Structure and dynamics of stock market in times of crisis

Physics Letters A 380 (2016) 654–666

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Physics Letters A www.elsevier.com/locate/pla

Structure and dynamics of stock market in times of crisis Longfeng Zhao ∗ , Wei Li, Xu Cai Complexity Science Center & Institute of Particle Physics, Hua-Zhong (Central China) Normal University, Wuhan 430079, China

a r t i c l e

i n f o

Article history: Received 4 March 2015 Received in revised form 15 October 2015 Accepted 8 November 2015 Available online 26 November 2015 Communicated by F. Porcelli Keywords: Correlation-based network Financial crisis Market structure Market dynamics Power-law distribution Market backbone

a b s t r a c t Daily correlations among 322 S&P 500 constituent stocks are investigated by means of correlation-based (CB) network. By using the heterogeneous time scales, we identify global expansion and local clustering market behaviors during crises, which are mainly caused by community splits and inter-sector edge number decreases. The CB networks display distinctive community and sector structures. Graph edit distance is applied to capturing the dynamics of CB networks in which drastic structure reconfigurations can be observed during crisis periods. Edge statistics reveal the power-law nature of edges’ duration time distribution. Despite the networks’ strong structural changes during crises, we still find some longduration edges that serve as the backbone of the stock market. Finally the dynamical change of network structure has shown its capability in predicting the implied volatility index (VIX). © 2015 Elsevier B.V. All rights reserved.

1. Introduction Recent subprime mortgage crisis has attracted very much attention on studying the relationship between financial market structure and economic crises [1–10]. Apparently new studies are needed towards such a purpose [11]. Benefiting from the works on correlation-based (CB) network of financial asset return time series [12–16], we can apply the methodology of network theory [17] to analyzing the financial market. As a powerful tool to understand the properties of stock return time series [12], the CB network method facilitates the stock market research by techniques of complex network analysis [18]. Some useful methods to construct CB networks are minimum spanning tree (MST) [12,19–21], threshold cutting procedure (the winners-take-all strategy) [22–25], planar maximally filtered graph (PMFG) [14], dependency network [26,27], bootstrap method [28], etc. The researches on extreme events and economic crises have been very intriguing. Black Monday crash has been analyzed by means of asset tree with MST method. The shrink in asset tree length and degree distribution difference are observed after the crash [20]. The asset tree research on financial indices also reveals the topological structure changing from star-like to chain-like during crisis [5]. The principal components analysis indicates an increase in the strength of the relationship between several different

*

Corresponding author. E-mail address: [email protected] (L. Zhao).

http://dx.doi.org/10.1016/j.physleta.2015.11.015 0375-9601/© 2015 Elsevier B.V. All rights reserved.

markets during subprime mortgage crisis [2] and is also employed as a measure of systemic risk [29,30]. By analyzing the eigenvalues and eigenvectors of correlation matrices, direct links between high volatility and strong correlations can be recognized, which means the market tends to behave like the one during big crashes [3]. Regression techniques have been used to identify the strong clustering behavior distributed by geographic differences [31] for stocks from different countries and sectors. A newly introduced measure named sector dominance ratio is capable of capturing the economic sectoral activities [32]. The main information that can be inferred from the previous works mentioned above is that the clustering and correlation strengthen behaviors that can be observed during crises. However, detailed information about structure and dynamics of stock market in times of crisis is still lacking. Here in this article, we aim at exploring the detailed structure and dynamics of stock market during crisis periods from the aspect of CB networks with planar maximally filtered graph (PMFG) method [14]. We investigate the correlations among 322 S&P 500 constituent stocks by using the stocks’ daily adjusted closure price return time series between January 1994 and January 2014. In order to filter the influences of economic crises, we first use the entire historical data records (heterogeneous time scales) to calculate the correlation coefficients between pairs of stocks. The PMFG method is then adopted to construct CB networks, which yields very clear modular structures. We argue that the heterogeneous time scales can characterize the slow dynamics of the market [33] by reducing the fluctuations more efficiently as compared to the traditional truncated method. We hereby identify two of the most serious

L. Zhao et al. / Physics Letters A 380 (2016) 654–666

crisis periods of US stock market during recent twenty years, dotcom bubble and subprime mortgage crisis. We observe very similar trends of the market during these two periods. Namely, the clustering coefficients and the shortest-path lengths are always positively correlated under almost all time scales ranging from 1 month to 100 months, as well as under the heterogeneous time scales. The key result obtained from the PMFG topological analysis is that during the crisis periods both shortest-path length and clustering coefficient increase, a clear signal of global expansion and local clustering behaviors. This finding can be well explained by network community detection [34] and sector relationship networks. The community splits and decline of the overlap between communities have been unfolded during the crisis periods. It has been noticed that the inter-sector edge decrease and the intra-sector edge increase happen simultaneously. The statistical quantities such as modularity of PMFGs and the edges of the sector relationship networks have been used to evaluate the structural evolution. We then use graph edit distance [35] and edge statistical analysis to investigate the dynamics of PMFGs. There exist abrupt increases in edit distance at the onset time points of crises. We show that the edges’ duration time follows a power-law distribution. A set of edges with very long duration time have been observed which serve as the backbones of the stock market. As an illustration of applications of our results, the successive edit distance has proved to Granger cause the implied volatility index (VIX). The successive edit distance is capable of helping predict the implied volatility. We further use the cross correlation analysis to study the correlation between successive edit distance and VIX. It turns out that VIX is similar to successive edit distance in the previous month. The paper is organized as follows. In Sec. 2, we describe the data, the methodology and the selection of proper time scales by using different quantities in both short and long time scales. In Sec. 3, we analyze the evolution of topology parameters of CB networks under the heterogeneous time scales. We also show the topological structural evolution filtered by PMFG method. In Sec. 4, we discuss the community and sector structural evolution. In Sec. 5 we analyze the dynamics of PMFGs by means of edit distance, as well as the statistical properties of edges. In Sec. 6 an application of our results is demonstrated. The last section is discussion and summary. 2. Data, correlation-based networks, and time scales 2.1. Data Our data sets include 322 stocks (see Appendix A). They are the constituent stocks of S&P 500 between 1st January 1994 and 1st January 2014. We adopt the logarithm return defined as

r i (t ) = ln p i (t + 1) − ln p i (t ),

(1)

where p i (t ) is the adjusted closure price of stock i at time t. We then compute the mutual correlation coefficients between any pair of return time series at time t by using the past return records sampled with different length  ranging from 1 month to 100 months (amounting to nearly 2500 trading days). Following Ref. [20] one can evaluate the similarity between stocks i and j at time t with the Pearson correlation coefficient by,

