Complexity in economic and social systems: cryptocurrency market at around COVID-19
Stanisław Drożdż, Jarosław Kwapień, Paweł Oświęcimka, Tomasz Stanisz, Marcin Wątorek
eentropy
Article
Complexity in economic and social systems:cryptocurrency market at around COVID-19
Stanisław Dro˙zd˙z , Jarosław Kwapie ´n , Paweł O´swi˛ecimka , Tomasz Stanisz andMarcin W ˛atorek Complex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences,ul. Radzikowskiego 152, 31-342 Kraków, Poland Faculty of Computer Science and Telecommunication, Cracow University of Technology, ul. Warszawska 24,31-155 Kraków, Poland Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. prof. StanisławaŁojasiewicza 11, 30-348 Kraków, Poland * Correspondence: [email protected]: date; Accepted: date; Published: date
Abstract:
Social systems are characterized by an enormous network of connections and factors that caninfluence the structure and dynamics of these systems. Among them the whole economical sphere ofhuman activity seems to be the most interrelated and complex. All financial markets, including theyoungest one, the cryptocurrency market, belong to this sphere. The complexity of the cryptocurrencymarket can be studied from different perspectives. First, the dynamics of the cryptocurrency exchangerates to other cryptocurrencies and fiat currencies can be studied and quantified by means of multifractalformalism. Second, coupling and decoupling of the cryptocurrencies and the conventional assets can beinvestigated with the advanced cross-correlation analyses based on fractal analysis. Third, an internalstructure of the cryptocurrency market can also be a subject of analysis that exploits, for example, anetwork representation of the market. In this work, we approach the subject from all three perspectivesbased on data from a recent time interval between January 2019 and June 2020. This period includes thepeculiar time of the Covid-19 pandemic; therefore, we pay particular attention to this event and investigatehow strong its impact on the structure and dynamics of the market was. Besides, the studied data coversa few other significant events like double bull and bear phases in 2019. We show that, throughoutthe considered interval, the exchange rate returns were multifractal with intermittent signatures ofbifractality that can be associated with the most volatile periods of the market dynamics like a bull marketonset in April 2019 and the Covid-19 outburst in March 2020. The topology of a minimal spanning treerepresentation of the market also used to alter during these events from a distributed type without anydominant node to a highly centralized type with a dominating hub of USDT. However, the MST topologyduring the pandemic differs in some details from other volatile periods.
Keywords: complex systems; cryptocurrencies; multifractal analysis; detrended cross-correlations;minimal spanning tree
1. Introduction
Whether complexity of a system is viewed in the purely intuitive sense of a nontrivial order thatemerges spontaneously from an overall disorder or it is grasped more formally using one of several dozenmathematical, physical, and information-theoretic measures, we are surrounded by its signatures and face
Entropy , xx a r X i v : . [ q -f i n . S T ] S e p ntropy , xx , 5 2 of 28 its manifestations almost everywhere. We are complex ourselves: We live in a society that is complex andwe interact with others in a complex way. There is no exaggeration in a statement that our society is the mostcomplex structure known to us in the universe. Social phenomena like the emergence of communicationand cooperation, build-up of hierarchies and organizations, opinion formation, the emergence of politicalsystems, and the structure and dynamics of financial markets are all among the iconic examples of thereal-world complexity [1–3].Specialists from such disciplines like mathematics, physics, information theory, and data scienceworking together with econometrists, sociologists, quantitative linguists, and psychologists for morethan a quarter century have already been dealing with such phenomena trying to describe them in alanguage of exact science, and to model and explain them using methods and tools that had earlier beenapplied successfully to natural systems. while much has already been done and much has been achieved,the complexity of the social and economic systems is still far from being properly understood. This iswhy every possible effort and every meaningful contribution is welcome as it can bring us closer tothe ultimate goal of understanding complexity both in reference to these systems in particular and as aphysical phenomenon in general. It is also important to approach the problem from different angles bycollecting many interdisciplinary works and views in one place like this Special Issue as human societyeludes any narrow-scope, single-discipline analysis. Among a variety of emergent phenomena that we observe in human society, one of the most importantis money. It appeared spontaneously and independently in many cultures and, although it used to havedifferent material forms in different regions, it always served the same purpose: To facilitate trade byavoiding a problem of double coincidence of needs that restricts barter trading severely and inherently.According to economical models, a status of money is acquired in a process of the spontaneous symmetrybreaking by a commodity that is the most easily marketable or, in other words, that is the most liquidone [4,5]. After receiving such a status by some commodity, its liquidity is amplified by a kind ofself-propelling mechanism, because everybody desires to have an asset that is considered as the mostdesirable by others. However, there is another condition for a commodity to be used as money: Its valueexpressed in other assets has to be viewed as stable. Sometimes it happens that current money loses itsvalue which causes people to withdraw themselves from using it and to replace it with some other, morestable asset. Thus, for a given asset its status of money may either be durable or temporary. This is animportant issue in contemporary economy based on fiat money that does not have any intrinsic valueunlike the assets that used to play a role of money earlier in history. Value of the fiat currencies dependscrucially on policies of the central banks, which can be subject to change. Moreover, the central banks mayincrease money supply at any time, which can lead to inflation rate increase. This undermines confidencein the official currencies and became the ignition to introduce cryptocurrencies over a decade ago.The first cryptocurrency was proposed in 2008—Bitcoin (BTC) [6]. The idea behind it was todecouple a currency from any institution or government, while preserving its status of a universalmeans of exchange, and to base a trust in this currency solely on a technology that supports it. Such acurrency had to combine the advantages of both cash and electronic money: Anonymity of use (like cash)and capability of being transferred immediately to any place in the world (like electronic money). Thealready-existing technologies of asymmetric cryptography and distributed database (with a new consensusmechanism—“proof of work”) were linked into a decentralized secure register—blockchain [7] that formsa staple of BTC. Unlike traditional currencies, Bitcoin has inherently limited supply to prevent any loss ofits value due to inflation. ntropy , xx , 5 3 of 28 The first widely recognized exchange enabling bitcoin to be exchanged for traditional currencies,Mt. Gox, was launched in July 2010 followed by the first online (black) market—Silk Road. The latterwas a place where one could anonymously buy anything and pay with bitcoins, which was the firstpractical application of a cryptocurrency. It significantly increased the demand and contributed to thefirst speculative bubble on BTC [8]. A subsequent crash occurred after closing Silk Road and suspendingtrade on Mt. Gox between October 2013 and February 2014. As Bitcoin’s recognition increased, the useof blockchain technology became more popular and it turned out that it can also be used for trustfulprocessing of computer codes in a decentralized way. In 2015 the Ethereum distributed computing networkwas launched [9], which allows one to issue private tokens through a so-called Initial Coin Offer (ICO) andto raise capital in a simplified way for various projects. An ICO boom that contributed to next speculativebubble on cryptocurrencies that occurred in 2017 (the ICO-mania [10]). At that time the number of issuedcryptocurrencies doubled from 700 to 1400 and the market capitalization reached 800 billion USD. A crashin January 2018, in which BTC lost over 80% of its value and other cryptocurrencies lost even 99%, may becompared with the dot-com bubble crash in 2000 that ended the most euphoric phase of investor attitudetowards the Internet-related companies. At present the market is more consolidated and shows signaturesof maturity [11].
