### Abeer ElBahrawy

1 Department of Mathematics—,City, University of London—,Northampton Square, London EC1V 0HB, UK

### Laura Alessandretti

1 Department of Mathematics—,City, University of London—,Northampton Square, London EC1V 0HB, UK

### Anne Kandler

Two Max Planck Institute for Evolutionary Anthropology, Department of Human Behaviour, Ecology and Culture, Leipzig, Germany

### Romualdo Pastor-Satorras

Trio Departament den Fí,sica, Universitat Politè,cnica den Catalunya, Campus Nord B4, 08034 Barcelona, Spain

### Andrea Baronchelli

1 Department of Mathematics—,City, University of London—,Northampton Square, London EC1V 0HB, UK

Four UCL Centre for Blockchain Technologies, University Collegium London, London, UK

## Abstract

The cryptocurrency market surpassed the barrier of $100 billion market capitalization ter June 2018, after months of sustained growth. Despite its enhancing relevance te the financial world, a comprehensive analysis of the entire system is still lacking, spil most studies have focused exclusively on the behaviour of one (Bitcoin) or few cryptocurrencies. Here, wij consider the history of the entire market and ontleden the behaviour of 1469 cryptocurrencies introduced inbetween April 2013 and May 2018. Wij expose that, while fresh cryptocurrencies emerge and vanish continuously and their market capitalization is enhancing (super-)exponentially, several statistical properties of the market have bot stable for years. Thesis include the number of active cryptocurrencies, market share distribution and the turnover of cryptocurrencies. Adopting an ecological perspective, wij vertoning that the so-called neutral proefje of evolution is able to reproduce a number of key empirical observations, despite its plainness and the assumption of no selective advantage of one cryptocurrency overheen another. Our results shed light on the properties of the cryptocurrency market and establish a very first formal verbinding inbetween ecological modelling and the investigate of this growing system. Wij anticipate they will spark further research te this direction.

## 1. Introduction

Bitcoin is a digital asset designed to work spil a medium of exchange [1,Two]. Users can send and receive native tokens, the ‘,bitcoins’,, while collectively validating the transactions te a decentralized and semi-transparent way. The underlying technology is based on a public ledger, or blockchain, collective inbetween participants and a prize mechanism ter terms of Bitcoins spil an incentive for users to run the transaction network. It relies on cryptography to secure the transactions and to control the creation of extra units of the currency, hence the name ‘,cryptocurrency’, ,[Trio,Four].

After Bitcoin appeared te 2009, approximately 1500 other cryptocurrencies have bot introduced, about 600 of which are actively traded today. All cryptocurrencies share the underlying blockchain technology and prize mechanism, but they typically live on isolated transaction networks. Many of them are basically clones of Bitcoin, albeit with different parameters such spil different supplies, transaction validation times, etc. Others have emerged from more significant innovations of the underlying blockchain technology ,[Five] (see electronic supplementary material, §,S3).

Cryptocurrencies are nowadays used both spil media of exchange for daily payments, the primary reason for which Bitcoin wasgoed introduced, and for speculation ,[6,7]. Other uses include payment rail for non-expensive cross-borders money transfer and various non-monetary uses such spil time stamping ,[Two]. The self-organization of different usages both within a single cryptocurrency and spil an factor of differentiation inbetween cryptocurrencies makes the market of cryptocurrencies unique, and their price utterly volatile ,[8–,Ten].

Inbetween Two.9 and Five.8 millions of private spil well spil institutional users actively exchange tokens and run the various transaction networks ,[Five]. Te May 2018, the market capitalization of active cryptocurrencies surpassed $91 billion ,[11]. Bitcoin presently predominates the market but its leading position is challenged both by technical concerns ,[12–,16] and by the technological improvements of other cryptocurrencies ,[17].

Despite the theoretical and economic rente of the cryptocurrency market ,[Two,Four,Legal,Nineteen], however, a comprehensive analysis of its dynamics is still lacking. Existing studies have focused either on Bitcoin, analysing, for example, the transaction network ,[20–,24] or the behaviour and fate of its price [9,25–,30], or on a restricted group of cryptocurrencies (typically Five or Ten) of particular rente ,[Five,17,32,31]. But even ter this case, there is disagreement spil to whether Bitcoin’,s superior position may be te peril ,[Five] or its future dominance spil leading cryptocurrency is out of discussion ,[31].

