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The impact of financial contagion on real economy-An empirical research based on combination of complex network technology and spatial econometrics model

This study presents financial network indicators that can be applied to inspect the financial contagion on real economy, as well as the spatial spillover and industry aggregation effects. We propose to design both a directed and undirected networks of financial sectors of top 20 countries in GDP based on symbolized transfer entropy and Pearson correlation coefficients. We examine the effect and usefulness of the network indicators by newly using them instead of the original Dow Jones financial sector as explanatory variables to construct the higher-order information spatial econometric models. The results demonstrate that the estimated accuracies obtained from both the two networks are improved significantly compared with the spatial econometric model using the original data. It indicates that the network indictors are more effective to capture the dynamic information of financial systems. And meanwhile, the accuracy based on the directed network is a little higher than the undirected network, which indicates the symbolized transfer entropy, i.e. the directed and weighted network, is more suitable and effective to reflect relationships in the financial field. In addition, the results also show that under the global financial crisis, the co-movement between financial sectors of a country/region and the global financial sector as well as between financial sectors and real economy sectors is increased. However, some sectors in particular Utilities and Healthcare are impacted slightly. This study tries to use the financial network indicators in modeling to study contagion channels on the real economy and the industry aggregation effects and suggest how network indicators can be practically used in financial fields.


Introduction
The complex network theory has been rapidly applied to various fields such as biology, physics, medicine and especially finance since it greatly promotes the development of economic physics, provides people with deeper understanding on complex financial system [1]. As we know, with the continuous development of the modern financial industry, the increasing participants adaptive segmentation method proposed by Wu et al. (2013) [28], which addresses the high demand for parameters compatibility in it.
At present, most of the scholars use the original data of financial markets to construct econometric models to research the contagion of financial crisis on real economy, which may necessarily be characterized by redundant and ineffective information caused by data noise in financial markets. Especially in the current era, with the rise of the internet financial and big data, financial data rapidly begin to emerge with the blowout trend, which includes not only structured data in digital form in traditional sense but also unstructured data such as texts, images and videos. This furtherly makes the financial markets generate more and more redundant and ineffective data, leading to deviations on model estimation results. Aiming to overcome this problem, we newly propose a method to construct an econometric model with financial network indicators called higher-order information in this study instead of original data, since it can be seen that from previous studies the network indictors can capture the effective information reflecting the internal structured characteristics of financial markets, which is the key factor to eliminate the redundant and ineffective information from original data. Although there have been many academics to research the importance of networks analysis [29], however, there have been no attempts to utilize them to construct models in the global financial market outlook. Recently, there have been only a few studies that use financial indictors to construct investment strategies [30][31][32].
On the other hand, with the increasingly economic globalization and integration, there also have been increasingly strong spatial correlation among regions, countries, markets and industries across the world [33,34]. However, in the assumptions of the traditional econometric model, the variables are independent of each other, without spatial effects of interaction. This is seriously inconsistent with the actual characteristics of economic activity [35]. Therefore, we combine complex network and spatial econometric theory, i.e., we use the network indictor as the explanatory variable to construct the specific spatial econometric models to research the contagion of financial crisis on real economy and spatial spillover and industry aggregation effects, which not only plays the merits of network indictor, also takes the spatial effects into consideration.
Accordingly, taking the European debt crisis as the research background, we expand the model that studies the contagion of financial crisis on real economy proposed by Baur (2012) [36] to a higher-order information spatial econometric model newly proposed using network indictors as explanatory variables extracted from the directed and weighted network constructed using transfer entropy based on the dynamic adaptive segmentation method. And aiming to study from a global perspective, we choose the stock indices of financial sector and nine industry sectors of real economy (including energy, raw material, industry, consumption service, consumer goods, health and medical service, information and technology, telecommunication and public services) from US, Europe and China, respectively. In addition, we also study the financial contagion using the newly proposed spatial econometric model between global financial sector represented by Down Jones financial sector and those of the three economies during the European debt crisis period. And to compare the empirical results with the undirected network, we also construct the undirected and weighted networked based on Pearson correlation coefficients most commonly used to extract network indictors to construct the higher-order information spatial econometric model. Finally we summarized the results of the three types of contagion: the global financial sector contagion on the financial sectors of the three economies, the global financial sector contagion on the nine industry sectors of them, and the domestic financial sector contagion on the nine industry sectors of them, respectively.
Our contribution in this study is mainly two-fold: first, we adopt the symbolic transfer entropy based on the dynamic adaptive segmentation method to construct the directed and weighted complex network. And second we use the indictors extracted from the newly constructed network as the explanatory variables to construct the higher-order information spatial econometric model, which provides a new perspective for the application of the complex network methodology in the financial field.
The paper is organized as follows. Section 2 presents methodologies applied to construct the directed and weighted network and calculate whose various indictors, and the construction of the proposed spatial econometric models based on the higher-order information. Section 3 reports our empirical results. Finally, concluding remarks are stated in Section 4.

