# Dynamic Analyses of Contagion Risk and Module Evolution on the SSE A-Shares
Market Based on Minimum Information Entropy

Muzi Chen  Yuhang Wang  Boyao Wu  and Difang Huang

###### Abstract

The interactive effect is significant in the Chinese stock market,
exacerbating the abnormal market volatilities and risk contagion. Based on
daily stock returns in the Shanghai Stock Exchange (SSE) A-shares, this paper
divides the period between 2005 and 2018 into eight bull and bear market
stages to investigate interactive patterns in the Chinese financial market. We
employ the LASSO method to construct the stock network and further use the Map
Equation method to analyze the evolution of modules in the SSE A-shares
market. Empirical results show: (1) The connected effect is more significant
in bear markets than bull markets; (2) A system module can be found in the
network during the first four stages, and the industry aggregation effect
leads to module differentiation in the last four stages; (3) Some stocks have
leading effects on others throughout eight periods, and medium- and small-cap
stocks with poor financial conditions are more likely to become risk sources,
especially in bear markets. Our conclusions are beneficial to improving
investment strategies and making regulatory policies.

Keywords: Map Equation; minimum information entropy theory; module detection;
LASSO method; industry aggregation; network analysis

##  I Introduction

From June 2014 to June 2015, the Shanghai Stock Exchange (SSE) A-shares index
increased by $57\%$. The market then experienced three large-scale collapses
during the following half year from June 2015 to January 2016, where the index
decreased by $49\%$ (Darby et al.,, 2021). During these events, the SSE
A-shares market plummeted with high volatility and lost about 36 trillion RMB.
Such abnormal fluctuations in the stock market were also accompanied by the
highly synergistic effect of the rise and fall of the stock market, further
increasing the stock market’s volatility (Ding et al.,, 2020).

Although the China Securities Regulatory Commission has conducted a
comprehensive reform since 2005, there are repeated abnormal fluctuations over
the transition period between the bull and bear market in the SSE A-shares
market partly due to the immaturity of the Chinese stock market in the aspects
of traders, trading system, market system, and regulatory system (Zhang et
al.,, 2021). During expansions (bull markets), stocks in the A-shares market
display blow-out increases, accumulating bubbles and financial risks. During
recessions (bear markets), the fire sale trading triggers the declines in
stock liquidity and the spreads the financial risk throughout the entire
financial system (Liu et al.,, 2020).

As the second-largest economy globally, China devotes itself to integrating
into the global finance market. Specifically, a series of policy measures,
including establishing the Renminbi Qualified Foreign Institutional Investor
(RQFII) program, opening the Shanghai-Hong Kong Stock Connect, and
continuously raising quotas in the Qualified Foreign Institutional Investor
(QFII) program and the Qualified Domestic Institutional Investor (QDII)
scheme, are adopted to significantly strengthen the connections between China
and the rest of the world (Jiang et al.,, 2019). The evolution of the SSE
A-shares market over good and bad periods is related to the reform of the
Chinese stock market and has profound influences on the international capital
and cross-border spillover risk (Wu,, 2019; Yu et al.,, 2020).

Traditional econometric measurements, including the Pearson correlations and
Granger causalities, qualify the pair-wise relationship of two concerned
stocks in a financial market without considering the potential influence from
rest ones in the same system (Billio et al.,, 2012; Engle,, 2002; Brownlees
and Engle,, 2011). Although the multivariate regression model can characterize
interactions between equities from a systemic way, this framework may fail to
effectively fit the real data due to the high-dimensional problem that limited
observations are used to estimate a significant number of parameters
reflecting relationships of stocks (Acemoglu et al.,, 2015; Acharya et al.,,
2012; Diebold and Yilmaz,, 2008, 2011). To overcome these above issues, we use
the Least Absolute Shrinkage and Selection Operator (LASSO) method to model
the SSE A-shares market returns and measure statistically significant
connections between the equities system, and shrink those insignificant ones
into zeros (Wu,, 2019; Yan et al.,, 2020). More importantly, we also use the
Map Equation method to conduct a dynamic analysis of financial contagion
patterns. Our empirical results show that the SSE A-shares market’s contagion
pattern reveals the industry differentiation after 2014, and compared with
large-cap stocks, medium- and small-cap stocks react to financial risks more
distinctly and function as financial contagion channels.

The interactions between stocks and contagion risks are essential to
understand the stock market fluctuations and global financial crisis (Girardi
and Ergün,, 2013; Hautsch et al.,, 2015; Kyrtsou et al.,, 2020). The classical
econometric methods rely on pair-wise measurements including the Pearson
correlations to describe the relationships within the network system (Adeloye
et al., 2015a, ; Adeloye et al., 2015b, ; Papana et al.,, 2013; Rudan et al.,,
2015). Naoui et al., (2010) use the DCC-GARCH model to study the pair-wise
relationships of stock index returns of different regions over the subprime
mortgage crisis and find that the United States is an essential source of
contagion during this crisis. Selecting three fields–industry, banking, and
public utilities–as research objects, Grout and Zalewska, (2016) show that
industrial market risks increase during the crisis. Bernal et al., (2014)
introduce the CoVaR method to measure the relationships among stock returns in
the banking, insurance, and other financial sectors during the financial
crisis. Das et al., (2019) propose a mixed-frequency based regression
approach, derived from functional data theories, to analyze the influence of
global crises on stock market correlations between G7 countries.

Classical econometric approaches mainly focus on the direct relationship
between two financial agents but fail to reflect potential influences from the
direct connection’s complex systems (Battiston et al.,, 2012; Bisias et al.,,
2012; Huang et al.,, 2009; Kritzman et al.,, 2011). However, such underlying
interactions can be well revealed under the networking framework by
investigating financial networks’ topological properties and statistical
characteristics. Liu and Tse, (2012) use five years of stock index data from
67 countries and use Pearson correlations to generate a complex network. Gong
et al., (2019) employ the transfer entropy method to analyze interactions
between national stock markets and discover that countries affected by the
crisis become closer to each other and the total network connectedness rises
during the crisis. Chen et al., (2020) use complex network theories to measure
systemic risks in the stock market and develop dynamic topological indicators
to analyze financial contagion and qualify the magnitude of systemic risks.
Constructing undirected and directed volatility networks of the global stock
market, Lee et al., (2019) apply machine learning methods to study network
indicators for establishing an international financial portfolio management
approach. Liu et al., (2017) base on 6,600 banks’ decision rules and behaviors
reflected in quarterly balance sheets to construct interbank networks and
further examine how financial shocks spread through financial contagion. Kumar
and Deo, (2012) apply random matrix theories to study topological properties
of the network consisting of 20 nations and analyze communities in the
generated network under different thresholds. Li et al., (2017) discuss the
relationships between listed energy companies and their shareholders under the
networking framework. Empirical studies show that most energy investment is
concentrated in a few countries, and some islands or regions play
irreplaceable roles in tax avoidance. Paltalidis et al., (2015) employ the
maximum entropy method to study the systemic risk and analyze the
vulnerability of the Euro area’s financial network.

Using networking methodologies to study stock markets may suffer from the
high-dimensional problem such that traditional estimations are not consistent.
The number of stocks ($N$) in a market is comparable to that of observations
($T$) over a specific period (e.g., the bull and bear market), and hence the
size of unknown parameters ($O(N^{2})$) is comparable to that of data
($O(NT)$). The LASSO method provides a promising solution to alleviate the
high-dimension problem when building financial networks. Xu et al., (2019)
utilize the LASSO-CoVaR model to construct a financial network for the Chinese
stock market between 2010 and 2017 and analyze financial institutions’ status
and role in crises. Using the data on the subprime mortgage crisis, Demirer et
al., (2018) adopt the LASSO-VAR method to analyze the static and dynamic
connectedness in the global system.