In order to create an electronic “currency” that can easily be exchanged for goods and operated withoutany central authority, while at the same time that cannot be multiplied indefinitely like electronic files, it isrequired that all transactions involving that “currency” have to be registered publicly, which ensures thatno registry can be modified afterwards. The Bitcoin network register consists of a sequence of block filesbuilt one upon another (a blockchain) containing information about past transactions and the instances ofnew Bitcoin unit creation. A new network participant has to enter the network directly via a network clientor via an external wallet and must send information about the client’s address and a specified Bitcoin sumit owns. This information is then distributed to all other network nodes but, in return, the new participantis granted access to the complete information about other network node addresses and how many BTCunits belong to these addresses. Thus, credibility of the system is provided by the technology itself byimposing certain set of rules each network participant must obey and by allowing the network participantsto control each other. However, since the Bitcoin blockchain is public, one can trace the transaction historyof each unit, which in theory might compromise transaction anonymity.The transaction correctness is guaranteed with the help of the asymmetric cryptography. Private keysof a sender and a receiver are used to encode and to decode a transaction (i.e., to send and to receive coins),while their public keys are used as their public addresses allowing for their network identity verification.while such a transaction is visible to any other network participant, nobody can effectively alter andre-encode it as they do not know the private keys of the involved parties. For the network, in order tofunction correctly, the key implemented feature is a consensus mechanism that ensures that all participantsagree upon ownership of the cryptocurrency units and how many units total circulate. while collectinginformation from many transactions taking place on the Bitcoin network, the consensus mechanism hasto overcome a problem that some information sources can be unreliable. It is done by the so-calledproof-of-work (PoW) protocol used by miners, i.e., the network nodes with dedicated software that collecttransactions, verify their correctness, and integrate them into blocks. This is a resource-consuming taskso the miners are got to perform it by receiving new coins in exchange for sharing their resources withthe network. The new Bitcoin unit is generated only after majority of the miners agree upon correctnessof the new block and it has been distributed over the network. The block has to meet relevant criteriaexpressed by a specific form of the hash function to be considered as a valid one and included in the ntropy , xx , 5 4 of 28 blockchain. Each miner decides to include a given block into its own blockchain copy individually and theconsensus is settled in a kind of game with a Nash equilibrium state. One has to believe that majority ofother miners agrees on the specific block’s validity and adds it to its own blockchain or will not receive theprofit otherwise.Mining a new Bitcoin unit requires much energy to be spent so the very process demands optimizationof the resources used and discourage padding the blocks with fictitious information as a rejectionprobability for such a block by other miners is too large. Therefore it serves as a proof of work thata participant made the effort of maintaining the network, indeed. The employed solution that eachnew block contains a header of the previous one practically eliminates a problem of potential modifyingthe past transactions—it is not viable economically since it would require rebuilding of the entire chain.The Bitcoin protocol was designed in such a way that new blocks are formed with constant frequency,which is achieved by adjusting the amount of the corresponding calculations needed to the network’sactual computing power. Moreover, the reward for forming a new block is halved every 210,000 blocks inorder to approach quasi-asymptotically an impassable limit of 21 million Bitcoin units.The Bitcoin protocol is not static and undergoes constant modifications. A reason for this is thatthe protocol in its original design has some drawbacks that can challenge its security and lower comfortof its use. Among the pivotal issues is low performance (the network can handle only 5 transactionsper second on average, compared to 1700 transactions per second in the Visa network), high operatingcosts that equal the amount of electric energy consumed by small industrialized countries (like Irelandor Denmark [12]), and formidable computer facility. Moreover, one of the blockchain technologyadvantages—the inability of making changes—may sometimes be viewed as its disadvantage if oneconsiders the protocol correcting since it requires cloning of the entire network and abandoning theoriginal blockchain. Up to now a mechanism of reducing transaction size and allowing to pack moretransactions in a single block (called “segregated witness”, SegWit) has already been implemented andwork on another mechanism—“Lightning Network”—that allows for micropayments outside the mainblockchain and increasing the bandwidth, is currently underway.However, such changes are viewed by inefficient by many who prefer building alternative networksfrom scratch or by using only certain features of the Bitcoin protocol, while replacing other with bettersolutions. Thus, over the last decade, a multitude of different protocols were proposed and implemented,which led to introduction of new cryptocurrencies. Most of them still exploit the PoW protocol, but its themost popular alternative is the proof-of-stake (PoS) [13]. In this algorithm miners do not exist and the blockvalidation process is granted to some randomly chosen network nodes. Consistently, the block formationis not rewarded with new units but rather the validator nodes are rewarded with transaction fees. Fraudis discouraged by excluding the fraudulent participants from the network and securing that, in such acase, the reward for forming a new block is smaller than possible loss in already owned units. The mainadvantage of PoS is efficiency: Because of a lack of the complicated and long calculations, no specializeduser group is needed to confirm blocks and everything can be done faster than in the case of PoW. Thereare various versions of the PoS protocol, like the “delegated proof of stake” (DPoS) based on voting systemengaging trustful delegated network nodes or the “proof-of-authority” (PoA) based on granting reputationto the validator nodes instead of cryptocurrency units and abandoning the decentralization paradigm.Main advantage of both protocols (together with their hybrid versions) is scalability—more participantsmean larger transaction capacity of the related network. The first Bitcoin alternative that was introduced in 2011 and managed to survive until today wasLitecoin (LTC). Basically, this is a Bitcoin’s clone that differs from its parent in that it has a higher average ntropy , xx , 5 5 of 28 creation frequency (4 min) and a higher prospected total number of units (84 million) as well as it usesdifferent hash function (script instead of dSHA-256 used by Bitcoin). These changes allowed LTC formuch smaller resource demand than BTC and made LTC be computable on standard CPUs. The