Methodology and modeling
Various methods have been proposed in the literature to study the contagion effect of financial crisis on real economy. These include GARCH model [36][37][38], Copula model [39], event study methodology [40], and error correction model [41]. In this paper, we firstly propose the higherorder information spatial econometric model based on the symbolic transfer entropy. This model mainly has three advantages over other estimation methods. First, the model accounts for the spatial effects presented by spatial weight matrix which have been proved to exist commonly in real world [42], financial markets are no exception [43][44][45], and thus it can make the estimation results more accurate. Second, the transfer entropy method fits well with the complexity and nonlinear characteristics of the financial economic system, as well as the new features of trading pattern, such as electronization and informatization. Therefore, it may probably be more appropriate and effective to measure the relationship between nodes than other correlation coefficients. Third, the model accounts for the great amount, redundancy and complexity by using the indictors extracted from the constructed network instead of original data in modeling, which may can more effectively capture the inner information of financial data and thus better reflect the structure feature of financial markets.

The symbolic transfer entropy based on the dynamic adaptive segmentation method
In this work, we use the symbolic transfer entropy calculated from stock indices, it is a directed measurement on information delivery among data as the edges to construct complex network. This method improves the defect on directionality of information transfer; besides, it can be utilized in nonlinear systems. This makes it accord with the practical activity characteristics and nonlinear feature of financial systems [1]. And compared with the linear correlation coefficient, Pearson correlation coefficient, Kendall correlation coefficient and mutual information, transfer entropy eliminates the interference between common historical information such as common external factors. Therefore, what is derived from transfer entropy is the statistical correlation of the source self-sequences. This property makes the method a more practical and adaptable analysis tool when studying the interrelationships between complex systems.
Transfer entropy is a method that obtains the causalities of time series based on the probability distribution, Shannon entropy and statistical method. It was first proposed by German academic Schreiber (2000) [46] based on the works of Shannon and Weaver (1949) [47] and Kolmogorov (1993) [48]. Academically speaking, it is the additional information needed to predict the system status in the situation of wrongly assuming that the transfer probability is pði tþ1 ji ðkÞ t Þ and not pði tþ1 ji ðkÞ t ; j ðhÞ t Þ. It has been successfully applied to biological systems which can been seen in Vicente et al. (2011) [49] and Faes et al. (2013) [50], financial markets system which can been seen in Baek et al. (2005) [51], Shi et al. (2013) [52], Li et al. (2013) [53] and Liang et al. (2014) [54] and other fields such as Lizier et al. [55] (2013), Prokopenko et al. (2013) [56] and Liu et al. (2014) [57].
In the assumption of two discrete time sequences I and J, the transfer entropy that measures the directionality information and dynamic information of systems is defined as where i ðkÞ t and j ðhÞ t denote the states of system I and system J at time t, respectively. Parameters k and h denote the time delay length of systems I and J, respectively, and are commonly set as k = h = 1, meaning that both I and J are first order Markov processes. The joint probability density function (hereafter, PDF) pði tþ1 ; i ðkÞ t ; j ðhÞ t Þ denotes the probability that i tþ1 ; i ðkÞ t and j ðhÞ t are in particular states. The conditional PDF pði tþ1 ji ðkÞ t ; j ðhÞ t Þ and pði tþ1 ji ðkÞ t Þ denote the probability that i t+1 is in a particular state when the states of i ðkÞ t , j ðhÞ t , and i ðkÞ t are uncertain. From Eq (1) we know that the transfer entropy from I to J and that from J to I is asymmetric, and the transfer entropy from I (J) to J(I) measures the influence of system I (J) on system J (I). The larger the transfer entropy is, the greater the influence. Eq (1) can be rewritten as where H(i t+m |i t )represents the conditional information entropy in which the uncertainty of i t+m is under the condition that i t has a particular value. Therefore, the transfer entropy can also be understood as the information of system I at time t+1 included by the system J which is known at time t, and is excluded from the information of system I at time t. From what described above, it can be seen that transfer entropy not only provides a way to measure the directionality information and dynamic information between systems but can also address the relationships between nonlinear systems. In addition, in essence, the transfer entropy is defined based on the information entropy, but it improves the lack of information entropy that which cannot measure the directionality of the information transfer.
However, the transfer entropy has high demands for parameter compatibility which greatly increases the difficulty of calculation and is also sensitive to noise. To address this problem, Staniek (2008) [58] propose symbolic transfer entropy (hereafter, STE) by statically dividing the data into symbolic values, but this inevitably leads to information loss. Therefore, in this study, we adopt the way in the literature Chen et al. (2017) [59] which proposes an improved dynamic adaptive segmentation method. This effectively improve the computational complexity.

