dynamic hybrid copula-extreme value theory (EVT) model. The investigation spanned from January 1, 2019, to March 27, 2022. Investor behavior, including investor attention, sentiment, and fear, was measured using Google search volumes. They found that investor behavior plays an important role in explaining pandemic-driven financial contagions.

Although the above studies examined global stock market co-movement during the COVID-19 epidemic using a complex network approach (Aslam et al. 2020; Samitas et al. 2022), they only utilized the return information of the close price. However, in the financial market, stock price index data take the form of OHLC data (see Fig. 2). In addition to close price, other intraday trading information includes open, high, and low prices (Huang et al. 2022a).

Fig. 2

A graphical representation of OHLC data

The correlation coefficient measure that considers only the close price loses essential trading information. In many situations, it does not accurately reflect the similarity between pairs of stock price indices. Figure 3 shows two toy cases. In Fig. 3a, b, the returns of stock price indices i and j are the same in period t. However, Fig. 3a shows that stock price index i is a bull market in periods (t − 1) and t, while stock price index j belongs to a bear market in the same period. In Fig. 3b, the stock price index i surges and then falls back, while there is a sharp dip in the stock price index j and then a rebound. Additional intraday trading information provides evidence of the significantly different gaming dynamics between market buyers and sellers. There should be similarity differences between the two stock price indices i and j in period t in both situations, as illustrated in Fig. 3, where the stock price indices i and j show the potential for rising and falling trends, respectively. In contrast, the two stock price indices have perfect similarities if the calculation is solely based on returns, which does not align with the economic implications. In contrast, the other method can measure the difference between i and j.

Fig. 3

Toy cases for the inadequacy of similarity measure based only on returns

In conclusion, the existing literature has three main shortcomings related to stock market co-movement based on complex networks. First, the existing literature lacks an analysis of stock market co-movement in the context of the COVID-19 pandemic, and the sample countries and time horizons investigated are inadequate. Second, the complex global stock market networks constructed in the literature solely consider close prices. This approach essentially loses important intraday trading information (e.g., open, high, and low prices). This does not correctly reflect the similarity between pairs of stock price indices in some cases (see Fig. 3). Third, existing literature uses artificially specified thresholds for selecting edges incorporated in complex networks that lack credibility. To fill these gaps, this study constructs complex networks of 70 worldwide stock markets from 2017 to 2019 as the pre-COVID-19 outbreak period and from 2020 to 2022 as the post-COVID-19 outbreak period. A new network construction method was proposed based on OHLC data and hypothesis testing for edge selection. A complex network analysis was conducted to investigate global stock market network changes according to the network basis and centrality indicators. Stock market conditions by year, market segmentation, and continent are discussed separately to provide different analytical perspectives. This study provides a new approach for studying global stock market co-movement using complex networks that can fully use intraday trading information, enrich the relevant literature, and have broad applications. Government regulators can use this analysis to monitor the core nodes and ensure a stable overall market. Government regulators can also consider the national stock market’s ability to resist epidemics and develop relevant response mechanisms. Investment institutions and individual investors can use this analysis to improve portfolio allocation and make better investment decisions.

Data and method

Data

This study uses OHLC data for major stock price indices worldwide from January 1, 2017, to December 31, 2022. The data covered 70 countries and regions from six continents and were sourced from the Wind database (https://www.wind.com.cn/). These countries were selected based on data availability and GDP size. The countries selected for this study account for more than 98% of global GDP. The dataset considered in this study examines a larger number of countries. It has a longer time horizon than most existing studies and provides detailed and reliable insights into global stock market co-movement observations during COVID-19.

Table 1 lists the specific countries and regions and the corresponding stock price index codes in the Wind database. Among the selected stock price indices, 2 were from Oceania, 4 were from North America, 5 were from South America, 7 were from Africa, 23 were from Asia, and 29 were from Europe. The sample countries are concentrated in Asia and Europe because of their different levels of geographical aggregation and economic development.

Table 1 Summary of selected stock price indices

Method

Correlation coefficient based on OHLC data

The existing literature tends to measure the similarity between different stock markets based on close price returns, with the close price return of the i-th stock price index in period t calculated according to the following formula (Liu and Tse 2012; Roy and Sarkar 2013; Nobi et al. 2014; Li and Pi 2018; Aslam et al. 2020).

$$R_{it} = \ln x_{it}^{\left( c \right)} - \ln x_{{i\left( {t - 1} \right)}}^{\left( c \right)} = ln\frac{{x_{it}^{\left( c \right)} }}{{x_{{i\left( {t - 1} \right)}}^{\left( c \right)} }},$$

(1)

where $x_{it}^{\left( c \right)}$ and $x_{{i\left( {t - 1} \right)}}^{\left( c \right)}$ represent the close price of the ith stock price index in periods $t$ and (t − 1), respectively.

The similarity between stock markets i and j is then measured based on the Pearson correlation coefficient (Liu and Tse 2012; Roy and Sarkar 2013; Li and Pi 2018).

$$\begin{aligned} \rho_{ij} = & \frac{{Cov\left( {R_{i} ,R_{j} } \right)}}{{\sqrt {Var\left( {R_{i} } \right)Var\left( {R_{j} } \right)} }} = \frac{{E\left( {R_{i} R_{j} } \right) - E\left( {R_{i} } \right)E\left( {R_{j} } \right)}}{{\sqrt {Var\left( {R_{i} } \right)Var\left( {R_{j} } \right)} }} \\ = & \frac{{T\mathop \sum \nolimits_{t = 1}^{T} \left( {R_{it} R_{jt} } \right) - \mathop \sum \nolimits_{t = 1}^{T} R_{it} \mathop \sum \nolimits_{t = 1}^{T} R_{jt} }}{{\sqrt {\left( {T\mathop \sum \nolimits_{t = 1}^{T} R_{it}^{2} - \left( {\mathop \sum \nolimits_{t = 1}^{T} R_{it} } \right)^{2} } \right)\left( {T\mathop \sum \nolimits_{t = 1}^{T} R_{jt}^{2} - \left( {\mathop \sum \nolimits_{t = 1}^{T} R_{jt} } \right)^{2} } \right)} }} \\ \end{aligned}$$

(2)

