We focus our research on China’s stock markets. First, according to the latest Standard Industrial Classification (SIC system), we randomly selected 150 A-share stocks in various industries and collected the closing prices of Shanghai and Shenzhen A-share component stocks between 2005/01/01 and 2022/03/31 from the Wind Financial Database. Considering factors such as long-term suspension and delisting, we cleaned the data and derived a sample of 100 stocks to examine their performance in terms of various stock market indicators during typical major events in the sample period.

There is a small amount of missing data among the daily closing price series of stocks. We interpolate the closing price and finally obtain that of 4,189 trading days. We use their daily log returns; that is, r _i (t) = lnP _i (t) − lnP _i (t − 1), i ∈ 1, 2, …, 100, where P _i (t) is the closing price of the index on trading day t.

We applied the sliding window method to process the data as follows. For each trading day t, we selected the daily return series (r _i (t − ω + 1), r _i (t − ω + 2), …, r _i (t)) and (r _j (t − ω + 1), r _j (t − ω + 2), …, r _j (t)) for each pair of stocks (i, j) based on the sliding window size ω (i.e., from trading day t − ω + 1 to trading day t). We then obtained the corresponding correlation coefficients C _t (i, j) and transformed into the distance measure d _t (i, j) using d t ( i , j ) = ( 2 ( 1 − C t ( i , j ) ) ) .

To establish a threshold network, we first construct an adjacency matrix A = (A(u, v)) based on the distance matrix D = (d(u, v)). When d(u, v) is less than or equal to a given threshold θ, the corresponding element A(u, v) = 1 in the adjacency matrix, and then the vertices u and v are connected by an edge (also called correlated); otherwise, A(u, v) = 0, and thus there is no edge connecting u and v directly. Considering the importance of the adjacency matrix, we present its definition below, A u , v = 0 , d t u , v > θ 1 , d t u , v ≤ θ . (4)

Note that the edges in the network have neither direction nor weight, and we refer to it as a non-directional and unweighted network. Such a complex network constructed in the above way is called a threshold network. To observe the features of such networks, we introduce four topological properties, including degree, density, clustering coefficient, and average path length.

Suppose the network contains N vertices. The degree of a vertex u is the number of edges connected to that vertex. The average degree of a network is the average of degrees for all vertices in the network [26], referred to as the degree of the network. Denote by M the actual number of edges. The maximum possible number of edges is obviously N ( N − 1 ) 2 in the network. Hence, the network density is defined and denoted by ρ = M N N − 1 2 = 2 M N N − 1 . (5)

Denote by k _u the degree of the vertex u, that is, there are k _u vertices connecting the vertex u. E _u is the number of edges between these k _u vertices. The clustering coefficient of vertex u, denoted by C _u , is then defined in the following way when k _u ≥ 2, C u = E u 1 2 k u k u − 1 = 2 E u k u k u − 1 . (6)

When k _u is equal to 0 or 1, the clustering coefficient C _u of vertex u is specified as 0. The clustering coefficient of the network is the average clustering coefficient of all vertices in the network, that is, C = 1 N Σ C u .

In the network, the number of edges contained in the shortest path from the vertex u to the vertex v is referred to as the shortest path length between u and v, denoted by D _uv . The average path length of the network, denoted by L, is defined as the average of the shortest path length between each pair of vertices in the network, that is, L = Σ u ≥ v D u v 1 2 N N − 1 = 2 Σ u ≥ v D u v N N − 1 . (7)

By the definitions of the above four topological properties, we know that the larger they are, the closer the vertices in the network are connected, except for the average path length. By contrast, for the average path length of a network, the smaller it is, the more closely the vertices are connected.