Indicators of the directed and weighted network based on the STE
In our work, the top 20 countries in GDP are selected as the network nodes, and the symbolic transfer entropy of aggregate stock indexes between them is taken as the edge, to construct the time-varying directed and weighted network with the sliding window technology.
For a complex network, character path length, weighted clustering coefficient, global efficiency and local efficiency are the four common indictors. The following shows the calculation method of them, in which N is the set of all the nodes in the network, and n is the number of nodes [60].

Character path length:
The minimum value of the sum of weight of connection edges in all the possible paths from node i to node j in the directed weighted network is known as the shortest weighted path d ! ij from i to j, and the reciprocal 1=d ! ij is known as the transfer efficiency from i to j, i.e., x ! ij , which is utilized to measure the information transfer speed. The shortest path length from node i to node j is defined as follows: where g i!j is the directed shortest path from i to j. The character path length of the directed weighted network is defined as the average value of the shortest path length among all the orderly node pairs: 2. Weighted clustering coefficient: Considering of the difference on the weight of the weighted network connection edges, it is needed to take the influences of related correlation degrees into consideration when calculating the clustering degree among nodes. As for the given node i, the weighted clustering coefficient is defined as follows: where t ! i is the number of directed triangles composed by node i: Similar to the definition of the network character path length, the network weighted clustering coefficient WC is the average value of the weighted clustering coefficient of various nodes: 3. Global efficiency: The global efficiency of the directed weighted network is defined as the reciprocal of the shortest path length of all the orderly node pairs, i.e., the average value of the delivery efficiency x ! ij from i to j: where E ! glo;i is the global efficiency of the node i.

Local efficiency:
In the graph theory, the subgraph of node i refers to the graph composed by all the nodes connected to the node directly. The local efficiency of the network is the average value of the global efficiency of the subgraph composed by all the nodes:

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The impact of financial contagion on real economy hj ðN i Þ) are the shortest path length from node j to h and from node h to j in the subgraph constituted by node i.
In modeling, before choosing the specific indictor as the explanatory variable, it is first needed to calculate the correlation coefficients between the Dow Jones financial sector and each of the ten sectors including the financial sector and the 9 real industries sectors, as well as between the four indicators and each sector. Thus, for each country/region, there are totally five kinds of average correlation coefficients, then choose the one corresponding to the biggest average as the explanatory variable. If one of the indictors is chosen, the higher-order information spatial econometric model will be constructed. Otherwise, a common spatial econometric model will be constructed.

Model construction.
According to the GARCH model proposed by Baur (2012) [36], contagion from financial sector to real industry sector is estimated as follows: where R S,i,t is the return of stock index of a specific sector of country i at time of t, R Fin,t is the return of global financial sector portfolio at the time of t. D t is a dummy variable, which is 1 when t is within the crisis period; or it is taken as 0. β 1 is utilized to measure the co-movement of R Fin,t and R S,i,t during the benchmark period, i.e., the spillover effect of R Fin,t to R S,i,t . β 2 measures the changes on the co-movement of R Fin,t and R S,i,t during the crisis period, i.e., the changes of spillover effect of R Fin,t to R S,i,t . The model is also used to analyze contagion across aggregate stock market indices. In this case S represents the sum of all sectors and is substituted by market. Thus the model can be used to test four alternative channels of contagion: aggregate stock market contagion, financial sector contagion and real industry sector contagion spread through the global financial system or the domestic financial system. To differentiate between global and domestic contagion, Eq (10) is augmented as follows: The Eq (11) can be used to estimate changes in the co-movement of a specific sector S (R S,i, t ) with the global financial system (R FIN,W,t ) or with the domestic/regional financial system (R FIN,W,t ). Based on the above two models, we combine the spatial econometrics and complex network, expand the contagion model described by Eqs (10) and (11) to Eqs (12) and (13) respectively, newly construct the higher-order information spatial econometric model shown as follows: where W is the spatial weight matrix, ρ is the spatial correlation coefficient, Pro FIN,W,t is a specific network indictor. And larger absolute value of ρ indicates stronger positive/negative spatial correlation. The representation of the rest parameters are the same as those of Eqs (10) and (11). And correspondingly Eq (12) is utilized to test the contagion effects of the global financial sectors on a specific sector (financial sector or non-financial sector) of country/region i, Eq (13) is utilized to differentiate between global and domestic contagion.
To compare with the common spatial econometric model based on the original data, we also construct the following models, and the difference of them with the models proposed by Baur (2012) [36] is the addition of spatial effects.

