Research on energy stock market associated network structure based on financial indicators

Physica A 490 (2018) 1309–1323

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Physica A journal homepage: www.elsevier.com/locate/physa

Research on energy stock market associated network structure based on financial indicators Xian Xi, Haizhong An * School of Humanities and Economic Management, China University of Geosciences, Beijing, 100083, China Key Laboratory of Carrying Capacity Assessment for Resource and Environment, Ministry of Land and Resources, Beijing, 100083, China Key Laboratory of Strategic Studies, Ministry of Land and Resources, Beijing, 100812, China

highlights • The structural similarity of stocks is described by multiple financial indicators. • The stock associated network model based on financial indicators is created. • Set threshold value and provide a detailed analysis of the topological features.

article

info

Article history: Received 16 July 2016 Received in revised form 12 May 2017 Available online 14 September 2017 Keywords: Complex networks Stocks Similarity Financial indicators

a b s t r a c t A financial market is a complex system consisting of many interacting units. In general, due to the various types of information exchange within the industry, there is a relationship between the stocks that can reveal their clear structural characteristics. Complex network methods are powerful tools for studying the internal structure and function of the stock market, which allows us to better understand the stock market. Applying complex network methodology, a stock associated network model based on financial indicators is created. Accordingly, we set threshold value and use modularity to detect the community network, and we analyze the network structure and community cluster characteristics of different threshold situations. The study finds that the threshold value of 0.7 is the abrupt change point of the network. At the same time, as the threshold value increases, the independence of the community strengthens. This study provides a method of researching stock market based on the financial indicators, exploring the structural similarity of financial indicators of stocks. Also, it provides guidance for investment and corporate financial management. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Energy financial market is a complex system consisting of many interacting units. Energy is a core component of a country’s economic and social development [1], which is of great significance to the development of energy financial markets. In recent years, the energy stock market, which means the stocks belonging to energy industry sector in the stock market, has developed rapidly. Complex network theory is becoming a very popular theory and method to analyze the topological features and problems of stock markets [2]. In order to better promote the development of the stock market, the definition of the relationship between the stocks became a research focus point. Previous studies largely relied on the correlation [3,4], such as, Pearson and partial correlation [5–7], gray correlation [8] and price fluctuation correlation [9] between stock prices to

*

Corresponding author at: School of Humanities and Economic Management, China University of Geosciences, Beijing, 100083, China. E-mail address: [email protected] (H. An).

http://dx.doi.org/10.1016/j.physa.2017.08.114 0378-4371/© 2017 Elsevier B.V. All rights reserved.

