The degree of node is a basic parameter to describe the local characteristics of the network, degree of node i is denoted as K_i. It is defined as follows [38]:

(1)

where n is the total number of node, X_ij and is defined as 1 if node i is connected to node j, and 0 otherwise. i is the focal node, j shows all other nodes.

From complex networks, probability of degree distribution of node with degree K is denoted as P_K and defined as follows:

(2)

where N_k is the number of nodes with degree k. For average degree , we have:

For random network, its degree distribution is as follows [38L.C. Freeman, "Centrality in social networks conceptual clarification", Soc. Networks, vol. 1, no. 3, pp. 215-239, 2012.
[http://dx.doi.org/10.1016/0378-8733(78)90021-7] ]:

However, degree distribution of scale-free network is as follows [39A. L. Barabási, R. Albert, and H. Jeong, "Mean-field theory for scale-free random networks", Physica A:, vol. 272, no. 1-2, pp. 173-187, 1999.]:

The small-world networks have the higher clustering coefficient and shorter average path length. For given node i, its clustering coefficient C_i is the probability of the connection between any two points connected with node i. It is defined as follows [5]:

(3)

where E_i represents the number of edges of connected neighboring nodes i.

The average coefficients C is defined as follows:

(4)

For fully-connected network, the value of C = 1. For scale-free network, the value of average coefficients is close to 1.

For unweighted complex networks, denoting l_ij as path length of between node i and node j, which satisfies:

(5)

The average path length of complex unweighted networks is given as follows:

(6)

In order to characterize time series from associated network topology, the visibility graph method was first proposed by L. Lacasa et al. in 2008 [29]. In the visibility graph method, the values of time series are plotted using vertical bars. A vertical bar links with others which can be seen from the top of itself. The visibility criterion is established in the literature as follows:

Definition 2.1 Two arbitrary data values (t₁, y₁) and (t₂, y₂) have visibility, and consequently they become two connected nodes of the associated graph [29], if any other data (t₃, y₃) placed between them fills the requirement:

As shown in Fig. (1), the histogram shows a time series with 11 data values, and the associated graph is obtained according to the visibility algorithm. In the histogram, if a bar can be observed from the top of considered one, it will found to be linked. If two bars are linked in the histogram, the vertices which represent them will be linked in the associated graph.

Fig. (1) The visibility graph.

The associated graph extracted from a time series is follows:

1. Connected: Each node sees at least its nearest neighbors.
2. Undirected: The associated graph extracted from a time series is undirected.
3. Invariant under affine transformations of the series data: the visibility criterion is invariant, when horizontal and vertical axes are rescaled.

Extraction of stock network from the stock market is an important task. The stock market is described as time series by many researchers [40-42]. However, they usually considered income of a stock company in a specific time period or in a trading day. These references do not portray the situation of stock companies of other nations. In this paper, all stock companies of other nations are considered and modeled as stock networks. The main steps of modeling are as follows:

• Step 1: The values of closing price of all stock companies are collected through software of stock.
• Step 2: The closing price of each stock company is described as a vertical bar. The value of close price is represented by height of vertical bar. The bigger the value is the higher the bar will be.
• Step 3: For each year, data of these vertical bars are arranged randomly.
• Step 4: Two vertical bars are joined by a straight line, if their direct linking is not crossing the other bars.
• Step 5: These vertical bars and joints are represented by nodes and edges of network, respectively.

According to the above five steps, a stock network is obtained. For each year, data of these vertical bars are randomly arranged a number of times. For each time, the range can be converted into complex networks based on the visibility graph method.

The application of complex networks in Chinese stock market is considered in this section. The basic data is the year-end closing price of all public companies in China, taken from the year 2012-2013 of Chinese stock market. The date derived from Straight flush, an online stock trading securities analysis software. According to the proposed method, every year, the data of these vertical bars are arranged randomly four times. And then, four stock networks are obtained and denoted as and, where is shows the times of random permutation. The four complex networks of 2012 and 2013 are shown in Figs. (2 and 3) respectively. The degree distribution and clustering coefficient of these complex networks are shown in Table 1.

Table 1
The average degree, shortest past length and clustering coefficient of networks.

From Table 1, the values of are always between 10 and 11, the values of are always less than 4 and the values of are always around 0.83. The values of average path length of these complex networks are small and the values of clustering coefficient are big, although these networks have a great number of nodes. From Figs. (2 and 3), the degree distribution is fitting with, the values of in these complex networks are approximate 3.5.

Table 2
The rank of top 10 of company according with their average degree.

Fig. (2) The degree distribution in 2012.

Fig. (3) The degree distribution in 2013.

The average degree of every company is calculated each year, and then the companies are sorted from high to low according to the value of average degree. The companies ranking in top 10 of every year are shown in Table 2. Table 2 shows that these companies are always banks and oil companies.

From previous section, although the number of companies is more than 2000, the average degree and the average shortest length of these complex networks are always about 10 and 4, respectively. The values of average degree in top 10 companies are always more than 95. Indeed, the No.1, The Industrial and Commercial Bank of China is 648.75 in 2013. In these complex networks, there are few Hob nodes, such as “The Industrial and Commercial Bank of China”, “The Agricultural Bank of China” and “Bank of China”. Meanwhile, from Figs. (2 and 3), the stock-market networks of Chinese stock market show scale-free property. From Table 2, ranking with average degree of companies, there are almost state holding pattern companies in the top 10 list.The result described that public sector of the economy is very important in the Chinese stock market.

Another interesting result here is that the verification of the Pareto principle. As per the ranking of average degree of company, the sum of 20 percent of companies having general income is 81.16% and 80.05% in 2012 and 2013, respectively. It reflects a market phenomenon that minority companies are occupied numerous markets. The result described the disequilibrium of income of every company in the Chinese stock market. Moreover, these stock networks follow the power-law distribution, which also explains these networks fit in the Pareto Principle. For Chinese stock market in 2012 and 2013, the Pareto Principle and the scale-free property are found to be consistent.