firstcryptocurrency that was not based on the Bitcoin’s PoW protocol was Ripple (XRP) [14] introduced in 2012.It was intended to be used as a method of transferring money between banks and stock markets in real timeeven outside national borders. In August 2020 XRP was the third cryptocurrency in terms of capitalization.A related cryptocurrency, Stellar (XLM), also offers transactions between financial institutions, but unlikeRipple based on a proprietary code its code is open source. Both XRP and XLM do not have a fixed supplylimit and, thus, they are subject to inflation.A separate group of cryptocurrency protocols was designed to ensure user anonymity.The corresponding cryptocurrencies are called “private coins”: Dash (DASH), Monero (XMR), Zcash (ZEC),and many others. Dash uses a two-layer network with PoW and miners in the first layer and PoS and“masternodes” in the second one. Monero, being considered as the most secure private coin and often usedby the criminal world [15], provides anonymity thanks to a Ring Confidential Transactions (RingCT) wherethe public keys (addresses) are hidden in the blockchain [16]. Zcash is based on a solution that allowsone to confirm information without having to disclose it. Zcash allows for perfect anonymity of both thesender and the recipient as well as transaction size. Since the anonymous addresses are compatible withthe public ones, transactions can be made between public and hidden wallets and vice versa. DASH andZEC have a maximum supply set in advance, while XMR does not.Apart from the cryptocurrencies, another important category of blockchain applications iscryptocommodities (together with the former called cryptoassets). They are automatically executedcomputer codes that perform certain actions if certain conditions are met. Cryptocommoditiesenable payments for using a decentralized computing network. The first such cryptoasset wasEthereum—an open-source computing platform designed for programming decentralized applicationsand smart contracts that was launched in 2015 [9]. This platform has its own programming language andits own cryptocurrency, Ethereum (ETH), that serves as a payment unit for carrying out computationaloperations on the platform. Ethereum is based on PoW consensus mechanism, but it uses anotherhash function (Ethash) supporting use of GPUs in the mining process and there is no upper limit onmining. Instead of fixed block size, here each block requires a specific number of “Gas” units related tothe computing power needed to complete the transactions it contains. The average block-completionfrequency is 15 s and the maximum transaction number per second is around 25. The Ethereum conceptgained quickly high popularity among the cryptocurrency community and, currently, ETH is the secondcrytocurrency in terms of capitalization. The success of smart contracts (i.e., computer codes allowing forautomatic execution and control of transaction agreement actions) and possibility of collecting funds underInitial Coin Offers on the Ethereum platform, gave a boost to the emergence of similar platforms offeringpossibility of creating applications in a decentralized environment. Major projects of this type include EOSand Cardano; both have their own cryptocurrencies and both allow for collecting funds under ICOs.Yet another group of cryptoassets are tokens, which are means of payment in decentralizedapplications built on platforms like Ethereum or contracts that are issued within ICOs for development ofblockchain ventures. They usually don’t have their own blockchain. In general, the blockchain technology,thanks to elimination of the need to trust individual participants of a given system and ensuring security,can satisfactorily be used wherever there is a central intermediary connecting sellers and buyers who earnson commissions (for example, Uber and Airbnb). Some of the already introduced applications in the tokenform are Augur (a platform enabling creation and participation in plants from any thematic range), Filecoin(a decentralized file storage system based on the PoW system that rewards users for sharing their computerstorage devices), and IOTA (a project of a partially decentralized, open settlement platform for the needsof the so-called “Internet of things”), Basic Attention Token (a project designed to connect advertisers and ntropy , xx , 5 6 of 28 content creators with users that rewards the creators for attracting users with the content they provide).Finally, the so-called “stable coins”—a combination of the token and cryptocurrency assets—allow one torelate their value to some other, more conventional asset like US dollar (e.g., USDT, USDC, TUSD, or PAX). Cryptocurrency trading is possible, because they are easily convertible to traditional currencieslike USD or EUR and to other cryptocurrencies. This possibility is provided by 330 trading platforms(August 2020) open 24 h a day, seven days a week. This, together with a fact that the most investors areindividuals, distinguishes the cryptocurrency market from Forex, where trading takes place from Mondayto Friday essentially on the OTC market where mainly banks and other financial institutions participate in.Another peculiarity of the cryptocurrency market is that there is no reference exchange rate unlike Forex,where such reference rates are provided by Reuters. The sole exception is Bitcoin, whose exchange rateto USD is given by futures quoted on Chicago Mercantile Exchange [17]. Decentralization of the marketmeans that the same cryptocurrency pairs are traded on different platforms, which—if accompanied bylimited liquidity—can lead to sizeable valuation differences between platforms that produce arbitrageopportunities, both the dual and triangluar ones [11,18,19].The entire cryptocurrency market capitalization is around 350 billion USD, which is close to thecapitalization of a middle-size stock exchange and also comparable with the capitalization of the largestAmerican companies. There are 6500 different cryptocurrencies on the market right now, which gives atotal of nearly 26,500 cryptocurrency pairs [20]. Founded in 2017, Binance [21] is currently one of the largestcryptocurrency exchange in terms of volume. Binance offers trading on approximately 650 cryptocurrencypairs including pairs with its own cryptocurrency called binance (BNB), used to pay commissions on thisexchange.The spectacular development of a cryptocurrency market has attracted much interest of the scientificcommunity. The first Bitcoin-related papers were published already in 2013–2015 [22,23], but a realboom on cryptocurrency-related publications occurred after 2017. Initially, only bitcoin was of significantinterest [24–26], but soon also other cryptocurrencies went under investigation [27–29]. Then thereappeared studies reporting on correlations within the market [30–38], and its relationship with regularmarkets [39–43]. Recently, some researchers focused their attention on possible use of BTC as a hedginginstrument for Forex [44], for gold and other commodities [45], as well as for the stock markets [46,47].There is also a few review papers devoted to the cryptocurrency markets:[11,48,49].The cryptocurrency market has already gone through a long route from a mere curiosity and aplayground for the technology enthusiasts, via an emerging-market stage characterized by a relatively smallcapitalization, poor liquidity, large price fluctuations, short-term memory, frequent arbitrage opportunities,and weak complexity, to a more mature form characterized by medium capitalization, improved liquidity,inverse-cubic power-law fluctuations [50,51], long-term memory, sparse arbitrage opportunities, andincreasing complexity. This is the most interesting aspect of the cryptocurrency market route to maturity:The signatures of complexity that are best quantified in terms of the multifractal analysis. See Ref. [11] fora comprehensive study of this transition started in 2012 and ended essentially in 2018, as viewed from themultifractality perspective. Here we shall consider a more recent period of 2019–2020, which comprises,among others, two significant events, i.e., the bull market between April and July 2019 and the Covid-19pandemics (from March 2020). Based on high-frequency data covering a large number of cryptocurrencypairs and a few principal traditional-market assets, we investigate a potential impact of these events onthe cryptocurrency market structure and its relation to the traditional markets. ntropy , xx , 5 7 of 28