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define the relationship between stocks. Also, some scholars paid attention to the impact of external economic and financial factors on the stock market. They discussed the government policy and monetary policy [10,11], corporate performance [12], stock index [13] and financial indicators [14] on the impact of stock prices to characterize the development of the stock market and to guide investment. What is more, some scholars studied the relationship between individual financial factors [15,16] and stock prices to describe the relevance of the stock. However, the definitions of these relationships are all based on one single indicator, such as stock prices, yield or exchange rates. They are only indicative indicators of the company’s external financial performance. The value of a stock is not only indicated by the impact of stock prices, but also by many other factors, such as, co-attendance behavior of senior executive [17], shareholder relations [18], cross-holding relationships [19], geographic locations [20] and information networks [21]. These factors will have an impact on the capital structure of the stock market. Especially, financial indicators of listed companies can strongly prove the intrinsic value of the stock. If we depict the relationship between stocks through a series of financial indicators, we can understand the relationship of energy stock market in more detail. The financial indicators are the relative indicators of the business summary and evaluation of financial status and operating results, which have a significant relationship with firm performance [22]. They can objectively and comprehensively reflect a company’s financial ability [23]. Financial indicators consist of a complex series of index factors, such as debt paying ability, operation ability, profitability, development capacity index [24,25] and cash flow. The future profitability of the listed company is an important guarantee for the stock price rise, which represents the evaluation of the value of listed companies. However, profitability and other capabilities interact with each other to evaluate the potential benefits of Listed Companies in the future [26]. The previous study mainly focused on the correlation between the financial performance and stock price [27], the individual financial indicator and the price [28] and several financial indicators and stock return based on time scales [29]. Also, some scholars are concerned with the comparison of financial indicators of different enterprises, then they can observe differences in financial structure and detect financial risks or potential benefits planning [30–32]. The systematic analysis and evaluation of the financial indicators can help to fully understand the past and present operating results, financial conditions and changes of the enterprise [33], and provide guidance to understand the present and future development of stocks and help stakeholders to make decisions. However, few studies have considered the structural differences and topological features of the stock market based on financial indicators from a holistic perspective. The complex network has been widely and effectively used to present topological features in a holistic view. As a tool to analyze the entity structure and evolution from an overall perspective, complex network methodology is widely applied in the financial market. Currently, the study of complex network of energy economy mainly concentrates on price fluctuations [34], price index correlation structure [35], financial time series [36], and the structures of heterogeneous network and multi-layer network. Furthermore, with respect to the energy market, there are many scholars focusing on the energy stock market [37,38], energy trade [39], listed companies [40,41], etc. Also, some studied on the cross-correlations of 67 stock market indices [42] and the partial correlation network [43]. However, stock markets and financial indicators are intricate, usually, in the analysis of complex network, the concept of threshold value was well used. The threshold value method can result in different structure of network and better detect the essence of the network. For instance, some scholars analyzed the network topology characteristics of the financial market based on the threshold method [2,44], some researchers analyzed the global financial index correlation clustering network under different thresholds with stochastic matrix theory (RMT) [45]. Under different threshold, investors could make different investment options. And scholars have discussed portfolio choice based on the complex network and community structure [46,47]. In the face of a complex stock market and certain financial indicators, the threshold method eliminates the weak correlation, and the complex network better reflect the law of development of the stock market and community cluster characteristics from an overall perspective. On the basis of the complex network research, we put forward the energy stock associated network model, whose foundation is the financial indicators of listed companies. We select financial indicator data from the financial sector of all energy companies in the Shanghai composite index of wind information. First, the data are collated and we normalize every financial indicator of each listed company to the same dimension for the empirical analysis. Second, using the formula of Pearson correlation coefficient, we calculate the structural similarity of financial indicators of every two listed company. Thus we get the structure similarity coefficient matrix. Finally, the threshold situation is set and combined with the complex network theory, the stock associated network model under different threshold values is established. By analyzing the topology of the network and the community cluster characteristics, we understand the characteristics of the current stock market and propose a quantitative approach to investment and enterprise financial management. 2. Data and methods 2.1. Data The data used in this paper are mainly extracted from the Wind Information Database (http://www.wind.com.cn/). The selected data include the Listed Company List of all of the energy companies listed on the Shanghai composite index in 2014. The majority of the data used in this paper were collected on April 15, 2015. We select part of the financial analysis data in 2014 as sample data. Financial indicators include eight aspects, which are the share index (18 secondary indicators), profitability and earnings quality (35 secondary indicators), capital structure and debt paying ability (44 secondary indicators), operation ability (secondary indicators), growth ability (14 secondary indicators), cash flow (12 secondary indicators), DuPont analysis (9 secondary indicators) and warning Z values (7 secondary indicators).

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Table 1 Financial indicators structural similarity coefficient.

Stock1 Stock2 Stock3

Stock1

Stock2

Stock3

1 0.283 0.976

0.283 1 0.332

0.976 0.332 1

Fig. 1. Financial indicators structure radar chart.

We then establish a financial indicator system, which contains 152 financial indicators. The data dimensionless processing mainly solves the comparability of data. Therefore, we use the min–max method to map the data uniformly to [0, 1]. So each sub-indicators has been normalized and we get a unified dimension data. 2.2. Methods 2.2.1. The similarity definition The correlation coefficient, i.e., the Pearson correlation coefficient, is an indicator that measures the degree of correlation between changing trends among variables and has a range of [−1, 1]. In fact, it is the covariance of the two variables divided by the standard deviation of the two. Since it is the normalized covariance, the more important feature is that it can simply reflects the degree of similarity of the two variables per unit change. It can well characterize the structural similarity between variables from a multi-dimensional perspective. With reference to the Pearson correlation coefficient, we calculate the structural similarity of financial indicators of every two listed companies. The data are normalized and the similar coefficient has a range of [−1, 1]. The greater the absolute value of the similarity coefficient is, the higher the degree of similarity between the variables is. The formula is as follows:

