Fig 1.
List of 48 countries with their stock-market indices and currencies (with gray hexagonal background).
These markers (symbols) are used in the subsequent figures throughout the paper. For the top ten countries (2012 GDP, less China), the markers are given individually with flag motifs, while for others the markers reflect their regionality.
Fig 2.
Significant eigenvalues identified by the CHPCA with RRS method.
The blue dot denoted “n” shows the n-th eigenvalue λ(n) (x-axis) and the eigenvalue rank (y-axis). The gray small dots and the lighter gray area show the average RRS and the 99% range. The six largest eigenvalues are clearly outside of their RRS ranges, and show significant relationships in the interdependent network.
Table 1.
List of CHPCA Eigenvalues, their 99% RRS Range, and the mean contribution rate .
Although the seventh eigenmode is outside the RRS range, we exclude this mode as insignificant as it is very close to the boundary.
Fig 3.
Comparison between the PCA and CHPCA eigenvalues.
In each case, the partial sum of the eigenvalues, (y-axis) versus K (x-axis), for PCA (blue and dashed) and for CHPCA (green and solid). The fact that the CHPCA sums of eigenvalues are always above PCA-based eigenvalue sums shows that CHPCA is a stronger analytic tool in identifying important co-movements.
Fig 4.
The first eigenvector (whose eigenvalue is the largest) components on a complex plane.
The flags without the gray background represent the equity markets, while the dark gray hexagons behind the flags represent the currencies as shown in Fig 1. The nodes with absolute value larger than 0.04 (which is indicated with a dashed circle) are grouped so that nodes with phase gap smaller than 0.2 [rad] are in the same group. Panel (a) for the entire period and (b)–(d) for the three subperiods. See main body of the text for interpretations of the results.
Fig 5.
The components of the second eigenvector.
We multiply the second eigenvector components by the ratio of strength of the corresponding mode-signals, , reflecting the fact that the contribution of the n-th eigenmode to the time series is proportional to
. For the entire period, the currency markets are distributed to the left of the origin, mainly close to the negative real axis, while equity markets are along the positive real axis and also in the first quadrant, indicating negative equal-time correlation between the equity and foreign exchange markets. Again, these features are almost the same for Period 3, the severe crisis period, and differ in other periods.
Fig 6.
The third eigenvector components.
For the entire period and all subperiods, we multiply the third eigenvector components by and observe two large clusters, one containing most of the European currencies and the other the Asian, South American, and Middle Eastern currencies, largely dominated by the US dollar. Note that here the correlations among stock market indices are almost non-existent.
Fig 7.
The sixth eigenvector components.
We multiply the sixth eigenvector components by . We include the names for the nodes that have large absolute values for the entire period. In this eigenmode the largest contribution comes from the Iceland’s stock market, followed by the Icelandic krona. Closely following Iceland, are Polish and Russian stock markets with positive correlation and oil exporting countries’ stock markets with negative correlation.
Fig 8.
Log-returns and mode-signals of the sixth eigenvector selected components.
(a) Absolute values of the mode-signals for the eigenmodes 1 to 6 from September 1st to December 1st of 2008 (The ticks for the x-axis are given only for Mondays). The red dots connected with red dash-dot lines show that the sixth eigenmode, evidently significant on October 13th at the height of the Icelandic banking crisis, exceeds the contribution of the first eigenmode. (b) Behavior of the log-returns of the time series that have large absolute values in the sixth eigenvector. This behavior of the actual time series is consistent with the large mode-signal of the sixth eigenmode observed in panel (a).
Fig 9.
Top 10 and bottom 10 nodes by kout−kin for the entire period (1999–2012).
Note that node indices larger than 48 are for currencies; for example, node index 74 is the currency for Mexico, or country #26 (= 74−48) in Fig 1.
Fig 10.
Stock markets and currencies with the lowest absolute differences |kout−kin| for the entire period (1999–2012).
We have ordered the components in descending order of (kout + kin) to show the relative position of stock markets and currencies in the coupled network.
Fig 11.
Results of the community detection for the entire period (1999–2012) and for the three characteristic periods (period 1: 1999–2002, period 2: 2003–2006, and period 3: 2007–2012).
The stock markets and currencies of 48 countries are decomposed into four co-moving communities, designated by numbers (“IN” means an independent node), in each period.
Fig 12.
Geographical distribution of the community structure for the entire period given in Fig 11 including the communities dominated by stock markets with red and orange colors and communities dominated by mainly currencies with blue and green colors.
Fig 13.
Community structure for the financial network for the entire period (1999–2012).
We constructed the network with θc/π = 0.1. The left panel shows the adjacency matrix sorted according to the classification into four communities of synchronizing nodes: the first community (C1) is a group of stock markets mainly in European and American countries; the second community (C2), a euro-based currency group; the third community (C3), a group of currencies represented by the U.S. dollar; the fourth community (C4), an Asian stock market group surrounding Japan. The right panel shows an optimized layout of the network in an spring-electrical model, with boundaries separating the communities.
Fig 14.
Same as Fig 13, for period 1 (1999–2002).
The network consists of four communities as it does in the entire period. The major equity community, C1, has the lowest strength of synchronization in this period while one of the two currency communities, C2, is relatively isolated from the rest of the network; in contrast, the other currency community, C3, is closely related to C1; the Asian equity community C4 is not so strongly connected.
Fig 15.
Same as Fig 13, for period 2 (2003–2006).
The network is decomposed into four communities to the largest extent in this period; we especially observe that the formation of community C2 is very tight.
Fig 16.
Same as Fig 13, for period 3 (2007–2012).
The community structure observed here is quite similar to that obtained in the entire period. This indicates that the global crisis brings a profound influence on the global financial network. Community C1 has the highest degree of synchronization in this period; C2 is now strongly connected to C1 while C3 is further apart from C1 compared to previous sub-periods; C4 has been well established as a group of synchronizing nodes.
Table 2.
Magnitude of the complex correlation coefficients averaged over all pairs of nodes within each of the four communities, C1, C2, C3, and C4, and across the communities for the entire period (1999–2012) and the three sub-periods, period 1 (1999–2002), period 2 (2003–2006), and period 3 (2007–2012).
These results measure how tightly the communities are synchronized and to what extent residual coupling still remains across the communities.
Fig 17.
Scatter plots of the complex correlation coefficients across the four communities on the complex plane.
(a) for C1-C2, (b) for C1-C3, (c) for C1-C4, (d) for C2-C3, (e) for C2-C4, and (f) for C3-C4. If the correlation coefficients between Cm and Cn have a positive median in regards to the distribution of their phases weighted by the associated magnitudes, we infer that Cn leads Cm; if the median is negative, then we infer that Cm leads Cn.
Fig 18.
Lead-lag diagram for the four communities inferred from the results in Fig 17.
The three communities, C1, C2, and C4, are steadily aligned in the time direction (C2 leading C1 and C4), while C3 has three possible positions depending on which binary correlation is emphasized, C1-C3, C2-C3, or C3-C4.