2. Methods and Results
For this study we collected high-frequency recordings of X/BTC and BTC/USDT exchange rates,where X is one of 128 cryptocurrencies traded on Binance platform [21] and USDT is related to USD by a 1:1peg [52]. The exchange rates P ( t ) were sampled every 1 min. We calculated their normalized logarithmicreturns r ∆ t defined by r ∆ t = ( R ∆ t − µ R ) / σ R , R ∆ t ( t ) = log ( P ( t + ∆ t )) − log ( P ( t )) , (1)where µ R and σ R are mean and standard deviation of R ∆ t ( t ) , respectively, and ∆ t is sampling interval. Wealso collected 1-min quotes of several conventional assets expressed in US dollar—13 currencies: AUD,EUR, GBP, NZD, CAD, CHF, CNH, JPY, MXN, NOK, PLN, TRY, ZAR, three stock market indices: DowJones Industrial Average (DJI), Nasdaq100, S&P500, and four commodities: XAU (gold), CL (crude oil),XAG (silver), and HG (copper). They all come from Dukascopy platform [53], so do the BTC/USD andETH/USD exchange rates. These quotes were also transformed into time series of returns. Multifractal analysis is one of the most promising methods of studying empirical data representingnatural and social systems as it is able to quantify complexity of such systems and express it in a relativelysimple way with a small set of associated quantities. It has already been applied in many works tounivariate and multivariate data sets from a number of different systems: Physics [54], biology [55],chemistry [56], geophysics [57], hydrology [58], atmospheric physics [59], quantitative linguistics [60],behavioral sciences [61], cognitive structures [62], music [63], songbird rhythms [64], physiology [65],human behaviour [66], social psychology [67] and even ecological sciences [68], but especially financialmarkets [69–77].Let us consider two time series of the same length: x i , y i , where i =
1, ..., T ( T has to be large enoughto overcome statistical uncertainties). Signal profiles are created from these time series by integrating andsubtracting their mean: X ( j ) = j ∑ i = [ x i − (cid:104) x (cid:105) ] , Y ( j ) = j ∑ i = [ y i − (cid:104) y (cid:105) ] . (2)These signal profiles are then divided into segments ν of length s . They may be separate or partiallyoverlapping; if they are separate, their number is M s = (cid:98) T / s (cid:99) . A local trend is then removed from eachsegment by fitting the data with polynomials P ( m ) X , ν , P ( m ) Y , ν of degree m (typically, it is m = F xy is determined from the residual signals for each segment [81,82]: F xy ( ν , s ) = s s ∑ k = { (cid:104) X (( ν − ) s + k ) − P ( m ) X , ν ( k ) (cid:105) (cid:104) Y (( ν − ) s + k ) − P ( m ) Y , ν ( k ) (cid:105) } (3)and then it is used to calculate the q -th order fluctuation function [83]: F qxy ( s ) = M s M s ∑ ν = sign ( F xy ( ν , s )) | F xy ( ν , s ) | q /2 , (4)where sign ( F xy ( ν , s )) means a sign function. If F xy ( ν , s ) is considered as a value of a random variable,the parameter q resembles an exponent specifying the order of the moment: Its large positive values ntropy , xx , 5 8 of 28 favour segments characterized by large variance by increasing their relative magnitude with respect tosmall-variance segments, while negative values of q do the opposite. Thus, by applying different values of q , one can construct effective filters that select the segments of a certain variance range.The fluctuation function (4) has to be calculated for different segment lengths s . If F qxy ( s ) is of apower-law form, i.e., F qxy ( s ) q = F xy ( q , s ) ∼ s λ ( q ) , (5)where q (cid:54) =
0, the original time series x i and y i are fractally cross-correlated. If λ ( q ) = const, this ismonofractal cross-correlation, otherwise it is multifractal one.A special case is x i ≡ y i for all i (one signal). In this case we have F ( q , s ) = (cid:104) M s M s ∑ ν = [ F ( ν , s )] q (cid:105) q (6)and the fractal case corresponds to F ( q , s ) ∼ s h ( q ) , (7)where h ( q ) is the generalized Hurst exponent. For h ( q ) = const the signal is monofractal, otherwise it ismultifractal [80]. A useful measure of fractal properties is singularity spectrum f ( α ) defined by α = h ( q ) + qh (cid:48) ( q ) , f ( α ) = q [ α − h ( q )] +
1, (8)where α is the Hölder exponent. f ( α ) can be interpreted as a fractal dimension of the singularitiescharacterized by a given α . In the monofractal case it consists of a single point, while in the multifractalcase it can have a shape of inverted parabola or some asymmetric concave function.Width of f ( α ) can be interpreted as a measure of a signal’s complexity, because the wider it is,the more singularity types can be identified in this signal. This width depends on a range of q and it isquantified by ∆ α = α max − α min , (9)where α min = α ( q max ) and α max = α ( q min ) are the minimum and maximum value of α that have beencalculated for different values of q . Another important feature of the f ( α ) its left-right asymmetry [84]. Aleft-hand-side asymmetry corresponds to more diverse multifractality (stronger correlations) at the largeamplitude level, while a right-hand-side asymmetry indicates that signal parts with small amplitude are adominant source of multifractality.As F qxy ( s ) denotes the q th-order detrended covariance, one can define the q th-order detrendedcorrelation coefficient [85,86]: ρ ( q , s ) = F qxy ( s ) (cid:113) F qxx ( s ) F qyy ( s ) , (10)in analogy to the q th-order Pearson correlation coefficient. Here F xx and F yy are calculated from Equation (6).The coefficient ρ ( q , s ) can assume values in a range [ −
1, 1 ] provided q >
0. For q ≤ ρ ( q , s ) may fall outside that range, which requires more delicate interpretation [85].Therefore, many studies in which ρ ( q , s ) is used are carried out with a restriction q >
0. The coefficient ρ ( q , s ) describes detrended cross-correlations between two signals on different scales s after amplifyingdata points within a given amplitude range. This filtering ability of ρ ( q , s ) is its advantage over morestandard correlation measures, because the cross-correlation strength among empirical time series can besize-dependent [87]. The coefficient ρ ( q , s ) may be used for any two signals without a requirement thatthey have to be fractal. ntropy , xx , 5 9 of 28 We start our analysis by taking a look at the BTC/USDT exchange rate from 01/2019 to 06/2020. Thisperiod shown in Figure 1 (top panel) starts near the lowest point of the bear market ( ∼ Figure 1.