∑n

m=1

rij = √

∑n

m=1

i

(

(

xim − x

xim − x

)(

j

xm − x

j

) (1)

i 2

) ∑n

m=1

(

j

xm − x

j

)2

where xim is the value of the mth financial indicator variable of the ith listed company, xi is the value of the average financial j indicator variable of the ith listed company, xm is the value of the mth financial indicator variable of the jth listed company, j x is the value of the average financial indicator variable of the jth listed company, n is the number of items in the financial indicator series, and rij = rji . For example, Table 1 shows the structural similarity coefficient of three stocks, and Fig. 1 shows the radar chart of the financial structure of them. From the above chart, we find that the similarity coefficient of stock 1 and stock 3 is about 0.976; the similarity coefficient of stock 1 and stock 2 is about 0.283; the similarity coefficient of stock 2 and stock 3 is about 0.332. Combined with the radar chart, it is find that the financial indicator structures of stock 1 and stock 2 are quite different. The financial indicator structures of stock 1 and stock 3 are similar. It confirms that this method is applicable to explain the structural similarity of financial indicators between stocks. What is more, by the structural similarity analysis, we can find stock 1 and stock 3 have a prominent value in the indicator 5, and stock 2 have a prominent value in the indicator 2 and 7. Similar financial structure shows that the two have a similar development model, which means that in the future they may face the same benefits and similar risks. This is of great guidance for business managers to optimize the financial structure, and for investors to identify and avoid the investment risk, which is the purpose of this paper. Accordingly, we calculate the structural similarity of financial indicators of 74 listed company. Then we obtain a similar coefficient matrix as shown below. The number of coefficients in the matrix is 5476 (74 * 74), rni is equal to rin . We delete

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Fig. 2. Similarity coefficient descending order distribution.

duplicate data and data which are equal to 1, leaving 2701 at the end. 1

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ R=⎢ ⎢ ⎢ ⎢ ⎣

r1,2 1

··· ... 1

r1,j−1 r2,j−1

r1,j r2,j

1

ri,j

.. .

.. .

1

... ... .. . ... .. . 1

r1,n r2,n ⎥

⎤

.. ⎥ ⎥ . ⎥ ⎥ ri,n ⎥ , .. ⎥ ⎥ . ⎥ ⎦ r

n = 74

(2)

n−1,n

1 where r12 represents the similarity coefficient between the first stock and the second stock, and rij represents the similarity coefficient between the ith stock and the jth stock. What is more, we test the significance of the data, delete the data that failed the test, and get the 2653 similarity coefficients in descending order, as shown in Fig. 2. 2.2.2. Complex network model In order to better understand the features of the similar structure of financial indicators, we establish the weighted stocks associated networks using complex network theory. A network is a collection of nodes and edges, G = (N , E). In the network, we represent every stock as a node and the structural similarity of financial indicators between two stocks as an edge. As a result, we turn the study of the similar structure of financial indicators into the study of stock associated networks (SANs). The set of nodes N in the SAN is expressed as N = {n1 , n2 , . . . ni . . . n74 } ,

(3)

where ni represents the ith stock. The set of edges E in the SAN is expressed as E = e(1,2) , e(1,3) , . . . e(i,j) . . . e(73,74) ,

{

}

(4)

where e(i,j) represents the similarity between the ith stock and the jth stock. If all of the similarity coefficients are reflected in the network with weights, the network is approximate completely connected, with 74 nodes and 2653 edges. It is unavailable for topology analysis. Therefore, in order to more accurately portray the relationship and structure between stocks, it is necessary to filter some of the data and choose the threshold, which is denoted by r. Accordingly, we select r = 0.1, 0.2, 0.3 . . . 0.9 as the threshold value to filter networks and enlarge the network structure. Thus, to gain a better understanding of the internal characteristics of the network, we observe the network structure and community cluster characteristics of different threshold situation. 2.2.3. The topological features of SAN Node degree refers to the number of nodes that have a direct link with a certain node, and thus, node degree is used to represent the quantity of the stock, whose similarity with another stock is greater than or equal to the threshold value. The definition of node degree is as follows: ki =

N ∑

aij

(5)

j=1

where ki is the value of degree, N is the total number of stocks, and aij is the number of edges between stock i and stock j.