 R ti R tj  −  R ti  R tj   , 2 2  R ti  −  R ti 2  R tj  −  R tj 2

ρitj, = 

(2)

where  is the estimation interval, and . . . is the sample mean over co-trading days of stocks i and j in the logarithm return series vector R ti = {r i (t )} and R tj = {r j (t )}. Then we obtain the N × N

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matrix C t , at time t with estimation interval , and N is the number of stocks. The matrix entries of C t , are the correlation t , coefficients ρi j between all pairs of stocks. 2.2. Correlation-based networks and time scales The planar maximally filtered graph (PMFG) method [14] is employed to construct networks based on the correlation matrices C t , . The algorithm is implemented as follows, t , (i) Sort all of the ρi j at time t with time scale  in descending order to obtain an ordered list lsort . (ii) Add an edge between nodes i and j based on the order in lsort only if the graph is still planar after edge addition. single

t ,

= 3( N − 2) (iii) A graph, G P ( V , E ), is formed with NUMe edges under the constrain of planarity. As described in Ref. [14] PMFGs not only keep the hierarchical organization of minimum spanning tree (MST) but also generate some cliques. We calculate the basic topological parameters such as clustering coefficient C , shortest-path length L, as well as average correlation coefficient ρ . In addition, we adopt a quantity γ [36] to measure the heterogeneity of PMFGs, whose definition is given by, N −2

γ=

 i j ∈e

(ki k j )−1/2

√

N −2 N −1

,

(3)

where ki is the degree of vertex i. The magnitude of  is crucial for analyzing the dynamics of financial market [33,37,38]. Short estimation interval is suitable for analysis of fast dynamics but with relatively large statistical uncertainty [33]. This means the influence of financial crises on structure of CB networks could be indistinguishable under improper estimation intervals. In Figs. 1 and 2 we present the influences of different estimation intervals varying from 1 month to 100 months via the evolution of four quantities ρ , C , L and γ . Here we only show four quantities during the time window from January 1999 to January 2014. If the time window is enlarged, the estimation interval cannot be extended to 100 months. We notice that when the estimation interval is shorter than 10 months, the fast dynamics can be observed. With the increase of , fluctuations of quantities decrease and the influences of major economic crises can be recognized. The influences of small shocks will be smeared out under large estimation intervals. We use different slide window sizes in Figs. 1 and 2, which is 1 week and 1 month respectively. This setting is a tradeoff between accuracy and computation load due to the algorithm complexity of planarity test [39]. When the estimation interval is large, slow dynamics will dominate the evolution of those quantities. So larger slide window size is more proper for capturing the evolution of the market under larger time scales. In Figs. 1 and 2, the average correlation ρ behaves very differently during dot-com bubble and subprime mortgage crisis. No abrupt increase of average correlation can be observed during dot-com bubble even under very short estimation intervals. This might be caused by the bankruptcy of many internet companies during dot-com bubble. However, ρ increases very fast at Lehman’s failure in September 2008. The clustering coefficient C and shortest-path length L show clear signals about major crises under estimation interval larger than 60 months in Fig. 2. Figs. 3(a)–(d) are the topological structures of PMFGs under different estimation intervals at 01\01\2014. The Fruchterman– Reingold layout [40] has been used to demonstrate the structures of networks. Stocks from different sectors do not show clear structures under estimation interval from 10 months to 100 months.

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Fig. 1. (Color online.) The basic parameters calculated by using estimation interval  varying from 1 month to 60 months. ρ is the average correlation coefficient, namely, mean value of the upper triangle entries of correlation matrix C t , . C is the clustering coefficient. L is the shortest-path length and γ is the heterogeneity of network as defined in Eq. (3).

Fig. 2. (Color online.) The basic parameters calculated by using estimation interval  varying from 61 months to 100 months.

We cannot identify distinctive sector structures under truncated estimation intervals. According to our time scale analysis above, the evolution tendencies of C and L are similar under different estimation intervals. As shown in Figs. 1 and 2, we have two time series of C and L under each time scale . We calculate the correlations of these two quantities under different estimation intervals. Figs. 4(a) and (b) show that C and L are positively correlated under estimation interval varying from 1 month to 100 months. The correlation strength ρC , L is always very strong (bigger than 0.8) even under very short estimation intervals less than 10 months. This interesting finding may reveal some unique properties of the stock market. We summarize this section here. In this section, we introduced our data and the methodology and then analyzed the effects of es-

timation intervals. Fast dynamics and slow dynamics under small and large estimation intervals can be observed. Although the behaviors of CB networks vary under different estimation intervals, the strong positive correlation between C and L is always discernable. However, the fluctuations of those parameters, especially C and L, are very prominent. We notice the disorganized topological structure under truncated estimation intervals, which may not lead to a unified conclusion about the structural evolution of PMFGs. One question still remains: If we intend to analyze the influences of major crises on the stock market, how shall we choose the estimation interval? In fact, the truncated method adopted in this section will always lose some information about the market and therefore might not be appropriate for analyzing the effects of major crises. Why can’t we use the entire historical records to eval-

L. Zhao et al. / Physics Letters A 380 (2016) 654–666

t ,

Fig. 3. The topological structures of G P with t equaling 2014\01\01 and  equaling (a) 10 months, (b) 40 months, (c) 70 months and (d) 100 months, respectively.