Time evolution of the BTC/USDT exchange rate (top) together with the corresponding logarithmicreturns (bottom). Several interesting events can be distinguished like start of a bull market in April 2019and its end in July 2019, a sudden decrease and then an equally sudden increase in October and November2019, the Covid-19 pandemic outbreak and related panic in March 2020 and the pandemic’s 2 wave in June2020. Local extrema of P ( t ) are indicated by the vertical (time) and horizontal (price) dotted lines. Despite of the fact that BTC/USDT rate is the most important observable on the cryptocurrencymarket since BTC has the largest capitalization, it cannot be used as a proxy allowing one to describedynamics of the whole market, which is in fact much richer. Thus, in order to express the evolution of asignificant part of the market in terms of a single quantity, a market index was created from the exchangerates X/USDT (with X standing for a cryptocurrency) for 8 the most capitalized cryptocurrencies: BTC,ETH, XRP, BCH, LTC, ADA, BNB, and EOS. In 2020, these assets stand for 88% of the market capitalization.In order to create the index, the exchange rates were summed with the same weight despite the differencein capitalization. A parallel, weighted index would predominantly reflect the dynamics of BTC, ETH, andXRP, so we prefer the unweighted version as more a diversified one. ntropy , xx , 5 10 of 28 Figure 2 shows results of the multifractal analysis of the cryptocurrency index returns and theBTC/USDT returns performed by using a moving window of 30 days with a 5-day step. Instead ofpresenting the singularity spectra f ( α ) for each window position, temporal evolution of the key quantitiesdescribing shape of these spectra is shown: α min ( t ) , α ( t ) , and α max ( t ) (see right panel of Figure 3 for theexamples). These quantities allow for inferring about the singularity spectrum localization, width, andpossible asymmetry of its shoulders [88]. We restricted the applied values of q to [ −
3, 3 ] for a reason thatwill be explained later. By looking at the spectra for BTC/USDT (the second topmost panel in Figure 2),one sees that a difference ∆ α = α max − α min describing the spectrum width is sufficient to infer aboutmultifractality of the data under study. This agrees with results of our previous study [11]. Figure 2. (Top) Characteristic values of the Hölder exponent: α min (green line, bottom), α (red line,middle), and α max (blue line, top)—see Equation (9) in Section 2.2 and Figure 3—describing the singularityspectra f ( α ) for the index returns representing 8 the most capitalized cryptocurrencies, calculated in a30-day-long moving window with a step of five days and for − ≤ q ≤
3. Each date represent a windowthat ends on that day. (Upper middle) The same quantities as in the top panel, but here calculated for theBTC/USDT exchange rate returns. Three interesting cases of small α min are indicated by dashed circles.(Lower middle) Scaling exponent γ of the cumulative distribution function fitted to tails of the empiricalcdf in each moving window position. Values equal or below γ = Except for July-August 2019, when f ( α ) is left-right symmetric ( α max − α ≈ α − α min ), throughoutthe remaining part of the analyzed period there is significant asymmetry with the left-hand shoulder( q >
0) being much longer than the right-hand one ( q < f ( α ) became extreme and revealed a bifractal-like ntropy , xx , 5 11 of 28 shape (see also [89]). Mathematical bifractals are characterized by the existence of only 2 singularity typeswith α = < α <
1. However, in practical situations, the finite-size effects smear the spectraso that in such a case there is a continuous transition between both singularity types and a spectrumconsists of a long left shoulder reaching a vicinity of α = α [84,90].Two characteristic cases of f ( α ) (symmetry and bifractal-like asymmetry) are shown in Figure 3 (rightpanel). Figure 3. (Left) Cumulative distribution function P ( X > | r ∆ t | ) calculated in 30-day windows. Two extremecases of power-law tail are shown with the scaling exponent γ ≈ γ ≈ f ( α ) calculated in the same windows as above. An example of asymmetric, bifractal-like spectrum (midFebruary - mid March 2020) and an example of symmetric spectrum (July 2019) are shown together withcharacteristic values of the Hölder exponent: α min , α , and α max (see Equation (9) in Section 2.2). On the probability distribution function level, the actual bifractal spectra occur if a signal under studyhas a heavy-tailed pdf in the Lévy-stable regime ( p ( | r ∆ t | ) ∼ | r ∆ t | γ + , where γ ≤ P ( X > | r ∆ t | ) ∼ | r ∆ t | γ (left panel) and f ( α ) (right panel) for two time windows thatshow clearly different properties of both cdf and f ( α ) —a symmetric f ( α ) corresponding to a steep cdfwith γ ≈ f ( α ) corresponding to aheavy-tail cdf with γ ≈ γ point to theaforementioned restriction − ≤ q ≤ F qxy and ∆ α : For | q | > p ( | r ∆ t | ) can diverge, so can F qxy ( s ) especially for small scales s .The BTC/USDT return distribution function reflects a combination of two factors: (1) How fast theinformation spreads over the market—the heavier tails are, the slower this spreading proceeds, and(2) how volatile is the market—periods that cover turmoils with high volatility also result in heaviertails of pdf/cdf. It was documented in Ref. [11] that along with a process of the cryptocurrency market ntropy , xx , 5 12 of 28 maturation the scaling exponent γ increases with time. This happens because as recognition of themarket and its capitalization increase, more and more transactions take place, which decreases the averageinter-transaction waiting time and allows the market participants to react faster. Faster reactions arecrucial for the market to become efficient, which means more Gaussian-like fluctuations (larger γ ). Onthe other hand, extremely large fluctuations and amplified volatility are characteristic for the periodswith negative events, which decrease γ . Figure 2 (the 3rd panel from top) shows a scaling exponent γ obtained by fitting a power-law function to the BTC/USDT returns cdf in each position of the 30-daymoving window. Indeed, such events like a bear market after July 2019 and the Covid-19 outbreak inMarch 2020 resulted in relatively small values of γ , while a bull market between April and July 2019 andan escape from conventional assets to alternative ones observed between January and February 2020 led tolarger values of γ .Even if BTC is only one of many actively traded cryptocurrencies on the Binance platform, its strongestposition due to the largest capitalization (between 50% and 70% of total market capitalization in theconsidered period) causes other cryptocurrencies to evolve accordingly. This‘observation comes from thetopmost panel of Figure 2 presenting α min , α , and α max for the 8-cryptocurrency index. Qualitatively,the temporal course of these quantities does not differ much from the temporal course of their counterpartsfor BTC/USDT (the 2nd panel from top). The only significant difference is that for the index a transitionto a bifractal-like f ( α ) spectrum in March 2020 was sharp and it was not preceded by its gradual changestarting from January 2020 as it was the case with BTC/USDT.By looking at the bottom panel of Figure 2, where total market capitalization is plotted as a functionof time together with the Covid-19 pandemic severity parametrized by the number of daily new cases,and by comparing this plot with the remaining three, one can infer about how various market eventsand the pandemic influenced complexity of the market dynamics. The main events are denoted byRoman numerals: The beginning of the bull market in April 2019 (event I), its end in July 2019 (event II),the Covid-19 panic in March 2020 (event III), and the second pandemic wave that started in May 2020 (eventIV). These events could be distinguished because they were associated with particularly large fluctuations(Figure 1). Among them, the events I, III, and IV had a significant impact on the multifractal properties ofthe exchange rate fluctuations by sizeable decreasing of α min (visible both for the cryptocurrency index andthe BTC/USDT exchange rate). However, the event II did not have such an impact. In contrast, the pdf/cdftails reflected overall market phase more than specific events except for the Covid-19 panic in March 2020. From a practical point of view, among the most interesting issues related to any asset and any marketis how much it is related to other assets or markets, and, in other words, whether it can be exploited forportfolio diversification and hedging [91–93]. As the investors may be interested in different time horizonsand may want to hedge against events of different magnitude, the q -dependent detrended cross-correlationcoefficient ρ ( q , s ) defined by Equation (10) is a measure that is particularly useful in this context since it issensitive to both scale and amplitude of the asset price returns. We choose the BTC/USD exchange rateas a representative of the whole cryptocurrency market—it is the most frequently traded asset, the mostcapitalized asset, and the most mature one (based on our previous results [11]). We calculate ρ ( q , s ) forthis rate and each of the remaining conventional assets listed in Section 2.1. However, we observe that thismeasure behaves similar for S&P500, Nasdaq100, and DJI, so we abandon the latter two indices and showonly the results for S&P500. In parallel, we neglect AUD, NZD, ZAR, CHN, MXN, EUR, GBP, NOK, TRY,and PLN as their correlations with BTC were close to zero throughout the period under consideration. Weconsider two temporal scales that correspond to different horizons: s =
10 min, which is the shortest scaleavailable provided we use 1-min returns, and s =
360 min that represents approximately a trading day in ntropy , xx , 5 13 of 28 the US stock market. The latter value means that in a moving widow there was only 10 segments overwhich the averaging was carried out in F qxy ( s ) (see Equation (4)), so we could not look at longer scales. Asregards the parameter q , we focused on q > q , but here we shallreport only the results for q = q =
4. The former choice did not favour any value range of thefluctuation function F xy since, for each segment ν in Equation (4), it was counted with the same weight.Therefore q = q = q were also investigated, but the related results fell between these twocases and, thus, they are not presented here. Moreover, the calculations for q > q as the event statistics became poor.Figure 4 displays temporal course of ρ ( q , s ) for a combination of the above-described cases of s and q .In each panel the cross-correlation coefficients for BTC and each of the 8 other assets are shown. Curiously,we do not observe any statistically significant values of ρ ( q , s ) during the whole year 2019 even thoughthere were then important events on the cryptocurrency market, like the bull and the subsequent bearmarket. However, these events were not related to any of the conventional assets considered here. Wesee that even the periods of high volatility in April and July-August 2019 did not cause any action thatcould potentially be sensed by the regular markets. We can explain this lack of reaction by a relativelysmall capitalization of the cryptocurrency market—far too low for the other markets to detect a possibleinflux of a capital withdrawn from cryptocurrencies (if such an influx actually took place). In 2019 therewas no turmoil in the conventional markets, thus nothing could correlate the cryptocurrencies with theconventional assets from this direction, too. ntropy , xx , 5 14 of 28 Figure 4.