Applying the LASSO method, we generate the stock networks and community
structures to analyze the evolution of financial systems. We further introduce
the Map Equation method to study the dynamic changes of the SSE A-shares
market and its differentiation in industries. The Map Equation method is based
on information theories and further improved in subsequent studies (Rosvall
and Bergstrom,, 2007, 2008; Rosvall et al.,, 2009), widely used in biological,
information, and social networks (Kim and Jeong,, 2011). This method utilizes
the probability flow of random walks on a network as a proxy for information
flows in the entire system to decompose the network into different modules by
compressing the probability flow description. The Map Equation approach is
also adopted to investigate risk transmission in financial settings, which
makes it possible to analyze the overnight market risk path of commercial
banks (Bech et al.,, 2015), the centrality of financial network institutions
and measurements on systemic risks (Chan-Lau,, 2018), and the financial
integration of banks in developed regions before the subprime crisis (Garratt
et al.,, 2014).

##  II Methodologies

###  A LASSO Estimation and Network Construction

Traditional econometric methods use pair-wise measurements, such as Pearson
correlations and Granger causalities, to qualify interactions between stocks.
However, these approaches only measure the direct relationship between the two
stocks without considering the potential influence from the rest stocks
through a systematic perspective. When using Pearson correlations, the
correlation between stock A and B may indirectly derive from stock C, which is
highly correlated with stock A and B separately. Granger causalities are also
inappropriate for describing sophisticated linkages in financial markets due
to theoretical reasons. For any given stock pair, the white noise assumption
in the Granger causality test implies that there are no connections between
the concerned two stocks and the rest. In other words, these two stocks are
presumed to be isolated from the entire system, and this may contradict the
networking structure in financial markets. Although classic multivariate
regression models can overcome the above two drawbacks, they fail to fit the
data in the high-dimensional situation that the number of stocks ($N$) is
proportional to that of observations ($T$) (i.e., $N=O(T)$). As a promising
solution, the LASSO method chooses an absolute value function as the penalty
term to screen significant variables and shrinks insignificant ones into
zeros, which can solve the fitting problem in high-dimensional cases.

We consider a multivariate linear regression model to reveal relationships in
the stock market from a systemic way. For the stock $i$, the model is

(1)
$r_{it}=r_{1t}\beta_{i1}+r_{2t}\beta_{i2}+\cdots+r_{i-1,t}\beta_{i,i-1}+r_{i+1,t}\beta_{i,i+1}+\cdots+r_{Nt}\beta_{iN}+\varepsilon_{it},$

where $r_{it}=\ln P_{it}-\ln P_{i,t-1}$ is the log return of stock $i$,
$P_{it}$ is the stock price of $i$ at time $t$, $N$ is the number of stocks
and $\varepsilon_{it}$ is the error term. Since unknown parameters
$\bm{\beta}_{-i}=\\{\beta_{ij},j=1,\cdots,i-1,i+1,\cdots,N\\}$ qualify the
stock relationships in the financial market, the LASSO method is adopted to
estimate those statistically significant parameters and shrink those
insignificant ones into zeros. The LASSO estimate is

(2)
$\widehat{\bm{\beta}}_{-i}=\operatorname*{arg\,min}_{\bm{\beta}_{-i}}\left\\{\frac{1}{2T}\sum_{t=1}^{T}\left(r_{it}-\sum_{j\neq
i}r_{jt}\beta_{ij}\right)^{2}+\lambda\sum_{j\neq
i}\big{|}\beta_{ij}\big{|}\right\\},$

where $T$ is the number of observations and $\lambda$ is the tuning parameter
preset by the cross-validation method.111This paper uses the glmnet package in
R to obtain LASSO estimations. Repeat the above procedure for all stocks.
Then, the adjacency matrix $\bm{A}=\\{a_{ij}\\}$ for the financial market is
define as

(3) $a_{ij}=\left\\{\begin{array}[]{cc}1,&\mathrm{if}\
\widehat{\beta}_{ji}\neq 0;\\\ 0,&\mathrm{if}\
\widehat{\beta}_{ji}=0.\end{array}\right.$

Equation (3) suggests that a direct link from stock $i$ to $j$ exists if and
only if its corresponding LASSO estimate $\widehat{\beta}_{ji}$ is non-zero.

###  B Module Detections Based on Information Entropy Methods

This paper introduces the Map Equation method to detect modules in the SSE A
marker network over different periods and further explores the evolution of
modules. The Map Equation algorithm (InfoMap algorithm), initially proposed by
Rosvall and Bergstrom, (2007), is based on a formula to evaluate the
effectiveness of a module structure in describing the path of a random walker
around the network. The random walk is used to simulate the information (risk)
transmission in the entire system (the stock market). Based on the Huffman
coding, the Map Equation algorithm adopts a two-level code to describe the
random walk path: the high-level codes distinguish modules in the network
(i.e., index codes), and the low-level codes represent node names that are
unique in the same group (i.e., module codes). The Map Equation algorithm aims
to discover an information (risk) map that gets rid of unnecessary details by
minimizing the amount of information needed to describe the random walk path
and highlights critical modules where nodes in the same group develop stronger
interior relationships than outside nodes.

Since the real relationships between stocks are not observable, we first base
on the discovered network structure in Eq. (3) to simulate the real one and
calculate the visit frequencies of a random walker traveling in the system.
Similar to the used method in Garratt et al., (2014), we convert the adjacency
matrix $\bm{A}$ defined in Eq. (3) to the Markov transition probability matrix
$\varPi$ to depict the random walk path. Since $\bm{A}$ may be non-symmetric,
we consider the follow systemic matrix

(6)
$\displaystyle\bm{V}=\\{v_{ij}\\}=\left[\begin{array}[]{cc}\bm{0}_{N}&\bm{A}\\\
\bm{A}^{\top}&\bm{0}_{N}\\\ \end{array}\right]\in\mathbb{R}^{2N\times 2N},$

where $\bm{0}_{N}\in\mathbb{R}^{N\times N}$ is a matrix with zeros. Then, the
Markov transition probability matrix is defined as

(7) $\displaystyle\varPi=\left[\pi_{ij}\right]_{2N\times
2N}=\left[\frac{v_{ij}}{\sum_{k=1}^{2N}v_{kj}}\right]_{2N\times 2N}.$

Let $p_{i}$ be the visit frequency of a random walker to the node $i$ .
Mathematically, we can calculate $p_{i}$ by considering the dominant
eigenvector of the Markov transition probability matrix

(8) $\displaystyle\bm{P}=\varPi\bm{P},$

where $\bm{P}=\left(p_{1},\cdots,p_{2N}\right)^{\top}$.

Given by the visit frequencies $\bm{P}$ and a module structure $\bm{M}$ with
$m$ modules, the exiting frequency of the traveler from module $\alpha$ is
given by

(9) $q_{\alpha\curvearrowright}=\sum_{i\in module\ \alpha}\sum_{j\notin
module\ \alpha}\pi_{ij}p_{i},$

and the exit frequency of the travel from _any_ module is given by

(10) $\displaystyle
q_{\curvearrowright}=\sum_{\alpha=1}^{m}q_{i\curvearrowright}.$

Moreover, the frequency that the random walker use module $\alpha$’s codes is
given by

(11) $p_{\circlearrowleft}^{\alpha}=q_{\alpha\curvearrowright}+\sum_{i\in
module\ \alpha}p_{i}.$

The probabilities $p_{\circlearrowleft}^{\alpha}$ and
$q_{\alpha\curvearrowright}$ measure the frequency of using module codes.
Next, we need to know the costs to access these codes. According to the
Shannon’s coding theorem, for a random variable $z$ having $n$ states with
probabilities $p_{k}$, the average length of the code word cannot be less than
the entropy of $z$, defined by

(12) $H(z)=-\sum_{k=1}^{n}p_{k}\log(p_{k}).$

Then, the minimum average length for index codes and module codes are given by

(13)
$H(Q)=-\sum_{\alpha=1}^{m}\frac{q_{\alpha\curvearrowright}}{q_{\curvearrowright}}\log\left(\frac{q_{\alpha\curvearrowright}}{q_{\curvearrowright}}\right)$

and

(14)
$H(P^{\alpha})=-\frac{q_{\alpha\curvearrowright}}{p_{\circlearrowleft}}\log\left(\frac{q_{\alpha\curvearrowright}}{p_{\circlearrowleft}}\right)-\sum_{i\in
module\
\alpha}\frac{p_{i}}{p_{\circlearrowleft}^{\alpha}}\log\left(\frac{p_{i}}{p_{\circlearrowleft}^{\alpha}}\right),$

where $H(\cdot)$ is the entropy function defined in Eq. (12), $H(Q)$ and
$H(P^{\alpha})$ reflect the encoding effectiveness of the entire module
structure $M$ and the specific module $\alpha$, respectively.

