bitcoin is a digital currency alternative to the legal ones , as any other crypto currency . nowadays , bitcoin is the most popular cryptocurrency .it was created by a cryptologist known as `` satoshi nakamoto '' , whose real identity is still unknown . like other cryptocurrencies, bitcoin uses cryptographic techniques and , thanks to an open source system , anyone is allowed to inspect and even modify the source code of the bitcoin software .the bitcoin network is a peer - to - peer network that monitors and manages both the generation of new bitcoins and the consistency verification of transactions in bitcoins .this network is composed of a high number of computers connected to each other through the internet .they perform complex cryptographic procedures which generate new bitcoins ( mining ) and manage the bitcoin transactions register , verifying their correctness and truthfulness .mining is the process which allows to find the so called `` proof of work '' that validates a set of transactions and adds them to the massive and transparent ledger of every past bitcoin transaction known as the `` blockchain '' .the generation of bitcoins is the reward for the validation process of the transactions .the blockchain was generated starting since january 3 , 2009 by the inventor of the bitcoin system himself , satoshi nakamoto .the first block is called `` genesis block '' and contains a single transaction , which generates 50-bitcoin for the benefit of the creator of the block .the whole system is set up to yield just 21 million bitcoins by 2040 , and over time the process of mining will become less and less profitable . the main source of remuneration for the miners in the future will be the fees on transactions , and not the mining process itself .  in this work , we propose an agent - based model with the aim to study and analyse the mining process and the bitcoin market starting from september 1 , 2010 , the approximate date when miners started to buy mining hardware to mine bitcoins .the proposed model simulates the mining process and the bitcoin transactions , by implementing a mechanism for the formation of the bitcoin price , and specific behaviors for each typology of trader .we try to reproduce the generation of bitcoins , the main stylized facts present in the real bitcoin market and the economy of the mining process .the model described is built on a previous work of the authors , which modeled the bitcoin market under a purely financial perspective . in this work ,we fully consider also the economics of mining .the paper is organized as follows . in section _ related work _ we discuss other works related to this paper , in section _ mining process _we describe briefly the mining process and we give an overview on the mining hardware and on its evolution over time . in section _ the model _ we present the proposed model in detail . section _simulation results _ presents the values given to several parameters of the model and reports the results of the simulations , including an analysis of bitcoin real prices , and a robustness analysis .the conclusions of the paper are reported in section _conclusions_. finally , appendix deals with the calibration to some parameters of the model .the study and analysis of the cryptocurrency market is a relatively new field . in the last years, several papers appeared on this topic given its potential interest and the many issues related to it ( see for instance the works ) .however , very few works were made to model the cryptocurrencies market .we can cite the works by luther , who studied why some cryptocurrencies failed to gain widespread acceptance using a simple agent model ; by bornholdt et al . , who proposed a model based on moran process to study the cryptocurrencies able to emerge ; by garcia et al . , who studied the role of social interactions in the creation of price bubbles ; by kristoufek who analysed the main drivers of the bitcoin price ; and by kaminsky et al . who related the bitcoin market with its sentiment analysis on social networks .  in this paperwe propose a complex agent - based model in order to reproduce the economy of the mining process and the main stylized facts of the bitcoin price series .our model is inspired by business , economic and financial agent - based models that depict how organizations , or in general the economy of a country , create , deliver , and capture value .  as regards the business models , amini et al . presented a agent - based model with the aim to analyze the impact of alternative production and sales policies on the diffusion of a new product ; cocco et al . proposed agent - based models to simulate the software market and analyze the business processes and policies adopted by proprietary software firms and open source software firms ; li et al . researched the dominant players behavior in supply chains and the relationship between the selling prices and purchasing prices in supply chains by using a multi - agent simulation model ; rohitratana et al . studied the pricing schemes of the market of the software as - a - service and on the market of the proprietary or traditional software ; finally , xiaoming et al . studied how a firm maximizes its profit by determining the production and sales policies for a new product during the lifetime of the product .  