Suppose that data can be divided into K classes (K is known). The data are a set of n observations, that is, {x ₁, …, x _n }. The observations are divided into K non-intersecting sets {C ₁, …, C _K }, and all clusters are concatenated for the whole sample. i ∈ C _k means that the i-th observation x _i belongs to the k-th cluster. We want the “in-group variation” of each cluster to be as small as possible, and the mean or center position of cluster k is given by: c k ≡ 1 | C k | ∑ i ∈ C k x i , (8) where |C _k | denotes the number of observations in cluster k. There are K means in the sample, so we refer to the algorithm as K-means clustering. For an observation x _i (i ∈ C _k ) in cluster k, the deviation (x _i − c _k ) from the center of the cluster is called the “error.” The sum of squares of all the errors in cluster k is the sum of squares error (SSE) of cluster k: S S E k ≡ ∑ i ∈ C k ‖ x i − c k ‖ 2 , (9) where ‖x _i − c _k ‖ is the Euclidean distance. The SSE of all clusters is summed to obtain the SSE of the full sample; the objective is to find a division C ₁, …, C _K for the set of sample subscripts 1, …, n that minimizes the SSE of: min C 1 , … , C K S S E ≡ ∑ k = 1 K ∑ i ∈ C k ‖ x i − c k ‖ 2 . (10)

Because it is difficult to determine the global minimum solution to the problem, K-means clustering generally uses an iterative algorithm to find the local minimum solution. The results obtained will depend on the initial (random) cluster assignment of each observation. For this reason, it is important to run the algorithm multiple times from different random initial configurations. Another key issue in using K-means clustering is how the number of clusters K should be chosen.

The elbow method as a heuristic is a commonly used cluster analysis method, which helps determine the optimum number K of clusters. The method can improve the accuracy and reliability of cluster analysis. The basic principle of the elbow method is to determine the optimal number of clusters by calculating the sum of SSE for different numbers of clusters. When the number of clusters increases, the SSE decreases, but the rate of decrease slows. When the number of clusters increases to a certain point, the decrease in SSE slows rapidly, forming an elbow. This elbow is the key to the K-means elbow rule, which indicates the optimal number of clusters.

For each trading day t, we consider all stocks as points to construct the point cloud data and carry out TDA on the complexes, mainly based on the persistence landscapes. We consider the 0- and 1-dim barcodes and calculate the L ^p -norms of the persistence landscapes both together. We first choose the length of the sliding window to obtain reasonable parameters for subsequent analysis. After comparison, the L ^p -norms clearly exhibit better results with a sliding window ω = 100.

In Figure 3, we present an overall description of 100 stocks and want to compare the general indicators with the L ^p -norms to illustrate the significance of using the TDA measure. We plot the average of 100 stocks’ return series and the average of their standard deviations in Figures 3A,B, respectively. The standard deviation, a common method for calculating volatility, measures the degree of volatility of a particular return series. Thus, the average standard deviation of the 100 stocks in Figure 3B can approximately depict the state of our stock market and also capture the turbulence in the market (manifested as an increase in the standard deviation).

The L ¹-and L ²-norms of persistence landscapes of the point cloud data are depicted in Figure 3C. In detail, the norms are based on the simplicial complex of high-dimensional point clouds, so the values of the norms are closely related to the distances and the structures of the point clouds. The smaller the norms correspond, the closer the distances; that is, the stronger the correlation between the stocks. We can observe the following phenomenon. Compared to the norms we obtained, the valleys of the norms are significantly less than the peaks of the standard deviation. This is because if the log return persists at a high level of volatility for a long period, its corresponding norms are small. Conversely, if the log return is very high in volatility at a time point or sits at a high level of volatility for a short period, its corresponding norms are at normal levels. This illustrates that the norms calculated from the topological perspective are very robust to outliers.

