**4. Results**

The moving time window technique must be used to study the time evolution of cross-correlation. The results of the analysis depend on the correlation time window length. The long time window smooths the fluctuations and it can hide important system features. On the other hand, the short time window does not provide a good quality fit of the power law, and the fluctuations are more apparent in the analysis. Therefore, PLCS algorithm was applied for three time window lengths: *Tc* ∈ (20, 60, <sup>120</sup>), which correspond to a month, quarter, and half of the year period.

#### *4.1. Month Time Window*

The frequency of connection is the first parameter investigated here. This parameter informs how often the correlation strength was converging or diverging, so how stable was the correlation in the analysed period. In the case of the diverging correlation strength network, the result is presented in Figure 2.

**Figure 2.** The frequency of connection presented in descending order. The time window *Tc* = 20 days. The blue line denotes a group of currencies of similar frequency of being connected on the network.

Applying the linear fit to the frequency rank allowed for distinguishing three groups of currencies. The first group is marked by the red line: UAH, RUB, and IDR. The second group is marked by the blue line: CHF, EGP, DKK, MYR, NOK, CNY, HKD, SGD, BRL, AUD, KRW, NZD, and HUF. The third group is marked by the green line: RON, CZK, PLN, JPY, SEK, CAD, TWD, PHP, THB, BGN, AR, USD, NAD, MXN, TRY, ISK, and ZAR.

In the case of the network construction based on the converging time series, i.e., the correlation strength *α* < 0 the frequency of connection ranks are presented in Figure 2 and denoted as the converging network. In this case, six groups can be distinguished. Taking more detailed analysis into account, the following groups can be pointed out: the first, marked by the red line AR, ISK, and TRY, and the second, denoted by the blue line, consists of CZK, NAD, DKK, HKD, and MXN. The third group, marked by the green line consists of two members BGN, ZAR. The fourth group is the biggest EGP, USD, SEK, RON, HRK, PLN, JPY, CAD, CNY, SGD, MYR, BRL, PHP, THB, and AUD. The two other groups

are formed by HUF, TWD, CHF, KRW, NZD and IDR, NOK, RUB, UAH. Although both graphs are, in some sense, complementary, divergent correlation graphs are constructed under the condition that on the graph there are currencies with *α* > 0, while the divergent graph under condition *α* < 0 the graphs in Figure 2 are not simple mirror images of each other. This is because, in the analysis, the whole correlation matrix is investigated and a given currency may be present on both types of graphs at the same time (it might be convergen<sup>t</sup> with respect to one time series and divergent with concerning another). Particularly interesting are the groups denoted by the blue lines. These groups consist of currencies with similar frequency of being present in the network (divergent or convergen<sup>t</sup> respectively), so the method introduces a natural categorization of time series.

Clique size evolution. In the context of correlation strength network, the cliques are special formations. The cliques are the fully connected group of currencies, with the same type of correlations. Figure 3 presents the clique size evolution graphs for both types of networks. The main advantage of the clique size evolution analysis is the possibility to observe the clique formation in time. The converging time series network that is presented in Figure 3 shows that the biggest clusters were formed in the fourth quarter in 2014, which can be interpreted as the moment when most of the time series were converging, so the differences were decreasing. The clique was formed by 24 currencies. At the other maxima, the formed clusters were not so large and they were in the interval 17–10 currencies. The local maxima were observed in mid-2015, the second and third quarter of 2016, the first quarter of 2017, the second and third quarter of 2018, and the second quarter of 2019. It is also worth noticing that the average level of clique size before 2017 was on the level of 12 currencies, whereas, afterwards, the average value becomes about five time series. Thus, the significant decrease of the clique size is noticeable.

**Figure 3.** The biggest clique size evolution. Time window size *Tc* = 20 days.

The changes in the average size of the clique that are observed for the converging time series graph are supported by the analysis of the clique size evolution for the diverging time series graph Figure 3. In this case, the initial average size of the clusters was increased from the size of about 10 currencies to more than 23 currency time series. In the high frequency (short time window) analysis, the clique size in the diverging time series network is of high variance, which means that there is no stable tendency. The clusters are formed temporarily. However, the significant value of the cluster size suggests that the majority of the time series are divergent.

The structure analysis of the network was continued by calculating the number of communities that formed on the network. This structure community analysis is based on a weaker constraint than the clique search. Another difference to the biggest clique size is the number of communities is analysed instead of the biggest clique size. The number of communities algorithm looks for the subgraph group with nodes with a higher density of connections than the other part of the network (indifference to the clique that is a fully connected subgraph). Figure 4 presents the results of the number of community analysis.