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The average degree is the average value of all node degrees, that is:

∑N

j=1

⟨k⟩ =

aij

.

n The average path length is defined as the average distance between any two stocks in a network, L=

1 1 N 2

∑

(N − 1)

dij

(6)

(7)

i≥j

where N is the total number of stocks in a network and dij is the distance of stock i and stock j. The closeness centrality is the sum of the shortest distance from a node to another node. It can examine the extent to which a node does not rely on other nodes to propagate information. The definition is as follows: CCi =

1 di

N −1 . = ∑N j=1

(8)

dij

Where N is the total number of stocks in a network. The dij is the distance of stocki and stock j. The clustering coefficient of network assumes that a node i has ki edges that connect other nodes to it, and the ki is the number of neighbor nodes of node i. Among the ki nodes, there are at most ki (ki − 1) /2 edges, but the number of real edges between the ki nodes is Ei . The ratio between Ei and ki (ki − 1) /2 is then defined as the clustering coefficient of node i, that is: Ci = 2Ei /ki (ki − 1) .

(9)

If Ci = 0, the neighbors of a node are not connected; however, if Ci = 1, all the neighbors are connected with each other. A high concentration indicates that the neighbors of this node has have good connectivity. The average clustering coefficient is the average clustering coefficient of all stocks, that is: CW −S =

n 1∑

n

Ci .

(10)

i=1

Betweenness centrality measures the stock roles as mediators. If the stock occupies an important position of two other stocks on their contact path for many times, the betweenness centrality of the stock is high. If a stock is the connection between two separate communities, it is called a bridge. The formula is as follows: Bi =

∑ gjk (i)

2

(N − 1) (N − 2)

j
(11)

gjk

where gjk is a shortcut from stock 1 to 2, gjk (i) is the number of stock 3, which is on the contact path of stock 1 to 2, and N is the number of stocks in the network. Because a bridge node often has the greatest number of shortcuts through it, the bridge node’s betweenness centricity is always the highest and accordingly, it is much higher than other stocks. The algorithm of modularity includes steps that are repeated iteratively. First, we assume each node of the network consisting of N nodes is a community. We then consider the neighbors j of each node i to calculate the gain in the modularity when i moves from its own community to community j. If the gain is positive, we move i to community j. If there is no positive gain, node i stays in its own group. This step is carried out repeatedly and sequentially for all nodes until a local maximum of the modularity is attained. We then move to the next phase where the gain in modularity 1Q obtained by moving node i to community C can be computed by the following equation:

1Q =

[∑

in + ki,in 2m

(∑ −

tot + ki 2m

)2 ]

[∑ −

in

2m

(∑ −

tot

2m

)2

( −

ki 2m

)2 ] (12)

where in is the sum of degrees of all of the links inside C . tot is the sum of the degrees of all of the nodes in C . ki is the sum of the degrees of i. ki,in is the sum of the degrees of the links from i to all of the nodes in C , and m is the sum of the degrees of the network.

∑

∑

3. Results 3.1. The overall network Based on the structural similarity of financial indicators and the theory of complex networks, we establish the stock associated networks under different thresholds. In order to better understand the stock network construction process, we give a simple example, as shown in Fig. 3. We construct a complex network with thresholds of 0.7 and 0.8, using the stock as the node and the similarity coefficient of the two stocks as the edge. For better understanding, edges are connected by solid lines, and no edges are connected by dashed lines. When the threshold value is 0.7, stock 1 and stock 2, stock 2 and

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Fig. 3. Schematic diagram of complex stock associated network.

(a) r = 0.20.

(b) r = 0.50.

(c) r = 0.80.

Fig. 4. The SAN model under different threshold.

stock 3 have edges, and there is no edge between stock 1and stock 3, because the similarity coefficient does not meet the requirement. They are in the same community with the same color. When the threshold value is 0.8, the edge between stock 1 and stock 2 disappears because the similarity coefficient is 0.71, which is less than 0.8. What is more, stock 2 and stock 3 make up a new community and the color turns pink. On this basis, we establish SAN models with different thresholds. And then we select the low, medium and high degree of similarity network as representatives, which are shown in Fig. 4. The size of each node is dependent on its node degree and the thickness of the edge is dependent on its weight. Different colors represent different communities. As evidenced by diagram, community cluster characteristics are obvious. In the low and medium similarity degree, the networks are divided into two communities, and there is no big difference between them. However, the network has a significant trend of socialization with high degree of similarity. What is more, the edges are thicker and the similarity degrees are higher among stocks in the same community. With the increase of the threshold value, the number of network nodes and edges gradually declines, and the network structure becomes more distinct. 3.1.1. Topological characteristics We analyze the network characteristics of the energy stock market under different threshold values. For example, the topological features of network scale, clustering, betweenness centrality and other features, which is described in details as follows. From Fig. 5, we can determine that, as the threshold value increases, the numbers of network nodes and edges decline. When the threshold value is less than 0.7, the number of nodes and edges decreases gently, and then falls sharply when the threshold value equals or exceeds 0.7. The results indicate that the threshold value of 0.7 is the abrupt change point of the network. The node number is relatively stable and the number of edges is slightly reduced when the threshold value is less than 0.7. It shows that in the low and middle degree of similarity, the vast majority of stocks in energy industry have similar financial structure. When the threshold is more than 0.7, there are only a few stocks which have financial structural similarity.