Stock price indices are available as OHLC data for financial markets. Therefore, measuring the similarity between stock price indices using only close-price returns may lead to a loss of intraday trading information. To utilize information from a full range of financial data, this study measured the similarity between different stock price indices based on OHLC data. For the OHLC data of the i-th stock price index in period t, that is, ${\varvec{x}}_{it} = \left( {x_{it}^{\left( o \right)} ,x_{it}^{\left( h \right)} ,x_{it}^{\left( l \right)} ,x_{it}^{\left( c \right)} } \right){\prime}$, this study first divides its portions by the previous day’s close price to obtain the normalized data:

$${\varvec{x}}_{it}^{*} = \left( {x_{it}^{{\left( {o*} \right)}} ,x_{it}^{{\left( {h*} \right)}} ,x_{it}^{{\left( {l*} \right)}} ,x_{it}^{{\left( {c*} \right)}} } \right){\prime} = \left( {\frac{{ x_{it}^{\left( o \right)} }}{{ x_{{i\left( {t - 1} \right)}}^{\left( c \right)} }},\frac{{ x_{it}^{\left( h \right)} }}{{ x_{{i\left( {t - 1} \right)}}^{\left( c \right)} }},\frac{{ x_{it}^{\left( l \right)} }}{{ x_{{i\left( {t - 1} \right)}}^{\left( c \right)} }},\frac{{ x_{it}^{\left( c \right)} }}{{ x_{{i\left( {t - 1} \right)}}^{\left( c \right)} }}} \right){\prime} ,$$

(3)

where * is the mark of the normalized data; $x_{it}^{\left( o \right)}$, $x_{it}^{\left( h \right)}$, $x_{it}^{\left( l \right)}$ and $x_{it}^{\left( c \right)}$ represent the open, high, low, and close prices of the ith stock price index in period t, respectively; $x_{{i\left( {t - 1} \right)}}^{\left( c \right)}$ stands for the close price of the ith stock price index in period t. There are two reasons for the normalization of Eq. (3). First, similar to taking the natural logarithm when calculating daily returns, dividing ${\varvec{x}}_{it}$ by the previous day’s close price narrows the value range, thus enhancing the stability of the data. Second, the stock price index is the ratio of the data relative to the basement period, and its absolute value size of the stock price index may vary significantly from one stock price index to another. The relative position of quaternary price data, which can reflect intraday gaming dynamics, is more important than the absolute numerical size of the OHLC data (Huang et al. 2022a). Therefore, stock price indices with similar quaternary price location structures should share high similarities. Dividing ${\varvec{x}}_{it}$ by the previous day’s close price eliminates the effect of the absolute value of the stock price index and highlights the importance of the relative position of its quaternary prices (Tao et al. 2017).

Although ${\varvec{x}}_{it}^{*}$ eliminates the effect of the absolute value of the stock price index, using ${\varvec{x}}_{it}^{*}$ directly as the basic unit to measure the correlation coefficient is still unreasonable. There are two reasons for this finding. (1) First, the quaternary components of ${\varvec{x}}_{it}^{*}$ do not differ significantly in value, and the difference between using its quaternary components directly and considering the close price four times is slight. Thus, the economic implications implied by the OHLC data in the relative positional relationship of its quaternary components cannot be adequately examined by ${\varvec{x}}_{it}^{*}$ (Huang et al. 2022b). (2) Three constraint relationships exist among the quaternary components of ${\varvec{x}}_{it}^{*}$: 1. $x_{it}^{{\left( {l*} \right)}} > 0$, 2. $x_{it}^{{\left( {l*} \right)}} < x_{it}^{{\left( {h*} \right)}}$, 3. $x_{it}^{{\left( {o*} \right)}} ,x_{it}^{{\left( {c*} \right)}} \in \left( {x_{it}^{{\left( {l*} \right)}} ,x_{it}^{{\left( {h*} \right)}} } \right)$. These constraints limit the range of values of the internal components. Therefore, a method is required that can effectively unconstrain ${\varvec{x}}_{it}^{*}$ and extract meaningful financial information.

Referring to Huang et al. (2022a), we conducted an unconstrained transformation method on ${\varvec{x}}_{it}^{*}$ and derived ${\varvec{y}}_{it}$, which has no more constraints and represents the financial characteristics of the OHLC data well. The transformation formula is as follows:

$${\varvec{y}}_{it} = \left( {\begin{array}{*{20}c} {y_{it}^{\left( 1 \right)} } \\ {\begin{array}{*{20}c} {y_{it}^{\left( 2 \right)} } \\ {y_{it}^{\left( 3 \right)} } \\ {y_{it}^{\left( 4 \right)} } \\ \end{array} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\ln x_{it}^{{\left( {l*} \right)}} } \\ {{\text{ln}}\left( {x_{it}^{{\left( {h*} \right)}} - x_{it}^{{\left( {l*} \right)}} } \right)} \\ {\begin{array}{*{20}c} {{\text{ln}}\frac{{\lambda_{it}^{\left( o \right)} }}{{1 - \lambda_{it}^{\left( o \right)} }}} \\ {{\text{ln}}\frac{{\lambda_{it}^{\left( c \right)} }}{{1 - \lambda_{it}^{\left( c \right)} }} } \\ \end{array} } \\ \end{array} } \right),$$

(4)

where $\lambda_{it}^{\left( o \right)} = \frac{{x_{it}^{{\left( {o*} \right)}} - x_{it}^{{\left( {l*} \right)}} }}{{x_{it}^{{\left( {h*} \right)}} - x_{it}^{{\left( {l*} \right)}} }}$ and $\lambda_{it}^{\left( c \right)} = \frac{{x_{it}^{{\left( {c*} \right)}} - x_{it}^{{\left( {l*} \right)}} }}{{x_{it}^{{\left( {h*} \right)}} - x_{it}^{{\left( {l*} \right)}} }}$.

