**2. Methods**

Data from the cryptocurrency market, which is characterized by volatility that exceeds volatility of the traditional markets, are not well-suited to being studied by means of the standard correlation formalism based on the Pearson correlation [60] that requires data stationarity. Thus, methods based on signal detrending are advised [56,61].

The *q*-dependent detrended correlation coefficient *<sup>ρ</sup>q*(*s*) was proposed in the Ref. [59] to quantify the detrended cross-correlations between two, typically non-stationary time series {*x*(*i*)}*<sup>i</sup>*=1,...,*<sup>T</sup>* and {*y*(*i*)}*<sup>i</sup>*=1,...,*<sup>T</sup>* of length *T*. Let these time series be divided into *Ms* boxes of length *s* starting from its opposite ends (thus, there are 2 *Ms* boxes total). In each box, the data points are subject to integration and polynomial trend removal:

$$X\_{\nu}(s,i) = \sum\_{j=1}^{i} \mathbf{x}(\nu s + j) - P\_{X,s,\nu}^{(m)}(i), \qquad \mathbf{Y}\_{\nu}(s,i) = \sum\_{j=1}^{i} \mathbf{x}(\nu s + j) - P\_{Y,s,\nu}^{(m)}(i), \tag{1}$$

where the polynomials *P*(*m*) of order *m* are applied. The next step is calculation of the local residual variances and covariance:

$$f\_{\mathbf{X}\mathbf{X}}^{2}(\mathbf{s},\boldsymbol{\nu}) = \sum\_{i=1}^{s} (\mathbf{X}\_{\boldsymbol{\nu}}(\mathbf{s},i) - \bar{\mathbf{X}}\_{\boldsymbol{\nu}}(\mathbf{s}))^{2}, \qquad f\_{\mathbf{Y}\mathbf{Y}}^{2}(\mathbf{s},\boldsymbol{\nu}) = \sum\_{i=1}^{s} \left(\mathbf{Y}\_{\boldsymbol{\nu}}(\mathbf{s},i) - \bar{\mathbf{Y}}\_{\boldsymbol{\nu}}(\mathbf{s})\right)^{2}, \tag{2}$$

$$f\_{\mathbf{X}\mathbf{Y}}^2(\mathbf{s},\boldsymbol{\nu}) = \sum\_{i=1}^s (X\_{\boldsymbol{\nu}}(\mathbf{s},i) - \bar{X}\_{\boldsymbol{\nu}}(\mathbf{s})) (Y\_{\boldsymbol{\nu}}(\mathbf{s},i) - \bar{Y}(\mathbf{s})),\tag{3}$$

where *X* ¯ and *Y* ¯ denote the local mean of *X* and *Y*, respectively. These quantities are used to define a family of the fluctuation functions of order *q*:

$$F\_{\rm XY}^{(q)}(\mathbf{s}) = \frac{1}{2M\_{\rm s}} \sum\_{\nu=0}^{2M\_{\rm s}-1} \left[ f\_{\rm XY}^2(\mathbf{s}, \nu) \right]^{q/2}, \qquad F\_{\rm YY}^{(q)}(\mathbf{s}) = \frac{1}{2M\_{\rm s}} \sum\_{\nu=0}^{2M\_{\rm s}-1} \left[ f\_{\rm YY}^2(\mathbf{s}, \nu) \right]^{q/2}, \tag{4}$$

$$F\_{\chi\chi}^{(q)}(s) = \frac{1}{2M\_s} \sum\_{\nu=0}^{2M\_s-1} \text{sign}\left[f\_{\chi\chi}^2(s,\nu)\right] |f\_{\chi\chi}^2(s,\nu)|^{q/2}.\tag{5}$$

The sign function in Equation (5) preserves the information that is otherwise lost after taking the modulus of *f* 2XY(*<sup>s</sup>*, *<sup>ν</sup>*), while the modulus itself excludes a possibility of obtaining complex values of the covariance *f* 2XY raised to a real power *q*/2 [59,62]. The *q*-dependent detrended correlation coefficient is defined by the following formula:

$$\rho\_q^{\chi\chi}(s) = \frac{F\_{\chi\chi}^{(q)}(s)}{\sqrt{F\_{\chi\chi}^{(q)}(s)F\_{\chi\chi}^{(q)}(s)}},\tag{6}$$

which generalizes for any *q* the standard (*q* = 2) detrended correlation coefficient *ρ*DCCA [58].The parameter *q* plays the role of a filter weighting the boxes *ν* in the sums in Equations (4) and (5) by their variance/covariance magnitudes. For *q* > 2, the boxes with large signal fluctuations are given higher weights with respect to the *q* = 2 case, while for *q* < 2 the boxes with small fluctuations contribute more than for *q* = 2. Therefore, by applying *ρq*, one can learn which fluctuations are the source of the observed detrended correlation of the time series.

For a set of *N* parallel time series indexed by *i*, the *q*-dependent correlation coefficient can be calculated for each time series pair (*i*, *j*) (*i*, *j* = 1, ..., *N*), and a *q*-dependent detrended correlation matrix **<sup>C</sup>***q*(*s*) with the entries *ρ*(*<sup>i</sup>*,*j*) *q* (*s*) can be created, as well as a *q*-dependent metric distance matrix **<sup>D</sup>***q*(*s*) whose entries are

$$d\_q^{(i,j)}(s) = \sqrt{2(1 - \rho\_q^{(i,j)}(s))}.\tag{7}$$

The matrix **<sup>D</sup>***q*(*s*) can then be used to create a weighted graph, where nodes labelled by *i* = 1, ..., *N* represent the time series and *N*(*N* − 1)/2 edges connecting the nodes *i*, *j* are attributed the weights equal to *<sup>d</sup>*(*<sup>i</sup>*,*j*) *q* (*s*). A subset of the complete graph, consisting of all *N* nodes and only *N* − 1 edges that minimize the weight sum, is a *q*-dependent detrended minimum spanning tree (*q*MST) [63]. This tree can be constructed by means of the Prim algorithm, for instance [64]. However, although the very algorithm is the same, such a tree differs from the standard approach that uses the Pearson correlation coefficient and a corresponding Pearson correlation matrix (see, for example, [55,65] for such a standard approach applied to the cryptocurrency market).