Consequently, the minimum average length of the random-walk path under the
given module structure $M$ is given by

(15)
$L(M)=q_{\curvearrowright}H(Q)+\sum_{\alpha=1}^{m}p_{\circlearrowleft}^{\alpha}H(P^{\alpha}).$

Here, $L(M)$ is the weighted sum of two information entropy parts. The one is
the average code length of the module name (index codes), and the other one is
the average code length of node names in different modules (module codes). The
weights are proportions of average code lengths of module and node names. When
minimizing the $L(M)$, the network achieves the minimum entropy, and the
corresponding module division is stable. The map equation algorithm uses these
criteria to compare different module divisions in practice and select the
optimal one.

###  C Network topological indicators

Based on the detected module structures, we further investigate the
topological properties of generated networks to study the contagion effect in
the SSE A market.

Average shortest path length

$L=\frac{2}{N(N-1)}\sum_{j\neq i}d_{ij}$, where $i$ and $j$ are two stocks
(nodes) in the SSE A network and $d_{ij}$ is the shortest path between nodes
$i$ and $j$. A smaller length means faster information or risk transmission in
the network.

Clustering coefficient

$C_{i}=\frac{2n_{i}}{k_{i}(k_{i}-1)}$, where $k_{i}$ is the number of nodes
directly connecting to node $i$ and $n_{i}$ is the number of edges between
$k_{i}$ neighbours of node $i$. A higher value implies better network
connectivity.

Network diameter

$\mathrm{Diameter}=\max_{1\leq i,j\leq N}d_{ij}$. A smaller value implies
faster information or risk transmission speed.

Network density

$\mathrm{Density}=\frac{\sum_{i,j}a_{ij}}{N(N-1)}$, where $a_{ij}$ is defined
in Equation (3). A higher density implies closer relationships between nodes.

Relative degree centrality

$C_{RD}(i)=\frac{k_{i}}{N-1}$. The high relative degree centrality implies the
important influence from the corresponding node on the network.

Relative betweenness centrality

$C_{RB}(i)=\frac{2}{(N-1)(N-2)}\sum_{j<k}\frac{g_{jk}(i)}{g_{jk}}$, where
$g_{jk}(i)$ is the number of shortest paths connecting nodes $j$ and $k$ and
passing through node $i$. This indicator measures the “bridge” role of node
$i$ in the network.

Relative closeness centrality

$C_{RC}(i)=\frac{N-1}{\sum_{j=1}^{N}d_{ij}}$ that measures how close node $i$
is to all other nodes in the network. The high value of relative closeness
centrality implies close connections between node $i$ and other nodes.

Degree centralisation

$C_{D}=\frac{\sum_{i=1}^{N}\left(C_{RD}(\max)-C_{RD}(i)\right)}{\max\left[\sum_{i=1}^{N}\left(C_{RD}(\max)-C_{RD}(i)\right)\right]}$,
where the numerator is the sum of differences between the maximum degree
centrality $C_{RD}(\max)=\max_{1\leq i\leq N}C_{RD}(i)$ and the degree
centrality of each node $C_{RD}(i)$ and the denominator is the maximum value
of numerator in theories. This indicator describes the centrality of the whole
network.

Betweenness centralisation

$C_{B}=\frac{1}{N-1}\sum_{i=1}^{N}\left(C_{RB}(\max)-C_{RB}(i)\right)$, where
the numerator theoretically represents the sum of the difference between the
maximum intermediate centrality and the intermediate centrality of each node,
and the denominator represents the maximum of the sum of the differences. This
indicator describes the degree to which the network relies excessively on a
node to transfer relations.

Closeness centralisation

$C_{C}=\frac{2(N-3)}{(N-1)(N-2)}\sum_{i=1}^{N}\left(C_{RC}(\max)-C_{RC}(i)\right)$
that describes the centralized trend in the network.

##  III Empirical Results

###  A Stock Market Data

This paper uses the SSE A-shares market’s weekly closing prices from 2005 to
2018 and divides the entire period into eight stages according to bull and
bear markets.

Table 1: The period division from 2005 to 2018 for the SSE A-shares market according to bull and bear markets. Eight stages | Code | Start time | End time | Reasons
---|---|---|---|---
Stage 1 | BULL1 | 2005-06 | 2007-10 | Reforms in the market and other capital dividends.
Stage 2 | BEAR1 | 2007-10 | 2008-10 | The subprime mortgage crisis and other external factors.
Stage 3 | BULL2 | 2008-10 | 2009-07 | Rescue policies from the government.
Stage 4 | BEAR2 | 2009-07 | 2014-03 | Combined influences from the crisis and stimulus policies
|  |  |  | result in market ups and downs.
Stage 5 | BULL3 | 2014-03 | 2015-06 | Deepen reforms of state-owned enterprises and arising
|  |  |  | financial leverages increase the market.
Stage 6 | BEAR3 | 2015-06 | 2016-01 | Deleveraging and other factors cause the collapse of
|  |  |  | the stock market.
Stage 7 | BULL4 | 2016-01 | 2018-01 | The slight rise in the market due to factors like stable leverage
|  |  |  | and financial stimulus.
Stage 8 | BEAR4 | 2018-01 | 2018-12 | Overseas factors like the trade war leads to the fluctuating
|  |  |  | decline in the stock market.

As shown in Table 1, stages 1 and 5 are typical rapid-rising bull markets, and
stages 2 and 6 are steeply declining bear markets. By contrast, stages 3 and 7
witness moderate increases in the market, and the market experiences a
fluctuating decline in stages 4 and 8. Overall, the surge and plummet can be
found in the bull market (stages 1 and 5) and the bear market (stages 2 and
6), respectively, whereas the stock market consistently fluctuates in stages
3, 4, 7, and 8. Based on the above division, we aim to utilize the information
entropy to discover the evolution of modules in the SSE A-shares market in
expansions and recessions.

The SSE A-shares backward closing prices from 2005 to 2018 are downloaded from
the wind database. Given the long period of the research data, some stocks are
out of the discussion in this paper because of the following reasons.

  1. 1.

Missing values. Until 18 December 2019, 1,547 stocks are traded on the SSE
A-shares market. Due to the late listing and suspension of trading, some of
these stocks are removed in advance to compare stock market networks in
different stages.

  2. 2.

Stocks prefixed with “ST” or “*ST”. Designated as Special Treatment (ST) by
the stock exchanges for warning investors, these stocks face delisting risks
and show distinct patterns from normal stocks.

  3. 3.

Stocks whose returns maintain at zeros over a long period for the long-term
suspension or other reasons are excluded in this paper to prevent misleading
information.

As a result, 716 stocks are selected to construct the SSE A-shares network.
For the stock $i$, its log return is calculated by

(16) $r_{it}=\ln P_{it}-\ln P_{i,t-1},$

where $P_{it}$ is the price of stock $i$ at time $t$.

###  B Module Analysis of the SSE A-shares Network Based on the Minimum
Entropy

####  B.1 Network Module Visualisation

To analyze the characteristics of modules in the SSE A-shares market, we
employ the Map Equation method to detect modules in the financial system in
different periods and present visualization results in Figure 1.222Figures B
and C in the Appendix provide the full-size version of Figure 1. Each module
is represented by a node whose area is proportional to its information flow,
reflecting its status in the network. The information flow of a module
includes two parts: the information flow out of the module, which is
proportional to its boundary thickness and can represent the probability of
risk passing from the module to other modules; and the information flow
staying in the module, which is proportional to the area of the inner circle
area that can indicate the probability that the risk remains in the module.
The connection thickness among modules suggests the probability of risk
transmission in different sectors. The thicker the connection, the greater the
contagion probability. Besides, the arrow indicates the direction of the risk
of contagion.

