**4. Conclusions**

In this paper, we studied the high-frequency time series of price returns representing 80 cryptocurrencies that were the most actively traded on the Binance platform. We focused on the detrended cross-correlation structure of the cryptocurrency market at different time intervals and calculated the *q*-dependent detrended cross-correlation coefficient *<sup>ρ</sup>q*(*s*) for all the cryptocurrency pairs and in different moving window positions. Based on these coefficients, we analyzed the spectral properties of the detrended correlation matrix and topology of the minimal spanning trees calculated from this matrix.

The main issue that has been pointed out is that our analysis comprises only a small fraction of all traded cryptocurrencies, whose number exceeds 7500 [78]. However, the less well-known and less capitalized a cryptocurrency is, the less liquid and less reliable are the related data. This is why restricting our analysis to the most capitalized ones was crucial. Another related issue was the MST construction, and it has already been mentioned in Section 3 that the exact connectivity of the MST links is prone to noise effects, which is the most significant source of possible errors. Fortunately, the more important these errors are, the weaker the correlations, while they are less effective if the correlations are strong (this is an issue that should be addressed independently in future work).

Our principal result is the observation that, over the last year, the cryptocurrency market has gradually become more compact from a topological perspective. This was achieved by the increasing market cohesion expressed by the rising average cross-correlation strength among the cryptocurrencies. Spectrally, it was manifested by the elevated magnitude of the largest correlation matrix eigenvalue *λ*1 after mid-2020, as compared with the earlier periods. *λ*1 is associated with an eigenvector that becomes more and more delocalised with time (as detected by the increasing entropy of its components). The largest component of this eigenvector is suppressed by the delocalisation, and its absolute value decreases significantly. These effects are observed if either the large or the small fluctuation intervals are filtered out by tuning the parameter *q* in *<sup>ρ</sup>q*(*s*). In addition, the detrended cross-correlations saturated faster than before (small difference between *λ*1 for different time scales). This is a detrended counterpart of the classic Epps effect, which describes a process of the market consolidation due to the cross-correlations among the assets [69–71].

The topological properties of the MSTs are in agreemen<sup>t</sup> with the outcomes of the spectral analysis and show that the market becomes more centralized with time. On the short scales, the most connected node nowadays develops more connections to other nodes than it used to have before. The MST topology in this case is centralized and close to a star-like structure. Usually, the role of a stable central hub is played by BTC or ETH on the short time scales, but on the longer scales (e.g., an hour or longer), the hub is unstable and it frequently switches among the most liquid cryptocurrencies. The corresponding MST topology is distributed without any central hub. By increasing the scales, the mean path length also increases and it indicates that the structure for the longer time scales is more distributed and random than for the short scales. In this case, the market consolidates quickly on the short time scales (e.g., 10 min), but then the fine-grained community structure develops itself owing to the increasing cross-correlations and the average cross-correlation level rises across the network. The structure becomes less centralized, but at this point the market is already strongly coupled and compact.

We also calculated the detrended cross-correlation coefficients for BTC and some selected traditional assets like the stock market indices, commodity prices, and the regular currency exchange rates. We found that during the periods associated with the strongly correlated cryptocurrencies, the inter-market cross-correlations are also stronger than usual. Typically, the inter-market couplings rise in the periods of market instability like the COVID-19-pandemic-related events and fall in the more quiet times. However, even in such periods, the cryptocurrency market is more independent from the other markets than those markets are independent among themselves. As the pandemic becomes a normal component of our reality, the cross-correlations between the cryptocurrency market and the other markets tend to decrease, but this process is more prolonged now than the

opposite process that occurred suddenly in early 2020. It is an open issue now whether the cryptocurrencies will at some point return to be an entirely independent market or the correlations that can occur from time to time will remain observable.

The main issue that has to be pointed out is that our analysis comprises only a small fraction of all traded cryptocurrencies, whose number exceeds 7500 [78]. However, the less well-known and less capitalized a cryptocurrency is, the less liquid and less reliable are the related data. This is why restricting our analysis to the most capitalized ones was crucial. Another issue is related to the MST construction, which has already been mentioned in Section 3: the exact connectivity of the MST links is prone to noise effects, which is the most significant source of possible errors. Fortunately, these errors are the more important, the weaker are the correlations, while they are less effective if the correlations are strong (this is an issue that should independently be addressed in future work).

**Author Contributions:** Conceptualization, S.D. and M.W.; methodology, S.D., J.K., and M.W.; software, M.W.; validation, S.D., J.K., and M.W.; formal analysis, M.W.; investigation, S.D., J.K., and M.W.; resources, M.W.; data curation, M.W.; writing—original draft preparation, J.K.; writing—review and editing, J.K. and M.W.; visualization, M.W.; supervision, S.D. and J.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

 **Data Availability Statement:** Data available freely from Binance [66] and Dukascopy [77].

**Conflicts of Interest:** The authors declare no conflict of interest.