The four components of ${\varvec{y}}_{it}$ have explicit and fruitful economic implications. The first component, $y_{it}^{\left( 1 \right)}$, is a measure of the absolute size of the stock price index. Given that the difference between $x_{it}^{\left( l \right)}$ and $x_{it}^{\left( c \right)}$ is not significant, $y_{it}^{\left( 1 \right)} = \ln x_{it}^{{\left( {l*} \right)}} = {\text{ln}}\frac{{ x_{it}^{\left( l \right)} }}{{ x_{{i\left( {t - 1} \right)}}^{\left( c \right)} }}$ is approximately equal to the widely used close-price return, $R_{it}$. This means that this study extracts three other characteristic indicators from intraday trading prices in addition to the returns considered in other studies. The second component, $y_{it}^{\left( 2 \right)}$ reflects the range of fluctuations in stock price indices. The third and fourth components of ${\varvec{y}}_{it}$ represent the relative positions of the open and close prices in the stock price index, respectively. A similarity measure between stock markets based on ${\varvec{y}}_{it}$ instead of ${\varvec{x}}_{it}$ can examine the original price information and the intraday gaming process between buyers and sellers (Huang et al. 2022a).

For multiple stock markets, the sample set we consider is an $n \times p$ dimensional matrix ${\varvec{Y}} = \left( {{\varvec{y}}_{ij} } \right)_{n \times p}$ containing n time points and p variables, where each element ${\varvec{y}}_{ij}$ represents the unconstrained stock price index OHLC data. Remark ${\varvec{Y}}$ as:

$${\varvec{Y}} = \left( {\begin{array}{*{20}c} {{\varvec{y}}_{11} } & {{\varvec{y}}_{12} } & {\begin{array}{*{20}c} \cdots & {{\varvec{y}}_{1p} } \\ \end{array} } \\ {{\varvec{y}}_{21} } & {{\varvec{y}}_{22} } & {\begin{array}{*{20}c} \cdots & {{\varvec{y}}_{2p} } \\ \end{array} } \\ {\begin{array}{*{20}c} \vdots \\ {{\varvec{y}}_{n1} } \\ \end{array} } & {\begin{array}{*{20}c} \vdots \\ {{\varvec{y}}_{n2} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} \ddots \\ \cdots \\ \end{array} } & {\begin{array}{*{20}c} \vdots \\ {{\varvec{y}}_{np} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right) = \left( {{\varvec{Y}}_{1} ,{\varvec{Y}}_{2} , \ldots ,{\varvec{Y}}_{p} } \right),$$

(5)

where ${\varvec{Y}}_{j} = \left( {{\varvec{y}}_{1j} ,{\varvec{y}}_{2j} , \ldots ,{\varvec{y}}_{nj} } \right){\prime}$ denotes the jth column of matrix ${\varvec{Y}}$ $\left( {j = 1,2, \ldots ,p} \right)$, which is composed of n observations ${\varvec{y}}_{ij} \in {\mathbb{R}}^{4}$ $\left( {i = 1,2, \ldots ,n} \right)$ corresponding to the stock price index of a country or region. To calculate the correlation coefficient between stock price indices of two countries or regions, the sample mean and covariance are first defined as follows:

• (1) For ${\varvec{Y}}_{j} \in {\mathbb{R}}_{n}^{4}$, define its sample mean as

  $$\overline{\user2{Y}}_{j} = \frac{1}{n}\left( {{\varvec{y}}_{1j} + {\varvec{y}}_{2j} + \cdots + {\varvec{y}}_{nj} } \right) \in {\mathbb{R}}^{4} .$$

  (6)

• (2) For any pair of ${ }{\varvec{Y}}_{j} ,{\varvec{Y}}_{k} \in {\mathbb{R}}_{n}^{4}$, define their sample covariance as

  $$S_{jk} = {\text{Cov}}\left( {{\varvec{Y}}_{j} ,{\varvec{Y}}_{k} } \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left\langle {{\varvec{y}}_{ij} - \overline{\user2{Y}}_{j} {,}\quad {\varvec{y}}_{ik} - \overline{\user2{Y}}_{k} } \right\rangle_{{{\mathbb{R}}^{4} }} \in {\mathbb{R}}.$$

  (7)

• (3) For ${\varvec{Y}}_{j} \in {\mathbb{R}}_{n}^{4}$, define its sample variance as

  $$S_{j}^{2} = Var\left( {{\varvec{Y}}_{j} } \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left\langle {{\varvec{y}}_{ij} - \overline{\user2{Y}}_{j} ,\quad {\varvec{y}}_{ij} - \overline{\user2{Y}}_{j} } \right\rangle_{{{\mathbb{R}}^{4} }} \in {\mathbb{R}}{\ominus }.$$

  (8)

Then, the correlation coefficient of any pair of ${\varvec{Y}}_{j} ,{\varvec{Y}}_{k} \in {\mathbb{R}}_{n}^{4}$ can be deduced by

$$r_{jk} = \frac{{S_{jk} }}{{S_{j} S_{k} }} \in {{\mathbb{R}}\ominus }.$$

(9)

In line with the Pearson correlation coefficient, the t test statistic for $r_{jk}$ can be constructed in the context of a large sample (Hollander and Wolfe 1973; Press et al. 1992).

$$t_{{r_{jk} }} = \frac{{r_{jk} }}{{\sqrt {\frac{{1 - r_{jk}^{2} }}{4n - 2}} }}\sim t\left( {4n - 2} \right).$$

(10)

The corresponding p value of the two-tailed t-test statistic $t_{{r_{jk} }}$ is given by Eq. (11), where ${\Gamma }\left( \cdot \right)$ is the gamma function.

$$p_{{t_{jk} }} = 2*\left( {1 - \mathop \smallint \limits_{ - \infty }^{{t_{{r_{jk} }} }} \frac{{{\Gamma }\left( {\frac{4n - 1}{2}} \right)}}{{\sqrt {\left( {4n - 2} \right)\pi } {xt{ \Gamma }}\left( {2n - 1} \right)}}\left( {1 + \frac{{x^{2} }}{4n - 2}} \right)^{{ - \frac{4n - 1}{2}}} dx} \right).$$

(11)

The null hypothesis ($H_{0}$) and alternative hypothesis ($H_{1}$) of the t test are given by Eq. (12). When the derived $p_{{t_{jk} }}$ is greater than 0.05, we consider that the null hypothesis cannot be rejected, and the correlation coefficient $r_{jk}$ between ${\varvec{Y}}_{j}$ and ${\varvec{Y}}_{k}$ equals zero; that is, there is no linear correlation. Accordingly, no connected edges exist from nodes $j$ to $k$ in a complex network. When the calculated $p_{{t_{jk} }}$ was less than 0.05, the alternative hypothesis was accepted instead of the null hypothesis. This indicates that $r_{jk}$ is not equal to zero; that is, there is a significant linear correlation between ${\varvec{Y}}_{j}$ and ${\varvec{Y}}_{k}$. Accordingly, a connected edge exists between nodes $j$ and $k$ in the complex network, indicating stock market co-movement.

$$H_{0} : r_{jk} = 0\quad {\text{and}}\quad H_{1} : r_{jk} \ne 0.$$

(12)

Basic indicators of complex network

The world stock market complex network is constructed in the following manner: each country or region is used as a node, and the correlation coefficient between the two stock price indices of the corresponding stock markets is calculated using Eq. (9), and the correlation coefficient that is significant at the 0.05 level is taken as the connection weight between nodes. The basic metrics of the network are as follows:

• (1) Number of nodes (N): number of nodes in the network.