concerning economic models , in the authors presented one of the most significant agent - based model developed to date in order to study the european economy .in particular , they show how monetary policies , i.e , credit money supplied by commercial banks as loans to firms , influence the economy of a country . in agentbased keynesian models are presented in order to investigate the properties of macro economic dynamics and the impact of public polices on supply , demand and the fundamentals of the economy , and to study the interactions between income distribution and monetary and fiscal policies .as regards artificial financial market models , they reproduce the real functioning of markets , trying to explain the main stylised facts observed in financial markets , such as the fat - tailed distribution of returns , the volatility clustering , and the unit - root property . for a review , see works and .raberto et al . and cincotti et al . proposed the genoa artificial stock market ( gasm ) an agent - based artificial financial market characterized by actual tracking of status and wealth of each agent , and by a realistic trading and price clearing mechanisms .gasm is able to reproduce some of the main stylised facts observed in real financial markets .this paper is built on gasm , adding specific features and a mix of zero - intelligence and trend - following traders with the aim to model the bitcoin exchange market and the economics of mining .today , every few minutes thousands of people send and receive bitcoins through the peer - to - peer electronic cash system created by satoshi nakamoto . all transactions are public and stored in a distributed database called blockchain which is used to confirm transactions and prevent the double - spending problem .people who confirm transactions of bitcoins and store them in the blockchain are called `` miners '' . as soon as new transactions are notified to the network, miners check their validity and authenticity and collect them in a block .then , they take the information contained in the block of the transactions , which include a variable number called `` nonce '' and run the sha-256 hashing algorithm on this block , turning the initial information into a sequence of 256 bits , known as hash .there is no way of knowing how this sequence will look before calculating it , and the introduction of a minor change in the initial data causes a drastic change in the resulting hash .  the miners can not change the data containing the information of transactions , but can change the `` nonce '' number used to create a different hash .the goal is to find a hash having a given number of leading zero bits .this number can be varied to change the difficulty of the problem . the first miner who creates a proper hash with success ( he finds the `` proof - of - work '' ) , gets a reward in bitcoins , and the successful hashis stored with the block of the validated transactions in the blockchain .  in a nutshell ,_ _ _ _ _ _ _ _ _ _ _ " , ._ _ _ _ _ _ _ _ _ _ _  the steps to run the network are the followings :  _ _ _ _ _ _ _ _ _ _ , ._ _ _ _ _ _ _ _ _ _  producing a single hash is computationally very easy , consequently in order to regulate the generation of bitcoins , over time the bitcoin protocol makes this task more and more difficult .the proof - of - work is implemented by incrementing the nonce in the block until a value is found that gives the block s hash with the required leading zero bits .if the hash does not match the required format , a new nonce is generated and the hash calculation starts again .countless attempts may be necessary before finding a nonce able to generate a correct hash .the computational complexity of the process necessary to find the proof - of - work is adjusted over time in such a way that the number of blocks found each day is more or less constant ( approximately 2016 blocks in two weeks , one every 10 minutes ) . in the beginning , each generated block corresponded to the creation of 50 bitcoins , this number being halved each four years , after 210,000 blocks additions .so , the miners have a reward equal to 50 bitcoins if the created blocks belong to the first 210,000 blocks of the blockchain , 25 bitcoins if the created blocks range from the 210,001th to the 420,000th block in the blockchain , 12.5 bitcoins if the created blocks range from the 420,001th to the 630,000th block in the blockchain , and so on .  over time , mining bitcoin is getting more and more complex , due to the increasing number of miners , and the increasing power of their hardware .we have witnessed the succession of four generations of hardware , i.e. cpu s , gpu s , fpga s and asic s generation , each of them characterized by a specific hash rate ( measured in h / sec ) and power consumption . with time , the power and the price of the mining hardware has been steadly increasing , though the price of h / sec has been decreasing . to face the increasing costs , miners are pooling together to share resources .   