A data set of interest is the 1 min price quotations of the 80 cryptocurrencies that were among the most actively traded ones on the Binance platform [66] over the period from 1 January 2020 to 1 October 2021. The quotes are expressed in USD Tether (USDT) that is a stablecoin linked to the US dollar and its value is \$1.00 by design [67]. Each time series of the price quotations is 921,600 points long and covers 640 trading days (the Binance platform is active 24 hours a day and 7 days a week). All the assets used in this study are listed in Appendix A (Table A1).

#### **3. Results and Discussion**

The price quotation time series *pi*(*tm*), where *m* = 1, ..., *T* and *i* stands for a given cryptocurrency ticker, were first transformed to the time series of logarithmic returns *<sup>R</sup>*X(*tm*) = ln *pi*(*tm*+<sup>1</sup>) − ln *pi*(*tm*) and then normalized to zero mean and unit variance, which is a standard procedure. Then, for each pair of cryptocurrencies (*i*, *j*), the *q*dependent detrended cross-correlation coefficient *ρ*(*<sup>i</sup>*,*j*) *q* (*s*) given by Equation (6) was determined for a number of time scales *s* from *s* = 10 min to *s* = 360 min and different values of the filtering parameter *q*. In what follows, we will present results obtained for *q* = 1, which corresponds to a situation where the small fluctuation period variances in Equations (4) and (5) are amplified relatively to the large ones, and for *q* = 4, which corresponds to the opposite situation. Thus, we can consider the asset cross-correlations for the quiet and turbulent periods in a separate manner.

Before we start a presentation of our results, in Figure 1 we show the historical data of the BTC price in USD in the years 2020–2021 together with the BTC share in the total cryptocurrency market capitalization over the same period. Among the most characteristic events for BTC was the crash on 13 March 2020 related to the COVID-19 pandemic onset in the United States, when BTC surged below 4107 USD, a long rally that started in October 2020 and ended on 14 April 2021 with then the all-time-high equal to 64,830 USD, a subsequent drop-down phase that ended on 20 July 2021 at 29,324 USD, and the next all-time-high on 20 October 2021 equal to 66,961 USD. As the BTC has been priced higher and higher, its share in the total market capitalization drops down steadily from about 70% in January 2020 to below 45% in October 2021, which seems to be inevitable if the number of the actively traded cryptocurrencies grows quickly.

**Figure 1.** Price evolution of bitcoin (BTC) expressed in US dollars (black) and the BTC share in the total cryptocurrency market capitalization (magenta) over the period from 1 January 2020 to 31 October 2021. Characteristic events are indicated by vertical dashed lines and Roman numerals: COVID-19 crash in March 2020 (event I), strong bull market on cryptocurrency valuation October 2020–April 2021 (event II), all-time high on 14 April 2021 (event III), the May crash on the cryptocurrency market (event IV), and recent rally with new all-time high on 20 October 2021 (event V).

Since for *N* = 80 cryptocurrencies there are N = *N*(*N* − 1)/2 = 3160 cryptocurrency pairs that have to be considered, it is convenient to analyze the whole set collectively by means of the spectral analysis of the *N* × *N q*-dependent detrended correlation matrix **<sup>C</sup>***q*(*s*), whose entries are the coefficients *ρ*(*<sup>i</sup>*,*j*) *q* (*s*). We can diagonalize it and calculate its eigenvalues *λi* and eigenvectors **<sup>v</sup>***i* (with *i* = 1, ..., *N*):

$$\mathbf{C}\_{\emptyset}(s)\mathbf{v}\_{i}^{(q)}(s) = \lambda\_{i}^{(q)}(s)\mathbf{v}\_{i}^{(q)}(s). \tag{8}$$

The eigenvalues are ordered typically from the largest one (*i* = 1) to the smallest one (*i* = *N*). (For simplicity, from now on we will omit the parameters *q* and *s* when dealing with the eigenvalues and eigenvectors of **<sup>C</sup>***q*(*s*). Their value will be known from the context.)

For the financial markets, a typical eigenvalue spectrum of the Pearson-coefficientbased correlation matrix consists of a large *λ*1 that is separated from the remaining eigenvalues by a considerable gap and corresponds to the average behaviour of the considered assets (the so-called market factor), a few elevated non-random eigenvalues that correspond to subsets of related assets (e.g., representing companies from the same industry or currencies from the same geographical region), and a bulk of mean eigenvalues that correspond to random fluctuations and, essentially, carry no genuine information. Here we use the detrended correlation coefficient *ρq* instead of the Pearson coefficient [60], but our experience shows that the corresponding matrix **C***q* reveals similar spectral properties [63]. The largest eigenvalue *λ*1 is associated with a maximally delocalized eigenvector **v**1 with many significant components, while the eigenvectors representing smaller eigenvalues are more localized, that is, few components are significant. The eigenvector structure is usually expressed by the inverse participation ratio or the localization length [68], but here we apply the Shannon entropy defined by

$$H(\mathbf{v}\_i) = -\sum\_{j=1}^{N} p\_i(j) \ln p\_i(j),\tag{9}$$

with *pi*(*j*) = *v*2*i* (*j*) (the eigenvectors are normalized to unit length, so that <sup>∑</sup>*Nj*=<sup>1</sup> *v*2*i* (*j*) = 1). If the eigenvector is maximally delocalized and all its components are equal to each other, the Shannon entropy assumes its maximum value: *<sup>H</sup>*(**<sup>v</sup>***i*) = ln *N*, while if there is only a single non-zero component, the entropy vanishes: *<sup>H</sup>*(**<sup>v</sup>***i*) = 0. Entropy can thus serve as a measure of vector localization.

