Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 9731219, 11 pages

https://doi.org/10.1155/2017/9731219

## Structure Characteristics of the International Stock Market Complex Network in the Perspective of Whole and Part

^{1}School of Economics and Management, Nanjing University of Information Science & Technology, Ningliu Road 219, Nanjing 210044, China^{2}Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science & Technology, Ningliu Road 219, Nanjing 210044, China^{3}School of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China

Correspondence should be addressed to Guangxi Cao; nc.ude.tsiun@ixgnaugoac

Received 23 September 2016; Accepted 12 February 2017; Published 6 March 2017

Academic Editor: Juan Pavón

Copyright © 2017 Guangxi Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

International stock market forms an abstract complex network through the fluctuation correlation of stock price index. Past studies of complex network almost focus on single country’s stock market. Here we investigate the whole and partial characteristics of international stock market network (ISMN) (hereinafter referred to as ISMN). For the analysis on the whole network, we firstly determine the reasonable threshold as the basic of the following study. Robustness is applied to analyze the stability of the network and the result shows that ISMN has robustness against random attack but intentional attack breaks the connection integrity of ISMN rapidly. In the partial network, the sliding window method is used to analyze the dynamic evolution of the relationship between the Chinese (Shanghai) stock market and the international stock market. The connection between the Chinese stock market and foreign stock markets becomes increasingly closer, and the links between them show a significant enhancement especially after China joined the WTO. In general, we suggest that transnational investors pay more attention to some significant event of the stock market with large degree for better risk-circumvention.

#### 1. Introduction

The international stock market network (ISMN) can be regarded as a complex network. In studies of securities markets to judge the method of constructing networks, scholars usually use correlation analysis to construct securities market networks, in which the nodes are stocks and the edges between nodes are the price fluctuation relationships of stocks.

Methods in designing networks include the minimum spanning tree (MST), planar maximally filtered graphs (PMFGs), and correlation threshold method. Mantegna was the first to use the correlation between stocks to build the stock market network [1]; he selected the main connections between the nodes and generated a tree graph to reveal the hierarchy of the network by adopting MST. Kim et al. also used the MST to study the topology of the network structure [2–5]. Tumminello et al. analyzed the portfolio of 300 most capitalized stocks traded at the New York Stock Exchange during the period 2001–2003 and used the statistical properties of the portfolio, such as the average path length to derive the PMFGs [6], which are based on MST but carry more information than the tree graphs. Boginski et al. all used the correlation threshold method to construct stock networks [7–11]. Clearly, the correlation method is widely used in building networks and is thus used in this paper. All aforementioned studies adopt the Pearson correlation coefficient; however, the statistical tests show that the sample data do not follow a normal distribution. Accordingly, we use Spearman rank correlation coefficient to describe the inherent relationships between stock indices.

Drawing from the conclusions of studies on stock market networks, Tse investigated all US stocks and found that the US stock network is scale free [9]. Gałązka studied the stock market of Poland by constructing a weighted complex network and MST [12]; the results showed that the Polish stock market is scale free. Namaki et al. constructed the Iran stock network using the threshold method and found that the network is scale free under particular conditions [13]. Caraiani investigated the emerging European stock markets, incorporated fractal theory into the complex network theory, found that the network is scale free [14], and identified multifractal characteristics of clustering coefficient. Ma et al. investigated the dynamics of Chinese stock market from 2005 to 2012 by using sliding window and found that the Chinese stock market is scale free when it experiences a bear market [15]. Tan and Ding applied the visibility graphs to the US stock market and found that the degree distribution agrees with the scale-free property during the Global Financial Crisis in 2007 [16]. In general, the aforementioned studies all concentrated on single national stock network and the results show that most of the single national stock networks are scale free. However, the topology graph constructed in this paper shows that the ISMN composed of the nodes we selected is not scale free. Rare literature of complex network refers to the whole international stock market, and it is circumscribed for transnational investors. But this paper is based on the international stock market whose conclusion can be a reference to transnational investors.