Figure 1: Module divisions in eight periods based on the Map Equation
algorithm. Nodes represent modules, directed links indicate directions of the
information flow and the thickness of links demonstrates the transmission
probability of risk between different modules. The node size is proportional
to the information flow in a module that includes the information flow out of
and within the module. Specifically, the node’s boundary thickness is
proportional to the information flow out of the module, corresponding to the
probability that risks transmit to other modules. The interior area of a node
is proportional to the module’s information flow, corresponding to the
probability that risks staying in the module.

As shown in Figure 1, the module numbers in bear markets are generally smaller
than those in bull markets, and more stocks belong to the same module in bear
periods, indicating closer interior connections and higher internal contagion
in modules. The number of modules in the stock market gradually increases,
implying the continuous development of the capital market leads to the
differentiation in the patterns of stock returns and the risk of contagion.

####  B.2 Module Analysis Based on Map Equation Algorithm

Based on the Map Equation method and the minimum information entropy, we
discuss the module divisions in eight periods and further analyze the risk
spillover among modules.

Table 2 compares the top five modules with the highest proportion of
information in bull and bear periods. The first column of Table 2 shows the
modules ranked by information flow from high to low, the second column
represents the number of stocks (nodes) involved in the module, and the third
column represents the number of information transmission linkages (links)
within the module. Table 2 suggests significant structural differences exist
between the first four stages and the latter four stages. In the first four
stages, the proportions of stock and link numbers within the largest module
account for more than $70\%$ of the entire SSE A-shares network, whereas the
proportions in other modules are low. In other words, the SSE A-shares market
does not show significant differentiation in the first four stages: the
largest module having the highest information flow largely represents the
entire network, and more than $70\%$ of the stocks belong to this module.
However, the SSE A-shares market experiences significant structural
differentiation in the last four stages. Specifically, the largest five
modules in BULL3, BULL4, and BEAR4 share comparable sizes in node and link
numbers. Compared with the other three stages, the BEAR3 period is closer to
the first four stages because the rapid deleveraging effect in this stage
leads to abnormal fluctuations and increases the system’s connected effect.

Table 2: Node and link numbers of the top five modules with the highest proportion of information in eight periods. BULL1 | BEAR1
---|---
Module name | Node numbers | Link numbers | Module name | Node numbers | Link numbers
M1 | 488 | 9637 | M1 | 608 | 10887
M2 | 86 | 688 | M2 | 44 | 307
M3 | 61 | 468 | M3 | 33 | 224
M4 | 26 | 161 | M4 | 19 | 49
M5 | 11 | 33 | M5 | 4 | 8
BULL2 | BEAR2
Module name | Node numbers | Link numbers | Module name | Node numbers | Link numbers
M1 | 528 | 11755 | M1 | 612 | 14331
M2 | 47 | 329 | M2 | 52 | 802
M3 | 51 | 797 | M3 | 15 | 50
M4 | 43 | 533 | M4 | 8 | 18
M5 | 20 | 166 | M5 | 10 | 23
BULL3 | BEAR3
Module name | Node numbers | Link numbers | Module name | Node numbers | Link numbers
M1 | 53 | 231 | M1 | 306 | 3750
M2 | 47 | 264 | M2 | 119 | 1147
M3 | 31 | 119 | M3 | 70 | 500
M4 | 27 | 119 | M4 | 72 | 503
M5 | 27 | 123 | M5 | 69 | 504
BULL4 | BEAR4
Module name | Node numbers | Link numbers | Module name | Node numbers | Link numbers
M1 | 40 | 256 | M1 | 95 | 668
M2 | 33 | 211 | M2 | 68 | 476
M3 | 30 | 122 | M3 | 52 | 328
M4 | 39 | 201 | M4 | 22 | 61
M5 | 29 | 108 | M5 | 36 | 126

We use the information flow to illustrate the spillover effect of risks within
and between modules in Table 3: The first column represents the modules ranked
by information flow from high to low; the second column represents the
proportions of information within modules to the total information; the third
(fourth) column represents the proportion of information flowing in (out) each
module. Table 3 presents similar results as Table 2. In the first four stages,
the information in the largest module accounts for at least $70\%$ of the
entire market, indicating that most stocks can be grouped and insignificant
differentiation exists in the stock market. Conversely, in the last four
stages, except for the BEAR3 stage, the most extensive module merely contains
approximately $10\%$ information, suggesting significant differentiation in
the stock market. Such differentiation derives from industry agglomeration
that stocks from the same industry are more likely to form modules.333See
details in Tables 5 and 6. Moreover, the BEAR3 can be viewed as a transition
period between the first and last four stages since the information proportion
of M1 is about $50\%$. During this period, given the excessive accumulation of
pre-leverage and the speed of the deleveraging process, abnormal fluctuations
occurred in the stock market, and the strengthened connected effect in the
system further increases the abnormal volatility of the stock market,
resulting in a more significant loss in the financial system.

Table 3: Information flows within and between modules over eight periods. BULL1 | BEAR1
---|---
Module | Within | Flow in | Flow out | Module | Within | Flow in | Flow out
Name | Modules | Modules | Modules | Name | Modules | Modules | Modules
M1 | 66.13% | 8.58% | 10.82% | M1 | 85.87% | 5.29% | 5.94%
M2 | 13.32% | 6.00% | 5.41% | M2 | 6.80% | 3.14% | 2.75%
M3 | 10.91% | 3.38% | 2.29% | M3 | 4.65% | 2.60% | 2.47%
M4 | 3.57% | 1.99% | 1.87% | M4 | 1.89% | 1.22% | 1.10%
M5 | 2.03% | 1.32% | 1.14% | M5 | 0.58% | 0.45% | 0.41%
BULL2 | BEAR2
Module | Within | Flow in | Flow out | Module | Within | Flow in | Flow out
Name | Modules | Modules | Modules | Name | Modules | Modules | Modules
M1 | 71.35% | 9.17% | 10.60% | M1 | 82.16% | 6.24% | 7.36%
M2 | 11.49% | 4.95% | 3.57% | M2 | 6.92% | 2.90% | 3.11%
M3 | 6.82% | 2.85% | 3.08% | M3 | 3.91% | 2.40% | 1.83%
M4 | 5.04% | 2.43% | 2.56% | M4 | 2.13% | 1.40% | 1.14%
M5 | 1.92% | 1.11% | 1.23% | M5 | 2.01% | 1.29% | 1.05%
BULL3 | BEAR3
Module | Within | Flow in | Flow out | Module | Within | Flow in | Flow out
Name | Modules | Modules | Modules | Name | Modules | Modules | Modules
M1 | 8.34% | 5.05% | 5.08% | M1 | 46.54% | 11.75% | 11.49%
M2 | 6.58% | 3.81% | 3.95% | M2 | 18.76% | 8.26% | 8.20%
M3 | 4.59% | 3.18% | 3.30% | M3 | 10.13% | 5.21% | 5.08%
M4 | 4.41% | 2.85% | 2.85% | M4 | 8.70% | 4.27% | 4.31%
M5 | 3.15% | 1.87% | 1.98% | M5 | 7.09% | 3.17% | 3.43%
BULL4 | BEAR4
Module | Within | Flow in | Flow out | Module | Within | Flow in | Flow out
Name | Modules | Modules | Modules | Name | Modules | Modules | Modules
M1 | 7.45% | 4.30% | 4.43% | M1 | 13.18% | 7.55% | 8.91%
M2 | 6.46% | 2.60% | 2.17% | M2 | 12.64% | 4.35% | 3.85%
M3 | 5.49% | 3.72% | 3.71% | M3 | 7.57% | 3.72% | 3.83%
M4 | 4.47% | 2.45% | 2.63% | M4 | 4.91% | 3.02% | 2.58%
M5 | 4.28% | 2.71% | 2.77% | M5 | 4.53% | 2.21% | 2.05%

To investigate how modules develop and evolve over the eight periods, we
consider stock categories in different modules and present results of BEAR1,
BULL4, and BEAR4 in Tables 4 to 6, respectively.