• (2) Number of edges (E): number of edges in the network.

• (3) Average degree (AD): number of edges connected by a node. The directed network is divided into in-degree and out-degree networks, but not into undirected networks. Average degree is the average number of edges connected to a node. In an undirected network,

  $$AD = \frac{2 \times E}{N}.$$

  (13)

• (4) Average weighted degree (AWD): the average degree weighted by the weights of the edges. Note the average correlation coefficient as $\overline{r}$ and we have

  $$AWD = AD \times \overline{r} = AD \times \frac{{\mathop \sum \nolimits_{{j \ne k, p_{{r_{jk} }} < 0.05}} r_{jk} }}{2 \times E}.$$

  (14)

• (5) Network diameter: the maximum of all shortest paths between two connected nodes.

• (6) Network density (ND): Ratio of the actual number of edges to the maximum possible number of edges. The calculation formula is as follows:

  $$ND = \frac{E}{{N \times \left( {N - 1} \right)/2}}.$$

  (15)

• (7) Average clustering coefficient: A ratio measurement of whether two different nodes that connect to a common node also have a connection.

• (8) Average path length: The shortest path length between two nodes.

Complex network centrality analysis

Centrality is an essential concept in complex network analysis, which describes the degree of importance of individual nodes in a complex network. Existing studies have defined different centrality measures that characterize the potential importance, influence, and prominence of network nodes from different perspectives. The centrality indicators selected for this study are as follows:

• (1) Degree centrality (D(x)): For a node x, its degree centrality denotes the number of edges it connects to. By denoting the set of edges connected by node x as $e\left( x \right)$, we obtain

  $$e\left( x \right) = \left\{ {e_{x1} ,e_{x2} , \ldots ,e_{xD\left( x \right)} } \right\}.$$

  (16)

For a world stock market network, the higher the degree of centrality of a node, the more significantly the stock price indices of other countries are correlated with the stock price index of that country or region.

• (2) Weighted degree centrality (WD(x)): For node x, the weighted degree centrality is calculated by weighting its connected edges based on their weights. We obtain:

  $$WD\left( x \right) = \frac{1}{D\left( x \right)}\mathop \sum \limits_{i = 1}^{D\left( x \right)} w_{xi} ,$$

  (17)

  where $w_{xi}$ is the weight of edge $e_{xi}$ connected to node x. Degree centrality can only measure the number of stock markets in other countries that are significantly correlated with a country or region’s stock market but not the strength of positive or negative correlations. The weighted degree centrality can compensate for the insufficient measurement of connection strength. Suppose that the weighted degree centrality of a node is high. In this case, other stock price indices are significantly and positively correlated with the country or region’s stock price index and the stock market co-movement phenomenon is more pronounced.

• (3) Closeness centrality (C(x)): In a network, the closeness centrality of a node is defined as the reciprocal of the sum of the shortest path lengths between that node and all the other connected nodes. Thus, a higher proximity centrality implies that a node is closer to all other nodes, indicating that the node occupies a central position in the network. The proximity centrality of node x was first defined by Bavelas (1950) and is expressed by the following equation:

  $$C\left( x \right) = \frac{1}{{\mathop \sum \nolimits_{y} d\left( {x,y} \right)}},$$

  (18)

  where d(x,y) denotes the shortest path between the nodes x and the node y connected to it. In practical applications, the normalized form of C(x) is commonly used to represent the average length of the shortest paths, rather than their sum. The normalized form of C(x) is generally obtained by multiplying the previous equation by (N-1). We obtain:

  $$\tilde{C}\left( x \right) = \frac{N - 1}{{\mathop \sum \nolimits_{y} d\left( {x,y} \right)}}.$$

  (19)

  The greater the closeness centrality of a node, the more rapid are the changes in the stock market of that country or region that can be transmitted to other stock markets.

• (4) Betweenness centrality (B(x)): In a fully connected network, the shortest path exists for any pair of nodes s and t. Betweenness centrality is a measure of the complex network centrality based on these shortest paths. The basic idea is to count the ratio of the number of nodes on the shortest paths of the other two nodes to the total number of shortest paths in the network. The first formal definition of intermediary centrality for node x was provided by Freeman (1977).

  $$B\left( x \right) = \mathop \sum \limits_{s \ne x \ne t} \frac{{\sigma_{st} \left( x \right)}}{{\sigma_{st} }},$$

  (20)

  where $\sigma_{st}$ denotes the number of shortest paths between any pair of nodes s and t, and $\sigma_{st} \left( x \right)$ denotes the number of nodes x passing through in $\sigma_{st}$. In this study, we used the centralized B(x), which was calculated as follows:

  $$\tilde{B}\left( x \right) = \frac{B\left( x \right) - \min }{{\max - \min }},$$

  (21)

  where max and min represent the largest and smallest betweenness centralities among all nodes, respectively. A country or region with high betweenness centrality can play an intermediary role in the correlation between the stock price indices of the other two countries and effectively transmit the fluctuations in the two stock markets.

