# Complexity in Economic and Social Systems: Cryptocurrency Market at around COVID-19

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## Abstract

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## 1. Introduction

#### 1.1. Money, Fiat Currencies, and Cryptocurrencies

#### 1.2. Basic Information on the Blockchain Technology

#### 1.3. Other Applications of the Blockchain Technology

#### 1.4. Cryptocurrency Market

## 2. Methods and Results

#### 2.1. Data

#### 2.2. Multifractal Formalism

#### 2.3. Multifractal Properties of the Cryptocurrency Market

#### 2.4. Cryptocurrency Market Versus Standard Markets

#### 2.5. Cryptocurrency Market Structure

## 3. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of the BTC/USDT exchange rate (top) together with the corresponding logarithmic returns (bottom). Several interesting events can be distinguished like start of a bull market in April 2019 and its end in July 2019, a sudden decrease and then an equally sudden increase in October and November 2019, the Covid-19 pandemic outbreak and related panic in March 2020 and the pandemic’s 2 wave in June 2020. Local extrema of $P\left(t\right)$ are indicated by the vertical (time) and horizontal (price) dotted lines.

**Figure 2.**(Top) Characteristic values of the Hölder exponent: ${\alpha}_{\mathrm{min}}$ (green line, bottom), ${\alpha}_{0}$ (red line, middle), and ${\alpha}_{\mathrm{max}}$ (blue line, top)—see Equation (9) in Section 2.2 and Figure 3—describing the singularity spectra $f\left(\alpha \right)$ for the index returns representing 8 the most capitalized cryptocurrencies, calculated in a 30-day-long moving window with a step of five days and for $-3\le q\le 3$. Each date represent a window that ends on that day. (Upper middle) The same quantities as in the top panel, but here calculated for the BTC/USDT exchange rate returns. Three interesting cases of small ${\alpha}_{\mathrm{min}}$ are indicated by dashed circles. (Lower middle) Scaling exponent $\gamma $ of the cumulative distribution function fitted to tails of the empirical cdf in each moving window position. Values equal or below $\gamma =2$ correspond to Lévy-stable distributions. (Bottom) Total cryptocurrency market capitalization and new Covid-19 cases in the world as function of time. Characteristic events are indicated by vertical dashed lines and Roman numerals: Start of a bull market in April 2019 (event I), its end in July 2019 (event II), the Covid-19 panic in March 2020 (event III), and start of the 2nd wave of the pandemic in May-June 2020 (event IV).

**Figure 3.**(Left) Cumulative distribution function $P(X>|{r}_{\mathsf{\Delta}t}\left|\right)$ calculated in 30-day windows. Two extreme cases of power-law tail are shown with the scaling exponent $\gamma \approx 1.8$ (mid February–mid March 2020) and $\gamma \approx 3.2$ (July 2019) representing stable and unstable distributions, respectively. (Right) Singularity spectra $f\left(\alpha \right)$ calculated in the same windows as above. An example of asymmetric, bifractal-like spectrum (mid February - mid March 2020) and an example of symmetric spectrum (July 2019) are shown together with characteristic values of the Hölder exponent: ${\alpha}_{\mathrm{min}}$, ${\alpha}_{0}$, and ${\alpha}_{\mathrm{max}}$ (see Equation (9) in Section 2.2).

**Figure 4.**Temporal evolution of the detrended cross-correlation coefficient $\rho (q,s)$ calculated for the BTC/USD exchange rate and the conventional assets expressed in US dollar: Japanese yen (JPY), Canadian dollar (CAD), Swiss franc (CHF), crude oil (CL), silver (XAG), gold (XAU), copper (HG), and the S&P500 index. The $\rho (q,s)$ coefficient was calculated in a moving 10-day-long window with a step of 1 day and its s and q parameters are represented by $s=10$ min (the shortest scale), $s=360$ min (approximately a trading day in the US stock market), $q=1$ (all data points are considered), and $q=4$ (only the data points with large amplitude are considered). In each panel events with the statistically significant, genuine cross-correlations are marked with dashed ellipses. The daily number of new Covid-19 cases in the United States is also shown for a comparison (bottom). The particular market events are indicated: (1) A sharp drop of the US stock market indices after the first case of Covid-19 had been identified in the United States; (2) a Covid-19 outburst related panic on the financial markets; (3) a bear market return on risky assets that was related to the 2nd wave of the pandemic.