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Fig. 5. Nodes and edges under different threshold.

Fig. 6. The average weighted degrees under different thresholds.

Fig. 7. The average path length and network diameter under different thresholds.

In Fig. 6, the average weighted degree has been reduced. When the threshold value is below 0.7, the average weighted degree changes gently and the slope tends to be more uniform. Once the threshold value reaches 0.7 and above, the average weighted degree of the network decreases rapidly as evidenced by the large change in the slope. This has a great relationship with the degree of variation of nodes and edges. When the similarity is in low degree, the stocks in the network are stable and the change of average weighted degree is small. When the degree of similarity is high, there are minority stocks has high financial structure similarity and edges reducing very fast, so the average weighted degree plummets. Fig. 7 presents the trends with respect to average path length and the network diameter. When the threshold value is less than 0.7, the average path length and network diameter demonstrate slowly increasing trend. The two lines rapid increase when the threshold is 0.7 and above. Although the degree of node and edge number decrease with the increased

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Fig. 8. The average network concentration coefficients under different thresholds.

Table 2 The betweenness centricity under different thresholds.

threshold, the trend of average path length and network diameter do not follow this trend. This is due to the change in the number of communities. The trend of community increases with the increase of the threshold value, though the node path within the community is unchanged, the path length between community increases, so the average path length of the network becomes larger. Also, because of the trend of community differentiation, the network diameter becomes larger. The network clustering coefficient reflects the close degree between neighboring nodes, as presented in Fig. 8. The network has been in a state of high average coefficient concentration with a threshold value less than 0.6, which means that the entire network’s connectivity is strong and the node neighborhood close degree is high. Once the threshold value exceeds 0.7, the change in the network’s average clustering coefficient is even larger than before with a sharp decreasing tendency. This indicates an even higher degree of similarity and looser neighbor network node. With respect to energy stocks, the enterprise growth ability, operation ability, profit ability, etc. all have a gap. As a result, when the structural similarity is high, the connectivity between the neighboring nodes of the stocks is greatly reduced. Table 2 shows the betweenness centricity of the top ten stocks. Stocks in the pink, blue, green, purple and orange sectors appear more often and they rank high, so they play a bridge role in the entire network. What is more, the stocks with higher levels of betweenness centricity are 600348 (YQMY), 600207 (ACHT), 600725 (YNYW) and 600721 (BHC). It show that the four stocks play an important bridge role in the network. They have the ability to control the interactions of others, so if the financial structure of the intermediary point changes, it is likely to make the connection between two communities or nodes disappear. This has a great impact on the network structure, which need to be concerned. When the threshold is greater than 0.7, there is no fixed intermediary point in the network, which indicating that the network is more dispersed.

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Fig. 9. The number of modularity and communities according to threshold values.

Fig. 10. The community characteristics of r < 0.7.

