*2.6. Cross-Correlations among Cryptocurrencies*

Information that flows into the market may have the same impact on certain assets that, for example, share similar characteristics, such as the market sector, the main shareholders, or, in the case of cryptocurrency, the type or consensus mechanism [86]. This can lead to the emergence of cross-correlation between such assets and to a certain hierarchy of cross-correlations (e.g., sector, subsector, and bilateral ones) [87]. The correlation structure is a dynamical property that can change suddenly and substantially over time as the market reacts to perturbations [88]. In quiet times, it is well-shaped, elastic, and hierarchical, whereas, during periods of turmoil, it becomes centralized and rigid. This dual behavior is characteristic for the developed markets, while a lack of cross-correlations or a persistent centralization may be attributed to immaturity.

The market cross-correlation structure can be concisely characterized by the matrix approach. For a set of *N* time series of log-returns representing different cryptocurrencies *N*(*N* − 1)/2, the *q*-dependent detrended cross-correlation coefficients *ρ ij <sup>q</sup>* (*s*) can be calculated, where *i*, *j* = 1, ... , *N* and *ρ ij <sup>q</sup>* = *ρ ji <sup>q</sup>* , which form a *q*-dependent detrended cross-correlation matrix **C***q*(*s*). Due to the fact that the cross-correlation strength increases typically with scale for all the asset pairs, the differences in *ρ ij <sup>q</sup>* (*s*) are, on average, minimal for the shortest studied scale of *s* = 10 min. However, even in this case, it is easy to observe that different cryptocurrency pairs reveal strong differences. Figure 9 presents the complete matrix **C***q*(*s*) with the cryptocurrencies ordered according to the average inter-transaction time *δt <sup>t</sup>*. The ordering allows one to find even by eye a significant cross-correlation between *δt <sup>t</sup>* and *ρ ij <sup>q</sup>* : the shorter this time is, the stronger the cross-correlations are. In full analogy to other markets, information needs time to propagate over the whole cryptocurrency market and the propagation speed is crucially dependent on the cryptocurrency liquidity, which can roughly be approximated by the transaction number per time unit. Based on the exact values of *ρ ij <sup>q</sup>* (*s*), one can notice that even the least frequently traded cryptocurrencies from the considered basket develop statistically significant dependencies among themselves. This, however, might not be true for even less capitalized tokens, which can have idiosyncratic dynamics.

**Figure 9.** The *q*-dependent detrended cross-correlation matrix entries *ρ ij <sup>q</sup>* (*s*) calculated from time series of log-returns representing 70 cryptocurrencies with *q* = 1 and *s* = 10 min. Cryptocurrencies have been sorted according to the average inter-transaction time *δt* in increasing order (top to bottom). The color-coding scheme is shown on the right.

The correlation matrix **C***q*(*s*) can be transformed into a distance matrix **D***q*(*s*) with the entries

$$d\_q^{ij}(\mathbf{s}) = \sqrt{2(1 - \rho\_q^{ij}(\mathbf{s}))},\tag{12}$$

which differs from the former in that its entries *d* (*ij*) *<sup>q</sup>* are metric. **D***q*(*s*) can be used for constructing a weighted graph with nodes representing cryptocurrencies and edges representing distances *d* (*ij*) *<sup>q</sup>* (*s*). Next, by using the Prim algorithm [89], one can construct the corresponding *q*-dependent detrended minimal spanning tree (MST), which can be considered as a connected minimum-weight subset of the graph containing all *N* nodes and *N* − 1 edges. The MST topology depends strongly on the cross-correlation structure of a market. A centralized market corresponds to a star-like MST, whereas a market containing idiosyncratic assets shows an MST with elongated branches and no dominant hubs. Figure 10 exhibits two MSTs created from all 70 cryptocurrencies for *q* = 1 (left) and *q* = 4 (right). The former involves cross-correlations between the fluctuations in all magnitudes, whereas the latter involves only the large fluctuations. For *q* = 1, the structure is dual-star with BTC and ETH as its central hubs. This is not surprising as both cryptocurrencies are distinguished by their fame and large capitalization, which makes them a kind of reference for the remaining cryptocurrencies. On the other hand, for *q* = 4, the structure is more distributed, with a primary hub, BTC, and a few secondary ones: LTC, XMR, and ONT. This means that the relatively large fluctuations are not collectively correlated, unlike the majority of fluctuations, and more subtle dependencies are present. This is in parallel with the conclusions based on the multifractal analysis, which were large fluctuations that carried clearly multifractal characteristics and long-term correlations, whereas the small fluctuations were much more noisy. It is worth mentioning that a similar behavior can be observed in the stock market, where the cross-correlation structure carried by the large fluctuations is much richer than that carried by the medium and small fluctuations [90]. However, in the stock market, the heterogeneous cross-correlation structure is more pronounced even in the latter case [86,90]. Since there is no clear division into market sectors [91], the cryptocurrency market appears to be less developed from this particular perspective.

**Figure 10.** Minimal spanning trees calculated from a distance matrix **D***q*(*s*) based on *ρq*(*s*) for *s* = 10 and for *q* = 1 (**left**) and *q* = 4 (**right**). Within each tree, the size of the vertex is proportional to the average value of the volume *W*Δ*<sup>t</sup>* for Δ*t* = 1 min, while the width of the edge is proportional to 1 − *d ij <sup>q</sup>* (*s*). The vertex sizes cannot be directly compared across the trees, however. Colors represent Groups I-III in terms of the trading frequency: *δt* < 1*s* (Group I, red), 1*s* ≤ *δt* < 2*s* (Group II, blue), and *δt* ≥ 2*s* (Group III, green).