Fig. 4. The correlations, ρC , L , between clustering coefficient C and shortest-path length L (a) under estimation interval  varying from 1 month to 60 months and (b) under estimation interval varying from 61 months to 100 months, respectively. As shown, C and L are positively correlated.

uate the correlations among stocks? We will answer this question in the next section. 3. Topological evolution of correlation-based networks In the previous section the effect of estimation interval has been checked. One conclusion is that larger estimation interval yields lower statistical uncertainty. Longer estimation interval can distinguish the major market crashes from random events. Even so the statistical fluctuations always exist at any time scale, which will be an obstacle for CB network study. No unified conclusion can be drawn from the time scale analysis in Figs. 1 and 2. Hence we switch to the heterogeneous estimation intervals, i.e., the entire historical data from first listing day of a stock until the evaluation time point t. The entire historical co-trading records are taken to compute the correlation coefficient ρi j between stocks i and j. The

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Fig. 5. (Color online.) The time-evolution of basic topological quantities of PMFGs computed by using the entire historical data records, i.e., the heterogeneous time scales. C is the clustering coefficient, L is the shortest-path length and γ is the heterogeneity of networks. The area between the blue lines represents the dot-com bubble from 1999 to 2002, and the area between the red lines, the subprime mortgage crisis from 2007 to 2009.

final time scales used to compute correlation coefficients between different pairs of stocks are not equal. This setting can utilize the complete information of the time series, as the entire historical data does assist in identifying the two major crises clearly. Based on the entire historical data the correlation coefficients are calculated, from which the correlation matrix Ct is composed. Then 241 PMFGs {G tP ( V , E )|t = 01\01\1994, . . . , 01\01\2014} are generated at the beginning of each month during the full time period between 1st January 1994 and 1st January 2014. Fig. 5 displays the evolution of C , L and γ during the whole time period. It should be noted that the fluctuations of the three quantities are smaller compared to the observations made in Sec. 2. The area between the two blue lines and the one between the two red lines are two most prominent crisis periods, the dotcom bubble and the subprime mortgage crisis, respectively. Both C and L increase during the two crisis periods, displaying rather similar trend. Such a trend implies local clustering and global expansion behaviors during the two crisis periods. In the meantime, the heterogeneity, γ , of the PMFGs decreases, which means the crises tend to make the market more homogeneous. We also find that C and L are positively correlated, with correlation coefficient ρC ,L being 0.95. The positive correlation between C and L is consistent with what has been found in the previous time scale analysis. We have seen clearly the signals of major crashes in Fig. 5. However, we still have to be cautious to say that the entire historical data records are better for the analysis of CB networks. Can this setting make the PMFG method filter the structure of the stock market clearly? We then display the PMFGs before and after subprime mortgage crisis to show whether the PMFG method under heterogeneous time scales can capture the structure of the stock market or not. We show the topological structures of the market before and after subprime mortgage crisis in Figs. 6 and 7, respectively. We mark the acronyms (see Appendix A) of those companies whose degrees are larger than 10, where different colors represent different sectors. We notice that PMFGs distinguish the sectors very well. This is quite different from the disorganized structure in

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Fig. 6. (Color online.) The network structure of the market before the subprime mortgage crisis. The sector names (colors) are Consumer Discretionary (red), Consumer Staples (orange), Energy (yellow), Financial (green), Health Care (light blue), Industrials (purple), Information Technology (pink), Materials (brown), Telecommunications Services (gray), Utilities (deep sky blue). The node size represents the magnitude of the degree. We notice that the PMFG distinguishes the sectors quite well.

Fig. 8. (Color online.) The community structures (a) before dot-com bubble and (b) after, dot-com bubble. The red edges are inter-community edges and the intracommunity edges are marked by black lines. The rainbow color bars represent different communities.

At the same instant, sectors such as Utilities (deep sky blue) and Information Technology (pink) are quite stable. In fact, those big companies serve as bridges between different sectors, such as GE (General Electric Company) and PPG (PPG Industries, Inc.) which also lose their links. By checking the variances of degree sequences of PMFGs before and after the crisis, one can find that it decreases from 30.8 to 25.7 during the subprime mortgage crisis and it also decreases from 60.39 to 32.8 during the dot-com bubble. (One should keep in mind that the number of total edges of a PMFG is always 960.) Those bridge companies’ huge degree drop weakens the global connections, which increases the shortest-path length L during the two crises. This also leads to the increase of both inner-sector edges and the clustering coefficient C . All the structural changes make the networks more homogeneous, as indicated by the decrease of the heterogeneity, γ , during the crisis periods. 4. Community structure and sector structure

Fig. 7. (Color online.) The network structure of the market after the subprime mortgage crisis. The notations of the colors are the same as in Fig. 6.

Fig. 3. We have carefully checked all 241 networks generated by heterogeneous time scales and the distinctive sector structures are consistent. The green nodes represent the stocks that belong to the financial sector. Before the crisis the company with the largest degree in the financial sector is AIG (America International Group), with degree being 32. After crisis AIG is no longer in the ranking list with degree greater than 10. Before the subprime mortgage crisis there are 12 companies with degree greater than 10 in the financial sector: AIG, CB, LNC, JPM, BAC, AXP, SCHW, PNC, BK, STI, CMA, KEY. After the crisis this number is 11: MTB, CMA, STI, BAC, LNC, TMK, JPM, AXP, NTRS, TROW, BEN. Only 5 companies, JPM, LNC, BAC, STI, CMA, are still in the top ranking list. The survival ratio of top ranked companies of the financial sector is merely 5/12.

As seen, the topological structures of CB networks display modular properties which imply the existence of very clear community structure. The modular structure is highly overlapped with sector structure. So the evolution behavior of topological parameters during crisis periods should be directly related to the structural change at community and sector levels. We now analyze the community and sector structures in this section. 4.1. Community structure We adopt the info map method [34] to detect the community structure of CB networks. This method is regarded as the best community detection algorithm so far according to a comparative analysis [41]. Figs. 8(a) and (b) show the community structures before and after dot-com bubble. The number of sub-communities is 20 before the dot-com bubble, and 21 after the crash. Although the community number only increases slightly, the number of intercommunity edges decreases from 244 to 177. Similar phenomenon has been observed and displayed in Figs. 9(a) and (b) for the subprime crisis period, where the community number is 20 and 22, respectively. To precisely quantify the community structure of networks, the quantity modularity [42,43] could be used.

L. Zhao et al. / Physics Letters A 380 (2016) 654–666

Fig. 9. (Color online.) The community structures (a) before and (b) after, subprime mortgage crisis. The color notations are the same as in Fig. 8.

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Fig. 11. (Color online.) Sector relationship networks (a) before dot-com bubble and (b) after dot-com bubble. The numbers on the edges are the connections between two sectors in PMFGs. The vertex labels represent acronyms of the sectors and the intra-sector edge numbers in PMFGs. The color symbols are the same as in Figs. 6 and 7.

4.2. Sector structure

Fig. 10. (Color online.) The modularity of CB networks as a function of time t. The regimes between the two pairs of blue and red lines correspond to the two biggest crises in the US stock market in our time window.