In order to track the evolution of the asset–asset detrended cross-correlations, we apply a moving window of size of 7 days (10,080 data points), which was shifted by a daily step (1440 data points) along the time series. For each window position *t*, based on the 80 time series of price returns, we create a detrended correlation matrix **<sup>C</sup>***q*(*<sup>s</sup>*, *t*) for a few selected values of *q* (*q* = 1 and *q* = 4) and *s* (*s* = 10 min, *s* = 60 min, *s* = 180 min, and *s* = 360 min). Next we diagonalize *Cq*(*<sup>s</sup>*, *t*) and derive a complete set of the eigenvalues *<sup>λ</sup>i*(*t*) and eigenvectors **<sup>v</sup>***i*(*t*). Figure 2 exhibits *<sup>λ</sup>*1(*t*), *<sup>H</sup>*(**<sup>v</sup>**1(*t*)), and the largest squared component *v*(max) 1 (*t*) of the eigenvector **<sup>v</sup>**1(*t*) for different time scales *s* and different values of the filtering parameter *q*. By increasing *s*, we also obtain a systematically increasing *<sup>λ</sup>*1(*t*), which reflects the increasing strength of the mean asset–asset detrended cross-correlations for the longer time scales *s*. This is a well-known property of the financial and commodity markets and it is called the Epps effect [41,69–71]. This effect has already been observed on the cryptocurrency market and reported, for example, in the Ref. [5]. It is a consequence of the fact that what dominates the price evolution on short time scales is noise: it takes time to spread a piece of information among the assets, especially if the asset liquidity is small like in the case of the cryptocurrencies. Therefore, only on the sufficiently long time scales, the cross-correlations are able to be built up to a full extent.

Another observation is that the difference in correlation strength between *s* = 10 min and *s* = 360 min is much stronger for *q* = 1 than for *q* = 4; the correlation strength for large scales is also significant then. The behavior of *λ*1 is also different: in the case of *q* = 1, periods with a large value of *λ*1 are accompanied by periods of moderate value, but there are also few periods with relatively small values of the largest eigenvalue. In turn, for *q* = 4 the *<sup>λ</sup>*1(*t*) evolution consists of large, but short "bursts" separated by small background values. In the latter case, *λ*1 is more sensitive to changes. Looking at the *<sup>λ</sup>*1(*t*) chart for *q* = 1 and the shorter *s* time scales, two characteristic epochs can be distinguished: (1) more or less until October 2020, we observe a horizontal trend, where the average value of *λ*1 does not change much, and (2) from October 2020 to mid-2021, a strong upward trend is noticeable. This is confirmed by looking at the Shannon entropy panel, where the behavior of this quantity is very similar. This means that from the fall of 2020 to mid-2021, there was a gradual increase in the strength of the market correlation and more cryptocurrencies began to behave in a similar way. It can be said that the market has consolidated. In the third quarter of 2021, this trend was halted, *λ*1 began to decrease slightly, and *<sup>H</sup>*(**<sup>v</sup>**1) was saturated close to its maximum allowed value of approximately 4.38. Understandably, as the delocalization of the vector **v**1 increases, the value of its largest component *v* (max) 1 decreases (see Figure 2).

Figure 3 shows the changes over time of the second largest eigenvalue *λ*2, the entropy of the components of the corresponding eigenvector **v**2 and the changes in the value of the largest component *v* (max) 2 of this vector. For both *q* = 1 and *q* = 4, the value of *λ*2 is much lower than the value of *λ*1, which results from a much smaller number of significant eigenvector components: entropy is lower than 4, and for *q* = 1, in the vast majority of windows, its value decreases as *s* increases, which is the opposite of the *λ*1 case. For *q* = 4, we do not observe such an effect. With *q* = 1, the global maximum of *λ*2 falls in July 2020, when its value more than doubled if compared to other time intervals. Simultaneously, *λ*1 reached one of its lowest values, as did *<sup>H</sup>*(**<sup>v</sup>**1). At the same time, the entropy for **v**2 did not change much from its typical value, but then and in the preceding period *<sup>H</sup>*(**<sup>v</sup>**2 was similar for different time scales. For large fluctuations (*q* = 4), the maximum *λ*2 also occurred in the same period, but was not as unique as for the smaller fluctuations (*q* = 1), because *λ*2 reached equally high magnitude in April and May 2021. However, *λ*1 for *q* = 4 also had its maxima at the same moments. This means that briefly in July 2020, there was a strong correlation of a small group of cryptocurrencies, and this mainly concerned small and medium fluctuations in their price, while the market as a whole was in a decoupling stage. In turn, in April and May 2021 there was a stronger than usual correlation of the entire market, with large fluctuations being particularly strongly correlated. As for the largest component of the vector **v**2 and *q* = 1, we do not observe systematic changes in its value for the short time scales, while for the long ones, starting from autumn 2020, there is a growing trend that ends in mid-2021. This increase in *v* (max) 2 suggests that one of the cryptocurrencies increased its dominance over other cryptocurrencies at that time. This behavior differs from the behavior of the analogous measure described above in the case of the vector **v**1, where there was a clear decrease.

**Figure 2.** Time evolution of the selected spectral characteristics of the *q*-dependent detrended correlation matrix **<sup>C</sup>***q*(*s*) for *q* = 1 (**a**) and *q* = 4 (**b**). A moving window of a length of 7 days shifted by 1 day was applied for sample values of the scale: *s* = 10 min (red), *s* = 60 min (blue), *s* = 180 min (green), and *s* = 360 min (orange). The largest eigenvalue *λ*1 (top panels in (**<sup>a</sup>**,**b**)), the Shannon entropy *<sup>H</sup>*(**<sup>v</sup>**1) of the squared eigenvector components *<sup>v</sup>*1(*j*) with *j* = 1, ..., *N* (middle panels), and the squared maximum component of the eigenvector **<sup>v</sup>**1 associated with *λ*1 (bottom panels) are shown. The cryptocurrency prices are expressed in USDT.

**Figure 3.** Time evolution of the selected spectral characteristics of **<sup>C</sup>***q*(*s*) (continuing). As in Figure 2, two cases are shown: *q* = 1 (**a**) and *q* = 4 (**b**). A moving window of length 7 days shifted by 1 day was applied for sample values of the scale: *s* = 10 min (red), *s* = 60 min (blue), *s* = 180 min (green), and *s* = 360 min (orange). The second largest eigenvalue *λ*2 (top panels), the Shannon entropy *<sup>H</sup>*(**<sup>v</sup>**2) of the squared eigenvector components *<sup>v</sup>*2(*j*) with *j* = 1, ..., *N* (middle panels), and the squared maximum component of the eigenvector **v**2 associated with *λ*2 (bottom panels) are shown.