Obviously, the L ¹-and L ²-norms were low and recovering from 2005, influenced by the SARS epidemic in 2003. Beginning at the end of October 2007, the return fluctuated rapidly and sharply, and the norms hit bottom in 2008 owing to the U.S. Subprime Lending Crisis along with the Wenchuan earthquake in China. The standard deviation continued to be at a high level for a long period of time and the L ¹-and L ²-norms declined significantly. The link among the stocks became closer, and the situation was severe. From December 2014 to May 2015, there was a fast-rising bull market in China’s stock market, during which the Shanghai and Shenzhen stock indices rose in turn (the Shanghai Composite Index and Shenzhen Stock Index accumulated 53.17% and 57.13%, respectively). Subsequently, the China Securities Regulatory Commission began to rectify illegal allocation and forced deleveraging in the short term, leading to a continuous decline in the A-share market. From June to September, the Shanghai Composite Index cumulatively declined −42.26% and the Shenzhen Composite Index declined −48.74%. The standard deviation rose to a record peak. This, in turn, led to successive plunges in various stock market-style assets. Hence, China’s 2015 stock market crash led to an unstable economic situation, and the norms rapidly fell to record valleys. The connections among stocks became increasingly close during the crisis period. Since 2018, China’s stock market has no longer been stable and flat due to the trade war between China and the U.S. and the COVID-19 pandemic. In general, when major events occur, the average standard deviation rises gradually. L ¹-and L ²-norms exhibit obvious drops and are at a low level, indicating that the structure of the data changes and the connections among stocks becomes close. Whenever there is a sudden structural change, our persistence landscapes change. This leads to changes in L ^p -norms. This is especially evident when the structure shifts to tighter situations. The use of persistence landscapes and L ^p -norms as barometers of shocks enables timely information to be detected.

We want to extract the feature of the returns using the aforementioned data processing method. Guo et al. [20] used TDA and detected the early warning signal of the 2008 crisis in the first half of 2007. The TDA model provides a better definition of market signals, especially during financial meltdown cycles [21]. The trend of the L ¹-norms is similar to that of L ²-norms, but the L ¹-norms change more frequently. We predict that using the L ¹-norms can detect more critical dates than the L ²-norms with the same other parameters and validate it through multiple tests.

We built a system to detect dates with sharp changes in point cloud data by referring to Guo et al. [20]. For every trading day t, we calculated and denoted by N(t) the L ^p -norms of the persistence landscape of the complex (we calculated the L ^p -norms of trading day t based on the sliding window from the point cloud data of trading day t − ω + 1 to trading day t), and by A(t) the L ^p -norms’ mean value of m consecutive trading days before day t. If the following two conditions are satisfied, we take trading day t as the critical date for a financial crisis or risk: i) The ratio N(t)/N(t − 1) ≤ α for a fixed α < 1. ii) The ratio N(t)/A(t) ≤ β for a fixed β < 1. For simplicity, let m = 100.

The first condition indicates the presence of risk on trading day t, whereas the second ensures that it is a true crisis-critical date. It is possible to identify multiple dates, which indicate significant stock market shocks, but some may be only brief declines caused by coincidental events that are not representative. Under the L ¹-and L ²-norms, some of the critical dates detected for the parameters at different values are list in Table 1.

From Table 1, the highest number of critical dates were detected for 2007–2008, with 2008/04/02 and 2008/06/10 appearing most frequently, followed by critical dates detected under different combinations of parameters for 2015 (2015/08/20, 2015/08/28, and 2015/11/02). The critical date in 2018 is 2018/12/03, those in 2020 are 2020/02/03 and 2020/04/01, and for 2022 is 2022/03/29. Considering that some events last longer and even had multiple rounds of shocks, we used the first identified dates in the critical years as representative critical dates for comparability: 2008/04/02, 2015/08/20, 2018/12/03, 2020/02/03,4 and 2022/03/29.

We use the same method to obtain the log return of the CSI 300 Index. Using the financial crisis-critical date detection system for its log returns, there are many sudden change dates, and it does not focus on identifying the real crisis-critical dates. In contrast, the norm series can better focus on and capture the critical dates of a real crisis, so it will be more effective to use this to perform region transformation in the future.