The modularity Q is defined as

Q =

1  2m

vw

 av w −

kv kw 2m

 δ(c v , c w ),

(4)

where a v w is the entry of adjacency matrix of the network. k v is the degree of node v, m is the number of total edges, and c v gives the community membership of node v. In practice a value of Q above 0.3 indicates significant community structure. Fig. 10 illustrates the evolution of modularity over time. The value of Q is remarkably larger than 0.3, which indicates a distinctive community structure. The modularity also increases dramatically during the two crisis periods. The increase of modularity Q shows us a quantitative way to understand the splits of community structure of PMFGs. Larger modularity means that the edges are not randomly distributed and in this sense the PMFG is highly modularized.

It is well known that sectors play very important roles in the stock market structure. For instance, the business sector has been identified as a indicator of crashes [44]. Here we introduce a new method, sector relationship networks, to study the dynamics of sectors. The vertices in CB networks are partitioned based on sectors, which leads to a coarse-grained sector relationship network with 10 (i.e., the number of real business sectors) vertices. The partition scheme is based on the principle of S&P 500. The sector relationship networks are displayed in Figs. 11 and 12 for periods before and after two crises, respectively. The acronyms of sectors are Consumer Discretionary (CD), Consumer Staples (CS), Energy (E), Financial (F), Health Care (HC), Industrials (I), Information Technology (IT), Materials (M), Telecommunications Services (TS), and Utilities (U). In Fig. 11 the sector edge numbers between different sectors are displayed as edge labels of sector relationship networks. Before the dot-com bubble there are totally 466 inter-sector edges in PMFG, but after the crisis this number drops to 328. Another phenomenon is the increase of intra-sector edge numbers for all the sectors. For example, the number of the inter-sector edges related to financial sector decreases from 161 to 103, but the number of its intra-sector edges increases from 90 to 109. The most dramatic change happening to the financial sector clearly indicates its importance among all the sectors [45]. Similar changes have been observed in other sectors as well. One very intriguing result is that the number of edges of sector relationship network decreases from 36 to 30, which means some pre-existed edges do not survive the crisis. For example, there is no edge between sectors energy (E) and consumer staples (CS) after dot-com bubble. Figs. 12(a) and (b) show the inter-sector edge number change during the subprime mortgage crisis. The overall inter-sector edge number decreases from 316 to 287, in which the financial sector contributes the most. Meanwhile the intra-sector edge numbers for most sectors increase after the subprime mortgage crisis. While the edge number of sector relationship network decreases from 31 to 26, which is consistent with the decrease of connections between different sectors.

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Fig. 12. (Color online.) Sector relationship networks (a) before subprime mortgage crisis and (b) after subprime mortgage crisis. The notations of vertices and edges are the same as in Fig. 11.

Fig. 14. (Color online.) The edit distances between all the PMFGs. The coordinates represent the PMFGs at different months. D t ,t  is the edit distance between G tP and 

G tP . The black lines are the contour lines of the edit distance values.

community and sector levels. It also clearly depicts the topological transition of the stock market during the two crisis periods. 5. PMFG dynamics Here in this section we investigate the overall relative structural change between PMFGs during both normal periods and crisis periods. To be more specific, we characterize the dynamics of all PMFGs by means of edit distance [46] and edge statistical analysis. 5.1. Edit distance The edit distance has been widely used to quantify the dissimilarity between strings in natural language process and gene sequence analysis [47]. The basic idea of graph edit distance is to define the dissimilarity of graphs by the amount of distortion that is needed to transform one graph into another [48]. Due to the high complexity of traditional edit distance algorithm for arbitrary graphs, we instead take an alternative approximation algorithm proposed in Ref. [35], that is, Fig. 13. (a) shows the inter-sector edges of PMFG at each month; (b) is the edge number of coarse-grained sector relationship network at each month.

In order to have a more quantitative knowledge of the sector structure evolution, we tracked the numbers of inter-sector edges of PMFGs and sector relationship networks as functions of time, as shown in Figs. 13(a) and (b), respectively. The local clustering and the global expansion of the stock market during the crisis periods can be perceived from the findings. Apparently the crisis weakens the correlations between sectors which can be revealed by the evolution of sector relationship networks. The above community detection and sector relationship network analysis display very similar evolution tendency. The modularity Q , inter-sector edges and edges of sector relationship network show the quantitative ways to measure the structural evolution of the stock market. We also calculate the correlation coefficient between the inter-sector edge number and the edge number of sector relationship networks, which is 0.95. And the correlation between modularity and inter-sector edge number is −0.88. These two correlations illustrate the structural coherence on both

Dij = 1 −

NUM( E i ∩ E j ) NUM( E mv ∪ E mv ) i j

,

(5)

where D i j is the edit distance between graph i and j. NUM( E i ∩ E j ) is the number of mutual edges between graphs i and j, and NUM( E mv ∪ E mv ) is the total number of unique edges connecti j ing only mutual vertices (mv) in both graphs. This algorithm is straightforward and easy to implement, with low computational complexity. In this paper, the vertices for all PMFGs are the same, so the mutual vertices are always the same with N = 322. It means only edge addition and deletion operations are needed to make two PMFGs the same. The edit distance quantifies how far two PMFGs are away from each other. Fig. 14 shows the edit distances among all the 241 PMFGs, where black curves are contour lines. During the two crisis periods the edit distances increase dramatically in very short time, which indicates strong structure reconfigurations. The shrinks of the contour lines around 2000 and 2008 demonstrate the PMFGs’ fast structural evolution during the two crisis periods. The edit distances exceed 0.5 near the two onset time points especially due to

L. Zhao et al. / Physics Letters A 380 (2016) 654–666

Fig. 15. The edit distance between successive PMFGs. D t −1,t is the edit distance between G tP−1 and G tP .