in january 3 , 2009 ,satoshi nakamoto created the first block of the blockchain , called `` genesis block '' , hashing on the central processing unit ( cpu ) of his computer .like him , the early miners mined bitcoin running the software on their personal computers .the cpu s era represents the first phase of the mining process , the other eras being gpu s , fpga s and asic s eras ( see web site https://tradeblock.com/blog/ + the - evolution - of - mining/ ) .each era announces the use of a specific typology of mining hardware .in the second era , started about on september 2010 , boards based on graphics processing unit ( gpu ) running in parallel entered the market , giving rise to the gpu era .  about in december 2011 ,the fpga s era started and hardware based on field programmable gate array cards ( fpga ) specifically designed to mine bitcoins was available in the market .finally , in 2013 fully customized application specific integrated circuit ( asic ) appeared , substantially increasing the hashing capability of the bitcoin network and marking the beginning of the fourth era .  over time , the different mining hardware available was characterized by an increasing hash rate , a decreasing power consumption per hash , and increasing costs . for example , nvidia quadro nvs 3100 m , 16 cores , belonging to the gpu generation , has a hash rate equal to 3.6 mh / s and a power consumption equal to 14 w ; modminer quad , belonging to the fpga generation , has a hash rate equal to 800 mh / s and power consumption equal to 40 w ; monarch(300 ) , belonging to the asic generation , has a hash rate equal to 300 gh / s and power consumption equal to 175 w ( see web site https://tradeblock.com / mining/.   the goal of our work is to model the economy of the mining process , so we neglected the first era , when bitcoins had no monetary value , and miners used the power available on their pcs , at almost no cost .we simulated only the remaining three generations of mining hardware .we gathered information about the products that entered the market in each era to model these three generations of hardware , in particular with the aim to compute :  * the average hash rate per us '' and a `` best power consumption function '' ( see table [ average ] ) .we call the fitting curves and , respectively .we used a general exponential model to fit the curve of the hash rate , obtained by using eq .[ r ] :    where and .the fitting curve of the power consumption is also a general exponential model :    where and .[ fig:1 ] ( a ) and ( b ) show in logaritmic scale the fitting curves and how the hash rate increases over time , whereas power consumtpion decreases .we used the _blockchain.info_ , a web site which displays detailed information about all transactions and bitcoin blocks , providing graphs and statistics on different data , for extracting the empirical data used in this work .this web site provides several graphs and statistical analysis of data about bitcoins .in particular , we can observe the time trend of the bitcoin price in the market , the total number of bitcoins , the total hash rate of the bitcoin network and the total number of bitcoin transactions .the proposed model presents an agent - based artificial cryptocurrency market in which agents mine , buy or sell bitcoins .we modeled the bitcoin market starting from september 1st , 2010 , because one of our goals is to study the economy of the mining process .it was only around this date that miners started to buy mining hardware to mine bitcoins , denoting a business interest in mining .previously , they typically just used the power available on their personal computers .the features of the model are :  * there are various kinds of agents active on the btc market : miners , random traders and chartists ; * the trading mechanism is based on a realistic order book that keeps sorted lists of buy and sell orders , and matches them allowing to fulfill compatible orders and to set the price ; * agents have typically limited financial resources , initially distributed following a power law ; * the number of agents engaged in trading at each moment is a fraction of the total number of agents ; * a number of new traders , endowed only with cash , enter the market ; they represent people who decided to start trading or mining bitcoins ; * miners belong to mining pools .this means that at each time they always have a positive probability to mine at least a fraction of bitcoin .indeed , since 2010 miners have been pooling together to share resources in order to be able avoiding effort duplication to optimally mine bitcoins .a consequence of this fact is that gains are smoothly distributed among miners .