Here, wij present a very first finish analysis of the cryptocurrency market, considering its evolution inbetween April 2013 and May 2018. Wij concentrate on the market shares of the different cryptocurrencies (see ,§,Four) and find that Bitcoin has bot steadily losing ground to the advantage of the instantaneous runners-up. Wij then voorstelling that several statistical properties of the system have bot stable for the past few years, including the number of active cryptocurrencies, the market share distribution, the stability of the ranking, and the birth and death rate of fresh cryptocurrencies. Wij adopt an ‘,ecological’, perspective on the system of cryptocurrencies and note that several observed distributions are well described by the so-called ‘,neutral prototype’, of evolution [33,34], which also captures the decrease ter Bitcoin’,s market share. Wij believe that our findings represent a very first step towards a better understanding and modelling of the cryptocurrency market.

## Two. Results

### Two.1. Market description

Our analysis concentrates on the market share of the different cryptocurrencies and is based on the entire history of the cryptocurrency market inbetween 28 April 2013 and 13 May 2018. Our dataset includes 1469 cryptocurrencies, of which around 600 were active by that time (see §,Four).

The total market capitalization C of cryptocurrencies has bot enlargening since late 2015 after a period of relative tranquillity ( figure ,1 ). Spil of May 2018, the market capitalization is more than four times its value compared to May 2018 and it exhibits an exponential growth C ? exp?(?t) with coefficient λ,=0.30±,0.02, where t is measured ter units of 15 weeks.

Evolution of the market capitalization. Evolution of the market capitalization overheen time (embarking from April 2013), for all cryptocurrencies (blue line, diamonds) and for Bitcoin (crimson line, dots). The dashed line is an exponential curve f(t)∼,e λ,t , with λ,=0.Three, shown spil a guide to the eye. Gegevens are averaged overheen a 15-week window.

### Two.Two. Decreasing Bitcoin market share

Bitcoin wasgoed introduced ter 2009 and followed by a 2nd cryptocurrency (Namecoin, see electronic supplementary material, §,S1) only te Legitimate April 2011. This first-mover advantage makes Bitcoin the most famous and prominent cryptocurrency to date. However, latest studies analysing the market shares of Bitcoin and other cryptocurrencies reached contrasting conclusions on its current state. While Gandal and Halaburda ter their 2018 investigate concluded that ‘,Bitcoin seems to have emerged, at least ter this stage, spil the clear winner’, ,[35], the 2018 report by Hileman and Rauchs noted that ‘,Bitcoin has ceded significant market cap share to other cryptocurrencies’, ,[Five].

To clarify the situation, wij consider the entire evolution of the Bitcoin market share overheen the past Four years. Figure ,Two a shows that Bitcoin’,s market share has bot steadily decreasing for the past years, beyond oscillations that might mask this trend to short-term investigations. The decrease is well described by a linear gezond f(t)=a+bt with angular coefficient b=−,0.035±,0.002 signifying the switch te market share overheen t=1 year. Neglecting the influence of nonlinear effects and potential switches te the competition environment, the specimen indicates that Bitcoin’,s market share can fluctuate approximately around 50% by 2025. Conversely, figure ,Two b shows that the top Five runners-up (see electronic supplementary material, §,S1) have gained significant market share and now account for more than 20% of the market.

Evolution of the market share of top-ranking cryptocurrencies. (a) The market share of Bitcoin across time sampled weekly (grey line) and averaged overheen a rolling window of Ten weeks (crimson line). The dashed line is a linear gezond with angular coefficient b=−,0.035±,0.002 (the rate of switch ter 1 year) and coefficient of determination R Two =0.63. The Spearman correlation coefficient is ρ,=−,0.8, exposing a significant negative correlation at a significance level of 1%. (b) Total market share of the top Five cryptocurrencies excluding Bitcoin sampled weekly (grey line) and averaged overheen a rolling window of Ten weeks (green line). The dashed line is a linear gezond with angular coefficient b=0.021±,0.002 (the rate of switch te 1 year) and coefficient of determination R Two =0.45. The Spearman correlation coefficient is ρ,=0.67, exposing a significant positive correlation at a significance level of 1%.

### Two.Three. Stability of the cryptocurrency market

To characterize the cryptocurrencies dynamics better, wij now concentrate on the statistical properties of the market. Wij find that while the relative evolution of Bitcoin and rival cryptocurrencies is tumultuous, many statistical properties of the market are stable.