We analyze the stocks’ relevance for all critical dates and the 50th days before. To save space, Figure 4A–D present only the persistent landscapes composed of point clouds before and at the time of two major events, depicting the evolution of data shapes before and during each critical date. Two endpoints of the intersection of each layer with the horizontal axis correspond to the left and right endpoints of each interval of the 1-dim barcode, respectively. The highest layer corresponds to the longest interval of the barcode. The comparison indicates different degrees of change in persistent landscapes before and after the first outbreak of the two events. Those in Figure 4B tend to become smaller in most triangles compared with those in Figure 4A; that is, the norms are significantly lower. Compared to Figure 4C, the persistent landscapes in Figure 4D exhibit a significant decrease in the highest layer, and the right side of the triangle is no longer concentrated in 0.6–1.1 but is more scattered in 0.4–1, thus limiting the total area of the inner layers to decrease. This reflects the decrease in the norm. Moreover, pre-crisis to crisis norms decrease more. We obtain good evidence that the stock market links strengthen when a crisis occurs.

We plotted threshold networks for each of the five pairs of critical dates identified. To obtain a reasonably uniform threshold θ, we calculated and compared the degrees of networks of the 10 aforementioned dates, and separately differed the degrees of networks of the critical dates from their corresponding previous 50th days. When θ = 1.1, the average difference of the network average degrees for the five pairs of critical dates reaches the maximum level. The comparison of the topological indicators of the return threshold network is presented in Table 2. Our stock market moved the most during the financial crisis triggered by the U.S. Subprime Lending Crisis in 2008 and the stock market crash in 2015, followed by the trade war between China and the U.S. in 2018 and the COVID-19 pandemic in 2020 and 2022. Before and after the U.S. Subprime Lending Crisis in 2008 and the stock market crash in 2015, the degree of the network increased from 26.58 to 41.68 to 87.52 and 82.88, the network density increased from 0.268 to 0.421 to 0.884 and 0.837, the clustering coefficient of the network changed from 0.61 to 0.749 to 0.945 and 0.931, and the average path length of the network from 1.85 to 1.602 to 1.098 and 1.146, respectively. Hence, the connection between stocks became stronger. The return threshold networks for 2015/06/10 and 2015/08/20 are presented in Figure 5. The number of connected edges in the network increases significantly before and after the event, indicating that the stock market is closely linked under major event shocks. This result is consistent with previous findings.

The correlation coefficients and corresponding frequent dynamic distribution, calculated from the daily log returns with a rolling window of 100 trading days, are illustrated in Figure 6. The correlation coefficients among stocks at normal times are mostly distributed in the interval of −0.25 to 0.50, and the corresponding probability distributions are between 0 and 0.1, which are approximately normal. This indicates that the correlations among stocks are not significant, and the clustering effect is not obvious. However, the peaks of the cross-sectional graphs of the frequency distribution of correlation coefficients in 2008 and 2015 are higher. Therefore, we think the correlation coefficients among stocks generally tend to rise and cluster during a sudden event. The changes in 2018 and 2020 are smaller than the previous two systemic risks, confirming that differences exist in the degree of impact of each major event on China. The following section analyzes the evolution of heterogeneity in the structure of the stock market under different events using the higher-order characteristics of stocks (i.e., neighborhoods).

After obtaining the distance between stocks in the overall sample interval, we first select the 30 stocks with the closest distance in the neighborhood for each stock to construct new 31 × 31 distance matrices D _i,total . The subscript i = 1, …, 100 represents one stock, and total represents the distance matrix we used for the full sample interval. Second, we consider 31 stocks as points to construct the point cloud data and conduct TDA on the complexes, mainly based on the persistence landscapes. Finally, we use the same method to calculate the L ^p -norms of the persistence landscapes, called neighborhood norms. To explore the dynamic evolution of the stock market structure before and during the event, after determining the number of clusters using the elbow method, we performed a cluster analysis using the K-means on the neighborhood norms. The elbow plot of the neighborhood norms during the full-time interval is illustrated in Figure 7. When the number of clusters gradually increases from one to four, the SSE decreases in a decreasing trend with a larger magnitude, and when the number of clusters continues to increase from four to five or more, the decrease in SSE becomes small. Using the elbow method, we selected the elbow corner of SSE in figure K = 4 for the number of clusters. Therefore, we obtained the L ^p -norms contained in each category, denoted L i j p -norms. The subscript j = 1, …, 4 denotes the category of clustering from the smallest to the largest based on the norms. To quantify the changes in the norms, let p = 1, that is, the L ¹-norms are considered as an example in the following.