Temporal evolution of the detrended cross-correlation coefficient ρ ( q , s ) calculated for theBTC/USD exchange rate and the conventional assets expressed in US dollar: Japanese yen (JPY), Canadiandollar (CAD), Swiss franc (CHF), crude oil (CL), silver (XAG), gold (XAU), copper (HG), and the S&P500index. The ρ ( q , s ) coefficient was calculated in a moving 10-day-long window with a step of 1 day andits s and q parameters are represented by s =
10 min (the shortest scale), s =
360 min (approximatelya trading day in the US stock market), q = q = ntropy , xx , 5 15 of 28 In contrast, there were 1 to 3 periods of significant inter-market cross-correlations in the first half of2020, dependent on s and q . The first important period in the end of January and the begin of February wasassociated with a sharp drop of S&P500 and other US stock market indices triggered by the first identifiedlocal case of Covid-19. The cryptocurrency market reacted rather moderately with only a short period oflarge and delayed fluctuations. This is why there are no significant elevation of | ρ ( q , s ) | for short scalesfor any return size. The cryptocurrency market must have been calm enough to delay reaction so longthat it is identifiable only on large scales (like s =
360 min). The cross-correlation is positive with the fiatcurrencies, while negative with the US stock markets. As all the considered assets are expressed in USD,the positive correlations of BTC with the fiat currencies in January/February 2020 mean that there was aglobal flee from US dollar to other major currencies that increased the corresponding exchange rates aswell as a flee from the US stock markets to the cryptocurrency market.Opposite situation took place during the pandemic’s 2nd wave in June 2020 (and, possibly, beyondthat month): The cross-correlations are stronger for q = q =
4. On the one hand, for s =
10 minmoderate values of ρ ( q , s ) , mainly positive ones, are seen for q =
1, but they are not seen for q =
4. On theother hand, for s =
360 min large values of ρ ( q , s ) are observed for q = q =
4. Therefore we still see that the correlations cannot be built in their full magnitude onshort scales and they need some time to develop completely. However, a larger ρ ( q , s ) for q = F qxy ( s ) are correlated in this case than in the case of q = s =
360 min than for s =
10 min. However, they are clearly evident even for s =
10 min. Interestingly, if we look at thelargest returns ( q = q =
1, the cross-correlations appearstrong between BTC and all other assets except for CHF (only small negative correlation) and gold (XAU).The corresponding values of ρ ( q , s ) are positive for S&P500, CAD, copper (HG), crude oil (CL), and silver(XAG), while they are negative for JPY. This cannot be viewed as a surprise since the Swiss franc andJapanese yen are considered safe assets together with gold and their pricing in USD behave differentlythan the remaining assets’ pricing did.For a comparison, Figure 5 shows ρ ( q , s ) calculated for the ETH/USDT exchange rate and the sameconventional assets as in the BTC/USDT case above. We see that the only qualitative difference betweenFigures 4 and 5 is a much smaller detrended cross-correlation coefficient value for q = ntropy , xx , 5 16 of 28 Figure 5.
Temporal evolution of ρ ( q , s ) calculated for the ETH/USDT exchange rate and the conventionalassets 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 toFigure 4. We have already discussed the fractal autocorrelations of the cryptocurrency exchange rates withrespect to US dollar and the cross-correlations between bitcoin and the assets representing conventionalmarkets. Now its time to look at the inner correlation structure of the cryptocurrency market itself.Our data set consists of 128 cryptocurrencies expressed in BTC. This effectively removes the impactof BTC on any other coin, so we have some insight into the market’s finer, secondary correlationstructure (the primary structure is such that all the cryptocurrencies are correlated with BTC and form themarket as a connected whole [11]). In our earlier work we identified that throughout short history of themarket, there were only two cryptocurrencies that played the role of the market’s center (in terms of thenetwork centrality): BTC for the most time and ETH in the first half of 2018. ETH, sometimes togetherwith USDT, was also identified as the most frequent secondary hub of the market, after BTC [11]. Here westudy the market’s structure between January 2019 and June 2020—a period that was not a subject of theprevious study.Minimal spanning tree is an acyclic spanning subset of a complete weighted network that is minimal interms of the total length of its edges. In a typical MST construction, the Pearson correlation coefficient [94]is used to form a correlation matrix that defines a complete network. Here we follow Refs. [11] and [87]and define the network based on the ρ ( q , s ) matrix. This matrix has entries equal to ρ ( q , s ) calculated forall possible pairs of the exchange rates X/BTC and Y/BTC, where X,Y denote any cryptocurrency from our N =
128 element set. By doing this, we obtain N ( N − ) /2 = ∗ = ρ ( q , s ) for ntropy , xx , 5 17 of 28 each choice of q and s (as before, here we restrict our discussion to q = q = s =
10 min, and s = d XY ( q , s ) = (cid:114) (cid:16) − ρ XY ( q , s ) (cid:17) . (11)Since − ≤ ρ ( q , s ) ≤ q >
0, we obtain limiting values for distance: 0 ≤ d XY ( q , s ) ≤
2, where d XY = d XY = d XY = √ d XY ( q , s ) we construct MST by using thePrim’s algorithm [95].Figure 6 shows q MSTs calculated for q = s =
10 min in Figure 6).However, no overwhelmingly dominant node was observed in MST corresponding to March 2020 and s =
360 min. In fact, the structure of the latter MST differs substantially from the structure of the remaining5 trees in Figure 6: It can be categorized as a distributed network in contrast to the generally centralizedform of the rest, where there is a clearly identifiable center (ETH or USDT) and the peripheries. Thereis a possible explanation why USDT becomes a central hub in turbulent periods, especially the suddendropdowns: Investors that want to close the cryptocurrency positions change them primarily to USDT,which is a stable coin pegged to USD [52] and only then to the proper US dollar. This manoeuvre canmutually correlate most cryptocurrencies via USDT. ntropy , xx , 5 18 of 28 Figure 6.