Table 4: Industry categories of stocks in top six modules during BEAR1. Module name | M1 | M2 | M3 | M4 | M5 | M6
---|---|---|---|---|---|---
Materials | 102 | 6 | 0 | 6 | 1 | 0
Telecommunication service | 2 | 0 | 0 | 0 | 0 | 0
Real estate | 27 | 6 | 21 | 3 | 1 | 0
Industrials | 146 | 7 | 7 | 4 | 1 | 0
Utilities | 38 | 0 | 1 | 2 | 0 | 0
Finance | 13 | 1 | 2 | 0 | 0 | 4
Consumer staples | 107 | 12 | 1 | 1 | 0 | 0
Energy | 20 | 1 | 1 | 1 | 1 | 0
Consumer discretionary | 54 | 2 | 0 | 0 | 0 | 0
Information technology | 45 | 4 | 0 | 2 | 0 | 0
Health care | 55 | 5 | 0 | 0 | 0 | 0

As shown in Table 4, the SSE A-shares network in the BEAR1 can be divided into
nine modules, and three modules are out of discussion due to only including
one stock. M1 has the largest size ($608$ stocks) in the network, includes,
and captures $85.87\%$ of the network’s information. Conversely, the remaining
modules contain less information, and the differentiation effect caused by
industry aggregation is not significant. More importantly, the industry
distribution of the M1 in the BEAR1 is analogous to that of the entire SSE
A-shares market distribution. Consequently, M1 can be viewed as a system
module, and other stocks that are not in the first module are peripheral in
the stock market. Similar patterns can also be found in the other first four
stages.

Unlike the first four stages, the last four stages except for BEAR3 present
the significant industry differentiation of the modules, as shown in Tables 5
and 6.

Table 5: Industry categories of stocks in top nine modules during BULL4. Module name | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9
---|---|---|---|---|---|---|---|---|---
Materials | 2 | 1 | 8 | 2 | 1 | 3 | 2 | 25 | 1
Real estate | 1 | 0 | 1 | 1 | 4 | 1 | 2 | 0 | 29
Industrials | 1 | 2 | 5 | 21 | 14 | 7 | 7 | 2 | 1
Utilities | 2 | 1 | 8 | 6 | 4 | 1 | 0 | 0 | 4
Finance | 0 | 0 | 1 | 6 | 0 | 2 | 1 | 0 | 0
Consumer staples | 5 | 4 | 2 | 1 | 1 | 3 | 11 | 5 | 3
Energy | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Consumer discretionary | 0 | 14 | 4 | 1 | 1 | 2 | 2 | 4 | 0
Information technology | 1 | 3 | 1 | 0 | 2 | 10 | 0 | 0 | 0
Health care | 28 | 8 | 0 | 1 | 2 | 1 | 2 | 0 | 0

Table 5 suggests that and nine largest modules in the BULL4 stage network and
the rest $42$ modules are out of discussion due to involving few stocks. The
differentiation of the modules displays a significant industry aggregation
phenomenon. Health care is the dominant industry category in M1 and is key to
the information flow and spillover risk. Given by Figure 1, the closed
connections between M1 and M2 implies the high potential risk of contagion
between the health care and the consumer discretionary industry because of the
distinct overlap of these two industries in the industry chain.

Table 6: Industry categories of stocks in top nine modules during BEAR4. Module name | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9
---|---|---|---|---|---|---|---|---|---
Materials | 15 | 3 | 1 | 1 | 1 | 18 | 5 | 0 | 1
Telecommunication service | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0
Real estate | 9 | 0 | 1 | 2 | 1 | 5 | 9 | 0 | 2
Industrials | 22 | 4 | 19 | 16 | 4 | 3 | 7 | 4 | 7
Utilities | 8 | 0 | 0 | 0 | 1 | 0 | 6 | 1 | 0
Finance | 1 | 0 | 0 | 0 | 0 | 4 | 0 | 1 | 0
Consumer staples | 20 | 10 | 3 | 1 | 18 | 2 | 1 | 10 | 0
Energy | 3 | 0 | 0 | 0 | 1 | 8 | 0 | 0 | 0
Consumer discretionary | 3 | 15 | 1 | 0 | 5 | 0 | 1 | 0 | 2
Information technology | 7 | 0 | 27 | 1 | 2 | 1 | 1 | 1 | 0

The BEAR4 stage can be divided into $46$ modules, and the top nine modules
with the largest sizes capture $57.44\%$ of the network information. The
industry categories of stocks in these nine modules are provided in Table 6.
M1 mainly consists of stocks from materials, industry, and consumer staples.
The dominant industries in M2, M3, M4, M5, and M6 are consumer discretionary
and consumer staples, information technology and industrials, industrials,
consumer staples, and materials.

Tables 5 and 6 demonstrate the significant industry aggregation effect in the
BULL4 and BEAR4 period. With the development of the stock market, the industry
aggregation results in module differentiation in SSE A-shares because stocks
from the same industry category share similar macro fundamentals.

In summary, more than $70\%$ of stocks in the SSE A-shares market belong to
the same module in the first four periods, which can be viewed as the system
module. Stocks in the system module are less connected with others, and those
outside the system module are pericardial in the network, implying the
relative low contagion risk and weak connections. As China’s financial market
develops, the system module is gradually differentiated into several small
parts based on industry categories. Stocks from the same or related industries
grouping in modules make the industry aggregation phenomenon more significant.
In an extreme situation like the BEAR3 stage, abnormally fluctuations and
large-scale declines in the market lead to the formation of a small system
module whose higher status accelerates risk contagion.

###  C Topological Properties of SSE A-Shares Networks

We utilize topological indicators in Section C to investigate the SSE A-shares
market’s characteristics over bull and bear markets. Details are presented in
Tables 7 and 8.

Table 7: Characteristics of the SSE A-shares networks during the first four stages. Topological properties | BULL1 | BEAR1 | BULL2 | BEAR2
---|---|---|---|---
Network diameter | 7 | 6 | 6 | 7
Network density | 0.0195 | 0.0252 | 0.02 | 0.0342
The average shortest path length | 2.72 | 2.7 | 2.85 | 2.59
Clustering coefficient | 0.071 | 0.078 | 0.055 | 0.097
Mean of relative degree centrality | 0.02 | 0.0252 | 0.02 | 0.0342
Mean of relative betweenness centrality | 0.23 | 0.234 | 0.255 | 0.22
Mean of relative closeness centrality | 0.3815 | 0.3909 | 0.3699 | 0.4138
Out-degree centralisation | 0.11404 | 0.05463 | 0.04857 | 0.11844
In-degree centralisation | 0.03841 | 0.13446 | 0.11719 | 0.06242
Betweenness centralisation | 0.012 | 0.0167 | 0.0308 | 0.0162
Out-degree closeness centralisation | 0.2976 | 0.1937 | 0.1891 | 0.275
In-degree closeness centralisation | 0.1742 | 0.345 | 0.336 | 0.1831
Table 8: Characteristics of the SSE A-shares networks during the last four stages. Topological properties | BULL3 | BEAR3 | BULL4 | BEAR4
---|---|---|---|---
Network diameter | 8 | 6 | 6 | 8
Network density | 0.0165 | 0.022 | 0.0193 | 0.0157
The average shortest path length | 3.1 | 2.84 | 3.08 | 3.23
Clustering coefficient | 0.066 | 0.076 | 0.084 | 0.075
Mean of relative degree centrality | 0.0165 | 0.022 | 0.0193 | 0.0157
Mean of relative betweenness centrality | 0.28 | 0.255 | 0.278 | 0.296
Mean of relative closeness centrality | 0.3326 | 0.3748 | 0.3389 | 0.3209
Out-degree centralisation | 0.09127 | 0.01859 | 0.15156 | 0.07114
In-degree centralisation | 0.04645 | 0.09201 | 0.06052 | 0.05434
Betweenness centralisation | 0.0361 | 0.0172 | 0.0448 | 0.025
Out-degree closeness centralisation | 0.2952 | 0.083 | 0.4127 | 0.2957
In-degree closeness centralisation | 0.2062 | 0.2378 | 0.2306 | 0.2169

As shown in Tables 7 and 8, network densities of bear markets are higher than
those of bull markets, and network diameters of bear markets are lower than
those of bull markets, suggesting that stocks have stronger connections with
others in the bear market. Affected by the subprime mortgage crisis and the
deleveraging of capital allocation, the SSE A-shares market experiences large-
scale collapses and shows the significant connected effect. Meanwhile,
investors are more sensitive to market information in these two periods, and
hence, tend to adopt similar strategies to avoid risks.