• (5) Eigenvector centrality (E(x)): Eigenvector centrality assigns more weight to a node’s connections with other high-centrality nodes when measuring the importance of a node in a complex network. A high eigenvector score implies that a node is closely connected to many nodes with high eigenvector centrality. For a given complex network G with N nodes and E edges, record $A = { }\left( {a_{x,y} } \right){ }$ as the adjacency matrix, where $a_{x,y} = 1$ if node x is connected to node y, and $a_{x,y} = 0$ otherwise. The eigenvector centrality of node x can be defined as

  $$E\left( x \right) = \frac{1}{{\uplambda }}\mathop \sum \limits_{y \in M\left( x \right)} E\left( y \right) = \frac{1}{{\uplambda }}\mathop \sum \limits_{y \in G} a_{x,y} E\left( y \right),$$

  (22)

  where E(x) and E(y) represent the eigenvector centralities of nodes x and y, respectively; λ is a constant; M(x) is a set of neighbors of node x. Equation (20) can be rewritten as the eigenvector equation Ax = λx. We can derive several different eigenvalues λ based on the eigenvector equation. However, the additional requirement that all entries in the eigenvector should be non-negative indicates that only the most significant eigenvalue outcome can be measured (Lohmann et al. 2010). Power iteration is one of the many eigenvalue algorithms that can be used to determine the principal eigenvector.

Given that multiple information flow mechanisms can coexist in the network (Borgatti 2005), it is difficult to determine which centrality measure to use to judge the importance of stock price indices in the financial market network. Identifying influential nodes in a network is an open problem, because a single centrality measure cannot account for all possible types of interactions between nodes in a network (Chen et al. 2012). Referring to Roy and Sarkar (2013), this study considers centrality indicators together, according to the idea of averaging. Specifically, for each centrality indicator, each stock price index was ranked first in descending order. The sorted stock price indices are then assigned ranks, with the first-ranked stock price index having a rank of one, the second-ranked stock price index having a rank of two, and so on. Stock price indices with the same centrality index were assigned the same ranks, whereas the ranks of the following lower-centrality stock price indices were adjusted according to the number. For example, if two stock price indices are tied for the first centrality, then they will both have a rank of 1, and the third stock price index will have a rank of 3. Given that the mechanism of the different centrality weights is unknown, the final ranking of each stock price index is in ascending order according to the average rank of these five centrality indicators. Therefore, for the centrality indicators, the most important stock price index will have the lowest average rank, and the least important stock price index will have the largest average rank.

Empirical analysis of global stock market complex network

This section first analyzes the overall correlation of global stock markets and divides them global stock market into developed, emerging, and frontier markets for separate discussions. Then, the power-law distribution of the degree centrality of the return-based network and OHLC data-based network is discussed. Finally, an analysis of the complex networks of global stock markets before and after the COVID-19 outbreak is conducted based on basic indicators of complex networks and centrality indicators.

Overall correlation analysis

Figure 4 shows global stock market correlations from 2017 to 2022. The correlation between any two stock markets [calculated using Eq. (9)] are represented by squares. Blue squares indicate positive correlations and red squares indicate negative correlations. Darker blue squares indicate stronger positive correlations, and darker red squares indicate stronger negative correlations.

Fig. 4

Overall correlation plots of global stock markets in different years

The overall correlation of the global stock market from 2017 to 2022 exhibits four patterns. First, the blue squares characterize the vast majority of positive correlations, whereas the red squares characterize a few negative correlations and are very light in color. Second, 2020 is the first year of the COVID-19 outbreak. The two stock price indices were positively correlated by 80.21%, which was 16.84%, 9.13%, and 13.27% higher than that in 2017, 2018, and 2019, respectively. This result indicates a significant strengthening of the overall positive correlation in the global stock markets in 2020, which is generally caused by the negative influence of COVID-19 on global stock markets. Third, in 2021, the second year of the COVID-19 outbreak, the number of blue squares decreases significantly lower among the investigated years, indicating a significant decrease in the overall positive correlation of the global stock market. Compared to 2020, the overall positive correlation will decrease by 12.49% by 2021.

The weakening of stock market co-movement in 2021 may be mainly due to the severe polarization of global stock markets. In 2021, the lockdown practices were gradually lifted in various countries, and the economy began to recover. However, owing to differences in the improvement of economic fundamentals in various countries, the traction for stock rebounds was also different. The more a country or region’s economy is on an upward trend, the more likely that investment institutions will become bullish in the stock market. Moreover, investment institutions are likely to bear stock markets when a country or region’s upward or downward economic trend is weak. Owing to the herding effect, investors amplify the impact of investment institutions on stock market movements. For example, Vietnam, a global hub for processing and manufacturing, received significant foreign investment in 2021 and experienced a rapid rise in its stock price index by 33.72%; the United States stock market increased by 28.79% due to accommodative monetary policy; Europe experienced substantial economic growth in early 2021 and fell back towards the end of the year due to the impact of the Omicron mutant strain, eventually reaching an increase of around 15%; South Korea and Japan experienced relatively weak economic growth, resulting in an increase of approximately 5.5% in their stock price indices; China’s stock index fell by 6.21% due to the ongoing lockdown, and the Hong Kong stock index fell by 14.83% due to the impact of the mainland. Fourth, the number of blue squares is expected to increase again in 2022 compared with 2021. This indicates an increase in the overall positive correlation of the global stock market by 2022, which is 5.66% higher than that in 2021. This enhanced co-movement is due to the worldwide recovery of economic fundamentals and the strengthening of economic trade.

According to the world-class financial services provider, the FTSE Group, the world stock market can be divided into three market segments. The first category includes developed markets dominated by developed capitalist countries. The second category comprises emerging markets dominated by developing countries in Asia, Africa, and South America. The third category is the frontier market, which mainly comprises countries in Eastern Europe and the Middle East. Table 2 lists countries in these three market segments.

Table 2 FTSE’s stock market segmentation

Figure 5 and Table 3 present the overall correlation results for the three market segments by year. The overall correlations of different market segments exhibited four characteristics. (1) The positive correlation in developed markets was significantly higher than that in emerging and frontier markets. From 2017 to 2022, the average positive correlation of developed markets reached 92.00%, compared to 78.22% in emerging markets and 57.23% in frontier markets, which were 17.62% and 60.75% higher, respectively. (2) In 2020, when COVID-19 broke out, the positive correlations in the developed, emerging, and frontier markets increased by 8.06%, 5.72%, and 30.00%, respectively, compared to 2019. (3) In 2021, the positive correlation ratios of the developed, emerging, and frontier markets decreased by 10.85%, 4.46%, and 12.09%, respectively, compared to 2020. However, the positive correlation ratios for emerging and frontier markets are still higher in 2021 than in 2017, 2018, and 2019. In contrast, the positive correlation ratios for developed markets in 2021 are lower than those in 2017, 2018, and 2019. (4) The same pattern was witnessed in 2022 as in 2021. The positive correlation ratios for emerging and frontier markets are still higher in 2022 than in 2017, 2018, and 2019. In contrast, the positive correlation ratios for developed markets in 2022 are lower than those in 2017, 2018, and 2019. This result suggests that stock market co-movement in the emerging and frontier markets strengthened after the COVID-19 outbreak. In contrast, stock market co-movement in developed markets strengthened significantly in the first year and weakened significantly in the second and third years. This provides evidence of the diversion of international investments in different developed markets during the post-pandemic period.