+ on july 18th , 2010 , + + and on september 18th , 2010 , + _ , _ + ( news from the web site http://historyofbitcoin.org/ ) .  since then , the difficulty of the problem of mining increased exponentially , and nowaday it would be almost unthinkable to mine without participating to a pool .  in the next subsectionswe describe in detail the model simulating the mining , the bitcoin market and the related mechanism of bitcoin price formation .agents , or traders are divided into three populations : miners , random traders and chartists .i_-th trader enters the market at a given time step , .such a trader can be either a miner , a random trader or a chartist .all traders present in the market at the initial time holds an amount of fiat currency ( cash , in us dollars ) and an amount of crypto currency ( bitcoins ) , where is the trader s index .they represent the persons present in the market , mining and trading bitcoins , before the period considered in the simulation .i_-th trader entering the market at holds only an amount of fiat currency ( cash , in dollars ) .these traders represent people interested in entering the market , investing their money in it .the wealth distribution of traders follows a zipf law .the set of all traders entering the market at time are generated before the beginning of the simulation with a pareto distribution of fiat cash , and then are randomly extracted from the set , when a given number of them must enter the market at a given time step .also , the wealth distribution in crypto cash of the traders in the market at initial time follows a zipf law .indeed , the wealth share in the world of bitcoin is even more unevenly distributed than in the world at large ( see web site http://www.cryptocoinsnews.com/owns-bitcoins-infographic-wealth-distribution/ ) .more details on the trader wealth endowment are illustrated in appendix .[ [ sec:4.1.1 ] ] miners + + + + + +  _ miners _ are in the bitcoin market aiming to generate wealth by gaining bitcoins . at the initial time, the simulated bitcoin network is calibrated respecting the satoshi original idea of bitcoin network where each node participates equally to the process of check and validation of the transactions and mining .we assumed that miners in the market at initial time ( ) own a personal pc such as core i5 2600k , and hence they are initially endowed with a hashing capability equal to 0.0173gh / sec , that implies a power consumption equal to 75w .core i5 is a brand name of a series of fourth - generation x64 microprocessor developed by intel and brought to market in october 2009 .miners entering the market at time acquire mining hardware , and hence a hashing capability , which implies a specific electricity cost , investing a fraction of their fiat cash .in addition , over time all miners can improve their hashing capability by buying new mining hardware investing both their fiat and crypto cash .consequently , the total hashing capability of trader at time , expressed in  per day , associated to her mining hardware units , are defined respectively as :    and    where :   (t)\ ] ]  * and are , respectively , the hash rate which can be bought with one us , considering the cost of 1 kwh equal to 0.14 for buy order and in bitcoins for sell order : the latter amount is a real number , because bitcoins can be bought and sold in fractions as small as a `` satoshi '' ; * residual amount ( bitcoins or & initial price .the average price as of september 2010 .+ & 23,274 & constant used in zipf s law ( ) , used to assign the initial cash for traders entering at .+ & 20,587 & initial bitcoin cash of the richest trader entering the simulation at .+  the model was run to study the main features which characterize the bitcoin market and the traders who operate in it . in order to assess the robustness of our model and the validity of our statistical analysis, we repeated 100 simulations with the same initial conditions , but different seeds of the random number generator .the results of all simulations were consistent , as shown in the followings .we started studying the real bitcoin price series between september 1st , 2010 and september 30 , 2015 , shown in fig .[ fig : realprice ] .the figure shows an initial period in which the price trend is relatively constant , until about day .then , a period of volatility follows between and day , followed by a period of strong volatility , until the end of the considered interval .the bitcoin price started to fall at the beginning fo 2014 , and is continuing on its downward slope until september 2015 .it is well known that the price series encountered in financial markets typically exhibit some statistical features , also known as `` stylized facts '' . among these ,the three uni - variate properties which appear to be the most important and pervasive of price series , are ( i ) the unit - root property , ( ii ) the fat tail phenomenon , and ( iii ) the _ volatility clustering_. we examined daily bitcoin prices and found that also these prices exhibit these properties as discussed in detail in .as regards the prices in the simulated market , we report in fig .