**Figure 5.**Temporal evolution of $\rho (q,s)$ calculated for the ETH/USDT exchange rate and the conventional assets expressed in US dollar: Japanese yen (JPY), Canadian dollar (CAD), Swiss franc (CHF), crude oil (CL), silver (XAG), gold (XAU), copper (HG), and the S&P500 index. For more description see caption to Figure 4.

**Figure 6.**Minimal spanning trees (MSTs) calculated based on the q-dependent detrended correlation coefficient $\rho (q,s)$ for the exchange rates of a form X/BTC, where X stands for one of 128 cryptocurrencies traded on Binance [21]. Each node is labeled by the corresponding cryptocurrency ticker. All trees correspond to $q=1$. On the left there are MSTs obtained for $s=10$ min, while on the right there MSTs obtained for $s=360$ min. Each row shows MSTs calculated in a different period (a 7-day-long moving window with a step of 1 day): January 2019 (top), July 2019 (middle), and March 2020 (bottom).

**Figure 7.**Minimal spanning trees (MSTs) calculated based on the q-dependent detrended correlation coefficient $\rho (q,s)$ for the exchange rates of a form X/BTC, where X stands for one of 128 cryptocurrencies traded on Binance [21]. Each node is labeled by the corresponding cryptocurrency ticker. All trees correspond to $q=4$. On the left there are MSTs obtained for $s=10$ min, while on the right there MSTs obtained for $s=360$ min. Each row shows MSTs calculated in a different period (a 7-day-long moving window with a step of 1 day): January 2019 (top), July 2019 (middle), and March 2020 (bottom).

**Figure 8.**Network characteristics describing minimal spanning trees (MSTs) calculated for $q=1$ and for the following scales: $s=10$ min, $s=60$ min, and $s=360$ min. The average path length $\langle L(q,s)\rangle $ between a pair of MST nodes (top), the average q-dependent detrended cross-correlation coefficient $\langle \rho (q,s)\rangle $ (upper middle), the maximum node degree ${k}_{\mathrm{max}}(q,s)$ (lower middle), together with the total market capitalization in US dollars and the daily number of new Covid-19 cases in the world (bottom). Several events related to a relatively strong cross-correlations are marked with vertical dashed lines, Roman numerals, and dashed ellipses: Start of a bull market in April 2019 (event I) and its continuation in May 2019 (event Ia), a peak of the bull market in July 2019 (event II), a local peak followed by a sharp drop of the market capitalization in November 2019 (event III), the Covid-19 panic in mid March 2020 (events IV-V), and the 2nd Covid-19 wave from May 2020 (event VI).

**Figure 9.**The same network characteristics describing MSTs as in Figure 8, but here calculated for $q=4$.

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**MDPI and ACS Style**

Drożdż, S.; Kwapień, J.; Oświęcimka, P.; Stanisz, T.; Wątorek, M.
Complexity in Economic and Social Systems: Cryptocurrency Market at around COVID-19. *Entropy* **2020**, *22*, 1043.
https://doi.org/10.3390/e22091043

**AMA Style**

Drożdż S, Kwapień J, Oświęcimka P, Stanisz T, Wątorek M.
Complexity in Economic and Social Systems: Cryptocurrency Market at around COVID-19. *Entropy*. 2020; 22(9):1043.
https://doi.org/10.3390/e22091043

**Chicago/Turabian Style**

Drożdż, Stanisław, Jarosław Kwapień, Paweł Oświęcimka, Tomasz Stanisz, and Marcin Wątorek.
2020. "Complexity in Economic and Social Systems: Cryptocurrency Market at around COVID-19" *Entropy* 22, no. 9: 1043.
https://doi.org/10.3390/e22091043