Lehman’s failure in September 2008. On the contrary, the edit distances are quite stable in times other than the two crisis periods, which imply very stable market structures at normal times. Fig. 15 shows the edit distances between successive PMFGs, namely, D t −1,t between G tP−1 and G tP . Hence we obtain an edit distances vector { D t −1,t |t = 01\02\1994, . . . , 01\01\2014} with 240 elements. The edit distances between successive PMFGs are relatively larger during the two crisis periods. However, between the two crises periods the edit distances are smaller than 0.1. The peak at September 2008 is larger than 0.35. And consequently 336 edges are rewired in the first month after the failure of Lehman Brothers. The evolution speed at the crisis and normal periods have significantly different impacts on the market. Therefore the stasis of the market most of the time is punctuated, however, by the bursts due to big crises. We then focus on the different dynamics of PMFGs at the two crises and normal periods. Fig. 16 provides two different evolution patterns of PMFGs. The red line represents edit distances { D t ,t  |t = 01\03\2000, t  = 01\01\1994, . . . , 01\01\2014} and t = 01\03\2000 is the onset time point of dot-com bubble. The blue line gives the edit distances { D t ,t  |t = 01\09\2008, t  = 01\01\1994, . . . , 01\01\2014} and t = 01\09\2008 is the onset time of subprime mortgage crisis. The green line represents the evolution pattern of edit distance at normal period of t = 01\01\2005. Obviously two distinctive evolution patterns can be recognized from Fig. 16. The red and blue lines indicate that the market escapes from its previous state very quickly after the onset of the crisis, and the green line shows its stabilized peculiarity during normal periods. Comparing Fig. 15 with Fig. 14, one can notice that the edit distances between successive PMFGs are not additive. (Additivity of edit distance means all newly introduced edges at time t are totally different from the counterparts at t − 1, so D t −2,t between G tP−2 and G tP is simply the sum of D t −2,t −1 and D t −1,t .) Moreover, two stable regimes in Fig. 14 indicate that the new edges created by the PMFG method do not increase very fast after the crisis single

periods. The edge number of a single PMFG is NUMe = 960 under the constrain of the planarity. The cumulated number of all the unique edges in those 241 PMFGs is NUMetotal = 6606, 6.88 single

times of NUMe . Suppose the successive edit distances are additive, the summation of all the elements of vector { D t −1,t |t = 01\01\1994, . . . , 01\01\2014} is 32.94, much greater than 6.88.

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Fig. 16. (Color online.) The edit distances between PMFG G tP at three dif

ferent time points and other PMFGs G tP . The red line is the edit distances { D t ,t  |t = 01\03\2000, t  = 01\01\1994, . . . , 01\01\2014} between {G tP |t = 01\03\2000} and other PMFGs. The blue curve corresponds to the edit distances { D t ,t  |t = 01\09\2008, t  = 01\01\1994, . . . , 01\01\2014} for {G tP |t = 01\09\2008}. The green line represents the edit distances { D t ,t  |t = 01\01\2005, t  = 01\01\1994, . . . , 01\01\2014}.

Fig. 17. (Color online.) Edge activity pattern of all unique edges. The active edges and the inactive ones are colored by white and red, respectively. The number of total unique edges is 6606.

This means the overall differences of PMFGs are not so big after the long-term evolution. This gives a hint of the existence of intrinsic stable structure, which is exactly our topic for the next subsection. 5.2. Edge statistics It is shown in Fig. 15 that the successive PMFGs are different from each other, which implies that the edges are not static. Edge e v w between stocks v and w may exist for some time and then vanish due to fluctuations or exogenous events. In this section we analyze the statistical properties of edges. Specifically, we study the activity pattern and duration time distribution of all the 6606 edges. Fig. 17 shows the edges’ activity pattern of all the unique 6606 edges. The white vertical lines specify the focal edges that are ac-

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Fig. 18. Duration time distribution of all the unique edges. The method proposed by Ref. [49] is used to fit the distribution, which yields a power-law with exponent α = 1.67.

tive during a period of time, while the red ones show that the edges are non-existent at the time of interest. Long duration times of edges might be interrupted by the crises especially around the subprime mortgage crisis. We notice that the activity pattern reveals some heterogeneity. Then we calculate the duration times td for all the unique edges, whose distribution is given in Fig. 18. In Fig. 18 we show that the duration time distribution of all unique edges. The duration time obeys a power-law distribution f (td ) ∼ td−α according to our fitting. The fitting method is the maximum likelihood estimator (MLE) – proposed by Clauset et al. [49]. Assume we have a series of empirical data x. The power law index estimated by MLE is given by



α 1+n

n  i =1

ln

−1

xi xmin −

1 2

,

(6)

where {xi , i = 1, 2, . . . n} are observed values of x such that xi ≥ xmin . xmin is determined by minimizing the distance between the power law function and the empirical data x, which is called Kolmogorov–Smirnov (KS) test

D = max | S (x) − P (x)| . x≥xmin

(7)

Here S (x) is the cumulative distribution function (CDF) for the observations of data with minimum value xmin , and P (x) is the CDF for the power-law model that best fits the data in the region xi ≥ xmin . The estimated xmin is the value of xmin that minimizes D. Our fitting yields a power-law with exponent α = 1.67 and p-value 0.89. This means not all the edges persist for a very long time. We also notice that there exists a point at the tail of the distribution in Fig. 18 with td = 241. The frequency value corresponding to that point is 107, meaning 107 very active edges. This can be verified in Fig. 17 that some edges are always active, specified by the longest white vertical lines. Their duration times span the whole observation period, namely, 241 months. Those edges are very stable even after experiencing so many small shocks and big crashes in twenty years. Fig. 19 shows the stocks connected by those edges of long duration time, as represented by the points at the tail of Fig. 18. We call them strongly connected clusters, which make up the most stable structure of the stock market. One interesting finding is that stocks

Fig. 19. (Color online.) Stocks connected by the edges with td = 241 months. Different colors represent different sectors, the same as described in Fig. 6. The stock acronyms of those connected clusters are marked. Those isolated stocks are shown as small points.

picked up in this way are nearly in the same sectors (specified by the same color). For instance, Energy (yellow), Information Technology (pink), Health Care (light blue) and Telecommunications Services (gray) clusters are not connected with stocks of any other sector. On the contrary, Industrials (purple), Materials (brown) and Financial (green) clusters are connected with stocks from other sectors. The stock of GE from industrial sector is connected with stocks from Consumer Discretionary (red) and Information Technology (pink) at the same time. Some stocks from Materials sector (brown) are connected with financial (green) and industrials (purple) sector stocks simultaneously. The strongly connected clusters shown in Fig. 19 again manifest the importance of sectors for the stock market structure research. From the previous edit distance and edge statistical analysis results, the overall dynamical evolution of market has been demonstrated. Although we only discuss the market dynamics during the two big crises, the effects of other shocks such as the Greek government-debt crisis in 2010 and the European debt crisis in September 2011 can still be recognized by checking the edit distance. The market reactions uncovered by the dynamical evolution of PMFGs are consistent. The sensitivity and stability of the market can be well described by the edit distance and edge statistics. Thus we conclude that the stock market is a combination of both vulnerability and stability. 6. Application Here in this section we discuss a possible application of our main results. In section 5, we have analyzed the dynamics of the market by the edit distance approach. The detailed structural evolution can be captured by the successive edit distance as shown in Fig. 15. We have also seen drastic market volatility stimulated by different events, which can be exhibited by the structure reconfigurations of PMFGs. Higher implied volatility index (VIX) implies higher volatility of returns in the near future. It is very natural for us to explore the connection between these two variables. So we