[ fig : price ] the bitcoin price in one typical simulation run .it is possible to observe that , as in the case of the real price , at first the price keeps its value constant , but then , after about 1000 simulation steps , contrary to what happens in the reality , it grows and continues on its upward slope until the end of the simulation period .    figs . [fig : averageprice ] ( a ) and ( b ) report the average and the standard deviation of the simulated price , taken on all 100 simulations .note that the average value of prices steadily increases with time , in contrast with what happens in reality .[ fig : averageprice ] ( b ) shows that the price variations in different simulation runs increase with time , as the number of traders , transactions and the total wealth in the market are increasing .  in the proposed model ,the upward trend of the price depends on an intrinsic mechanism in fact , the average price tends to the ratio of total available cash to total available bitcoins .since new traders bring in more cash than new mined bitcoins , the price tends to increase .  in reality ,bitcoin price is also heavily affected by exogenous factors .for instance , in the past the price strongly reacted to reports such as those regarding bitcoin ban in china , or the mtgox exchange going bust .moreover , the total capitalization of the bitcoin market is of the order of just some billions of us 1,000 at the beginning of the simulation and 1,000 at the beginning of the simulation and 1,000 and 1,150 in december 2013 . in 2014bitcoin price fell following the shutdown of historical mt .gox exchange site and reports regarding bitcoin ban in china .trying to reproduce this market trend , we introduced in the model a particular speculative behaviour by some traders .the speculative mechanism implemented stems from a report , called the `` willy report '' , published by an anonymous researcher , which alleges suspicious trading activity at mt .`` the willy report : proof of massive fraudulent trading activity at mt .gox , and how it has affected the price of bitcoin '' , was posted on may 25 , 2014 in web site https://willyreport.wordpress.com/.  the anonymous researcher claims to have noted a suspicious bot behavior on mt .gox , that spread its trading activity over many accounts , and how this fraudulent massive trading activity impacted on the price , causing bubble and crash .  in the reportthe researcher writes :  _ . _according to what just mentioned , we modeled a similar behaviour .we assumed that , until the end of january 2014 , 40% of random traders entering the market were mt .gox accounts .gox accounts have a behaviour equal to that of random traders described in paragraph _random traders _ until july 2012 .then , from august 2012 and until the end of january 2014 , they issue only buy orders .next , from february 2014 they issue only sell orders .their trading probability is set equal to 0.1 in every period .[ fig : price ] shows the bitcoin price in one typical simulation run , under these conditions . at first , the price keeps its value constant , then at about 700 simulation steps , it grows as happens in reality .the price maintains its value high for about 500 simulation steps , then its value falls down , but after a short delay it continues on its upward slope until the end of the simulation , due to the intrinsec mechanism of our model already previous described .the mtgox accounts behaviour has a key rule in the reproduction of the price that has a trend more similar to the real one ( shown in fig .[ fig : realprice ] ) than that described in section _ bitcoin prices in the real and simulated market_. figs .[ fig : averageprice ] ( a ) and ( b ) report the average and the standard deviation of the simulated price across all monte carlo simulations showing the consistency of the results .all the analysis described in the previous sections was performed also for the model including `` mt.gox '' accounts , producing results consistent with those studied in the previous sections .in this work , we propose an heterogenous agent model of the bitcoin market with the aim to study and analyse the mining process and the bitcoin market starting from september 1st , 2010 , the approximate date when miners started to buy mining hardware to mine bitcoins , for five years .the proposed model simulates the mining process and the bitcoin transactions , by implementing a mechanism for the formation of the bitcoin price , and specific behaviors for each typology of trader .it includes different trading strategies , an initial distribution of wealth following pareto law , a realistic trading and price clearing mechanism based on an order book , the increase with time of the total number of bitcoins due to mining , and the arrival of new traders interested in bitcoins .the model was simulated and its main outputs were analysed and compared to respective real quantities with the aim to demonstrate that an artificial financial market model can reproduce the stylized facts of the bitcoin financial market .  