Since a sum of all the eigenvalues must equal trace of **C***q* with Tr**C***q* = *N*, the high values of *λ*1 take a significant part of each time series variance. This can suppress all the other eigenvalues with *λ*2 in particular and can also have a strong impact on the eigenvector **v**2. We thus prefer to look at these quantities once more after removing the variance contribution of *λ*1 from the original time series of returns. In order to accomplish this, we created an eigensignal representing *λ*1 as a sum of the original time series weighted by the corresponding eigenvector components *<sup>z</sup>*1(*tm*) = <sup>∑</sup>*Nj*=<sup>1</sup> *<sup>v</sup>*1(*j*)*rj*(*tm*), where *rj*(*tm*) are the normalized returns of a cryptocurrency *j* at time *tm*, *m* = 1, ..., *T*. We then least-square fit the eigensignal {*<sup>z</sup>*1(*tm*)} to each original time series {*rj*(*tm*)} and subtract the fitted component from {*rj*(*tm*)}. What remains then is a residual signal {*r*(res) *j* }, which does not comprise any contribution from {*<sup>z</sup>*1(*tm*)} and, thus, also from *λ*1:

$$r\_i^{(\text{res})}(t\_m) = r\_i(t\_m) - \kappa\_i z\_1(t\_m) - \beta\_{i\prime} \tag{10}$$

where *αi*, *βi* are the parameters of a linear fit. Finally, we calculate the coefficients *ρ*(*<sup>i</sup>*,*j*) *q* (*s*) for all the cryptocurrency pairs (*i*, *j*) and form a residual *q*-dependent detrended correlation matrix **C**(res) *q* (*s*). After diagonalising it, we obtain its eigenvalues *λ*(res) *i* and eigenvectors **v**(res) *i* . We repeat this procedure a few times for different scales *s* and filtering parameters *q*. Figure 4 collects the results.

Now the largest eigenvalue *λ*(res) 1 , which inherits some information stored previously in *λ*2 but without the former clear impact of *λ*1, is not suppressed any more and, for *q* = 1, it shows richer behaviour with more fluctuations and more pronounced maxima (see Figure 4a). Interestingly, the large maximum of *λ*2 observed in Figure 3a in July 2020 disappeared almost completely here and was replaced by a series of pronounced maxima in February, March, September, and December 2020, and a smaller one in May 2021. They are the more visible the longer time scale is considered. From a present perspective, the unique maximum of *λ*2 in July 2020 might solely be a product of a relatively small value of *λ*1 in that moment, which was unable to suppress *λ*2 to its overall level of 4.

As regards the Shannon entropy, three phases can be distinguished: (1) from January to May 2020, (2) from May 2020 to April 2021, and (3) from May to October 2021. In the first and third phases there is no difference in *<sup>H</sup>*(**v**(res) 1 ) if we consider different scales *s*, while during the second phase, which largely overlapped with the bull market, the entropy fluctuates in time and increases with increasing *s*. However, its saturation level for *s* = 360 min in this phase is comparable with the analogous level in the other phases – this is because *<sup>H</sup>*(**v**(res) 1 ) for *s* = 10 min can be much smaller in phase (2) than in phases (1) and (3). Dissimilarity between the phases is observed also for *v*(res)(*max*) 1 : in phase (2) its value is substantially elevated as compared with the phases (1) and (2). These outcomes sugges<sup>t</sup> that the eigenvector **v**(res) 1 became delocalised and some cryptocurrency used to contribute more to this eigenvector during phase (2) than the other cryptocurrencies did.

For *q* = 4 (Figure 4b), both *<sup>H</sup>*(**v**(res) 1 ) and *v*(res)(max) 1 fluctuate over the whole analysed period more than it is observed for *q* = 1. The largest residual eigenvalue for *q* = 4 displays local maxima in the same moments as for *q* = 1, but their height varies. Apart from the maxima, typical fluctuations of *λ*(res) 1are smaller in 2021 than they used to be in 2020.

**Figure 4.** Time evolution of the selected spectral characteristics of the residual *q*-dependent detrended correlation matrix **C**(res) *q* (*s*) after filtering out the component corresponding to *λ*1. As in Figure 2, two cases are shown: *q* = 1 (**a**) and *q* = 4 (**b**). A moving window of length 7 days shifted by 1 day was applied for sample values of the scale: *s* = 10 min (red), *s* = 60 min (blue), *s* = 180 min (green), and *s* = 360 min (orange). The largest residual eigenvalue *λ*(res) 1 (top panels), the Shannon entropy *<sup>H</sup>*(**v**(res) 1 ) of the squared eigenvector components *v*(res) 1 (*j*) with *j* = 1, ..., *N* (middle panels), and the squared maximum component of the eigenvector **v**(res) 1 associated with *λ*(res) 1 (bottom panels) are shown.

Some deeper insight into the cross-correlation structure of the cryptocurrency market can be gained by transforming the *q*-dependent detrened correlation matrix **<sup>C</sup>***q*(*s*) into a related distance matrix **<sup>D</sup>***q*(*s*), whose elements are defined by Equation (7). The latter is used as a basis for creating a minimum spanning tree, in which each node represents a particular cryptocurrency and each weighted edge represent the metric distance between a pair of assets or, equivalently, the detrended cross-correlation coefficient. To facilitate comprehension of the MST pictures, the edge weights between the nodes (*i*, *j*) are proportional to the coefficients *ρ*(*<sup>i</sup>*,*j*) *q* (*s*) even though the metric distances *<sup>d</sup>*(*<sup>i</sup>*,*j*) *q* (*s*) were used to determine the MST edges in this work.

We created an MST for each moving window position and for the same values of *s* and *q* as before. Owing to this, we are able to observe the evolution of the MST topology along the considered time span. The first topological characteristics we discuss here is the probability that a given node has a degree *k*. Its cumulative distributions *P*(*X* ≥ *k*) for a few sample window positions are shown in Figure 5 for *q* = 1 (top) and *q* = 4 (bottom) and for *s* = 10 min (red line) and *s* = 360 min (blue line). The MST topology expressed by these characteristics varies between different time intervals from a centralized graph with a single dominant node playing the role of a hub, that is, when there is a significant gap between the largest degree *k*max and the second largest degree, to a distributed graph with a small *k*max and a small difference in the degrees of the most connected nodes. The former situation is more typical for the short time scales (*s* = 10 min) and the periods with small return fluctuations (*q* = 1), while the latter situation occurs frequently for the long time scales (*s* = 360 min) and both the small and large fluctuation periods (*q* = 1 and *q* = 4); see Figure 5.