**Figure 4.** Evolution of the communities number. Time window size *Tc* = 20 days.

Figure 4 presents the evolution of the community number that is observed for both types of networks. Intriguing is the evolution of community number in the case of the converging network (the correlation strength *α* > 0). Three levels of community number can be distinguished in Figure 4 for the converging network these are the ground level where a few communities are observed and two other states of 17 and 34 nodes. Such a big number of communities suggests that they are of very small size (one or at most a few nodes), additionally, the huge increment denotes that shift of the time window by one day has changed the situation significantly. This can be interpreted as either the period is extremely unstable or the correlation strength is approximately close to zero and small changes of the data set have affected the classification of the time series. This observation suggests that, in future applications of the method, it might be worth considering the introduction of an additional class of time series cross-correlation *α* ≈ 0. Besides the twostate period, the other local maxima are not spectacular, because they are not exceeding seven communities.

The graph presenting the evolution of the community size for the diverging network (Figure 4) differs significantly from the converging network. In this case, except for the initial part at the end of 2014, the two-level behaviour is not observed. Therefore, the diverging network seems to be more robust to the network switching effect. Similarly to the converging network, the "baseline" of the community number can be distinguished (2– 5 communities). Several clear maximums can be distinguished in the case of the diverging network quantity of community evolution: June 2015, April and July 2016, February 2018, and several maxima in 2019. 2019 was the most unstable year out of those analysed when many times the network was split into a big number of small communities.

The evolution of the community number for the converging network might sugges<sup>t</sup> that the time window size *Tc* is too short and fluctuations significantly influence the results of the analysis.

Figures 5 and 6 present the evolution of the node rank histogram for converging and diverging time series networks, respectively. When analysing the evolution of the node rank histogram for the converging time series network shown in Figure 5, it can be observed that in 2015 and 2016 the nodes with a significant number of links (*k* > 20) are dominating. Whereas, in 2017 and later, the nodes with the low number of links (*k* < 15) are dominating. A short exception is observed in 2018 (during the Chinese crisis) when nodes with a high number of links were clearly present in the network. In 2019 and later, the nodes with a small number of links are prevailing on the converging network.

**Figure 5.** Evolution of the rank nodes histogram for converging network. The time window size *Tc* = 20 days. The counts denote how many times the node of given rank (number of links) was observed on the network.

**Figure 6.** Evolution of the rank nodes histogram for diverging network. The time window size *Tc* = 20 days. Counts denotes how many times the node of given rank (number of links) was observed on the network.

The evolution of the diverging time series network histograms is presented in Figure 6. Initially, in 2015, the nodes with a small number of links are most evident, but, since 2016, the situation has changed and the nodes with a high number of connections are the most common on the network. It is particularly well seen at the end of 2019 and the beginning of 2020, when nodes with the degree *k* > 30 are dominating on the network.

Figure 7 presents the evolution of rank node entropy. There are no significant differences between the generated networks. Particularly interesting are the minima, which correspond to the situation where there is a significant group of nodes of the same rank. Although several minima can be distinguished, they do not form a clear evolution; this is due to the noise influence. This results indicate that the time window is rather too short to obtain a clear evolution of the system.

**Figure 7.** Evolution of the rank node entropy for diverging and converging networks. The time window size *Tc* = 20 days. The blue circles and green squares denote the entropy of diverging and convergen<sup>t</sup> network, respectively.

#### *4.2. Quarter Time Window*

Extending the size of the time window *Tc* to 60 days results in filtering high frequency changes, which were observed in the one-month time window. Following the same scheme of network feature analysis shown in Section 4.1, the discussion starts from the frequency of being connected. The results are presented in Figure 8. In the case of networks constructed with the constraint of the converging time series, the most frequent connections are ISK and TRY, while, for the divergent time series network, the most frequent observed currencies are UAH and RUB, which are present in 94% and 93% of the constructed network. The blue line denotes the group of currencies with similar frequency of the network member. For the converging time series network, the biggest group has a frequency in the interval 27–3%, being rather low, while, in the second type of network considered here, the frequency is in the interval 82–51%, so the probability of connection is significantly higher.