In the entire period, the lengths of the average shortest path in the SSE
A-shares networks are between two and three, meaning that approximately two or
three intermediate stocks can connect any stock pairs. The clustering
coefficients of bear markets are generally higher than those of bull markets,
reflecting the more distinct connected effect in recessions. The BEAR1 is
during the subprime mortgage crisis, the BEAR2 is affected by the European
debt crisis, and the BEAR3 experiences the “thousand-share limit-down” after
the stock market deleverages in 2015, which reflects a financial crisis
enhances the small-world effect of the SSE A-shares network.

Regarding the difference between bull and bear markets, the average degrees of
stocks in bear markets are greater than those in bull markets. In the bear
market, the core stocks have more leading influence, and more synchronous
changes appear in the market. Also, closeness centralities are relatively
high, which means that the reachable distances of risk propagation are
relatively short, and risks can transmit to most stocks from the source via a
short distance.

From the structural differences between bear markets, BEAR1 and BEAR3 show
rapid declines with large volatility, while BEAR2 and BEAR4 fall with
fluctuations and have small volatility. Therefore, structural differences
exist between these two types of bear markets. The low out-degree
centralizations of BEAR1 and BEAR3 reflect the marginal differences between
the out-degree centrality of each node and the maximum out-degree centrality.
Therefore, most stocks in these two periods have relatively high out degrees,
and risks are more likely to transmit to other stocks through these stronger
connections, accelerating the propagation of risks. By contrast, the BEAR2 and
BEAR4 markets show different patterns. The low in-degree centralization and
high out-degree centralizations reflect that most stocks have relatively high
in degrees but low out degrees, suggesting that stocks in the network absorb
risks and prevent the spread of risks. BEAR1 and BEAR3 are periods with
abnormal fluctuations due to the impact of the subprime mortgage crisis and
rapid deleveraging in 2015. In BEAR1 and BEAR2, it is hard to identify
critical stocks leading to the massive collapse, making the supervision on
risks more challenging. Conversely, the less connected structure of the SSE
A-shares networks in BEAR2 and BEAR4 lowers the transmission risks and
benefits of identifying the risk source and restraining the network’s spread
of risk.

###  D Analysis of Core Stocks in Bull and Bear Markets

We use three types of centralities to measure stocks’ influence in the SSE
A-shares market to identify core stocks in the network and investigate how
these stocks transmit risks over different periods. Tables 9 to 11 list stocks
with the top five relative degrees, betweenness and closeness centralities in
eight stages, respectively, where the first column reports stock codes in the
SSE A-share market and the second column gives the corresponding industry
categories. Stocks in these tables are referred to as core stocks due to their
distinct influence in specific periods.

Table 9: Stocks with Top Five Relative Degree Centralities in Eight Periods. BULL1 | BEAR1
---|---
600624 | Health care | 600340 | Real estate
600218 | Industrials | 600085 | Health care
600373 | Consumer discretionary | 600088 | Consumer discretionary
600172 | Materials | 600006 | Consumer discretionary
600626 | Consumer discretionary | 600590 | Industrials
BULL2 | BEAR2
600405 | Industrials | 600370 | Consumer discretionary
600967 | Industrials | 600853 | Industrials
600665 | Real estate | 600522 | Information technology
600692 | Real estate | 600590 | Industrials
600601 | Industrials | 600360 | Information technology
BULL3 | BEAR3
600460 | Information technology | 600410 | Information technology
600439 | Consumer staples | 600502 | Industrials
660360 | Information technology | 600271 | Information technology
600131 | Utilities | 600749 | Consumer discretionary
600345 | Information technology | 600531 | Materials
BULL4 | BEAR4
600229 | Consumer discretionary | 600168 | Utilities
600561 | Industrials | 600331 | Materials
600356 | Materials | 600292 | Industrials
600757 | Consumer discretionary | 600713 | Health care
600422 | Health care | 600269 | Industrials

As suggested in Tables 9 to 11, high influence stocks are mainly from four
industry categories: consumer discretionary, health care, materials, and
industrials, which corresponds to the system module and the industry
aggregation phenomenon demonstrated in Tables 4 to 6. Moreover, there exists a
great joint part in three stock lists measured by different centralities. A
few stocks are even included by three lists in the same stage, implying these
stocks play irreplaceable roles in the system during the particular time.
Also, bull and bear markets have noticeably different core stocks. About
$22.5\%$ of stocks are ranked in the top 20 stocks regarding one type of
centralities in a specific stage, meaning the diversity of core stocks in
eight stages and the complicated leading effect raised by these stocks.

Table 10: Stocks with Top Five Relative Betweenness Centralities in Eight Periods. BULL1 | BEAR1
---|---
600373 | Consumer discretionary | 600798 | Industrials
600353 | Information technology | 600088 | Consumer discretionary
600624 | Health care | 600811 | Consumer staples
600138 | Consumer discretionary | 600736 | Real estate
600426 | Materials | 600565 | Real estate
BULL2 | BEAR2
600692 | Real estate | 600480 | Consumer discretionary
600967 | Industrials | 600131 | Utilities
600665 | Real estate | 600370 | Consumer discretionary
600229 | Consumer discretionary | 600585 | Materials
600662 | Industrials | 600004 | Industrials
BULL3 | BEAR3
600131 | Utilities | 600719 | Utilities
600439 | Consumer staples | 600410 | Information technology
600460 | Information technology | 600533 | Real estate
600166 | Consumer discretionary | 600502 | Industrials
600879 | Industrials | 600501 | Industrials
BULL4 | BEAR4
600135 | Materials | 600331 | Materials
600561 | Industrials | 600594 | Health care
600422 | Health care | 600713 | Health care
600757 | Consumer discretionary | 600375 | Industrials
600730 | Consumer discretionary | 600390 | Finance
Table 11: Stocks with Top Five Relative Closeness Centralities in Eight Periods. BULL1 | BEAR1
---|---
600624 | Health care | 600085 | Health care
600172 | Materials | 600790 | Real estate
600983 | Consumer discretionary | 600006 | Consumer discretionary
600373 | Consumer discretionary | 600811 | Consumer staples
600985 | Energy | 600590 | Industrials
BULL2 | BEAR2
600405 | Industrials | 600370 | Consumer discretionary
600967 | Industrials | 600522 | Information technology
600692 | Real estate | 600853 | Industrials
600665 | Real estate | 600480 | Consumer discretionary
600787 | Industrials | 600861 | Consumer staples
BULL3 | BEAR3
600439 | Consumer staples | 600410 | Information technology
600460 | Information technology | 600749 | Consumer discretionary
600131 | Utilities | 600502 | Industrials
600345 | Information technology | 600271 | Information technology
600166 | Consumer discretionary | 600661 | Consumer discretionary
BULL4 | BEAR4
600229 | Consumer discretionary | 600331 | Materials
600561 | Industrials | 600168 | Utilities
600218 | Industrials | 600713 | Health care
600757 | Consumer discretionary | 600292 | Industrials
600422 | Health care | 600320 | Industrials

We present the averages of financial indicators of stocks with the top 20
centralities in different stages in Table 12 to further investigate
characteristics of core stocks. For comparisons, Table 12 also provides the
financial indicator averages of all SSE A-shares in parentheses.