From the analysis of the above network structure, we can understand the current situation of energy stock market. With the change of the threshold, the network structure presents different functional characteristics, which can give us a clearer understanding of the essential characteristics of the energy stock market. 3.2. Community cluster characteristics As shown in Fig. 9, before and after the threshold of 0.7, the modularity of the network and the number of communities are very different. When the threshold is less than 0.7, the modularity increases slightly from 0.01 to 0.03 and the number of communities remains unchanged. However, when the threshold is greater than 0.7, they increase rapidly. It indicating that the community phenomenon becomes more and more obvious as the threshold increases. What is more, at the low and medium degree of similarity, the vast majority of stocks have structural similarities in financial indicators, and they tend to be in the same community. When the degree of similarity is high, there is only a small number of stocks with financial structural similarity and they tend to gather in small communities, so the community differentiation is more obvious. 3.2.1. The community characteristics of r < 0.7 Based on the above, we know that the threshold 0.7 is the abrupt point of the network. Therefore, we use the threshold 0.7 as the dividing line and analyze the characteristics of network community under low, medium and high thresholds respectively. We remove the high degree of similarity of the stock nodes, and analysis the stock association network characteristics when the threshold is less than 0.7, as shown in Fig. 10. There are 74 nodes, 482 edges and the proportion of edges is 18%. The proportion of the edges is small, indicating that the financial structural similarity of the stock in energy industry is high,

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X. Xi, H. An / Physica A 490 (2018) 1309–1323 Table 3 The network characteristic value of top 10 stocks. Stock

Degree

Weighted degree

Betweenness centrality

968 300191 600725 835 554 601918 603003 600721 600179 300309

70 70 69 64 41 38 34 32 26 24

35.47 31.98 34.67 38.30 24.83 22.08 20.30 19.33 5.54 14.29

479.55 487.63 459.57 342.70 102.60 96.39 61.94 61.65 47.70 21.01

and there are fewer associated stocks whose similar degrees are less than 0.7. However, it has a great effect for us to explore their differences in the structure of financial indicators, discover the structural advantages of others and make up for our weaknesses. Therefore, we have carried on the detailed analysis of the network. The network is divided into three communities and there are some nodes with high degree values. So we sort the degree values in descending order and pick the top 10 stocks, which are shown in Table 3. Their degree values ranking high, and their weighted degree and betweenness centrality are high. This indicates that they are associated with a large number of stocks, and they also play a bridge role in the network. Also, we found that the main business of 000968 (MQH) and 600725 (STYW) are coal products industry and they belong to blue community. 601918 (SDIC XINJI) and 600721 (BHC) are coal mining industry belonging to green community, 603003 (Lonyer Fuels) and 300309 (GI-TECH) are service enterprises belonging to purple community. This shows that the division of the community and the main business has a great relationship. Because the main business similarity makes the financial indicators tend to be similar, and thus proves the necessity of building networks based on financial indicators. 3.2.2. The community characteristics of r > 0.7 When the threshold is greater than 0.7, there are 2171 edges between stocks. In order to better understand the topological features, we choose five thresholds r = 0.75, 0.8, 0.85, 0.9, 0.95. Fig. 11 shows the evolution of each community when r > 0.7. With the increase of the threshold, the trend of community is obvious. The numbers of nodes and edges decrease rapidly, and the network structure is more simple and clearer. There are three or four communities in the networks, and they all have connections. However, 601918(SDIC XINJI) and 600721(BHC) form an independent green community, and their financial structural similarity is 0.903. Their similarity with other stocks are less than 0.8, which indicates that the financial structures of the two are different with most of other stocks. We know they are both stocks in coal mining industry and their net profit declined seriously in 2014. What is more, they are state-owned enterprises restructuring concept stocks. This should be the reason why their financial indicator structures are similar. The nodes within the community have a high similarity, and the relationships between them are relatively stable. So the investor could focus on more stock relationships within the community. For example, 002490 (Shandong Molong), 600397 (Anyuan coal) and 600121(ZCE) belong to the red community. When the threshold value is less than 0.95, 600395(PJRC) and 000571(Sundrio) belong to the purple community; 600997(KEC) and 000983 (SXCE) belong to the blue community. As the threshold increases, the relationships between them are stable. The above pairs of stocks have high structural similarity in financial indicators and their main business is similar. This shows that the characteristics of the industry and the nature of the business have some relevance with the similarity of financial indicators. Therefore, we should pay close attention to the associated stock with similar structure, because the two have a similar business model and development direction, so they may face the same income, loss or risk. One changes in financial indicators may largely affect the other one, then the other one should remain vigilant because the same effects may occur. So when investing in stocks of the same business type, not only investors need to concern the relationship between them, managers should always pay attention to other similar company’s financial changes, to avoid their own financial risks. What is more, we focus on the situation of stocks within and outside the community in high-threshold scenarios. For example, node 000780(PingZhuang Energy) and 002700(Xinjiang Haoyuan) belong to a community, with the correlation coefficient of 0.923. They have a similar financial structure. 600348(YQMY) and 002700(Xinjiang Haoyuan) are not in a community. Their similar coefficient is 0.879, and the distribution trend with some differences, as shown in Fig. 12. The first two solvency and cash flow are better, but the operating capacity is relatively poorer compared with 600348. Furthermore, we have analyzed the changes of stocks within the community in detail, as shown in Fig. 13. We selected six to ten stocks that exhibit high volatility or a high degree of ranking as the representatives of the community. We analyze the evolution of communities at different thresholds and find that some stocks are highly volatile. For example, 002490(Shandong Molong), 000552(JYCE) and 600348(YQMY) move between different communities, indicating that their financial structure has its own characteristics and is not highly similar to the financial structure of the associated stock, and the relationship is not strong. The managers of these enterprises can examine the characteristics of their own development,