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Fig. 21. The cross correlation between VIX and successive edit distance D t −1,t . Fig. 20. Comparisons between successive edit distance D t −1,t and VIX on both monthly and yearly based time horizon. Table 1 This table presents the results of Granger causality test between successive edit distance D t −1,t and VIX. The significance level is 0.01 for all Granger causality tests. If F-statistic is lager than the critical value, the null hypothesis is rejected and the alternative hypothesis is accepted. Granger causality test

F-statistic

Critical value

D t −1,t Granger VIX (monthly) VIX does not Granger D t −1,t (monthly) D t −1,t does not Granger VIX (yearly) VIX does not Granger D t −1,t (yearly)

4.6148 0.7008 0.4691 1.1285

3.782 3.782 5.092 8.746

the market dynamics characterized by the successive edit distance D t −1,t may help predict the implied VIX to some extent. The market structure filtered by the PMFG algorithm is far more sensitive than what we have known before. We also perform a cross-correlation analysis between monthly successive edit distance D t −1,t and VIX. The cross correlation can quantify the level to which two series are correlated at different time lags. The correlation between series X and Y is defined as [32] N −h

[( X (i ) −  X ) · (Y (i + h) − Y )] XCF (h) =   N N   2 ( X (i ) −  X ) · (Y (i ) − Y )2 i =1

compare the successive edit distance D t −1,t to the VIX on monthly and yearly bases (see Fig. 20). Fig. 20 shows the evolution patterns of VIX and successive edit distance D t −1,t . They are very similar to each other on both monthly and yearly time scales. In order to explore the relationship between VIX and D t −1,t , we employ Granger Causality Analysis (GCA) [50–52] to probe the extent to which successive edit distance D t −1,t predicts VIX, or vice versa. Assume we have two stationary time series X and Y . One says that variable X Granger causes Y if the predictions of Y based on both its own past values and the past values of X are better than the counterparts based only on its own past values. We give our null hypothesis that X does not Granger cause Y and the alternative hypothesis is that X Granger causes Y . We perform two vector auto-regressions as follows

Y (t ) =

H 

ah Y (t − h) + 1 ,

(8)

h =1

i =1

i =1

h = 0, 1 , 2 , . . . , N − 1 N  i =1−h

XCF (h) = 

N 

[( X (i ) −  X ) · (Y (i + h) − Y )]

( X (i ) −  X )2

i =1

h = −1, −2, . . . , −( N − 1)

 ·

N 

(Y (i ) − Y )2

i =1

(10)

where the lag h = 0, ±1, ±2, . . . , ±6. The cross correlation result is shown in Fig. 21. Larger value for positive h means that the VIX is more similar to D t −1,t in the previous month. One can see that the cross correlation with positive lag is lager than its value with negative lag. The maximum value of cross correlation is XCF (h = 1), which is consistent with the Granger causality test results. 7. Discussion and summary

and

Y (t ) =

H  h =1



ah Y (t − h) +

H 



bh X (t − h) + 2 ,

(9)

h =1

where H is the maximal time lag. We can conclude that X Granger causes Y if Eq. (9) is statistically significantly better than Eq. (8). The results of GCA are presented in Table 1. From Table 1 one finds that the monthly successive edit distance D t −1,t Granger causes the monthly VIX index but not vice versa. The yearly successive edit distance and the yearly VIX do not Granger cause each other. These findings provide a hint that

In this paper, we have analyzed the structure and the dynamics of correlation-based networks of 322 constituent stocks of S&P 500 in times of crisis, by means of the PMFG method. We have performed systematic time scale analysis to determine a proper estimation interval in order to capture the effects of major crises. The truncated time scales display large fluctuations and disorganized structures and are obviously not suitable for dealing with major crises. Alternatively, we conduct our research by using the heterogeneous time scales. This setting can essentially capture the distinctive structure of the stock market. The evolution of topological quantities such as the clustering coefficient C and the shortest-

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path length L unveils the global expansion and local clustering behaviors of the stock market with small fluctuations during dotcom bubble and subprime mortgage crises. The unified evolution patterns can be well explained by the results of community structure and sector relationship networks. The evolution of community structure has been studied via info map community detection algorithm, which can be demonstrated by the quantity named modularity Q . High modularity of PMFGs shows the modular structures of the stock market. It is also noted that Q of PMFGs increases rapidly during the two crisis periods. We have also introduced a method called sector relationship networks to characterize the evolving of sector structures. The structure of sector relationship networks can exhibit the macrostructures of the stock market. It is found that edge number of the sector relationship networks decreases during the two crisis periods, which reveals the global expansion of the stock market. We have then adopted the edit distance and edge statistical analysis to characterize the dynamics of PMFGs. The edit distances between all 241 PMFGs give a comprehensive view of the dynamical evolution of the stock market. The edit distances become relatively larger during the crisis periods and the rapid increase of edit distance around onset time points of crises indicates strong structure reconfigurations. The edge statistical analysis shows the power-law distribution of edges’ duration time with index 1.67. This heterogeneous distribution displays the fragility of the correlations among stocks. Despite the unstable correlation structures, we have found some edges of long duration time serve as the backbones of the stock market. The existence of strongly connected stocks may shed some light on understanding of the real stable structure of the stock market. In order to further investigate the dynamics of PMFGs, we have compared the successive edit distance D t −1,t between G tP−1 and G tP with the VIX using both monthly and yearly averages. The Granger causality tests manifest that the monthly successive edit distance Granger causes the monthly VIX. The relation between D t −1,t and VIX can be specified by the cross correlation analysis (XCF). The maximum of XCF = 0.342 at lag h = 1, which means that the VIX would be more related to the successive edit distance in the previous month. These findings suggest that the successive edit distance D t −1,t can be helpful to predict the behavior of the implied volatility index VIX. This could be a very useful signal for the market analysis. In summary, we have given a comprehensive analysis of the structure and the dynamics of the stock market in times of crisis from the point view of PMFG networks. The local clustering and the global expansion of the market structure have been explained from the aspects of community and sector structures. The dynamics of stock market has been shown by the edit distance method and edge statistical analysis. The vulnerability and the stability of the stock market discovered in our paper may offer a guidance for portfolio management. The dynamical change of PMFGs has been proved to be useful for predicting the VIX index. This provides a new way to quantify the systemic risk from the network science perspective. Acknowledgement This work is supported in part by the Programme of Introducing Talents of Discipline to Universities under grant No. B08033. Appendix A. Information about selected stocks The acronyms and names of stocks are listed here. AA, Alcoa Inc.; AAPL, Apple Inc.; ABT, Abbott Laboratories; ADBE, Adobe Systems Inc.; ADI, Analog Devices Inc.; ADM, Archer–Daniels–Midland Co; ADP, Automatic Data Processing; ADSK, Autodesk Inc.; AEP,