Figure 9 shows the time evolution of the biggest clique size (so the clusters of a fully connected set of currencies). It can be noticed that the divergent and convergen<sup>t</sup> time series networks results are on average complementary—the size of the cliques in convergen<sup>t</sup> time series is growing in time, but in divergent time series are decreasing. Of course, the graphs differ in details. Moreover, the general similarity does not apply to the position and magnitude of extreme points. For the converging network, as in Figure 9, six local extremes can be distinguished. The local maxima are observed in April 2015, March 2016, May–June 2016, April–June 2017, January 2018 (which is the highest maxima of 30 nodes in one clique), and the local minimum in June 2018. The clique size evolution in the diverging time series network has approximately four local extremes. The first maximum is observed at the end of 2014, which is followed by a very deep minimum in April 2015. The decrease of the clique size is enormous, because, at the first maximum, there are 26 nodes in the clique, while at the minimum the biggest clique consists of 5 nodes, so the biggest clique size decreased by 21 nodes. Immediately after that minimum, the biggest clique is growing to achieve the size of 17 nodes in August 2015. Subsequently, the clique size is relatively slowly decreasing to the level of 4–7 nodes. The last maximum is observed in July 2018.

**Figure 8.** The frequency of connection presented in descending order. The time window *Tc* = 60 days. The blue line denotes group of currencies of similar frequency of being connected on the network.

**Figure 9.** The biggest clique size evolution. Time window size *Tc* = 60 days.

Figure 10 presents the evolution of the community number. In the evolution of community number of converging networks, one can distinguish three levels: the ground state, where the community approximately 3–6 communities, the second level of 16–17 communities, and the third level of 34 communities. Because the border between converging and diverging time series is *α* = 0, the bistable behaviour of the graph means that a significant group of currencies is at the border and a small shift of the time window position is changing their classification. A similar observation was made for the evolution of community number for *Tc* = 20 days. As it was already mentioned, the additional class of *α* ≈ 0 is not introduced here due to the clarity of the analysis, because the main aim of the study is to verify the properties of the algorithm. The additional class should be considered in such a case in, e.g., commercial analysis.

**Figure 10.** Evolution of the community number. The time window *Tc* = 60 days.

The bistable behaviour of the size of the community size is also observed in the diverging network shown in Figure 10 at the end of 2014. Afterwards, the bistable evolution is not observed and several clear maxima can be noticed: June 2016, at the end of 2017, and in April–May 2019. It can be observed that, due to the longer time window, the number of maxima has been reduced when compared to the previously discussed case, as in Figure 4.

The evolution of the node rank histograms for converging and diverging networks are presented in Figures 11 and 12, respectively. In both types of network, two periods can be distinguished: the most common is the high-rank nodes or the reverse situation—the low-rank nodes. The converging time series network, as in Figure 11, is, in general, complementary to the diverging network case, as in Figure 12. At the end of 2014, the low-rank nodes are dominating, while, in 2016, 2017, and 2019, the high-rank nodes are prevailing in the histograms. Combining the results of the rank node histograms evolution with the clique size analysis, where huge clusters are observed, as in Figure 9, it can be concluded that the generated networks are very close to a fully connected network.

**Figure 11.** Evolution of the rank nodes histogram for converging network. The time window size *Tc* = 60 days. Counts denote how many times the node of given rank (number of links) was observed on the network.

**Figure 12.** Evolution of the rank nodes histogram for diverging network. The time window size *Tc* = 60 days. Counts denote how many times the node of given rank (number of links) was observed on the network.

In the case of the diverging time series network, as in Figure 12, the nodes of high rank are observed at the end of 2014, at the end of 2015, and the beginning of 2016. A very special situation occurs at the beginning of 2015, when there is no dominating group of nodes, but all the ranks of nodes are present in the histogram. In 2017, at the end of 2018, and then the beginning of 2019, the networks are divided into small subgraphs. In the mid of 2018, the increase of high-order nodes is observed—this situation can be related to the Chinese crisis.

Figure 13 presents the entropy of the rank node distribution for the time window of *Tc* = 20 days. In this case, the influence of noise is significantly reduced. The different periods can be clearly distinguished. Initially, in 2015 the decrease of entropy is observed, which is the effect of domination of high rank nodes in the histograms. The period of stable high entropy follows, which lasts until the mid of 2016. Later, oscillation appears, which are combined with the decrease of the minimum value to achieve minimum in the beginning of 2018. In 2018, another period of maximum entropy is observed. It seems that level 1.4 is the maximum entropy observed in these networks and can be considered as a measure of the stability of the market. A significant lowering of the entropy may be considered as a signature of the crisis.

**Figure 13.** Evolution of the rank node entropy for diverging and converging networks. The time window size *Tc* = 60 days. The blue circles and green squares denote the entropy of diverging and convergen<sup>t</sup> network respectively.