While increasing the scale from *s* = 10 min to *s* = 360 min, for *q* = 1 we observe a systematic change of the MST topology from centralized towards more distributed. For *q* = 4 there is no such a change and the topology is largely preserved. From the network perspective, this means that the detrended cross-correlations during the strong volatility periods are already well-developed at the 10-min time scale and, possibly, one has to consider even shorter scales to detect any topological transition (this would require a higher frequency of the price quotations than 1 min considered here, however). It is also worth noting that the cumulative probability distributions in some windows show a scalefree decay with *k* (the almost-straight lines in double logarithmic plots). This conclusion supports the results reported earlier for the data covering the years 2016–2019 [5] and 2017–2018 [72].

Topological changes of the MSTs while going from past to present can be expressed by the time evolution of the node degree *ki*(*t*) for the most connected nodes representing the cryptocurrencies *i*. The results for the MSTs created based on three distinct data sets are presented in Figure 6: (1) the original time series of the price returns, (2) the residual time series obtained after filtering out the contribution of *λ*1 from the original data (both are based on the quotes given in USDT), and (3) the time series of the price returns based on the quotes given in BTC. The latter case allows us for effective filtering out the impact of BTC on the other assets' detrended cross-correlations.

**Figure 5.** Node degree cumulative distribution *P*(*X* ≥ *k*) of the MSTs created for the cryptocurrency prices expressed in USDT. Results for sample moving windows are shown for *q* = 1 (**a**) and *q* = 4 (**b**). In each panel the distributions for two temporal scales are displayed: *s* = 10 min (red) and *s* = 360 min (blue). The nodes with the highest degree *k* are labelled by the corresponding cryptocurrency ticker.

There are the following observations:


**Figure 6.** *Cont*.

**Figure 6.** Evolution of the node degree *ki* for the most connected nodes of the MST calculated in the seven-day-long moving window with a step of 1 day. For the prices expressed in USDT, two cases are shown: (**a**) the results for the complete data set without any filtering and (**b**) the results for the residual signals after filtering out a contribution from the component represented by the largest eigenvalue *λ*1. The results for (**c**)—the prices expressed in BTC , which corresponds to filtering out any BTC-related contribution to other assets' evolution, are also shown. In each case, three exemplary scales are shown: *s* = 10 min (top graph in each panel), *s* = 60 min (middle graph), and *s* = 360 min (bottom graph). Different colors and line styles denote the node degree for different cryptocurrencies.

A variety of the MST topologies that can be observed in the cryptocurrency market in different periods is illustrated in Figures 7 and 8. The top left MST in Figure 7 has largely a star-like structure with BTC being its central node and ETH being a secondary hub. All other nodes are peripheral in respect to these two. The bottom left tree is also significantly centralized but now the most connected node is ETH, while BTC, FTT, and BAT are secondary hubs. A mixed type of topology is shown in the bottom right MST, where there are two primary hubs that are almost equivalent topologically (BTC and ETH) and a single secondary hub (BCH). However, despite this interesting dual centrality, the network has a part that is rather distributed. A largely distributed structure can be seen in

the top right MST, in which only BTC possesses a significant number of the satellite nodes, while the overall network structure is distributed and almost random.

**Figure 7.** Minimal spanning trees calculated from a distance matrix **<sup>D</sup>***q*(*s*) based on *<sup>ρ</sup>q*(*s*) for *q* = 1 and *s* = 10 min. Each node represents a cryptocurrency and the edge widths are proportional to value of the corresponding coefficient *<sup>ρ</sup>q*(*s*). Each MST was created for moving window of length 7 days ended at specific dates: (**a**) 6 April 2020, (**b**) 1 August 2020, (**c**) 9 October 2020, and (**d**) 25 February 2021.

While the asset–asset correlation strength can be amplified by increasing scale *s*, Figure 8 shows that this operation weakens at the same time the centralized topology of the associated MST, which can show the signatures of a decentralized network. This can be seen by comparing the trees corresponding to the same windows in Figures 7 and 8.

**Figure 8.** Minimal spanning trees calculated from a distance matrix **<sup>D</sup>***q*(*s*) based on *<sup>ρ</sup>q*(*s*) for *q* = 1 and *s* = 360 min. Each node represents a cryptocurrency and the edge widths are proportional to value of the corresponding coefficient *<sup>ρ</sup>q*(*s*). Each MST was created for moving window of a length of 7 days ended at specific dates: (**a**) 6 April 2020, (**b**) 1 August 2020, (**c**) 9 October 2020, and (**d**) 25 February 2021.

This conclusion receives additional support from the top panels of Figure 9a,b presenting the mean path length as a function of time. It is defined by the following formula:

$$
\langle L(q, s, t) \rangle = \frac{1}{N(N-1)} \sum\_{i=1}^{N} \sum\_{j=i+1}^{N} L\_{ij}(q, s, t), \tag{11}
$$

where *Lij* is the length of the path connecting nodes *i* and *j*. The larger *L*(*q*,*s*, *t*) is, the more distributed is the corresponding MST. Indeed, by considering a given window, this quantity systematically increases with increasing *s*. The smallest values of the mean path length (2 < *Lij*(*q*,*s*, *t*) < 3) can be seen in April-May 2020 (see also [74]), in August– September 2020, between March and May 2021, in May 2021, and in September–October

2021 for *s* = 10 min. These are the periods of the most centralised market, where a vast majority of the nodes is connected to a central hub. In each of these periods, the maximum node degree *k*max assumes high values as well (see Figure 6a). In contrast, the elevated values of *Lij*(*q*,*s*, *t*) (*Lij*(*q*,*s*, *t*) > 5) are observed in February 2020, July 2020, and between February and May 2021.

The power-law exponent *<sup>γ</sup>*(*q*,*s*, *t*) describing slope of the cumulative probability distribution of the node degree is shown in the middle panel of Figure 9a for *q* = 1 and it is accompanied by the standard error of its least-square fit (the lower panel). It is an unstable quantity that fluctuates between 0.5 and 2 (see also Figure 5) for the results in sample windows. The smaller *<sup>γ</sup>*(*q*,*s*, *t*) is, the more distant *k*max can be from the smaller values of *ki*, but this relation does not always hold. The same quantities are shown in Figure 9b for the case of *q* = 4. Now we see smaller differences between the network characteristics for different time scales. This is the same rule as the one observed in Figure 5 for *q* = 4.