American Electric Power; AES, AES Corp; AET, Aetna Inc.; AFL, AFLAC Inc.; AGN, Allergan Inc.; AIG, American Intl Group Inc.; ALTR, Altera Corp; AMAT, Applied Materials Inc.; AME, AMETEK Inc.; AMGN, Amgen Inc.; AN, AutoNation Inc.; AON, Aon plc; APA, Apache Corporation; APC, Anadarko Petroleum Corp; APD, Air Products & Chemicals Inc.; APH, Amphenol Corp A; ARG, Airgas Inc.; AVP, Avon Products; AVY, Avery Dennison Corp; AXP, American Express Co; AZO, AutoZone Inc.; BA, Boeing Company; BAC, Bank of America Corp; BAX, Baxter International Inc.; BBBY, Bed Bath & Beyond; BBT, BB&T Corporation; BBY, Best Buy Co. Inc.; BCR, Bard (C.R.) Inc.; BDX, Becton Dickinson; BEN, Franklin Resources; BF-B, Brown–Forman Corporation; BHI, Baker Hughes Inc.; BIIB, BIOGEN IDEC Inc.; BK, The Bank of New York Mellon Corp.; BLL, Ball Corp; BMS, Bemis Company; BMY, Bristol–Myers Squibb; BSX, Boston Scientific; CA, CA, Inc.; CAG, ConAgra Foods Inc.; CAH, Cardinal Health Inc.; CAT, Caterpillar Inc.; CB, Chubb Corp.; CCE, CocaCola Enterprises; CCL, Carnival Corp.; CELG, Celgene Corp.; CERN, Cerner; CI, CIGNA Corp.; CINF, Cincinnati Financial; CL, ColgatePalmolive; CLX, Clorox Co.; CMA, Comerica Inc.; CMCSA, Comcast Corp.; CMI, Cummins Inc.; CMS, CMS Energy; CNP, CenterPoint Energy; COG, Cabot Oil & Gas; COP, ConocoPhillips; COST, Costco Co.; CPB, Campbell Soup; CSC, Computer Sciences Corp.; CSCO, Cisco Systems; CSX, CSX Corp.; CTAS, Cintas Corporation; CTL, CenturyLink Inc.; CVC, Cablevision Systems Corp.; CVS, CVS Caremark Corp.; CVX, Chevron Corp.; D, Dominion Resources; DD, Du Pont (E.I.); DE, Deere & Co.; DHI, D. R. Horton; DHR, Danaher Corp.; DIS, Walt Disney Co.; DNB, Dun & Bradstreet; DOV, Dover Corp.; DOW, Dow Chemical; DTE, DTE Energy Co.; DUK, Duke Energy; DVN, Devon Energy Corp.; EA, Electronic Arts; ECL, Ecolab Inc.; ED, Consolidated Edison; EFX, Equifax Inc.; EIX, Edison Int’l; EMC, EMC Corp.; EMR, Emerson Electric; EOG, EOG Resources; EQT, EQT Corporation; ESRX, Express Scripts; ESV, Ensco plc; ETN, Eaton Corp.; ETR, Entergy Corp.; EXC, Exelon Corp.; EXPD, Expeditors Int’l; F, Ford Motor; FAST, Fastenal Co; FDO, Family Dollar Stores; FDX, FedEx Corporation; FISV, Fiserv Inc.; FITB, Fifth Third Bancorp; FLS, Flowserve Corporation; FMC, FMC Corporation; FRX, Forest Laboratories; FTR, Frontier Communications; GAS, AGL Resources Inc.; GCI, Gannett Co.; GD, General Dynamics; GE, General Electric; GHC, Graham Holdings Co; GILD, Gilead Sciences; GIS, General Mills; GLW, Corning Inc.; GPC, Genuine Parts; GPS, Gap (The); GT, Goodyear Tire & Rubber; GWW, Grainger (W.W.) Inc.; HAL, Halliburton Co.; HAR, Harman Int’l Industries; HAS, Hasbro Inc.; HBAN, Huntington Bancshares; HCN, Health Care REIT; HCP, HCP Inc.; HD, Home Depot; HES, Hess Corporation; HOG, Harley–Davidson; HON, Honeywell Int’l Inc.; HOT, Starwood Hotels & Resorts; HP, Helmerich & Payne; HPQ, Hewlett–Packard; HRB, Block H&R; HRL, Hormel Foods Corp.; HRS, Harris Corporation; HST, Host Hotels & Resorts; HSY, The Hershey Company; HUM, Humana Inc.; IBM, International Bus. Machines; IFF, International Flav/Frag; INTC, Intel Corp.; IP, International Paper; IPG, Interpublic Group; IR, IngersollRand PLC; ITW, Illinois Tool Works; JCI, Johnson Controls; JEC, Jacobs Engineering Group; JNJ, Johnson & Johnson; JPM, JPMorgan Chase & Co.; JWN, Nordstrom; K, Kellogg Co.; KEY, KeyCorp; KLAC, KLA-Tencor Corp.; KMB, Kimberly–Clark; KO, Coca Cola Co.; KR, Kroger Co.; KSS, Kohl’s Corp.; KSU, Kansas City Southern Inc.; L, Loews Corp.; LB, L Brands Inc.; LEG, Leggett & Platt; LEN, Lennar Corp.; LH, Laboratory Corp. of America Holding; LLTC, Linear Technology Corp.; LLY, Lilly (Eli) & Co.; LM, Legg Mason; LMT, Lockheed Martin Corp.; LNC, Lincoln National; LOW, Lowe’s Cos.; LRCX, Lam Research; LUK, Leucadia National Corp.; LUV, Southwest Airlines; M, Macy’s Inc.; MAS, Masco Corp.; MAT, Mattel Inc.; MCD, McDonald’s Corp.; MDT, Medtronic Inc.; MHFI, McGraw Hill Financial Inc.; MHK, Mohawk Industries Inc.; MKC, McCormick & Co.; MMC, Marsh & McLennan; MMM, 3M Co.; MO, Altria Group Inc.; MRK, Merck & Co.; MRO, Marathon Oil Corp.; MSFT, Microsoft Corp.; MSI, Motorola Solutions Inc.; MTB, M&T Bank Corp.; MU,