Minimal spanning trees (MSTs) calculated based on the q -dependent detrended correlationcoefficient ρ ( q , s ) for the exchange rates of a form X/BTC, where X stands for one of 128 cryptocurrenciestraded on Binance [21]. Each node is labeled by the corresponding cryptocurrency ticker. All treescorrespond to q =
1. On the left there are MSTs obtained for s =
10 min, while on the right there MSTsobtained for s =
360 min. Each row shows MSTs calculated in a different period (a 7-day-long movingwindow with a step of 1 day): January 2019 (top), July 2019 (middle), and March 2020 (bottom).
Now let us consider MSTs constructed from the filtered signals, in which the largest returns wereamplified by taking q = s =
10 min and March 2020, though its central hub (USDT) does notdominate the networks unlike it was for q = s =
360 min and March 2020. All the remaining trees reveal intermediateform between the centralized and distributed ones: There are several nodes that can be called local hubs.This is the case of the hierarchical networks that sometimes are scale-free. For s =
10 min, such a situationwas present in January 2019 (ETH and USDT) and July 2019 (USDT, ONT, XLM, THETA, BCPT, and RVN),while for s =
360 min similar situations also occurred in January 2019 (ETH, EOS, and LTC) and in July2019 (XLM, THETA, LOOM, USDT, ADA, AION, and DAX). ntropy , xx , 5 19 of 28 Figure 7.
Minimal spanning trees (MSTs) calculated based on the q -dependent detrended correlationcoefficient ρ ( q , s ) for the exchange rates of a form X/BTC, where X stands for one of 128 cryptocurrenciestraded on Binance [21]. Each node is labeled by the corresponding cryptocurrency ticker. All treescorrespond to q =
4. On the left there are MSTs obtained for s =
10 min, while on the right there MSTsobtained for s =
360 min. Each row shows MSTs calculated in a different period (a 7-day-long movingwindow with a step of 1 day): January 2019 (top), July 2019 (middle), and March 2020 (bottom).
The trees shown in Figures 6 and 7 represent only a few periods, but in order to look at the marketstructure evolution over the whole considered interval of time, it is not convenient to look at the treesfor individual windows. Therefore, we calculated a few network characteristics that grasp the essentialproperties of the MST topology in each window. These are the mean path length (cid:104) L ( q , s ) (cid:105) between a pairof the MST nodes (the averaging is carried out over all possible pairs) describing how distributed (large (cid:104) L ( q , s ) (cid:105) ) or concentrated (small (cid:104) L ( q , s ) (cid:105) ) is a tree, the mean q -dependent detrended cross-correlationcoefficient (cid:104) ρ ( q , s ) (cid:105) (the averaging is carried out over all possible cryptocurrency pairs), describing howstrong are typical network edges, and the maximum node degree k max ( q , s ) , describing how central is themain hub. Time evolution of these quantities is shown in Figure 8 for q = q = ntropy , xx , 5 20 of 28 Apart from two scales considered in Figures 6 and 7, i.e., s =
10 min and s =
360 min, we added a mediumscale of s =
60 min.
Figure 8.
Network characteristics describing minimal spanning trees (MSTs) calculated for q = s =
10 min, s =
60 min, and s =
360 min. The average path length (cid:104) L ( q , s ) (cid:105) between apair of MST nodes (top), the average q -dependent detrended cross-correlation coefficient (cid:104) ρ ( q , s ) (cid:105) (uppermiddle), the maximum node degree k max ( q , s ) (lower middle), together with the total market capitalizationin US dollars and the daily number of new Covid-19 cases in the world (bottom). Several events related toa relatively strong cross-correlations are marked with vertical dashed lines, Roman numerals, and dashedellipses: Start of a bull market in April 2019 (event I) and its continuation in May 2019 (event Ia), a peak ofthe bull market in July 2019 (event II), a local peak followed by a sharp drop of the market capitalizationin November 2019 (event III), the Covid-19 panic in mid March 2020 (events IV-V), and the 2nd Covid-19wave from May 2020 (event VI). ntropy , xx , 5 21 of 28 Figure 9.
The same network characteristics describing MSTs as in Figure 8, but here calculated for q = While (cid:104) ρ ( q , s ) (cid:105) is largely a different measure than the two other ones, (cid:104) L ( q , s ) (cid:105) and k max ( q , s ) canbe related to each other: If k max ( q , s ) is large, a majority of the nodes is connected to it and (cid:104) L ( q , s ) (cid:105) canbe small; the opposite relation is also true. Figure 8 confirms these observations for q =
1: Typically,the elevated values of (cid:104) L ( s ) (cid:105) (top panel) are associated with the suppressed values of k max ( s ) (lowermiddle panel) no matter what was a particular cause of such a change of the MST structure. The mostimportant topological changes detectable by (cid:104) L ( s ) (cid:105) and k max ( s ) occurred after the end of ETHdomination in the market in January–February 2019 (topolgy changed from a highly centralized onewith ETH being the hub to a rather distributed one), after the end of the bull phase in July–August 2019(topology returned temporarily to a centralized form but with USDT as the central hub), during a localpeak and the subsequent decline of the market in November 2019 (another short period of a centralizedtopology with USDT domination), and during and after the Covid-19 outbreak March–May 2020 (anotherphase of USDT domination, but longer than the preceding ones). On the level of (cid:104) ρ ( s ) (cid:105) , there can be 7interesting periods pointed out (upper middle panel of Figure 8). As one might expect, the longer scale,the stronger are the mean cross-correlations; this is a systematical relation throughout the whole analyzedtime interval. This is a typical effect observed on many financial markets, which is related to the liquidityand capitalization differences among the assets. Since the cryptocurrencies with small capitalizationare traded less frequently than those with large capitalization, it takes more time for a piece of marketinformation to spread over such cryptocurrencies. Thus, the cross-correlations among them can only bebuilt and detected on longer scales.A more interesting situation as regards the different scales s can be found if one compares, on the onehand, k max ( s ) between these scales and, on the other hand, (cid:104) L ( s ) (cid:105) . Let us look at two events: A peakand decline of the bull market in July 2019 (event II) and the Covid-19 pandemic (events IV–VI). Duringthe former, k max ( s ) shows a standard behaviour, i.e., for s =
10 min and s =
60 min it is significantlylarger than for s =
360 min; the same can be said for the events IV-VI. According with what it has been ntropy , xx , 5 22 of 28 said above, we might expect that in both cases (cid:104) L (
1, 360 min ) (cid:105) should be larger than (cid:104) L (
1, 10 min ) (cid:105) and (cid:104) L (
1, 60 min ) (cid:105) . while this was the case, indeed, during the pandemic outbreak in March 2020, nothing likethis happened during the bull market peak in July 2019, when (cid:104) L ( s ) (cid:105) was comparable for all the scales.Such a deviation from the overall rule that a longer scale is associated with a better-developed hierarchicalor a more distributed MST topology (smaller k max ( q , s ) ) and a shorter scale is associated with either a morecentralized network topology (larger k max ( q , s ) ) was rather unusual as for the whole studied period.Figure 9 differs from Figure 8 only in that it shows the same quantities but for q = k max ( q , s ) for different scales s is less clear for q = q =
1. Only in July 2019 and inMarch-April 2020 there can be distinguished some characteristic structures in time evolution of k max ( s ) and (cid:104) L ( s ) (cid:105) . For the events that took place in July 2019, a relation between values of the maximum nodedegree for different scales and a relation between values of the mean path length also for different scalesresemble those identified for q =