X. Xi, H. An / Physica A 490 (2018) 1309–1323

(a) r = 0.75.

(c) r = 0.85.

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(b) r = 0.80.

(d) r = 0.90.

(e) r = 0.95.

Fig. 11. The community characteristics of r > 0.7.

Fig. 12. Distribution of financial indicators.

as well as the difference with the financial structure of the associated stock. Then they can learn from each other or develop their own advantage indicators. There are also some stable node pairs. For example, 601918(SDIC XINJI) and 600721(BHC), of which the financial structure similarity coefficient is 0.9. In September 2014 they faced the most serious loss, while in December both of them have a raising limit situation. This is related to the coal industry itself. Also, it is due to its similar financial structure. For instance, 600188 and 601088 are large coal enterprises in China. They have high degree of structural similarity in terms of profitability, operational capacity, solvency and cash flow and they are only different in growth capacity. In reality 600188 does continue to learn and follow the development of 601088. Thus we know that the financial structure between them is very similar, then there will be similar development potential in various indicators, and investors can reduce the risk to take a diversified

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Fig. 13. The changes in communities.

investment. We must consider the stock financial indicators structure and the location of the stock in networks under different threshold values. Based on the analysis above, we should pay attention to the stock with high similarity within the community and consider the financial structure characteristic of the stocks within different communities. Also, we should consider the stock with strong liquidity among the communities. 3.3. Regression model test We found that the influence of the network structure based on financial indicators on stock returns is significant. We confirmed this by building a regression model. First, we calculated the yield of every stock and set it as the dependent variable Y. Then, we calculated the value of the network parameters, such as weighted degree, closeness centrality, betweenness centrality and the clustering coefficient of every stock. And set them as the explanatory variables, respectively as X1(weighted degree), X2(closeness centrality), X3(betweenness centrality) and X4(clustering coefficient). These parameters objectively represent the structure of the network. At last, under different threshold values (r = 0.4, 0.5, 0.6, 0.7, 0.8, 0.9), we constructed six regression models to see whether the impacts of network parameters on stock returns are significant. These threshold values can represent low, medium and high similarity levels. The results of regressions are shown in Table 4. In general, the regression performs well in the various variables. when 0.4 ≤ r ≤ 0.7, the impacts of network structure parameters on stock returns are significant. When the threshold is 0.4, 0.5 and 0.6, the coefficients of some network structure variables are significant at the 0.01 level, such as, X1(weighted degree) and X2(closeness centrality). Especially when r = 0.7, the coefficients of each network structure variables are significant at the 0.01 level. It means that the impact of network

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Table 4 Regression models under different thresholds. (1) r = 0.4

(2) r = 0.5

(3) r = 0.6

(4) r = 0.7

(5) r = 0.8

(6) r = 0.9

−0.036*** (−2.861) −6.626*** (−3.023) −0.003 (−0.564) −5.593*** (−3.024)

−0.039*** (−3.356) −3.476*** (−3.628) −0.002 (−0.359) −2.799*** (−3.486)

−0.045*** (−3.514) −3.114*** (−3.689) −0.015 (−1.457) −2.826 (−0.864)

−0.073*** (−4.191) −3.093*** (−3.779) −0.022*** (−2.853) −8.942*** (−2.945)

−0.012 (−1.096) −0.285 (−0.555) −0.003 (−0.975) −0.175 (−0.292)

−0.002 (−0.235)

Constant

14.699*** (3.148)

8.889*** (3.843)

8.919* (2.275)

16.185*** (3.767)