Figure ,Three a shows the evolution of the number of active cryptocurrencies across time, averaged overheen a 15-week window. The number of actively traded cryptocurrencies is stable due to similar birth and death rates since the end of 2014 ( figure ,Three b). The average monthly birth and death rates since 2014 are 1.16% and 1.04%, respectively, corresponding to approximately seven cryptocurrencies appearing every week while the same number is abandoned.

Evolution of the number of cryptocurrencies. (a) The number of cryptocurrencies that everzwijn entered the market (packed line) since April 2013, and the number of actively traded cryptocurrencies (dashed line). (b) The birth and death rate computed across time. The birth (respectively, death) rate is measured spil the fraction of cryptocurrencies coming in (respectively, leaving) the market on a given week overheen the number of living cryptocurrencies at that point. Gegevens are averaged overheen a 15-week window.

Interestingly, the market share distribution remains stable across time. Figure ,Four a shows that forms obtained by considering different periods of time are indistinguishable. This is remarkable because the reported forms are obtained by considering gegevens from different years spil well spil gegevens aggregated on different time spans—,from one week to the entire approximately Four years of gegevens. The obtained distribution exhibits a broad tail well described by a power-law P(x)∼,x −,α, with exponent α,=1.58±,0.12 ( figure ,Four a), where the getraind coefficient is computed using the method detailed ter ,[36]. The expected relationship inbetween the probability distribution and the frequency rank distribution predicts the latter is a power-law function P(r)∼,r −,β, with exponent β,=1/(α,−,1) ,[37], yielding te our case β,=1.72 ( figure ,Four b). The empirical getraind coefficient β,=1.93±,0.23 is consistent with this prediction. This wasgoed also verified for each year individually (see electronic supplementary material, §,S4).

Stable properties of the cryptocurrency market. (a) Distribution of market share computed aggregating across a given year (grey packed lines), and overheen the week 6–,13 May 2018 (blue thick line). The dashed line is a power law P(x)∼,x −,α, with exponent α,=1.Five. (b) Frequency-rank distribution of cryptocurrencies, computed aggregating across a given year (grey packed lines), and overheen the week 6–,13 May 2018 (blue thick line). The dashed line is a power-law curve P(r)∼,r −,β, with exponent β,=Two. (c) Average amount of time (ter weeks) a cryptocurrency occupies a given rank, computed averaging across all years (blue line), and across given years (grey lines, inset). (d) Turnover of the ranking distribution, defined spil the total number of cryptocurrencies everzwijn occupying a rank higher than a given rank. The measure is computed averaging across given years (grey packed lines). The 2013 and 2018 forms voorwaarde be taken purely spil an indication spil they are computed on less than 12 months (approx. eight and four months, respectively). The dashed line has angular coefficient 1, and corresponds to the case ter which the ranking of cryptocurrencies is immobilized (i.e. the variable turnover captures only the initial size of the top-list).

Wij further investigate the stability of the market by measuring the average rank occupation time ( figure ,Four c), defined spil the amount of time a cryptocurrency typically spends ter a given rank before switching it. Wij find that the time spent te a top-rank position decays swift with the rank, while for low-rank positions such time approaches one week. Again, this behaviour is stable across years ( figure ,Four c, inset). Wij also consider the turnover profile defined spil the total number of cryptocurrencies everzwijn occupying a rank higher than a given rank te period t (see ,[38] for a similar definition). Figure ,Four d shows that also this quantity is substantially stable across time.

The very first rank has bot always occupied and proceeds to be occupied by Bitcoin, while the subsequent Five ranks (i.e. ranks Two–,6) have bot populated by a total of 33 cryptocurrencies with an average lifetime of 12.6 weeks. Thesis values switch rapidly when wij consider the next set of ranks from 7 to 12 to reach 70 cryptocurrencies and an average lifetime of Three.6 weeks. At higher ranks, the mobility increases and cryptocurrencies continuously switch position.

### Two.Four. A elementary prototype for the cryptocurrency ecology

To account for the empirical properties of the dynamics of cryptocurrencies wij have discussed above wij adopt the view of a ‘,cryptocurrency ecology’, and consider the neutral proefje of evolution, a prototypical prototype te population genetics and ecology ,[33,34].