1. For the Covid-19 outbreak period, the small difference is that now k max (
4, 60 min ) is comparable to its counterpart for s =
360 min instead of s =
10 min as for q =
1. Thereis no difference between q = q = (cid:104) ρ ( s ) (cid:105) : The longer the scale is, the stronger are thecross-correlations.To summarize observations related to the MST topology, in the analyzed period from January 2019to June 2020 this topology used to change substantially during periods of large volatility in such a waythat from a hierarchical or distributed network structure that was typical outside these periods it used totransform itself to a more centralized structure with a dominating hub and much stronger cross-correlationsbetween the nodes (see also [38]). The most interesting period was the Covid-19 pandemic, during whichon short and moderate scales for q = k max ( q , s ) ) and a subsequent slow return to a more distributed form (moderate k max ( q , s ) ) butstill with a distinguished central hub. However, on the longest scale this effect was not observed and k max (
1, 360 min ) was elevated only once in May 2020. This suggests that the most sudden and nervousmovements that correlate the market and centralize its topology on short time scales tend to be blurred astime passes and we go from short to long scales, where topology becomes much more of a distributed orhierarchical type. Such a behaviour observed recently during the pandemic, which can be considered asan external perturbation to the market, differs from the behaviour observed during the peak and collapseof the bull market in July 2019, which was no doubt a result of the internal evolution of the market.Whether this internal/external events may be source of the observed peculiarities of the Covid-19 period,one cannot state for sure as both events were unique during the analyzed time interval and cannot beconfirmed by other events of similar type.
3. Summary
In our work we focused on dynamical and structural properties of the cryptocurrency market. Weanalyzed empirical data representing the exchange rates of 129 cryptocurrencies traded on the Binanceplatform, including BTC. The analysis comprised three parts, each of which was intended for investigatinga different aspect of the market structure. We started from a multifractal analysis of the BTC/USDTexchange rate as the most important one together with a similar analysis of an artificial cryptocurrencyindex based on 8 the most capitalized coins. This analysis may be considered as an extension of theanalysis reported in Ref. [11] on the most recent time interval from January 2019 to June 2020. The resultsshowed that throughout this interval the cryptocurrency dynamics produces multifractal fluctuations(returns) with some intermittent signatures of bifractality that can be assigned to specific volatile periodslike the Covid-19 outburst in March 2020 or a bull market start in April 2019. Moreover, on a level of thereturn distributions such bifractal-like singularity spectra can be accounted for by the pdf/cdf power-law ntropy , xx , 5 23 of 28 tails that fall into the Lévy-stable regime [90]. Outside these volatile periods spectra are wide but withmuch smaller left-right asymmetry.The analysis of the cross-correlations between the cryptocurrency market represented by BTC/USDor ETH/USD and the conventional markets represented by the major fiat currencies, the most importantcommodities (e.g., crude oil and gold), and the US stock market indices brought us to an observation thatthe cryptocurrency market was decoupled from the remaining markets throughout the whole year 2019,but it used to couple temporarily to those markets during some events in the first half of 2020, like inJanuary when the first Covid-19 case was reported in the United States, in March during the pandemicoutbreak, and in May-July during the pandemic’s 2nd wave. In the first case, BTC was anticorrelatedwith the major stock market indices like S&P500 and Nasdaq100, but in the second and the third casesthe analogous cross-correlations were positive. Positive were then also the cross-correlations betweenBTC and several fiat currencies and commodities. A lack of the statistically valid cross-correlations in2019, when the conventional assets did not experience anything turbulent, was supposedly caused by theasymmetry in market capitalization between the cryptocurrency market and the conventional markets tothe disadvantage of the cryptocurrency market, which was too small to have any sizeable impact on theother markets. However, the conventional markets can easily influence the cryptocurrency market if theyare turbulent. This is exactly what was observed in March 2020 and June 2020. Except for January 2020,when, unlike BTC/USD, ETH/USD was not correlated with the conventional assets, both the exchangerates reveal a similar relation with these assets.A network representation of the cryptocurrency market can shed light on the market’s innercross-correlation structure. Our analysis based on the exchange rates of 128 coins with respect to BTCrevealed that turbulent periods on the market result in a sudden transition between different networktopology types. During the periods of normal dynamics, the market has a distributed-network topology ora hierarchical-network topology, in which no node dominates the network and there is a hierarchy of hubswith decreasing centrality (e.g., node degree). Typically, for long scales the hierarchical-network topologyis more pronounced than for short scales, where a centralized-network topology prevails. This is becausethe cryptocurrencies of small capitalization are less liquid, so a piece of information needs more time to befully processed by them and the cross-correlations, especially those more subtle, sector-like, and related toless prominent cryptocurrencies, can only build up on sufficiently long time scales. This picture is alteredif there comes a volatile period. During such periods the network becomes highly centralized with onedominating hub for all the scales. The most often the role of such a hub is played by USDT, because itis pegged to the US dollar and, thus, considered as more stable than other cryptocurrencies. If investorsflee the cryptocurrency market, they first change their assets to USDT, and only then to USD, which cancorrelate a majority of the cryptocurrencies together via USDT. However, a compact, star-like topologicalform exists shortly and soon it returns to a more distributed, more branched form.We also noticed that the most significant events as regards their impact on the market topology—thetransition from a bull market to a bear market in July 2019 and the Covid-19 pandemics that started inMarch 2020—differ in some details of that impact. During the pandemics, a transition from a centralizedform to a distributed form occurred predominantly on short and medium scales, while on long scales itwas less pronounced. Contrary to that, in July 2019 the topology shift was visible on all the scales. It is amatter of future analyses to address a question whether this difference can be related to endogenous (atrend reversal) vs. exogenous (the pandemic) origin of both events. Author Contributions:
Conceptualization, S.D., P.O., and M.W.; methodology, S.D., J.K., P.O., T.S., and M.W.; software,T.S. and M.W.; validation, S.D., J.K., P.O., T.S., and M.W.; formal analysis, P.O., T.S., and M.W.; investigation, P.O., T.S.,and M.W.; resources, T.S. and M.W.; data curation, T.S. and M.W.; writing–original draft preparation, J.K. and M.W.;writing–review and editing, J.K. and M.W.; visualization, T.S. and M.W.; supervision, S.D. and P.O. All authors haveread and agreed to the published version of the manuscript. ntropy , xx , 5 24 of 28 Conflicts of Interest:
The authors declare no conflict of interest.
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