the main result of the model is the fact that some key stylized facts of bitcoin real price series and of bitcoin market are very well reproduced .specifically , the model reproduces quite well the unit root property of the price series , the fat tail phenomenon , the volatility clustering of the price returns , the price peak in november 2013 , its next fall in april 2014 , the generation of bitcoins , the hashing capability , the power consumption , and the hardware and electricity expenses incurred by miners .the proposed model is fairly complex .it is intrinsically stochastic and of course it includes endogenous mechanisms affecting the market dynamics .the zipf distribution of traders wealth , that impacts to the size of the orders and the `` herding '' effect of chartists , when a price trend is established , play a key role in the distribution of the price returns , and hence in the reproduction of the fat tail phenomenon .the chartist behavior and also the variability of the spread of limit prices as a function of past price volatility contribute to the volatility clustering .the threshold of activation of chartists based on price relative variation , and the past price volatility used to determine the spread of limit prices impact on the unit - root property of the price series .the percentage of each trader s population , the choice of a trader that trades at a given time step , and the type of trading ( buy or sell ) , as well as the setting of the quantity to trade , impact on the price trend .the setting of the amount of cash to devote to buy new hardware impacts on the wealth and hashing capability of miners , and consequently on their hardware and electricity expenses .future research will be devoted to study in deeper detail the mechanisms impacting on the model dynamics .in particular , we will perform a comprehensive analysis of the sensitivity of the model to the various parameters , and will add traders with more sophisticated trading strategies , to assess their profitability in the simulated market .in addition , since the calibration of our model is based on very few specific real data , and on many assumptions aiming to derive the needed data from indirect real data , we plan to perform a deeper analysis of the block chain , and to gather financial data from the existing exchanges , in order to extract specific information needed for a better calibration of our model .the distributions of cash and bitcoins , for traders in the market at initial time , follow a power - law with exponent set equal to 1 , a value yielding the distribution known as zipf s law .this is the same assumption made in other papers .we assumed that 40% of the total initial crypto cash is not involved in the trading activity , given the initial miners hoarded a fraction of their bitcoins ( see web site http://www.coindesk.com / mit - report - bitcoin - more - likely - spent - hoarded/. ) .  to create the zipf s distribution we used the ranking property of the power - law .if the total number of traders is and the number of bitcoins owned by them , , follows a pareto law with exponent , it is well - known from harmonic - series theory that the total number of bitcoins , where is the euler - mascheroni constant and is the number of bitcoins owned by the richest trader .the number of bitcoins owned by -th trader is consequently : .we set the cash of each initial trader equal to five times the value of their crypto cash .a similar approach was followed to set the wealth of traders who enter the simulation at , but in this case the traders are only endowed with cash . in this case , we had no specific data to calibrate the wealth of these traders .we stipulated that the cash , , of the richest trader is about five times the cash owned by the richest initial trader , and that the exponent of the pareto law is in this case .overall , we performed various simulations varying these parameters , with no significant impact on the results .the set of `` new '' traders are generated before the simulation starts .when new traders are needed to enter the simulation , they are chosen randomly in this set .as mentioned in subsection _ the agents _ , only a given percentage of traders is active in the market , and hence enabled to issue orders . to compute this percentage we made some assumptions starting from the work from which we extracted table [ transactionnumber ] , which shows the distribution of the transactions number per entity ( _ entity _ means the common owner of multiple bitcoin addresses ) and per address on a period between january 3rd , 2009 and may 2012 .[ tab : transactionnumber ]  by analyzing table [ transactionnumber ] , we can observed that 97.37% of all entities had fewer than 10 transactions each , 2.27% of all entities had a number of transactions ranging from 10 to 100 , 0.34 % of all entities had a number of transactions ranging from 100 to 1,000 , and the remaining 0.02 % had a number of transactions higher than 1,000 .  