The Wright–,Fisher prototype of neutral evolution describes a fixed-size population of N individuals where each individual belongs to one of m species. At each generation, the N individuals are substituted by N fresh individuals. Each fresh individual belongs to a species copied at random from the previous generation, with probability 1−,μ,, or to a species not previously seen, with probability μ,, where μ, is a mutation parameter that does not switch overheen time ,[39]. Despite its simpleness, the neutral monster is able to reproduce the static patterns of the competition dynamics of many systems including ecological ,[40] and genetics ,[41] systems, cultural switch ,[42], English words usage ,[43] and technology patents citations ,[44].

Te our mapping of the ecological specimen to the cryptocurrency market, each individual corresponds to a certain amount of dollars, while species correspond to different cryptocurrencies (see electronic supplementary material, §,S2). The copying mechanism represents trading, with μ, denoting the probability that a fresh cryptocurrency is introduced. Our choice of μ, is informed by the gegevens to yield a number of fresh cryptocurrencies vanaf unit time corresponding to the empirical observation. Wij thus fix μ,=7/N, where N is the population size te the specimen. Thus, one proefje generation corresponds to one week of observations, the choice of μ, ensuring an average of seven fresh cryptocurrencies injecting the system every week, spil empirically observed. Ultimately, te tegenstelling with most neutral models, wij assume that a fresh species does not come in the system with a single individual but with a size proportional to the empirical average market share of a fresh cryptocurrency (see electronic supplementary material, §,S2).

The neutral specimen translates ter the simplest way three main assumptions ,[45]: (i) interactions inbetween cryptocurrencies are omschrijving on an individual vanaf capita poot (i.e. vanaf US dollar), (ii) the process is stochastic, and (iii) it is a sampling theory, where the fresh generation is the onderstel to build the following one. Ter other words, the neutral specimen assumes that all species/cryptocurrencies are omschrijving and that all individuals/US dollars are omschrijving.

Testing the consistency inbetween observed patterns of the cryptocurrency market and theoretical expectations of neutral theory exposed that neutrality captures well at least four features of the cryptocurrency ecology, namely:

Neutral monster for evolution and empirical observations. (a) Distribution of cryptocurrencies market shares aggregated overheen all years (grey line, dots) and the equilibrium distribution resulting from numerical simulations (blue line, squares) aggregated overheen 210 generations. The dashed line is the power-law curve P(x)∼,x −,α, predicted analytically with exponent α,=1.Five ,[46]. (b) Turnover of the ranking distribution computed considering 52 generations of the cryptocurrencies gegevens (grey lines, dots) and for numerical simulations (blue line). (c) Average number of generations of a cryptocurrency (grey lines) and a species te the neutral proefje (blue line) occupies a given rank. Averages are computed across 52 generations. (d) Evolution of the market share of Bitcoin (grey line) and the expected market share of the very first species ter numerical simulations (blue line). All simulations are run for N=Ten Five and μ,=7/N kicking off from one species te the initial state. The size of injecting species m, whose average m=15 is informed by the gegevens, is taken at random te the interval m=[Ten,20]. Error caf are standard deviations, computed across 100 simulations. (b) and (c) Measures embark at generation g1=105 (see electronic supplementary material, §,S2 for variations of this parameter).

The neutral monster generates ter fact an aggregated species distribution (i.e. obtained when all generations up to the ith are combined together and analysed spil a single population of size N*i ,[44,47]) that, at equilibrium, can be described by a power-law distribution P(x)∼,x −,α, with α,=1.Five ,[46], te agreement with the empirical value α,=1.58±,0.12 obtained by the fitting proces described ter ,[36]. Figure ,Five a shows the agreement inbetween simulations and gegevens (same behaviour of the long tail), where simulation results are aggregated overheen i=210 generations, corresponding to Four years of empirical observations under our choice of μ,. The existence of a power-law phase with exponent 1.Five ter the monster is independent of μ, (see electronic supplementary material, §,S2, and ,[46]).