Model hypotheses.
Following the testing framework of Baur (2012) [36], the tests and hypotheses are given below.

Test 1: (global financial contagion of financial sector)
Compared with tranquil period, the co-movement between the financial sector of country/ region i and the global financial sector increases during the crisis period. It is assumed that the European debt crisis is triggered by the crisis of the global financial sector.
Test 2: (global financial contagion of real economy) Compared with tranquil period, the co-movement between the real economy sector of country/region i and the global financial sector increases during the crisis period. It is assumed that the European debt crisis is triggered by the crisis of the global financial sector.
Test 3: (domestic financial contagion of real economy) Compared with tranquil period, the co-movement between the real industry sectors of country/region i and the domestic financial sector increases during the crisis period. It is assumed that the domestic financial sector is the source of contagion.
The null hypothesis (H 0 ) and alternative hypothesis (H 1 ) of tests 1-2 are listed as follows: The null hypothesis (H 0 ) and alternative hypothesis (H 1 ) of test 3 are listed as follows:

Data
The The development level of all the sectors is represented by the daily prices of stock indices first-order logarithm difference: where R t is the return of the data of t; P t and P t-1 are the prices on date of t and date t-1, respectively. The summary statistics of the Dow Jones financial sector and the financial sectors of US, Europe and China during the crisis period and the whole sample period are listed in Table 1.
Based on Table 1, the average of the daily return of the four financial sectors during both the crisis period and the sample period are close to zero, with certain differences on standard difference, deviation and kurtosis; especially, there are some differences on JB statistical magnitude.