Table 12: Financial characteristics of stocks in the SSE A-shares network over eight periods. Numbers in cells and parentheses report the financial indicator averages of stocks with the top 20 centralities and all SSE A-shares. Period | Current | Quick | Debt to asset | Total assets | Return on | Margin trading
---|---|---|---|---|---|---
| ratio | ratio | ratio | turnover ratio | equity | balance
BULL1 | 1.42 (1.38) | 1.02 (1.43) | 59.33 (43.03) | 0.69 (0.81) | 15.26 (6.05) | \- (-)
BEAR1 | 1.42 (1.22) | 0.98 (0.71) | 60.09 (55.94) | 0.78 (0.78) | 11 (11.23) | \- (-)
BULL2 | 1.45 (1.55) | 0.92 (0.85) | 62.47 (44.29) | 0.79 (0.86) | 5.30 (7.40) | \- (-)
BEAR2 | 1.71 (1.08) | 1.14 (1.51) | 58.95 (47.17) | 0.79 (0.60) | 10.18 (3.30) | 1.25 (0.48)
BULL3 | 1.65 (1.81) | 1.26 (1.03) | 54.93 (43.48) | 0.77 (0.57) | 10.86 (5.98) | 5.14 (2.31)
BEAR3 | 1.94 (1.83) | 1.48 (1.41) | 49.28 (55.78) | 0.71 (0.54) | -0.07 (-4.09) | 6.52 (6.64)
BULL4 | 1.95 (2.54) | 1.48 (2.16) | 49.84 (43.20) | 0.64 (0.71) | 6.82 (4.57) | 4.76 (4.00)
BEAR4 | 1.95 (1.66) | 1.50 (1.28) | 50.13 (48.31) | 0.66 (0.76) | -11.39 (-3.98) | 4.34 (3.08)

Table 12 suggests that most core stocks (i.e., stocks with the top 20
centralities) are medium- or small-cap stocks rather than large-cap ones.
Moreover, core stocks lack short-term solvency because about two-thirds of
them have lower current ratios and quick ratios than the average levels of all
SSE A-shares at the same stage, and the current ratio of nearly one-third of
core stocks is below one. Regarding profitability, The return on net assets of
core stocks declines annually and is far lower than the average value of SSE
A-shares in the same period. The return on equity of two-thirds of core stocks
is lower than $10\%$, reflecting low profitability.

In sum, core stocks that have the leading effect in the SSE A-shares network
are mainly issued by medium- or small-cap companies with relatively low
financial indicators. Conversely, the large-cap stocks weigh more in the stock
index but are less connected with other stocks. and the herding behavior can
explain this phenomenon in the stock market. In the bull market, individual
investors, accounting for a significant proportion in the Chinese stock
market, tend to follow the upward trend via buying medium or small-cap stocks.
Compared with large-cap stocks, these stocks have relatively high growth rates
in the short term because they have relatively small market values and are
easily influenced by capitals. However, these stocks’ rise does not derive
from fundamentals like technological breakthroughs but from short-term
arbitrage strategies, which is difficult to attract long-term investors’
attention. Consequently, such rapid increases caused by blindness only quickly
accumulate bubbles in a short period. Selling off these stocks at high prices,
speculators also release a signal that stimulates individual investors to sell
stocks with panic, which eventually leads to a steep decline in stock prices.
Besides, the immature capital market and irrational behaviors widely spreading
in the market result in widespread risks and the phenomenon of a sharp rise
and fall for the entire market.

##  IV Discussion

Based on the daily closing prices of SSE A-shares from 2005 to 2018, this
paper utilizes the minimum entropy method and topological properties of
networks to investigate the evolution of the SSE A-shares market from macro
and micro perspectives. The main research conclusions are listed as follows:

First, stocks in the SSE A-shares market are closely connected over all
periods, and the connected effect is more significant in bear markets. As a
result, the degree of declines in bear markets is much greater than that of
rises in bull markets. No apparent risk sources exist in the rapidly falling
bear markets, but those sources can be identified in the slowly falling bear
markets, which is beneficial to control.

Second, the SSE A-shares network shows the industry differentiation in the
last four stages. In the first four stages, most stocks belong to the same
module, referred to as the system module, which implies that risk contagion
mainly appears in this module and risks transmit from the central module to
peripheral modules. As the Chinese financial market develops, the growing
industry aggregation in the SSE A-shares market gives rise to the module
differentiation and gradually undermines the system module’s status.
Consequently, risks are more likely to spread in modules with similar industry
categories.

Third, some stocks have consistent influences on the SSE A-shares market over
eight periods, and most of them belong to health care, consumer discretionary,
industrials, and materials. Status analyses suggest that a few stocks have
leading effects on others and play irreplaceable roles in the network. Also,
medium- and small-cap stocks with poor financial conditions are more likely to
become risk sources in the SSE A-shares network, especially in the bear stage.

The Chinese financial system’s development increases investors’ risk
awareness, the number of institutional investors, fundamental analysis
abilities, and the sensitivity to industry policies. As a result, the SSE
A-shares market is expected to increase systematic differentiation and
industry aggregation. This paper helps investors better understand this
phenomenon and improve their portfolio strategies.