Topology of the MSTs representing the residual time series {*r* (res) *i* (*tm*)} differs from the original time series {*ri*(*tm*)} significantly. Because the removed component representing *λ*1 is connected with the strength of the average detrended cross-correlation coefficient *ρq*(*s*), a lack of this component weakens the detrended cross-correlations and can thus destroy the star-like structures within the MST. This must obviously lengthen many inter-node paths and increase *Lij*(*q*,*s*, *<sup>t</sup>*). In fact, Figure 10a,b shows that *Lij*(*q*,*s*, *t*) > 5 over almost the whole analyzed period for both *q* = 1 and *q* = 4. It happens sometimes that its value reaches 10, which indicates a distributed network topology. The slope exponent *<sup>γ</sup>*(*q*,*s*, *t*) behaves even more erratically than for the original, complete data in Figure 9, and the standard error of the fitted values is much larger.

The same topological characteristics for the MSTs created from the time series of price quotations expressed in BTC are presented in Figure 11a,b. Their temporal evolution seems to be less random than in Figure 10 and resembles the picture for the USDT-based data shown in Figure 9. For *q* = 1, the mean path length fluctuates along a horizontal line at *Lij* ≈ 5 until April 2021. Then the trend line starts to decrease towards a level of 4 or even below this value. This suggests that the MST topology has gradually become more centralized in the recent months. Such an effect is hardly visible for *q* = 4. A rather high values of *<sup>γ</sup>*(*q*,*s*, *t*) above 1.5 for *q* = 4 confirm a more compact topology of the corresponding MSTs than in the case of the prices expressed in USDT.

Our study of the cryptocurrency network topology can be completed with an analysis of the network cluster structure. Obviously, in this case we have to consider the complete weighted networks defined by the matrix **<sup>C</sup>***q*(*s*) instead of the MSTs. In order to identify node clusters, we exploit the Louvain algorithm of community detection, whose performance is counted among the best methods [75].

For the most moving window positions, the algorithm detects a few cryptocurrency clusters, but their composition fluctuates among the windows. To show how the clusters vary in time, we select a few significant nodes and associate them with a set of nodes they share a given cluster with. Among the distinguished nodes that frequently play a role of the MST cluster centers are BTC, ETH, LINK, TRX, ONT, BNB, and others. In the case of BTC, we consider a network of all 80 cryptocurrencies expressed in USDT, while for the other nodes, we consider a limited set of 68 cryptocurrencies expressed in BTC and that are not pegged to US dollar. Some of the related clusters consist of a few nodes only throughout the whole period under study, but there are also clusters consisting of a variable number of nodes. Here we show the examples of the latter group of clusters: the clusters to which BTC, ETH, BNB, or ONT belong. It should be noted, however, that (1) a node representing a given cluster might not necessarily be its center in the MST representation, (2) some clusters are merged in some windows, while they remain separate in the other windows, and (3) the nodes can jump between clusters.

**Figure 9.** Time evolution of the selected network characteristics of the MST created from a distance matrix **<sup>D</sup>***q*(*s*). Two cases are shown: *q* = 1 (**a**) and *q* = 4 (**b**). In each case, a moving window of length 7 days shifted by 1 day was applied for the scales: *s* = 10 min (red), *s* = 60 min (blue), *s* = 180 min (green), and *s* = 360 min (orange). The mean path length *L*(*q*,*s*, *t*) (top panels), the node degree cumulative probability distribution *P*(*X* ≥ *k*) power-law slope exponent *<sup>γ</sup>*(*q*,*s*, *t*) (middle panels) together with its standard error (SE, bottom panels). The cryptocurrency prices are expressed in USDT.

**Figure 10.** The same quantities as in Figure 9 but here obtained from the residual MSTs calculated for **D**(res) *q* (*s*) after filtering out the component corresponding to *λ*1. Two cases are shown: *q* = 1 (**a**) and *q* = 4 (**b**). The cryptocurrency prices are expressed in USDT.

**Figure 11.** The same quantities as in Figures 9 and 10 but obtained from the cryptocurrency prices expressed in BTC. Two cases are shown: *q* = 1 (**a**) and *q* = 4 (**b**).

In Figures 12–15 we present the time evolution of the cluster composition for different time scales: *s* = 10 min, *s* = 60 min, and *s* = 360 min, and for *q* = 1. For example, a full point in the plot depicting the BTC cluster indicates that a respective cryptocurrency shares a cluster with BTC in a particular time window. The more dense points are seen along a horizontal line representing that cryptocurrency, the more stable is the coexistence of these

two cryptocurrencies within the same cluster. On the other hand, the more numerous are the points along a vertical line, the larger is the cluster at that particular moment.

**Figure 12.** Composition of the BTC-related cryptocurrency cluster as a function of time for sample temporal scales: *s* = 10 min (**top**), *s* = 60 min (**middle**), and *s* = 360 min (**bottom**). Each point on the horizontal axis represents a non-overlapping seven-day-long moving window. Asset prices have been expressed in USDT.

Time

 Jan 2021  Apr 2021  Jul 2021  Oct 2021

 Oct 2020

 Jul 2020

Jan 2020

 Apr 2020

**Figure 13.** Composition of the ETH-related cryptocurrency cluster as a function of time for sample temporal scales: *s* = 10 min (**top**), *s* = 60 min (**middle**), and *s* = 360 min (**bottom**). Each point on the horizontal axis represents a non-overlapping seven-day-long moving window. Asset prices have been expressed in BTC, therefore any BTC-related contribution has been filtered out.

**Figure 14.** Composition of the BNB-related cryptocurrency cluster as a function of time for sample temporal scales: *s* = 10 min (**top**), *s* = 60 min (**middle**), and *s* = 360 min (**bottom**). Each point on the horizontal axis represents a non-overlapping seven-day-long moving window. Asset prices have been expressed in BTC, therefore any BTC-related contribution has been filtered out.

**Figure 15.** Composition of the ONT-related cryptocurrency cluster as a function of time for sample temporal scales: *s* = 10 min (**top**), *s* = 60 min (**middle**), and *s* = 360 min (**bottom**). Each point on the horizontal axis represents a non-overlapping seven-day-long moving window. Asset prices have been expressed in BTC, therefore any BTC-related contribution has been filtered out.