Contagion channels, higher-order spatial spillover and industry aggregation effects
As described in Section 2.2, before constructing the contagion model to study the financial contagion on the real economy and the global financial contagion, we firstly need to determine to choose either the original data, i.e., the Dow Jones financial sector, or a specific network indictor, i.e., the higher-order information, as the explanatory variable in the above equations. Therefore, we firstly calculate the correlation coefficients between the Dow Jones financial sector and each of the ten sectors, as well as the four indicators and each of the sectors, the final results are shown as in in Tables 2, 3  From Tables 2, 3 and 4, we can see although not all the average values of correlation coefficients between the network indictors and each sector are larger than that between the Dow Jones financial sector and each sector, there are at least two are larger than that from the original data for each country/region, preliminarily indicating that network indictors can better extract information of financial systems. In addition, it can be seen the network indicators with the largest average values of correlation coefficients in US, Europe and China are global efficiency, clustering coefficient and character path length, respectively. Therefore, the three network indicators will be adopted to replace the original data of Dow Jones financial sector in the following modeling by utilizing the improved higher-order information spatial econometric models represented by Eqs (12) and (13).
3.2.1 Global financial contagion and higher-order spatial spillover effects. The following Tables 5 and 6 show the estimation results of the model testing for global financial sector contagion on the financial sectors of US, Europe and China based on the Eqs (14) and (12).
Tables 5 and 6 display the results associated with Hypotheses1, thus testing the existence of contagion from the world financial sector and the financial sectors in each country/region. Firstly, from the results we can see that the two models have the same coefficient of estimation results, indicating that the empirical results are robust. Secondly, all the spatial correlation coefficients o both the two models are significant, which indicates the introduction of spatial  (14), the newly proposed Eq (12) shows certain improvement. Meanwhile, both AIC and SC show certain reduction, illustrating the higher-order information spatial econometric model is more effective. Moreover, we can also see that (1) Though both the ρ values of the two models are significantly larger than zero, China stands out as an example of a country whose spatial correlation is the smallest (around 0.2), the reason for this may lie in the limited degree of financial market openness, leading to relatively small spatial spillover effects. (2) As described in section 2.3, when β 2 is statistically and significantly larger than zero, there is contagion. Therefore, the estimated β 2 of both the two models indicate all the US, Europe and China exhibit an increased co-movement, indicating that all the financial sectors are infected by the global financial sector (i.e., the original hypothesis 1 is rejected). This is consistent with the Baur (2012) [36] and Kenourgios et al. (2014) [37]. Moreover, the Europe shows the highest linked in the crisis period (0.964), which is obtained by adding up the estimated coefficients β 1 and β 2 . This may be because the Europe is the source area of the European sovereign debt crisis. Tables 7-12 display the results associated with the hypothesis 3 testing for the existence of contagion from the domestic financial sector and the real economy sectors in each country/region. The test assumes that the local financial sector is infected by the global financial system firstly, and then spreads it to the domestic real economy sectors. And Tables 7, 9 and 11 represent the estimated results of the common spatial econometric models (Eq 15), whereas Tables 8, 10 and 12 represent the estimated results of the newly proposed higher-order information spatial econometric models (Eq 13), corresponding to US, Europe and China, respectively. And limited by the length of the article, we don't report the estimation results of the same model, i.e., the Eqs (15) and (13) testing for the global financial sector contagion on the real industry sectors of US, Europe and China, in which we assume  [37]. And for all the equations corresponding to a specific country/region, the accuracies of Eq (12) shows significant improvement compared to those of Eq (14). Anyone who need the results, please contact us. From the statistical results, firstly it can be seen the paired (Tables 7, 8, 9, 10, 11 and 12, corresponding to US, Europe and China respectively) of estimation results including R 2 , Loglikelihood, AIC and SC of a specific country/region obtained from the Eq (13) (the higher-order information spatial econometric model) show a significant improvement than those obtained from the Eq (15) (the common spatial econometric model), supporting the finding in Tables 5  and 6 and further indicating that replacing the original data using the network indictors is more applicable and effective. Secondly, consistent with the finding in Tables 5 and 6, all the spatial correlation coefficient ρ are significantly larger than zero, again proving the introduction of the spatial effect is necessary, and also illustrating the universality of it. Therefore, the combination of the network indictor and spatial effect proposed in our work is effective and useful. In addition, as described in section 2.3.2, in case of this type of contagion, we should focus on the coefficient θ 2 which represents the changes of degree of co-movement of the local financial sector and the real economy sectors during the crisis period, and a positive and statistically significant coefficient estimate θ 2 implies the existence of contagion. So the estimation results show the developed country/region exhibit no case of local financial sector contagion on the domestic real economy sectors including both US and Europe, supporting the findings of Baur (2012) [36] and Kenourgios et al. (2014) [37], indicating that the global financial system indeed has a direct impact on the real economy in developed country/region. Whereas, China, an emerging country, there are totally 5 real economy sectors including Basic Materials, Telecom, Industrials, Technology and Energy infected by domestic financial sector. i.e., for China, it's the local financial sector that is infected by the global financial system and spreads it to other domestic real economy sectors.  Table 13 summarizes the three different types of contagion. On one hand, Table 13 shows the fact that compared with the real economy sectors, there is stronger evidence for contagion in the financial sector for US, Europe and China, indicating that the financial sector is the epicenter of the crisis, supporting the findings of Baur (2012) [36] and Kenourgios et al. (2014) [37]. Meanwhile a total of 19 real economy sectors are affected, with a contagion rate of up to 70.37%, and the sectors most affected are Basic Materials (3), Industrials (3) and Energy (3) and the sectors least affected are Utilities (1) and Healthcare (1). Almost the same findings are provided by Baur (2012) [36], which indicates the industry aggregation effect.

The domestic financial contagion, higher-order spatial spillover effects and industry aggregation effects.
And on the other hand, Table 13 shows the sources of the financial contagion on the real economy sectors of US, Europe and China are different. All the affected real economy sectors of both US and Europe are infected by the global financial system, whereas all the affected real economy sectors China, an emerging country, is infected by domestic financial system. i.e., the increased return co-movement of the global financial system and real economy sectors often exist in the developed country/region, whereas the increased return co-movement of the domestic financial system and real economy sectors often exist in emerging country/region.