The contribution of our study is as follows. First, our study has been conducted on the stability of ISMN under random and intentional attacks and found that ISMN has robustness against random attacks, which no one has studied before. Second, in all the aforementioned studies, static networks are constructed even if a stock market is a changing complex system; thus, we adopt the sliding window method to study the dynamic law of the ISMN and we come to an interesting finding: there exists enhancement in the links between the Chinese stock market and the foreign stock markets after China joined the WTO. The rest of the paper is organized as follows. Section 2 provides formulas for the statistical characteristics of a complex network and illuminates how the network is constructed. Section 3 presents an empirical analysis of the characteristics of an ISMN. Section 4 summarizes the results and deficiencies of this work.

#### 2. Complex Network Model and Data

##### 2.1. Complex Network Model

Normally, size and density are selected to describe the macrocharacteristics of a network, while the average path length and clustering coefficient are used to measure both indicators.

The distance between any two nodes, and , in the network is defined as the edges of the shortest path between them. A random sample of the maximum distance between two nodes is called the network diameter, which is denoted by , .

The average path length of the network isIn the formula, is the number of nodes in the network, and the distance of the node itself is zero. The average path length represents the average distance between any two nodes and reflects the size of the network. The average path length is used to represent the transmission efficiency of the network.

Clustering coefficient analysis is conducted to assess if the nodes are closely interrelated and if the network is dense. If any one node, , in the network is connected to other nodes via edges, then nodes are neighbor nodes of and can each have up to edges. The ratio of the edges that exist between node neighbor nodes of , , to the maximum edges that may exist as the clustering coefficient is denoted by ; that is,The clustering coefficient of the entire network, , is equal to the arithmetic mean of clustering coefficients of all the nodes in the network; that is,Obviously, . When , all nodes in the network are isolated: that is, the network has no edges. However, when , any two nodes in the network are connected.

In many realistic complex networks, the connectivity probability between nodes is usually associated with the type of nodes. This selective connection between nodes is called assortativity. We can characterize assortativity in the network quantitatively by using an assortativity coefficient. The assortativity coefficient can be calculated in a variety of ways. In this study, we use the classic method proposed by Newman [17].

The nodes in the network are divided into types. Let be the number of edges of the node types and ; let the elements in matrix be ; let normal matrix be , where is equal to the sum of all elements in matrix. The assortativity coefficient is defined aswhere is the trace of the matrix, which is the sum of diagonal elements in a matrix, and is the sum of all elements in the matrix. When , the different types of nodes show no connective preference. That is, the nodes are connected completely and randomly. When , these nodes are connected only to the nodes of the same type, and the network has complete assortativity. When , the nodes in the network are more likely to connect to nodes of a different type. Conversely, when , the nodes are more likely to connect to the nodes of the same type.

##### 2.2. Data

For a dynamic research on the interactions and relationships between the stock markets in the world from January 1999 to December 2014, with 2369 observations being used, we use previous studies for reference and collect stock index data of 27 countries (regions). The data are extracted from “RESSET.” We choose the 27 indices which include six continents except Africa. The 27 indices abbreviations are listed as follows: HSI, KOSPI, NKY, FSSTI, TWSE, PCOMP, FBMKLC, SENSEX, JKSE, SZZS, AS51, UKX, CAC, DAX, AEX, BEL20, IBEX, BUX, ATX, OMX, SMI, DJIA, S&P 500, S&P/TSX, MERVAL, BVSP, and MEXBOL. We construct ISMN under different threshold, in which the stock indices are nodes and the price fluctuation relationships of stock indices, as edges. ISMN describes the price fluctuation relationships of stock indices with the daily log-return of the nodes. The log-return of index iswhere is the yield rate of index at time and is the closing price. Log-return is used as empirical analysis data to avoid the influence of exponential trend and can reduce nonstationarity of data.

Given the limited space, we select eight stock markets. The basic statistical characteristics of the return series are presented in Table 1.