A cluster, to which BTC belongs, is typically the largest cluster in the network. By looking at Figure 12, we see that, on the shortest time scale of 10 min, the BTC cluster's size increases substantially in March 2021 and remains such till the end of the analyzed time interval. This is in agreemen<sup>t</sup> with the increase of the Shannon entropy *<sup>H</sup>*(**<sup>v</sup>**1) observed in Figure 2 and it indicates that the market network has become more compact recently. A situation looks different for *s* = 60 min, because apart from the BTC cluster growth observed in the *s* = 10 min case, only a slightly smaller cluster structure was seen before mid-2020. Thus, for *s* = 60 min the BTC cluster shrunk considerably over the period from July 2020 to February 2021 and it was larger outside that period. There are nodes that accompany BTC regularly, like STX, RVN, NANO, and BEAM, and there are nodes that fall into the BTC cluster only few times, like TRX and ETH, or even never do this, like ETH. For *s* = 360 min we do not detect any comparably large cluster and the BTC cluster is much smaller. It also tends to shrink even more after mid-2020.

The ETH cluster is much less numerous that the BTC one, which is partially due to a smaller number of the analyzed assets, but also to the properties of this cluster. Despite this, however, some long-term trend can be seen for *s* = 10 min that resulted in the temporary cluster growth in the latter half of 2020 followed by its shrinking that lasts till the end of the analyzed period. Such an effect cannot be noticed for the longer scales, where the density of points remains at the same level throughout the years 2020-2021. Among the nodes that frequently accompany ETH are BNB, LTC, BCH, and LINK.

The BNB cluster can be counted among the most numerous clusters on a par with the ETH cluster. For *s* = 10 min we also observe its interim growth between September 2020 and January 2021, which overlaps with the ETH growth phase. It also overlaps with the BTC cluster shrinking phase, which suggests that these events can be related with each other. No significant trends can be seen for *s* = 60 min and *s* = 360 min. The nodes that share the cluster with BNB most frequently are FTT and ETH.

Finally, the ONT cluster also shows its specific growth phase between May and July 2021 (*s* = 10 min and *s* = 60 min), outside of which no significant trend can be seen. NEO and IOTA are the nodes that appear the most frequently in the same cluster with ONT. In general, Figures 12–15 show highly unstable composition of the analyzed clusters. This outcome differs from the results of some earlier studies based on data from more a distant past that reported stability of the cryptocurrency clusters (e.g., [76]). Additionally, the identified community structure of the market differs from the result of another study, where a core-periphery structure was identified instead [72].

Our discussion hitherto is focused on the simultaneous time series without delays between them. However, there is an interesting question whether the most capitalized and liquid cryptocurrencies like BTC and ETH drive the remaining ones, which can generate the delayed cross-correlations that can be observable. In order to address this question, we calculated the coefficients *ρ* (BTC,X) *q* (*s*, *τ*) for all the cryptocurrency pairs (BTC,X) and (ETH,X), where X stands for any cryptocurrency other than BTC and ETH. A time lag *τ* that can assume two values: *τ* = −1 min and *τ* = 1 min, defines whether the BTC (ETH) time series is advanced or lagged relatively to the second time series. For these two cases, we calculate the average coefficients *ρq*(*<sup>s</sup>*, *τ*) for BTC and ETH (the averaging is carried out over all other cryptocurrencies X).

Figure 16 shows the results for *q* = 1 and *q* = 4 and for the shortest scale *s* = 10 min (a potential effect of 1 minute delay can be too weak to be detectable on longer scales). If the time series of the BTC returns is considered, *ρq*(*<sup>s</sup>*, *τ*) is significantly larger for *τ* = 0 than for *τ* = ±1. For *q* = 1 the advanced BTC time series produces larger *ρq*(*<sup>s</sup>*, *τ*) than the lagged one. This difference is statistically significant. For *q* = 4 both shifted time series produce *ρq*(*<sup>s</sup>*, *τ*) with comparable magnitude for a vast majority of windows with a few exceptions, where the advanced BTC time series produces slightly stronger crosscorrelations than the lagged one does. The qualitatively similar results are obtained for the advanced and lagged ETH time series. We can therefore conclude that by shifting the time series representing BTC or ETH we still preserve some amount of the valid detrended cross-correlations. The relative dominance of the advanced (*τ* = −1 min) time series over the lagged (*τ* = 1 min) ones sugges<sup>t</sup> that the remaining part of the market absorbs information that occurred first in the price fluctuations of BTC and ETH with a time needed for this absorption being as long as a minute. An opposite process of information transfer from the less liquid cryptocurrencies to BTC and ETH cannot be detected based on our

data set. It must be noted, however, that both the BTC and ETH returns exhibit a detrended autocorrelation with the length of more than 1 min. Such an autocorrelation can artificially produce the delayed detrended cross-correlations which can manifest themselves in a way similar to that observed in Figure 16. We cannot therefore answer the formulated question decisively.

**Figure 16.** Mean lagged *q*-dependent detrended cross-correlation coefficient *<sup>ρ</sup>q*(*<sup>s</sup>*, *τ*) as a function of time after averaging over all the considered cryptocurrencies other than BTC and ETH. Time series representing BTC and ETH returns have been advanced (green) or delayed (red) by *τ* = 1 min and compared with the original non-shifted time series (orange). Two values of the filtering parameter *q* are shown: *q* = 1 (all fluctuations enter with the same weight, the first and third panels) and *q* = 4 (large fluctuations are amplified, the second and fourth panels).

Our former studies of the cryptocurrency market showed that, recently, it begun to be positively or negatively cross-correlated in some specific periods with the traditional financial markets like the stock market, the currency exchange market, and the commodity markets [5,13]. Among such periods of the statistically significant detrended cross-correlations there was the COVID-19 pandemic in the United States: the very first case on the US territory in the end of January 2020, the first COVID-19 wave outburst in April, and the second wave development in June–July, and the subsequent pandemic slowdown, which brought the across-market rally starting in September 2020. As we have already collected more contemporary data that end in October 2021, we are able to extend our analysis of the detrended cross-correlations between the cryptocurrencies and a few other financial assets. We consider the logarithmic price returns of a few basic cryptocurrencies (BTC, ETH, DASH, EOS, and XMR), the main regular currencies (AUD, CAD, CHF, CNH, CZK, EUR, GBP, JPY, MXN, NOK, NZD, PLN, and ZAR), sample commodities (crude oil, copper, silver, and gold), and the most important stock market indices (S&P500, NASDAQ100, Russel 2000, DJIA, FTSE, DAX, and NIKKEI). All the assets except the stock market indices are priced in US dollars (data from Dukascopy [77]).