#### *4.3. Half Year Time Window*

This subsection contains the results obtained for the longest time window *Tc* = 120 days. Figure 14 presents the results of the frequency of connection of nodes to the network for both types of networks. In the case of the converging network, AR is the most frequent currency, which is present in 83% generated graphs. This node is separated and does not belong to any group. The first group, which can be distinguished in this analysis, consists of five currencies: ISK, TRY, MXN, HKD, and NAD. Currencies of this group are connected to others in 58–54% of networks. The second group consists of two currencies: ZAR and BGN. The last group is the biggest one—26 currencies. Within this group, the frequency of being connected is rather low: from 32% to 3%.

**Figure 14.** The frequency of connection presented in descending order. The time window *Tc* = 120 days. The blue line denotes group of currencies of similar frequency of being connected on the network.

The frequency of being connected on a divergent time series graph is slightly different because only two groups of similar frequency, i.e., without significant differences between consecutive elements, can be distinguished. The first group consists of five currencies: NZD, MYR, BRL, SGD, and KRW, and their frequency is varying from 72% to 70%. This group is followed by the second one: JPY, PHP, CAD, HRK, PLN, SEK, EGP, and BGN with the frequencies from 68% to 57%.

Figure 15 presents the biggest clique size evolution for the time window *Tc* = 120 days. When comparing to the previously discussed cases, i.e., *Tc* = 20, 60 days, the smoothing effect of the time window size is clearly visible. In this case, the biggest clique size for the converging time series network is asymptotically increasing with the exception in the middle of 2018, which can be related to the Chinese stock market crisis. An analogous maximum is observed in the graph presenting the biggest clique size evolution for the diverging network shown in Figure 15.

**Figure 15.** The biggest clique size evolution. Time window size *Tc* = 120 days.

Figure 16 presents the evolution of the community number on the graph for the time window *Tc* = 120 days. In the case of the converging network, the observed previously switching effect between two states for shorter time windows is also present in this case. However, in difference to the previous analyses, there is a period when the network brakes into separate nodes. This is the second and third quarter of 2017. At this time, in the community number of the diverging network graph, the maximum is reaching the value of 20 nodes. Simultaneously, the high number of communities is observed in diverging and converging networks this suggests that no clear tendency (or significant correlation) is present in the market. This finding agreed with the fact that, at this time, there was no serious global crisis.

**Figure 16.** Evolution of the community number. The time window *Tc* = 120 days.

The node rank histogram evolution that were obtained for the time window *Tc* = 120 days are presented in Figures 17 and 18. In both graphs, the change node rank structure is clearly visible. In the case of converging time series network, as in Figure 17, at the beginning of the analysed period, i.e., at the end of 2014 and in the first quarter of 2015 the low-rank nodes are prevailing in the network, while, from 2016, the high-rank nodes are dominating. Differently from the already analysed rank histograms evolutions for shorter time windows (*Tc* = 20 and *Tc* = 60 days) in the case of *Tc* = 120 days, the process of network transition from domination of low-rank nodes to high-rank nodes, prevailing network is a kind of continuous process. The transformation process lasts approximately a year when the nodes are gaining connections. The significant shift of the maximum position of the low-rank nodes is observed in mid-2018, probably due to the Chinese stock market crisis.

**Figure 17.** Evolution of the rank nodes histogram for converging network. The time window size *Tc* = 120 days. Counts denotes how many times the node of given rank (number of links) was observed on the network.

**Figure 18.** Evolution of the rank nodes histogram for diverging network. The time window size *Tc* = 120 days. Counts denote how many times the node of given rank (number of links) was observed on the network.

The diverging network rank node histogram evolution, as shown in Figure 18, is complementary to the converging series network. At the end of 2014, the high-rank nodes are prevailing in the histogram. During 2016, the node rank frequency of occurrence is evolving from high node rank domination to low-rank nodes prevailing in 2016. Finally, since 2016, the low-rank nodes have dominated the network except for mid-2018.

The rank node entropy evolution that is observed in the case of the time window *Tc* = 120 days is presented in Figure 19. The long time window results in significant filtering of the time series. In this case, the most stable effects can be observed. In the presented results, there are two such events—one in 2017 and the second in 2019. The outcomes of the analysis for the half-year time window confirm the previous observations that the crisis is characterised by a low value of the entropy of the rank node distribution.

**Figure 19.** Evolution of the rank node entropy for diverging and converging networks. The time window size *Tc* = 120 days. The blue circles and green squares denote the entropy of diverging and convergen<sup>t</sup> network respectively.