Fig. 5

Correlation coefficient of global stock price index. Note The first row is 2017, the second row is 2018, the third row is 2019, the fourth row is 2020, the fifth row is 2021, and the sixth row is 2022; the first column is developed markets, the second column is emerging markets, and the third column is frontier markets

Table 3 Proportion of positive and negative correlation coefficients for market segments

Basic indicators of global stock market complex network

The global stock market complex networks are established by using each country or region as a node and the significant correlations at a significance level of 0.05 as the weights of the edges. Table 4 presents the base indicators of the world stock price index complex network for each year from 2017 to 2022.

Table 4 Basic indicators of the world stock market complex network during 2017–2022

According to Table 4 and Fig. 6, the following conclusions can be found.

• (1) The global stock market network was relatively stable from 2017 to 2019 before the COVID-19 outbreak. Stock market co-movement was weakest in 2017 and strongest in 2018. For these 3 years, each region’s stock price index is significantly correlated with the stock price indices of 25.143, 30.171, and 27.171 in other regions. The average weighted degrees are positive, indicating that the stock price indices of each country or region are mostly positively correlated, with an average correlation coefficient of around 0.14–0.17. The average path lengths are below 2, indicating a "small world" phenomenon in the global stock market network, which was also verified in the works of Tse et al. (2010), Li and Pi (2018), and Yang and Hou (2022).

• (2) In 2020, when the COVID-19 outbreak began, the global stock market co-movement became more pronounced. Compared to 2017, 2018, and 2019, there were significantly more stock markets with significant correlations in 2020, with a significant increase in the average degree, average weighted degree, network density, and average cluster coefficient, and a decrease in the average path length. Compared with 2019, the average degree, average weighted degree, average correlation coefficient, network density, and average clustering coefficient of the global stock market network in 2020 increased by 43.43%, 85.51%, 29.34%, 43.40%, and 25.66%, respectively. The year 2020 also witnessed an 11.56% decrease in the average path length compared with 2019. Figure 6 illustrates a comparison of the global stock market complex networks for 2019 and 2020. The size of the nodes in Fig. 6 is proportional to the degree of centrality, and the degree of centrality of each node in 2020 was significantly larger than in 2019. This phenomenon indicates a general downward trend of stock price indices in most countries under COVID-19, making stock market co-movement significantly more robust in 2021 than in 2020. These results are consistent with those reported by Aslam et al. (2020), Ashraf (2020), Gupta et al. (2020), Jackson (2021) and Samitas et al. (2022).

• (3) The 2021 global stock price index complex network exhibits more inconsistent characteristics, and only 920 pairs of stock markets are significantly correlated. Compared to 2020, the average degree decreased by 32.55%, average weighted degree decreased by 50.62%, average correlation coefficient decreased by 26.85%, network density decreased by 32.57%, average clustering coefficient decreased by 20.29%, and average path length increased by 16.88%. This finding illustrates the different ups and downs in stock price indices worldwide in 2021, and the weakening of stock market co-movement. For countries where the epidemic was under control and the lockdown was lifted, industrial production gradually recovered, consumer confidence increased, and stock price indices showed an upward trend. Their stock price indices tended to decline in countries where the epidemic persisted or where the lockdown persisted.

• (4) The year 2022 witnessed a strengthening of global stock market co-movement. The closeness of the global stock market network in 2022 is second only to that in 2020 during the entire observation period. Compared to 2021, the average degree, average weighted degree, average correlation coefficient, network density, and average clustering coefficient increased by 22.06%, 45.13%, 18.99%, 22.05%, and 11.57%, respectively, whereas the network diameter and average path length decreased by 25.00% and 9.38%, respectively. Due to the milder disease caused by the Omicron strain and the successful promotion of vaccines and potent drugs, the epidemic's impact on economic production activities has decreased (Antonini et al. 2022; Deb et al. 2022b). The year 2022 saw a consistent improvement in economic fundamentals across countries and more frequent import and export trade, strengthening economic ties between countries. Therefore, stock market co-movement is enhanced by 2022.

Fig. 6

A comparison of the global stock market complex network between 2019 and 2020

Power-law distribution of degree centrality

Aiello et al. (2001) argued that the degree centricity distributions of complex networks in the Internet, telecommunications, finance, biology, sociology, and other fields follow a power law model. Boginski et al. (2006) measured the similarity of listed stock returns in the financial sector in the United States from 1998 to 2002. They found that in the threshold network, the degree centricity showed a precise power-law distribution. Tse et al. (2010) constructed a complex network based on the close prices of all U.S. stocks from July 2005 to August 2007 and from June 2007 to May 2009. The results show that the stock market network’s degree distribution is scale-free and follows a power-law distribution. According to this pattern, the degree centrality distribution of the global stock market complex network constructed using an appropriate network construction method should be characterized by a power-law distribution.

The testing and characterization of power-law distributions are complicated because of fluctuations in the long-tail component and an uncertain range of applicability values. Therefore, the commonly used ordinary least squares method may perform poorly, leading to biased estimations and misleading conclusions. Clauset et al. (2009) proposed a framework for identifying and measuring power-law distributions. The model is based on the Kolmogorov–Smirnov statistic, which combines the maximum likelihood estimation method. They argued that random variables may obey power-law distributions in ranges larger than ${X}_{\mathrm{min}}$ instead of the full value range. Compared to the BIC, Kuiper, and Anderson–Darling statistics (D’Agostino and Stephens 1986), Clauset et al. (2009) proved that the Kolmogorov–Smirnov statistic is a better goodness-of-fit test method for determining ${X}_{\mathrm{min}}$. As the degree centricity distribution in this study did not have heavy tails, ${X}_{\mathrm{min}}=1$ was set to ensure the integrity of the data. The bootstrapping method based on the Kolmogorov–Smirnov statistic proposed by Clauset et al. (2009) is used to measure the extent to which the degree centricity distribution of the network obeys a power-law distribution.