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The results also indicate that compared with the listed real economy companies in emerging markets, those in developed countries/regions have stronger correlation with the global financial system, i.e., the listed real economy companies in developed markets are more vulnerable to the impact of the global financial sector. It also indicates the limited openness of China and relatively low connection degree with the global financial markets. And the empirical results illustrate the finding of previous study Kumar (2012) [61] which investigated the network property of 20 global financial indices and concluded that indices corresponding to Americas and Europe are combined together to form a cluster while the Asia/Pacific indices forms another cluster during the financial crisis. The conclusion might be connected to the different financial contagion channels concluded by this work: the industrial sectors of USA and Europe are infected by global financial system, so the two regions are combined together to form a cluster while China, belongs to Asia/Pacific area, is infected by domestic financial system not global financial system, therefore forms anther cluster. The results also support the findings of other previous studies such as Yang (2013) [62], Nobi (2014) [63]and Gidea (2017) [64], all of which indicates that the topological structure of industrial sectors network changes obviously during the financial crisis, supporting the results of our work that all the three regions are infected by the financial sector, whatever from the global or domestic financial system.

Conclusions
By combing the information theory, complex network and spatial econometrics, this paper empirically investigates the contagion effects of the European debt crisis from the financial sector to real economy sectors of US, Europe and China using a newly proposed higher-order information spatial econometric model based a time-varying and directed and weighted network. In addition, the symbolic transfer entropy based on the dynamic adaptive segmentation method is used to construct the network and the newly model uses a specific network indictor instead of the original return of the financial sector index as the explanatory variable. The purpose of the study is to examine the effectiveness and usefulness of the financial network based on the return of stock indices of top 20 countries in GDP. We test for contagion across stock markets on the global level and the domestic/regional level.
The results of the study show that the spatial effects widely exist among financial sectors and real economy sectors and the hypothesis of global financial sector contagion on a specific country/region financial sector holds for US, Europe and China, indicating the financial sector is the epicenter of the crisis. While in line with Baur (2012) [36], the global financial sector contagion on real economy sectors is evident only for US and Europe, not for China, whose affected real economy sectors are all infected by domestic financial sector. The finding illustrates that the real economy sectors in developed markets are more vulnerable to global financial system while those in emerging markets are more vulnerable to domestic financial sector. In addition, the affected real economy sectors of US, Europe and China show the aggregation effect, such as the Basic Materials, Industrials and Energy sectors in US, Europe and China are all affected during the crisis period.
Moreover, the effectiveness and usefulness of the network indictors can be confirmed based on the higher accuracies of the Eqs (12) and (13) as shown in Tables 6, 8, 10 and 12 (i.e., the higher-order information spatial econometric models) compared with the accuracies of the Eqs (14) and (15) as shown in Tables 5, 7, 9 and 11 (i.e., the common spatial econometric models) testing for the three different contagion channels. This illustrate that the network indictors may be more appropriate and effective to capture information of the financial market, a nonlinear and complex system.
Our work contributes to the literature by providing a new perspective for the utilization of complex network in the financial field, using the network indictors as the explanatory variables instead of the financial sector stock index data. The study use a novel higher-order spatial econometric model in combination spatial econometrics and complex network. In addition, our findings have important implications for policy makers, investors and international organizations, such as WTO, World Bank and IMF, with taking the linkages among the nonfinancial sectors and the markets into account during the crisis. Especially from the results the investors can learn that holding a portfolio from various sectors is less subject to the global crisis. Based on this model and the reported empirical findings, on one hand, the future study is to be mainly focused on utilizing financial network indicators instead of industrial sectors data to construct other financial contagion and real economy models, such as generally used financial contagion models of copula, VEC and CCA, etc. By analyzing the empirical accuracies obtained from these different models, the robustness of the proposed method could be future verified. On the other hand, the main limitation of this study is that the network indictors are only used as explanatory variables in modeling, they might be used to exhibit the topological structures of financial systems in the future, which can reveal the dynamic evolution of financial networks and then future study the contagion mechanism of financial risks from the perspective of it. In addition, we have only used the symbolic transfer entropy based method to construct the complex network, but we should develop network indictors based on other nonlinear measure method for capturing various market dynamic information, such as mutual information, permutation entropy and relative entropy. We hope these limitations will be addressed in future studies.
Supporting information S1 Dataset. The main daily data of the stock index of the Dow Jones financial sector and the financial sector and 9 major real industry sectors (energy, raw material, industry, consumption service, consumer goods, health and medical service, information and technology, telecommunication and public services) of US, Europe and China from April 15, 2008 through December 31, 2014 were obtained from the Wind database. (ZIP)