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Micron Technology; MUR, Murphy Oil; MWV, MeadWestvaco Corporation; MYL, Mylan Inc.; NBL, Noble Energy Inc.; NBR, Nabors Industries Ltd.; NE, Noble Corp; NEE, NextEra Energy Resources; NEM, Newmont Mining Corp. (Hldg. Co.); NI, NiSource Inc.; NKE, NIKE Inc.; NOC, Northrop Grumman Corp.; NSC, Norfolk Southern Corp.; NTRS, Northern Trust Corp.; NU, Northeast Utilities; NUE, Nucor Corp.; NWL, Newell Rubbermaid Co.; OI, Owens–Illinois Inc.; OKE, ONEOK; OMC, Omnicom Group; ORCL, Oracle Corp.; OXY, Occidental Petroleum; PAYX, Paychex Inc.; PBCT, People’s United Bank; PBI, Pitney–Bowes; PCAR, PACCAR Inc.; PCG, PG&E Corp.; PCL, Plum Creek Timber Co.; PCP, Precision Castparts; PDCO, Patterson Companies; PEG, Public Serv. Enterprise Inc.; PEP, PepsiCo Inc.; PFE, Pfizer Inc.; PG, Procter & Gamble; PGR, Progressive Corp.; PH, Parker–Hannifin; PHM, Pulte Homes Inc.; PKI, PerkinElmer; PLL, Pall Corp.; PNC, PNC Financial Services; PNR, Pentair Ltd.; PNW, Pinnacle West Capital; POM, Pepco Holdings Inc.; PPG, PPG Industries; PPL, PPL Corp.; PRGO, Perrigo; PSA, Public Storage; PVH, PVH Corp; PX, Praxair Inc.; QCOM, QUALCOMM Inc.; R, Ryder System; RDC, Rowan Cos.; REGN, Regeneron Pharmaceuticals Inc.; RF, Regions Financial Corp.; RHI, Robert Half International; ROK, Rockwell Automation Inc.; ROP, Roper Industries; ROST, Ross Stores Inc.; RRC, Range Resources Corp.; RTN, Raytheon Co.; SBUX, Starbucks Corp.; SCG, SCANA Corp; SCHW, Charles Schwab; SEE, Sealed Air Corp. (New); SHW, Sherwin–Williams; SIAL, Sigma-Aldrich; SLB, Schlumberger Ltd.; SNA, Snap-On Inc.; SO, Southern Co.; SPLS, Staples Inc.; STI, SunTrust Banks; STJ, St Jude Medical; STT, State Street Corp.; STZ, Constellation Brands; SWK, Stanley Black & Decker; SWN, Southwestern Energy; SWY, Safeway Inc.; SYK, Stryker Corp.; SYMC, Symantec Corp.; SYY, Sysco Corp.; T, AT&T Inc.; TAP, Molson Coors Brewing Company; TE, TECO Energy; TEG, Integrys Energy Group Inc.; TGT, Target Corp.; THC, Tenet Healthcare Corp.; TIF, Tiffany & Co.; TJX, TJX Companies Inc.; TMK, Torchmark Corp.; TMO, Thermo Fisher Scientific; TROW, T. Rowe Price Group; TRV, The Travelers Companies Inc.; TSN, Tyson Foods; TSO, Tesoro Petroleum Co.; TSS, Total System Services; TWX, Time Warner Inc.; TXN, Texas Instruments; TXT, Textron Inc.; TYC, Tyco International; UNH, United Health Group Inc.; UNM, Unum Group; UNP, Union Pacific; USB, U.S. Bancorp; UTX, United Technologies; VAR, Varian Medical Systems; VFC, V.F. Corp.; VLO, Valero Energy; VMC, Vulcan Materials; VRTX, Vertex Pharmaceuticals Inc.; VZ, Verizon Communications; WAG, Walgreen Co.; WDC, Western Digital; WEC, Wisconsin Energy Corporation; WFC, Wells Fargo; WFM, Whole Foods Market; WHR, Whirlpool Corp.; WM, Waste Management Inc.; WMB, Williams Cos.; WMT, Wal–Mart Stores; WY, Weyerhaeuser Corp.; X, United States Steel Corp.; XEL, Xcel Energy Inc.; XL, XL Capital; XLNX, Xilinx Inc.; XOM, Exxon Mobil Corp.; XRAY, Dentsply International; XRX, Xerox Corp.; ZION, Zions Bancorp. References [1] A.M. Petersen, F. Wang, S. Havlin, H.E. Stanley, Market dynamics immediately before and after financial shocks: quantifying the Omori, productivity, and Bath laws, Phys. Rev. E 82 (3) (2010) 036114, http://dx.doi.org/10.1103/ PhysRevE.82.036114, http://link.aps.org/doi/10.1103/PhysRevE.82.036114. [2] D.J. Fenn, M.A. Porter, S. Williams, M. McDonald, N.F. Johnson, N.S. Jones, Temporal evolution of financial-market correlations, Phys. Rev. E 84 (2) (2011) 026109, http://dx.doi.org/10.1103/PhysRevE.84.026109, http://link.aps.org/doi/ 10.1103/PhysRevE.84.026109. [3] L. Sandoval, I.D.P. Franca, Correlation of financial markets in times of crisis, Physica A: Stat. Mech. Appl. 391 (1–2) (2012) 187–208, http:// dx.doi.org/10.1016/j.physa.2011.07.023, http://linkinghub.elsevier.com/retrieve/ pii/S037843711100570X. [4] V. Filimonov, D. Sornette, Quantifying reflexivity in financial markets: toward a prediction of flash crashes, Phys. Rev. E 85 (5) (2012) 056108, http://dx.doi.org/10.1103/PhysRevE.85.056108, http://link.aps.org/doi/10.1103/ PhysRevE.85.056108. [5] S. Kumar, N. Deo, Correlation and network analysis of global financial indices, Phys. Rev. E 86 (2) (2012) 026101, http://dx.doi.org/10.1103/PhysRevE. 86.026101, http://link.aps.org/doi/10.1103/PhysRevE.86.026101.

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