For the return-based and OHLC data-based networks, the degree of centrality and corresponding number of nodes are summarized for the years from 2017 to 2022. Power-law distributions were fitted to the degree centrality of the two network types using the maximum-likelihood estimation method. The results are shown in Fig. 7. The estimation yielded a goodness-of-fit of 0.5369 for the return-based network and 0.5939 for the OHLC data-based network. The 500 bootstrapping of Kolmogorov–Smirnov statistics shows that the degree centricity of the return-based network has a 10.4% probability of obeying a power law distribution. In contrast, the probability of the OHLC data-based network was 78.6%. These results show that the proposed OHLC data-based network outperforms the traditional return-based network in terms of the scale-free power-law distribution properties of the global stock market network.

Fig. 7

Power-law distribution of return-based and OHLC data-based networks

Centrality analysis of the global stock market complex network

This section examines the five centrality indicators: degree, weighted degree, closeness, betweenness, and eigenvalue centrality. Table 5 summarizes the average rankings of the five degree centrality indicators for the different market segments and continents for each year from 2017 to 2022. Table 6 presents detailed results for each sample country or region. The individual rankings of the five centrality indicators are given in Table 8, 9, 10, 11 and 12 in the Appendix. Fruitful findings were derived from the figures in Tables 5 and 6. (1) In terms of market segmentation, developed markets occupied an overwhelmingly dominant position in the world stock market network, with an average centrality ranking of 24.68 for the 25 developed markets’ stock price indices from 2017 to 2022. Emerging markets were in second place, with an average centrality ranking of 34.18, for its 28 stock price indices. Frontier markets had the lowest importance, with an average centrality ranking of only 52.16 for the 17 stock price indices included. Liu and Tse (2012) and Aslam et al. (2020) similarly find that developed markets have more robust connectivity properties than other markets in the global stock market. (2) Regarding continents, Europe occupies the most critical position in the world stock market network. The average centrality ranking of the stock price indices in the European region from 2017 to 2022 is 25.79, followed by South America (average ranking of 32.92), North America (average ranking of 35.20), Oceania (average ranking of 40.07), Asia (average ranking 42.40), and Africa (average ranking 49.41). Roy and Sarkar (2013), Qiao et al. (2015), Wen et al. (2019), and Samitas et al. (2022) also find European countries dominate the global stock market network. They explained that the European countries' shared commercial trade and common currency meant that their links were intensely weighted. (3) Developed markets were centered on Austria (average ranking 9.3), Portugal (average ranking 9.7), the United Kingdom (average ranking 12.0), Ireland (average ranking 13.1), Sweden (average ranking 13.4), and Norway (average ranking 10.30); emerging markets were centered on South Africa (average ranking 9.7), the Czech Republic (average ranking 14.6), Poland (average ranking 15.9), Chile (average ranking 14.9), Brazil (average ranking 18.4), Mexico (average ranking 18.6), and Argentina (average ranking 20.3); and frontier markets were centered in Croatia (average ranking 37.8) and Romania (average ranking 38.1). Several other studies have reported similar findings. Examples include Aslam et al. (2020), who revealed the importance of Poland and the Czech Republic in the stock market network before and after the COVID-19 epidemic using a minimum spanning tree (MST); Memon and Yao (2021), who identified Austria and Sweden as super-hub nodes in Europe during the first wave of the COVID-19 epidemic based on MST; and Samitas et al. (2022), who found that South Africa and Sweden had high centrality in global stock markets from 2018 to 2020 based on dependency dynamics and network analysis.

Table 5 Average rankings of centrality in 2017 to 2022 by market segments and continent

Table 6 Each country or region’s average ranking of centrality indicators in 2017–2022

Table 7 shows the significant changes in the centrality rankings of the world stock market network in 2020, 2021, and 2022 after the COVID-19 outbreak. In this study, a country or region is considered significantly less or more critical if its stock price index has increased or decreased by at least ten places compared to the previous year’s centrality ranking. The centrality ranking in this study was calculated based on the average degree centrality, weighted degree centrality, closeness centrality, betweenness centrality, and eigenvector centrality. These five centrality indicators provide a comprehensive measure of the importance of a stock price index in a global stock market network from various perspectives (Roy and Sarkar 2013). When the centrality ranking of a stock price index decreases or increases significantly, it means that there are more or less other stock price indices with which it has a significant correlation, the strength of the correlation is greater or smaller, changes in that stock market are transmitted to other stock markets faster or slower, the efficiency of transmitting fluctuations between two other stock markets is higher or lower, and the connection to other important stock price indices is tighter or looser (Moghadam et al. 2019).

Table 7 Stock price indices with significant changes in centrality ranking

Several patterns were derived from the results presented in Table 7. First, the stock price indices for Norway, Russia, the United Kingdom, and Egypt witnessed significant centrality ranking changes for three consecutive years in 2020, 2021, and 2022. This indicates that the financial markets in these countries are more volatile and that attention should be paid to strengthening financial risk prevention during the epidemic. A typical example is the United Kingdom, which adopted a herd immunization policy at the beginning of the COVID-19 outbreak in 2020, resulting in a large population being infected (Burckhardt et al. 2022). The widespread infection caused a tight labor market and an economic downturn, corresponding to a decline in the importance of its stock price index in 2020. As vaccine promotion and herd immunity were achieved, the United Kingdom economy gradually recovered in 2021 and experienced an increase in the importance of its stock price index in 2021. In 2022, the United Kingdom’s economy was hit by an Omicron strain, with a record number of infections. Repeated epidemic outbreaks led to a lack of investor confidence and a renewed decline in the importance of the stock price index in 2021. Second, for countries that experienced only a drop in the stock price index centrality ranking, attention should be paid to encouraging production and economic recovery during the epidemic. For example, Ireland is the only country or region that has seen two drops in its centrality rankings in 2021 and 2022, respectively. In the general context of an epidemic, Ireland should pay particular attention to encouraging people to work, vigorously reviving production, and stabilizing economic levels to attract investment. Third, the centrality rankings of many established capitalist countries such as France, Germany, the Netherlands, Australia, the United States, Portugal, and Spain did not change significantly between 2020 and 2022. This result indicates that the stock markets of these countries were more resilient to financial volatility in an epidemic environment than those with significant centrality ranking changes. Fourth, Bulgaria, Switzerland, and Serbia experience two increases in their stock price index rankings from 2020 to 2022. International investors can focus on these markets for effective asset allocation.