Figure 17 shows the historical quotes of S&P500 and BTC together with the distinguished periods of the elevated detrended cross-correlations inside the cryptocurrency markets. One can see that these periods are associated with specific market events that are observed in the historical data: the all-market surge at the COVID-19 pandemic onset in March-April 2020, the second pandemic wave in June–July 2020, a market rally and the following drawdowns in September–October 2020, the cryptocurrency market rally in March–April 2021 and a surge and a subsequent rally in September–October 2021. Looking from a macroscopic perspective, in all these cases the coarse-grained behaviours of S&P500 and BTC were similar to each other at least for some period of time.

**Figure 17.** Temporal co-evolution of BTC price in USD (maroon) and the S&P500 index (blue) over the years 2020–2021. Periods, in which *<sup>ρ</sup>q*(*s*) calculated for these two assets exceed a threshold of 0.25 for *s* = 360 min and *q* = 1 (see Figure 18), are denoted by grey vertical strips. Specific market events are indicated by Roman numerals: I—the all-market surge at the COVID-19 pandemic onset in March–April 2020, II—the second pandemic wave in June–July 2020, III—a market rally and the following drawdowns in September–October 2020, IV—the cryptocurrency market rally in March–April 2021, and V—a surge and a subsequent rally in September–October 2021.

To inspect this issue in more detail, we calculated the *q*-dependent detrended crosscorrelation coefficients for all the possible pairs of the considered assets. Before we did this, we had to concord all the time series by eliminating the gaps caused by different trading hours. The results for *q* = 1 and *q* = 4 and for *s* = 10 min and *s* = 360 min are shown in Figure 18. For both values of the filtering parameter *q*, the cross-correlations are stronger on the long time scale and weaker on the short one. Except for the maximum of *<sup>ρ</sup>q*(*s*) that occurred for *q* = 4 and *s* = 10 min in the end of June 2020, which is not present at all for *q* = 4 and *s* = 360 min and for *q* = 1, all the other periods of the amplified cross-correlations can be observed in each case. The maxima of *<sup>ρ</sup>q*(*s*) calculated for BTC and the traditional assets occur, roughly, over the same periods than the maxima of the inner cross-correlations on the cryptocurrency market.

Different traditional assets reveal different levels of the detrended cross-correlation with BTC: the strongest correlations can be detected for S&P500 and other stock indices, while the weaker but also significant ones for crude oil, copper, CAD and other regular currencies except for JPY and, to a much smaller extent gold. The Japanese currency is significantly anticorrelated with BTC in the periods, in which the other assets are positively cross-correlated. This means that JPY can be used for the hedging purposes while investing on the cryptocurrency market. After comparing the cross-correlation strength for *q* = 1 with that for *q* = 4, we may conclude that, during the large fluctuation periods, the traditional assets are less strongly cross-correlated with BTC than during the smaller fluctuation periods. They also need rather long time scales to be fully built up. What can be inferred from these results is that the detrended cross-correlations are weaker in 2021 than they used to be in 2020, but they are still stronger than the corresponding cross-correlations before the COVID-19 pandemic.

**Figure 18.** The *q*-dependent detrended cross-correlation coefficient *<sup>ρ</sup>q*(*s*) calculated in 10-day-long moving windows with a 1-day step for BTC and the traditional market assets: the S&P500 index (blue), crude oil price (CL, black), copper price (HG, brown), gold price (XAU, yellow), and a few regular currencies expressed in the US dollars: euro (EUR, cyan), Swiss franc (CHF, orange), Canadian dollar (CAD, light green), Japanese yen (JPY, magenta), and Norwegian krone (NOK, red). Two temporal scales *s* (*s* = 10 min in the first and third panels, and *s* = 360 min in the second and fourth panels) and two filtering parameter *q* values (*q* = 1 in the first and second panels, and *q* = 4 in the third and fourth panels) are shown. The horizontal dashed line at *<sup>ρ</sup>q*(*s*) = 0.25 in the second panel denotes a discrimination threshold applied to determine the shaded regions in Figure 17.

Based on the coefficients *ρ*(*<sup>i</sup>*,*j*) *q* (*s*), where *i* and *j* labels the cryptocurrencies and traditional assets, we created the related minimal spanning trees. A few sample trees for specific moving window positions are presented in Figure 19. It is easy to notice that the detrended cross-correlation strength between BTC and the traditional markets, the closest ones being the stock markets and not the currency markets is much smaller than the analogous strength among the traditional assets representing the same market type and even different market types. Topology of the MSTs is heterogeneous with both the significant hubs (S&P500, AUD, EUR, and some cryptocurrency) and the long branches.

**Figure 19.** Minimal spanning trees calculated from a distance matrix **<sup>D</sup>***q*(*s*) based on *<sup>ρ</sup>q*(*s*) for *q* = 1 and *s* = 10 min. The data used to create MSTs consists of cryptocurrencies (BTC, ETH, DASH, EOS, and XMR), regular currencies (AUD, GBP, NZD, MXN, ZAR, CNH, EUR, CHF, JPY, CZK, NOK, CAD, and PLN), commodities (gold-XAU, silver-XAG, copper-HG, and crude oil-CL), as well as stock market indices (S&P500-SP, NASDAQ100-NQ, Russel 2000, FTSE, DAX, NIKKEI, and DJIA) in 10-day-long moving windows ended at specific dates: (**a**) 31 March 2020 (highly correlated markets during the pandemic onset in the United States), (**b**) 19 May 2020 (maximum cross-market correlations), (**c**) 28 January 2021 (the GameStop short squeeze related market turbulence accompanied by the cryptocurrency market decoupling), (**d**) 9 March 2021 (the elevated market cross-correlations), (**e**) 30 July 2021 (the cryptocurrencies starting a rally phase with minimum cross-market correlations), and (**f**) 4 October 2021 (the latest phase of the cross-market correlations).