## References

  * Acemoglu et al., (2015) Acemoglu, D., Ozdaglar, A., and Tahbaz-Salehi, A. (2015).  Systemic risk and stability in financial networks.  The American Economic Review, 105(2):564–608.
  * Acharya et al., (2012) Acharya, V., Engle, R., and Richardson, M. (2012).  Capital shortfall: A new approach to ranking and regulating systemic risks †.  The American Economic Review, 102(3):59–64.
  * (3) Adeloye, D., Basquill, C., Papana, A., Chan, K. Y., Rudan, I., and Campbell, H. (2015a).  An estimate of the prevalence of copd in africa: a systematic analysis.  COPD: Journal of Chronic Obstructive Pulmonary Disease, 12(1):71–81.
  * (4) Adeloye, D., Chua, S., Lee, C., Basquill, C., Papana, A., Theodoratou, E., Nair, H., Gasevic, D., Sridhar, D., Campbell, H., Chan, K. Y., Sheikh, A., and Rudan, I. (2015b).  Global and regional estimates of copd prevalence: Systematic review and meta-analysis.  Journal of Global Health, 5(2):20415–20415.
  * Battiston et al., (2012) Battiston, S., Gatti, D. D., Gallegati, M., Greenwald, B. C., and Stiglitz, J. E. (2012).  Liaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk.  Journal of Economic Dynamics and Control, 36(8):1121–1141.
  * Bech et al., (2015) Bech, M. L., Bergstrom, C. T., Rosvall, M., and Garratt, R. J. (2015).  Mapping change in the overnight money market.  Physica A-statistical Mechanics and Its Applications, 424:44–51.
  * Bernal et al., (2014) Bernal, O., Gnabo, J.-Y., and Guilmin, G. (2014).  Assessing the contribution of banks, insurance and other financial services to systemic risk.  Journal of Banking and Finance, 47(10):270–287.
  * Billio et al., (2012) Billio, M., Getmansky, M., Lo, A. W., and Pelizzon, L. (2012).  Econometric measures of connectedness and systemic risk in the finance and insurance sectors.  Journal of Financial Economics, 104(3):535–559.
  * Bisias et al., (2012) Bisias, D., Flood, M., Lo, A. W., and Valavanis, S. (2012).  A survey of systemic risk analytics.  Review of Financial Economics, 4(1):255–296.
  * Brownlees and Engle, (2011) Brownlees, C. T. and Engle, R. F. (2011).  Volatility, correlation and tails for systemic risk measurement.  SSRN Electronic Journal.
  * Chan-Lau, (2018) Chan-Lau, J. A. (2018).  Systemic centrality and systemic communities in financial networks.  Quantitative Finance and Economics, 2(2):468–496.
  * Chen et al., (2020) Chen, L., Han, Q., Qiao, Z., and Stanley, H. E. (2020).  Correlation analysis and systemic risk measurement of regional, financial and global stock indices.  Physica A-statistical Mechanics and Its Applications, 542:122653.
  * Darby et al., (2021) Darby, J., Zhang, H., and Zhang, J. (2021).  Institutional trading in volatile markets : evidence from chinese stock markets.  Pacific-basin Finance Journal, 65:101484.
  * Das et al., (2019) Das, S., Demirer, R., Gupta, R., and Mangisa, S. (2019).  The effect of global crises on stock market correlations: Evidence from scalar regressions via functional data analysis.  Structural Change and Economic Dynamics, 50:132–147.
  * Demirer et al., (2018) Demirer, M., Diebold, F. X., Liu, L., and Yilmaz, K. (2018).  Estimating global bank network connectedness.  Journal of Applied Econometrics, 33(1):1–15.
  * Diebold and Yilmaz, (2008) Diebold, F. X. and Yilmaz, K. (2008).  Measuring financial asset return and volatility spillovers, with application to global equity markets*.  The Economic Journal, 119(534):158–171.
  * Diebold and Yilmaz, (2011) Diebold, F. X. and Yilmaz, K. (2011).  On the network topology of variance decompositions: Measuring the connectedness of financial firms.  Journal of Econometrics, 182(1):119–134.
  * Ding et al., (2020) Ding, H., Jin, Y., Koedijk, K. G., and Wang, Y. (2020).  Valuation effect of capital account liberalization: Evidence from the chinese stock market.  Journal of International Money and Finance, 107:102208.
  * Engle, (2002) Engle, R. (2002).  Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models.  Journal of Business and Economic Statistics, 20(3):339–350.
  * Garratt et al., (2014) Garratt, R. J., Mahadeva, L., and Svirydzenka, K. (2014).  The great entanglement: The contagious capacity of the international banking network just before the 2008 crisis.  Journal of Banking and Finance, 49(12):367–385.
  * Girardi and Ergün, (2013) Girardi, G. and Ergün, A. T. (2013).  Systemic risk measurement: Multivariate garch estimation of covar.  Journal of Banking and Finance, 37(8):3169–3180.
  * Gong et al., (2019) Gong, C., Tang, P., and Wang, Y. (2019).  Measuring the network connectedness of global stock markets.  Physica A-statistical Mechanics and Its Applications, 535:122351.
  * Grout and Zalewska, (2016) Grout, P. A. and Zalewska, A. (2016).  Stock market risk in the financial crisis.  International Review of Financial Analysis, 46:326–345.
  * Hautsch et al., (2015) Hautsch, N., Schaumburg, J., and Schienle, M. (2015).  Financial network systemic risk contributions.  Review of Finance, 19(2):685–738.
  * Huang et al., (2009) Huang, X., Zhou, H., and Zhu, H. (2009).  A framework for assessing the systemic risk of major financial institutions.  Journal of Banking and Finance, 33(11):2036–2049.
  * Jiang et al., (2019) Jiang, T., Bao, S., and Li, L. (2019).  The linear and nonlinear lead–lag relationship among three sse 50 index markets: The index futures, 50etf spot and options markets.  Physica A-statistical Mechanics and Its Applications, 525:878–893.
  * Kim and Jeong, (2011) Kim, Y. and Jeong, H. (2011).  Map equation for link communities.  Physical Review E, 84(2):26110.
  * Kritzman et al., (2011) Kritzman, M., Li, Y., Page, S., and Rigobon, R. (2011).  Principal components as a measure of systemic risk.  The Journal of Portfolio Management, 37(4):112–126.
  * Kumar and Deo, (2012) Kumar, S. and Deo, N. (2012).  Correlation and network analysis of global financial indices.  Physical Review E, 86(2):26101–26101.
  * Kyrtsou et al., (2020) Kyrtsou, C., Mikropoulou, C. D., and Papana, A. (2020).  Exploitation of information as a trading characteristic: A causality-based analysis of simulated and financial data.  Entropy, 22(10):1139.
  * Lee et al., (2019) Lee, T. K., Cho, J. H., Kwon, D. S., and Sohn, S. Y. (2019).  Global stock market investment strategies based on financial network indicators using machine learning techniques.  Expert Systems With Applications, 117:228–242.
  * Li et al., (2017) Li, H., An, H., Fang, W., Wang, Y., Zhong, W., and Yan, L. (2017).  Global energy investment structure from the energy stock market perspective based on a heterogeneous complex network model.  Applied Energy, 194:648–657.
  * Liu et al., (2017) Liu, A., Paddrik, M. E., Yang, S. Y., and Zhang, X. (2017).  Interbank contagion: An agent-based model approach to endogenously formed networks.  Journal of Banking and Finance, 112(3):105191.
  * Liu et al., (2020) Liu, X., Wei, S.-J., and Zhou, Y. (2020).  A liberalization spillover: From equities to loans.  National Bureau of Economic Research.
  * Liu and Tse, (2012) Liu, X. F. and Tse, C. K. (2012).  A complex network perspective of world stock markets: Synchronization and volatility.  International Journal of Bifurcation and Chaos, 22(6):1250142.
  * Naoui et al., (2010) Naoui, K. N., Liouane, N., and Brahim, S. (2010).  A dynamic conditional correlation analysis of financial contagion: The case of the subprime credit crisis.  International Journal of Economics and Finance, 2(3):85.
  * Paltalidis et al., (2015) Paltalidis, N., Gounopoulos, D., Kizys, R., and Koutelidakis, Y. (2015).  Transmission channels of systemic risk and contagion in the european financial network.  Journal of Banking and Finance, 61(12):36–52.
  * Papana et al., (2013) Papana, A., Kyrtsou, C., Kugiumtzis, D., and Diks, C. (2013).  Simulation study of direct causality measures in multivariate time series.  Entropy, 15(7):2635–2661.
  * Rosvall et al., (2009) Rosvall, M., Axelsson, D., and Bergstrom, C. T. (2009).  The map equation.  European Physical Journal-special Topics, 178(1):13–23.
  * Rosvall and Bergstrom, (2007) Rosvall, M. and Bergstrom, C. T. (2007).  An information-theoretic framework for resolving community structure in complex networks.  Proceedings of the National Academy of Sciences of the United States of America, 104(18):7327–7331.
  * Rosvall and Bergstrom, (2008) Rosvall, M. and Bergstrom, C. T. (2008).  Maps of random walks on complex networks reveal community structure.  Proceedings of the National Academy of Sciences of the United States of America, 105(4):1118–1123.
  * Rudan et al., (2015) Rudan, I., Sidhu, S., Papana, A., Meng, S., Xin–Wei, Y., Wang, W., Campbell–Page, R. M., Demaio, A. R., Nair, H., Sridhar, D., Theodoratou, E., Dowman, B., Adeloye, D., Majeed, A., Car, J., Campbell, H., and Chan, K. Y. (2015).  Prevalence of rheumatoid arthritis in low- and middle-income countries: A systematic review and analysis.  Journal of Global Health, 5(1):10409–10409.
  * Wu, (2019) Wu, B. (2019).  The global stock network connected and resonance effect based on the time-zone VAR model with LASSO.  SSRN Electronic Journal.
  * Xu et al., (2019) Xu, Q., Li, M., Jiang, C., and He, Y. (2019).  Interconnectedness and systemic risk network of chinese financial institutions: A lasso-covar approach.  Physica A-statistical Mechanics and Its Applications, 534:122173.
  * Yan et al., (2020) Yan, Y., Wu, B., Tian, T., and Zhang, H. (2020).  Development of stock networks using part mutual information and australian stock market data.  Entropy, 22(7):773.
  * Yu et al., (2020) Yu, D., Huang, D., and Chen, L. (2020).  Stock return predictability and cyclical movements in valuation ratios.  SSRN Electronic Journal.
  * Zhang et al., (2021) Zhang, B., Chen, W., and Yeh, C.-Y. (2021).  Turnover premia in china’s stock markets.  Pacific-basin Finance Journal, 65:101487.

##  Appendix A Appendix

Figure B: Module divisions in eight periods based on the Map Equation
algorithm. Nodes represent modules, directed links indicate directions of the
information flow, and the thickness of links demonstrates the transmission
probability of risk between different modules. The node size is proportional
to the information flow in a module that includes the information flow out of
and within the module. Specifically, the boundary thickness of a node is
proportional to the information flow out of the module, corresponding to the
probability that risks transmit to other modules. The interior area of a node
is proportional to the information flow within the module, corresponding to
the probability that risks staying in the module. Figure C: Module divisions
in eight periods based on the Map Equation algorithm. Nodes represent modules,
directed links indicate directions of the information flow, and the thickness
of links demonstrates the transmission probability of risk between different
modules. The node size is proportional to the information flow in a module
that includes the information flow out of and within the module. Specifically,
the boundary thickness of a node is proportional to the information flow out
of the module, corresponding to the probability that risks transmit to other
modules. The interior area of a node is proportional to the information flow
within the module, corresponding to the probability that risks staying in the
module.