according to the insights coming from this work , we neglected the entities having fewer than 10 transactions each and the entities having more than 1,000 transactions each .this was done hypothesizing that the formers , typically involving a small number of bitcoins , refer to entities who made transactions by chance `` only to use '' this new coin , whereas the latters , involving a very high of transactions , are probably not linked to single traders , but are the addresses of exchange sites , or of retailers accepting bitcoins .we considered the remaining entities , that is the entities with a number of transactions in the range from 10 to 1,000 .they issue orders with a period ranging from about one day to about 122 days , because these data are computed on 1227 days . as a result ,the daily trading probability of an entity ranges from 0.008 to 1 .we set the values of trading probabilities for random traders and chartists in this range .specifically , we assumed that , random traders are active with a probability , whereas the chartists are active with a probability .this because the interest of chartists in purchasing or selling bitcoins is higher than that of random traders .random traders issue orders to satisfy their needs , whereas chartists issue orders for speculative reasons , study carefully the price variation over time and are readier to place orders .note that active chartists actually place orders only if the price variation is above a given threshold .one of the most attractive property of bitcoin is to provide quasi - anonymous transactions , so knowing the number of traders in the real market is very difficult .the bitcoin addresses used for the transactions are known , but a user can have , and typically has , more than one address .  at the moment of writing ( last quarter of 2015 ) we had three figures :  1 .the massive and transparent ledger of every bitcoin transaction was generated starting since january 3 , 2009 presumably by the inventor of the bitcoin system itself , nakamoto and so in january 2009 there was only one person owning bitcoins ; 2 .  __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ( http://www.whoishostingthis.com/blog/2014/03/03/who-owns-all-the-bitcoins/ ) ; _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 3 .  __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ( https://bitcointalk.org/index.php?topic=316297.0 ) ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  in addition to these figures , we have other data related to the period between january 2009 and september 2010 .these data were extracted from an analysis on the daily number of downloads of the official bitcoin software client from the sourceforge platform ( http://sourceforge.net/projects/bitcoin ) .  in the period just mentioned , the bitcoin network began to spread and bitcoin had no monetary value .so , for this period we assumed the number of downloads of the official bitcoin software client equal to the number of traders in the market .we made the assumption that a person who downloads the bitcoin software is mainly interested to use it to mine bitcoins .so , we extracted two figures from this data : the total number of downloads on may 1st , 2010 equal to 2,769 , and the total number of downloads on september 1st , 2010 , equal to 30,589 , and we set the number of downloads equal to the number of traders .so , we computed the number of traders in the market fitting the curve through the five figures available :  1 .1 people owned bitcoins on january 2009 .he was satoshi nakamoto ; 2 .2769 people had downloaded bitcoin mining software on may 2010 ; 3 .30589 people had downloaded bitcoin mining software on september 2010 ; 4 .280,000 people owned bitcoins at the end of 2013 ; 5 .1000000 people owned bitcoins on april 2014 .the fitting curve of the number of traders is defined by using a general exponential model :    where , .[ fig : tradprobbemin ] ( a ) show the fitting curve and how the number of traders increases over time .at first , users in the bitcoin network were mainly miners .subsequently , when the bitcoin network started to grow and bitcoins started to acquire a monetary value , new users entered the network .most of these users did not mine , but simply traded bitcoins .they are represented in the model as random traders and chartists .  since the percentages of the different trader populations is not known , to compute the probability of a trader to be a miner we performed an analysis of the bitcoin blockchain .we analysed the blockchain until may 1st , 2010 and then until september 1st , 2010 .each block in the blockchain contains a list of validated transactions .each transaction has in input all the addresses containing the amount of bitcoins to transfer , and in output all the addresses that receive the bitcoins .users can use multiple addresses .we assumed , as in , that the input addresses that send bitcoins to the same output address must belong to an unique owner , since to proceed with the transaction it is necessary to know the private keys of all input addresses .in addition , we considered the input addresses that transfer more than 20 bitcoins owned by the same owner of the corresponding outputs . in fact , in the period from may 1st , 2010 , to september 1st , 2010 bitcoins had no monetary value , and so it is acceptable to consider that miners exchanged only small amounts of bitcoins to test the operation of the network . of course , this last assumption can be valid only for the period under study .  