Furthermore, when wij account for the fact that Bitcoin wasgoed originally the only cryptocurrency by setting one species ter the initial state, the specimen captures also the remaining properties. Te figure ,Five b,c, wij compare the turnover profile and the ranking occupation times with the corresponding simulation results. Wij compute thesis quantities overheen a period of 52 generations, corresponding to 1 year of observations. The curve reported te figure ,Five b,c corresponds to measures performed inbetween generation g1=105 and g2=156, corresponding to year Three (2015) ter the gegevens. Crucially, however, both measures are stable te time, i.e. they do not depend on the choice of g1 (but for an initial period of high rank variability for the very very first generations see electronic supplementary material, §,S2). It is worth noting that the linearity of the turnover profile te figure ,Five b corresponds to a similar behaviour observed te ,[38] when the measure is performed inbetween two consecutive generations. Figure ,Five d shows the observed linear decrease of the leading cryptocurrency market share, indicating that newborn cryptocurrencies mostly harm the predominant one.

## Trio. Discussion and outlook

Ter this paper, wij have investigated the entire cryptocurrency market inbetween April 2013 and May 2018. Wij have shown that the total market capitalization has entered a phase of exponential growth 1 year ago, while the market share of Bitcoin has bot steadily decreasing. Wij have identified several observables that have bot stable since the beginning of our time series, including the number of active cryptocurrencies, the market share distribution and the rank turnover. By adopting an ecological perspective, wij have pointed out that the neutral proefje of evolution captures several of the observed properties of the market.

The prototype is elementary and does not capture the total complexity of the cryptocurrency ecology. However, the good match with at least part of the picture emerging from the gegevens does suggest that some of the long-term properties of the cryptocurrency market can be accounted for based on plain hypotheses. Te particular, spil the monster assumes no selective advantage of one cryptocurrency overheen the other, the gezond with the gegevens shows that there is no detectable population-level overeenstemming on what is the ‘,best’, currency or that different currencies are advantageous for different uses. Furthermore, the matching inbetween the neutral prototype and the gegevens implies that the observed patterns of the cryptocurrency market are compatible with a script where technological advancements have not bot key so far (see electronic supplementary material, §,S3) and where users and/or investors allocate each packet of money independently. Future work will need to consider the role of an expanding overall market capitalization and, more importantly, attempt to include the information about single transactions, where available, te the modelling picture.

Te the instant and mid-term future, legislative, technical and social advancements will most most likely influence the cryptocurrency market gravely and our treatment, together with latest results te computational social science dealing with the quantification of financial trading and bubble formation ,[48–,51], could help make sense of the market evolution. Ter April 2018, for example, Japan commenced treating Bitcoin spil a legal form of payment driving a unexpected increase ter the Bitcoin price te US dollars ,[52], while te February 2018 a switch of regulation te China resulted ter a $100 price druppel ,[53]. Similarly, the exponential increase te the market capitalization ( figure ,1 ) will most likely attract further speculative attention towards this market, at the same time enhancing the usability of cryptocurrencies spil a payment method. While the use of cryptocurrencies spil speculative assets should promote diversification ,[31], their adoption spil a payment method (i.e. the conventional use of a collective medium of payment) should promote a winner-take-all staatsbestel [54,55]. How the self-organized use of cryptocurrencies will overeenkomst with this pressure is an interesting question to be addressed te future studies.

## Four. Material and methods

### Four.1. Gegevens

Cryptocurrency gegevens were extracted from the webstek Coin Market Cap ,[11], collecting weekly gegevens from 157 exchange market platforms embarking from 28 April 2013 up to 13 May 2018. For all living cryptocurrencies, the webstek provides the market capitalization, the price te US dollars and the volume of trading te the preceding 24 ,h. Gegevens on trading volume were collected beginning from 29 December 2013.

The webstek lists cryptocurrencies traded on public exchange markets that are older than 30 days and for which an API spil well spil a public URL showcasing the total mined supply are available. Information on the market capitalization of cryptocurrencies that are not traded te the 6 ,h preceding the weekly release of gegevens is not included on the webstek. Cryptocurrencies inactive for 7 days are not included ter the list released. Thesis measures imply that some cryptocurrencies can vanish from the list to reappear straks on.

### Four.Two. Analysis

The following quantities characterize individual cryptocurrencies: The circulating supply is the number of coins available to users. The price is the exchange rate, determined by supply and request dynamics. The market capitalization is the product of the circulating supply and the price. The market share is the market capitalization of a currency normalized by the total market capitalization.

Most of our analyses consider the market capitalization and market share of cryptocurrencies. Thesis quantities neglect the demolished or dormant coins, accounting, for example, to 51% of mined Bitcoins based on gegevens from the period Eighteen July 2010 to 13 May 2012 ,[20].