1.363 (0.825)

0.069 (0.432) −0.001 (−0.282) −0.162 (−0.827) 0.276 (0.767)

Observations R-squared F -statistic Prob (F -statistic)

74 0.131 2.591 0.044

74 0.169 3.504 0.012

73 0.172 3.54 0.011

72 0.225 4.876 0.002

69 0.099 1.774 0.145

59 0.036 0.507 0.731

VARIABLES X1(weighted degree) X2(closeness centrality) X3(betweenness centrality) X4(clustering coefficient)

Note: t-statistics in parentheses. *** p<0.01, ** p<0.05, * p<0.1.

structure on the stock returns is significant when r = 0.7. When r = 0.8 and 0.9, Prob (F-statistic)>0.05 and the coefficients on each network structure variables are not significant at the 0.1 level. The network structure is not significantly related to the stock returns, because the number of nodes and edges in the network are significantly reduced, and the network tends to be more decentralized, so the effects of these parameters on stock returns are not significant. According to the analysis of the network structure, we also know that the threshold value of 0.7 is the abrupt change point of the network. This shows that the information in the network structure at this time is more complete and effective. We should consider the network structure at different thresholds when investing, and pay more attention to the network structure when the threshold is 0.7. Through the model, we can know that the stock associated network structure based on financial indicators has an impact on investment. 4. Discussion and conclusions We studied the infrequently considered but important topic of financial indicators. Based on this, the structural similarity matrix of the financial indicators between stocks was established, and the stock associated network model was established by combining the complex network method. In addition, we divided the network into communities and analyzed the network structure and community characteristics under different threshold. Through a series of detailed studies, we came to several conclusions. The threshold has a significant effect on the energy stock associated network structure. It filters network layers, like a magnifying glass in general, which makes us understand the nature of the network more clearly. The threshold value of 0.7 is the abrupt change point of the network. Different structures show different network functions. We made a regression model to confirm that when r = 0.7 the network structure has a very significant impact on the stock returns. The financial structural similarity of the stock in energy industry is high. There are only a few associated stocks whose similar degrees are less than 0.7. However, it has a great effect for us to explore their differences in the structure of financial indicators and to make up for our weaknesses. When managers optimize the financial structure and investors make investment decisions, they should give consideration to stocks with both strong and weak financial structure similarity. We find that with the increase of similarity, the trend of community is significant. Though the community structure is simple and clear, nodes in the same community have a tendency to transfer and separate. Community division is mainly affected by the main business and the network structure based on financial indicators also validates this. As the saying goes, ‘‘eggs cannot fit in the same basket’’, Investors should consider comprehensively and decentralize risk according to their preferences and the economic strength of the stock. Financial indicators objectively determine the intrinsic value of the stock. In the same community, the financial structure of the stock is similar, which indicates similar trend in future gains or losses. If investors invest the stocks in distinct communities, it will decrease the investment risk. What is more, for managers, they can compare the financial differences between enterprises in different communities, identify their weaknesses and learn from others’ advantages, so as to rationally change their strategic planning to avoid financial risks. Furthermore, there are some important nodes in the network serving as important bridges in the whole network that should be considered. When the threshold is less than 0.7, we find some influential nodes in the network. They belong to different communities because of their main business, and this network shows this accurately. This also proves the feasibility and necessity of building networks based on financial indicators. What is more, we find some pairs with strong financial structure similarity, and their relationship is stable. In general, the understanding of the similarity of financial indicators is of great importance. In this paper, we analyze the stock based on financial indicators, and propose a new perspective. We depicts the structural similarity of the financial indicators between stocks from a multi-dimensional perspective. Combining with

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complex network theory and threshold method, we excavate the essential characteristics of the network and discover the network mutation point, which can help us deeply understand situation of stocks in energy industry. We carry out detailed community analysis and find some important nodes and stable node pairs. Also, we established a regression model and found that the network structure has a significant impact on stock return. In future research, we will consider a more detailed explanation of the economic significance of financial indicators in order to further find out which indicators affect the financial structural similarity of the stock. Also, we will combine some investment models to provide more quantitative advice for investment. Acknowledgments This research is supported by grants from the National Natural Science Foundation of China (Grant No. 71173199). The authors would like to express their gratitude to Xiangyun Gao, Huajiao Li, Sida Feng and Qingru Sun who provided valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

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