The industry segmentation is determined by the main business of the listed company, which in turn influences the price through fundamental factors; therefore, so the volatility of stock prices in the same industry is more similar. Therefore, according to the latest SIC, we conduct in-depth research on 100 stocks by industry. Owing to the large number of industries involved, we represent them by industry codes ¹ . We can draw the following conclusions from Table 3. In general, the distribution of stocks in most industries is more uniform and scattered, and the owning industries of several mass organizations, including stock structures, are rich. This is due to the influence of financial market psychology, such as the herd effect and market expectations, which are benign and reflect the normal characteristics of China’s stock markets. The first category is closely related to other stocks; it basically covers the representative stocks in each industry, such as wholesale and retail. Financial sector stocks are distributed in the second and third categories, which indicates that most are at the midstream level with surrounding connections because these play a role in stabilizing the stock markets. The manufacturing industry and so on dominate in all categories.

Whether it is a complementary or a collaborative relationship, there are certain connections between industries. We want to provide a more accurate characterization of how the industry is changing and influenced by major events. Therefore, we use each of the aforementioned critical dates and the 50th day before them as the scope for the industry analysis. We then apply TDA to calculate the neighborhood norms separately for all stocks on a day t and use the K-means to cluster the norms, respectively. According to each elbow graph, K = 4 is always the optimal cluster number, and finally obtain the neighborhood norms L i j , t p of each category under critical dates. The subscript t = 1, …, 10 denotes the dates in order.

We present the changes in the neighborhood norms for the 100 stocks for each pair of dates in Figure 8. Although the L ¹-norms before and during events are different, they generally display a synchronous trend, that is, the neighborhood norms before events are always greater than those during events. This illustrates that when the event occurs, each stock is influenced by other stocks in all industries. It is more consistent with the norms of the overall sample interval, which indicates that there is risk linkage among stocks in various industries, and also demonstrates the stability and applicability of the norms and categorization.

We use K-means to cluster the neighborhood norms before and during events (Tables 4, 5, respectively). There are some industries where stocks are distributed and dispersed, both before and during events. For example, the manufacturing industry is in almost every category. This illustrates that China’s stocks in the manufacturing industry are in a relatively stable state before and during events. There are also some industries whose stocks have relatively significant changes in their distribution. For example, the mining and real estate industries have large changes in their categories in response to different shocks; that is, there is an increase in correlation of different magnitudes.

In 2008, the Subprime Lending Crisis caused significant changes in the stock markets. The stock correlation of building materials and furniture industries strengthened, indicating that the crisis had a very severe impact on export-dependent industries. There is also a significant increase in stock correlations in the mining industry. The crash in China’s stock markets in 2015 led to a significant increase in stock correlations in the farming, forestry, animal husbandry, and fishery industry and wholesale and retail trade, and so on. In 2018, industries were affected by the trade war between China and the U.S.; the changes in correlations are more consistent, and a few stocks have smaller changes. The COVID-19 pandemic in 2020 has dealt an irretrievable blow to the global real economy. Electrical, heat, gas, water production and supply industry, and so on. are mainly located in the first two categories, and wholesale and retail trade are mainly located in the first category; the correlation of all of them has increased significantly. In 2022, the COVID-19 outbreak was concentrated in Shanghai. The correlation between stocks in the electrician, heat, gas, and water production and supply industry and the real estate industry all increased significantly.