## 3 Results
The plots for \(\tau=0\) and \(\tau=\pm 1\) clearly differ from each other: for the shifted time series, \(\rho_{q}(s)\) is more variable than for the simultaneous ones. However, what is similar is that here, \(\rho_{q}(s)\) also shows steeper growth for \(q=2\) than for \(q=4\)--see Figure 5(top panels, green and red lines). For \(q=2\), there is no visible difference between the results if either BTC or ETH leads (top left). This changes after we move to higher \(q\)s: for \(q=4\) there is a noticeable difference between the corresponding lines, with higher values of \(\rho_{q}(s)\) if BTC leads. This difference, however, is much smaller that that between the results for \(\tau=0\) and \(\tau=\pm 1\). This result is somehow expected, because contemporary markets operate at scales that are much shorter than 1 min. However, due to the long-range temporal autocorrelation in volatility lasting up to a few trading days [44] even if we shift one time series with respect to the other by 1 min, the fractal structure of the detrended cross-correlations can still be observed for these time series. Figure 4: **(Main panels)** Fluctuation functions \(F_{q}^{\rm XY}(s)\) calculated for the time series of logarithmic price returns \(r_{\Delta t}(t)\) for X = BTC and Y = ETH. In each panel, vertical dashed lines denote a range of time scales \(s\) for which a power-law model can be fitted to \(F_{XY}^{q}(s)\). Extreme values of \(q\) are also shown. Three cases are considered: both time series are simultaneous (**top**), time series representing BTC is advanced by \(\tau=1\) min (**middle**), and time series representing ETH is advanced by \(\tau=1\) min (**bottom**). (**Insets**) The bivariate scaling exponent \(\lambda(q)\) vs. the mean univariate scaling exponent \(h_{XY}(q)\) calculated for the same time series. Error bars denote the standard errors. We apply the same formalism to the volume traded \(V_{\Delta t}(t)\) and plot the corresponding quantities in Figure 6. There is a slightly broader range of scales for which a power-law dependence can be seen (almost two decades for the simultaneous time series and two decades for the lagged ones) than was seen for \(r_{\Delta t}(t)\) in Figure 4. The range of \(q\) over which multiscaling can be detected is, however, narrower than for \(r_{\Delta t}(t)\) in each case, and is restricted to positive values of \(q\) only. Another significant difference concerns \(\lambda(q)\) and \(h_{XY}(q)\), which are sizeably separated even for \(\tau=0\). For \(\tau=\pm 1\) min, this difference is also more pronounced than for the price returns, but here, one can notice broken symmetry between the BTC- and ETH-led time series - even in the case of \(q=2\): for the advanced ETH, the difference \(d_{XY}(q)\) is larger than in the opposite case. This effect is also seen in Figure 5b (middle). Despite the fact that these gaps are larger, they asymptotically decrease with increasing \(q\), suggesting that in the case of volume, \(\rho_{q}(s)\) increases more slowly with \(s\) for larger \(q\). Examples of this increase in \(\rho_{q}(s)\) are plotted in Figure 5a for \(q=2\) (middle left) and \(q=4\) (middle right). For short scales, the detrended coefficient for the simultaneous time series (\(\tau=0\)) is substantially larger than for the lagged ones, while for \(s>1000\) (10,000 s), this difference vanishes in both cases (for both \(q=2\) and \(q=4\)). This effect can be understood to be such that, on sufficiently long scales, all the information that has any meaning to the market has already managed to be exchanged by the major cryptocurrencies. On the other hand, the larger values of \(\rho_{q}(s)\) for \(\tau=1\) min suggest that information is transferred slightly faster from BTC to ETH than in the opposite direction. This can be explained by the BTC dominance in the market.