Conclusions

This study investigates global stock market co-movements during the COVID-19 pandemic. This study has important implications for determining the impact of the COVID-19 outbreak on the topology of the global stock market network, government policies and regulations in financial markets, portfolio adjustments, and risk management by individual and institutional investors (Tang et al. 2018; Aslam et al. 2020). A novel complex network construction method is proposed based on OHLC data and hypothesis testing for edge selection. The degree distribution of the OHLC data-based network exhibited better power-law distribution properties than those of the return-based network, implying a more rational construction of the complex network. The topologies of the global stock market complex networks constructed using 70 important global stock price indices before (2017–2019) and after (2020–2022) the COVID-19 outbreak were examined using a fruitful dataset. Several important conclusions are drawn.

First, significant stock market co-movements occurred before and after the COVID-19 pandemic. This positive correlation is significantly higher in developed markets than in emerging or frontier markets. The positive correlation ratios between the two stock price indices in the global stock market complex network reached 68.65% in 2017, 73.50% in 2018, 70.81% in 2019, 80.21% in 2020, 70.19% in 2021 and 74.16% in 2022. In addition, the developed markets' average positive correlation ratio from 2017 to 2022 is 92.00%, 17.62%, and 60.75% higher than those of the emerging and frontier markets, respectively.

Second, stock market co-movement in emerging and frontier markets strengthened from 2020 to 2022, following the outbreak of the COVID-19 pandemic. In contrast, the stock market co-movement of developed markets strengthened in 2020 but weakened in 2021 and 2022. The results provide evidence of the diversion of international investments in different developed markets during the post-pandemic period.

Third, in the wake of the COVID-19 outbreak, the global stock market network became very dense in 2020, relatively sparse in 2021, and returned to a dense state by 2022. Compared with 2019, the average degree of the 2020 global stock market complex network increased by 43.43%. The year 2021 witnessed inconsistent characteristics in the complex network of the world stock market. Compared to 2020, the 2021 global stock market complex network shows a 32.55% decrease in the average degree. In 2022, the closeness of the global stock market network is second only to that of 2020 during the entire investigation period from 2017 to 2022. Compared to 2021, the average degree of the 2022 global stock market complex network increases by 22.06%.

Fourth, the stock price indices of developed markets and European countries occupy a dominant position in the world’s stock market complex network. The rankings based on the five centrality indicators indicated an average ranking of 24.68 for developed markets, 34.18 for emerging, and 52.16 for frontier markets 2017 to 2022. The centers of developed markets are recognized as Austria, Portugal, the United Kingdom, Ireland, Sweden, and Norway; the centers of emerging markets are South Africa, the Czech Republic, Poland, Chile, Brazil, Mexico, and Argentina; and the centers of frontier markets are Croatia and Romania. The European region occupies the most crucial position in the world’s stock market network, with an average centrality ranking of 25.79 from 2017 to 2022, followed by South America (32.92), North America (35.20), Oceania (40.07), Asia (42.40), and Africa (49.41).

Fifth, the centrality rankings of different countries showed different dynamics during the pandemic period. The stock price indices for Norway, Russia, the United Kingdom, and Egypt witnessed significant changes over three consecutive years from 2020 to 2022. Ireland was the only country or region with two drops in rankings in 2021 and 2022, respectively. The centrality rankings of established capitalist countries, such as France, Germany, the Netherlands, Australia, the United States, Portugal, and Spain, did not change significantly from 2020 to 2022. Countries such as Bulgaria, Switzerland, and Serbia only experienced an increase in their stock price index rankings from 2020 to 2022.

Based on the estimation results of the OHLC data-based complex network, an array of concrete recommendations is provided. First, developed markets enjoy better stock market co-movement characteristics than do emerging and frontier markets. This indicates that the economic fundamentals of developed countries are interconnected with those of other countries. As a result, developed market stock price indices have more stable real economic support and are suitable for long-term investors. Establishing capitalist countries with high financial risk resilience during an epidemic is a good choice. Second, the frontier and emerging markets are uncertain because of their weaker overall interconnectedness with global stock markets. More uncertainty indicates more significant rates of return and risk. Short-term investors seeking substantial profits can focus on frontier and emerging markets. Bulgaria and Serbia, which have consistently increased in terms of centrality in the global stock market network, can be considered. Third, as the COVID-19 pandemic progressed, the economic trends in each country or region varied depending on epidemic prevention and economic recovery. Greater volatility and uncertainty in the global equity markets are double-edged swords for investors. Investors should divest appropriately to control risk when an epidemic and production recovery are uncertain.

Moreover, additional investments can lead to substantial profits when epidemics and production recovery are inevitable. Fourth, for international investors, risk can be reduced by reducing their exposure to countries with strong economic correlations. One potential investment strategy is to choose several countries and regions with different co-movement patterns for diversification. Pairs trading strategies designed based on the movement of correlated stock price indices can also be considered (Elliott et al. 2005; Gatev et al. 2006; Mudchanatongsuk et al. 2008). Fifth, Norway, Russia, the United Kingdom, and Egypt continue to witness significant changes in their centrality rankings in 2020, 2021, and 2022, respectively. These countries should improve their public healthcare protection systems and enhance their epidemic risk resilience, thereby improving their financial risk-prevention capabilities and protecting their macroeconomic security. Finally, high-welfare countries, led by Ireland, must encourage the workforce and make solid efforts to restore production and stabilize economic levels.

The primary limitation of this study is the relatively macroscopic nature of the participants. Considering that most investors choose portfolios that tend to be on the same stock market, future studies could employ the proposed complex network construction approach to conduct an in-depth investigation of various stocks in a particular stock market. Thus, the network topology may be utilized to optimize portfolios and provide investors with practical trading strategies (Boginski et al. 2014; Tang et al. 2018).

Appendix

See Tables 8, 9, 10, 11 and 12.

Table 8 Degree centrality ranking in 2017 to 2022

Table 9 Weighted degree centrality ranking in 2017–2022

Table 10 Closeness centrality ranking in 2017 to 2022

Table 11 Betweenness centrality ranking in 2017–2022

Table 12 Eigenvector centrality ranking in 2017 to 2022