with these assumptions, we found 46,005 unique addresses on may 1st , 2010 and 82,294 unique addresses on september 1st , 2010 . with further analysis , we found that on may 1st , 2010 , out of 46,005 addresses , 43,389 were addresses that mined , and on september 1st , 2010 , out of 82,294 addresses , 55,974 were addresses that mined , so we computed the probability of an user to be a miner .this probability is equal to 0.94% on may 1st , 2010 and equal to 0.68% on september 1st , 2010 .[ fig : tradprobbemin ] ( b ) shows the fitting curve and how this probability decreases over time .we see that the probability of a trader to be a miner decreases over time , going about from 0.38 to less than 0.01 . of course ,defining this probability using a fitting curve computed from only two points is questionable . in the followings, we made some considerations to validate the adoption of this curve .at first , the number of bitcoin transactions was low because in the market there were mainly miners . over time , as bitcoin was acquiring monetary value , the number of users interested in exchanging bitcoins increased .so , while the percentage of miners in the market was decreasing , also due to the increasing difficult to mine bitcoins , the percentage of random traders and chartists greatly increased , according to the growth of the number of transactions , which slowly rose until a peak on may 2012 and then at the end of the 2013 ( see figure [ fig : tradprobbemin ] ( c ) ) . .we assumed that the random traders to chartist ratio is 7/3 , meaning that 30% of traders who are not miners are speculators , whereas the remaining 70% are non - speculative investors .these figures are consistent with recent data obtained for the foreign exchange market .the probabilities of non - miners to be a random trader or a chartist , and respectively , are defined as a function of , respectively as :     with these probabilities , we have at the end of the simulations a number of miners equal to about 1000 , corresponding to 100,000 miners in the real world .this is in agreement with what an australian bitcoin miner , andrew geyl , estimated ( see web site http://bravenewcoin.com/news/number-of-bitcoin-miners-far-higher-than-popular-estimates/ ) .amini m , wakolbinger t , racer m , and nejad m.g .alternative supply chain production sales policies for new product diffusion : an agent - based modeling and simulation approach .european journal of operational research .2012 ; 216 : 301311 .arthur w. b , holland j. h , lebaron b , palmer r , and tayler p. asset pricing under endogenous expectations in an artificial stock market .the economy as an evolving complex system ii , ser .sfi studies in the sciences of complexity , arthur w.b , durlauf s.n , and lane d.a .addison wesley longman .1997;1544 .cincotti s , focardi s , marchesi m , and raberto m. who wins ?study of long - run trader survival in an artificial stock market .physica a : statistical mechanics and its applications , elsevier .2003 ; 324(1 ) : 227 - 233 .cocco l , mannaro k , concas g , marchesi m. study of the competition between proprietary software firms and free/ libre open source software using a simulation model .software business lecture notes in business information processing , springer .2011 ; 80 : 5669 .cocco l , concas g , marchesi m. using an artificial financial market for studying a cryptocurrency market .journal of economic interaction and coordination , springer . 2015; doi:10.1007/s11403 - 015 - 0168 - 2 .garcia d , tessone c.j , mavrodiev p , and perony n. feedback cycles between socio - economic signals in the bitcoin economy the digital traces of bubbles : feedback cycles between socio - economic signals in the bitcoin economy .2014 ; 1408.1494.       lengnick m. agent - based macroeconomics - a baseline model .economics working paper / christian - albrechts - universitt kiel , department of economics , no . 2011,04 , provided in cooperation with : christian - albrechts - university of kiel , department of economics .ponta l , scalas e , raberto m. , and cincotti s. statistical analysis and agent - based microstructure modeling of high - frequency financial trading .ieee journal of selected topics in signal processing .2012 ; 6(4 ) : 381 - 387 .raberto m , and cincotti s , and dose c , and focardi s , and marchesi m. price formation in an artificial market : limit order book versus matching of supply and demand .nonlinear dynamics and heterogenous interacting agents , springer - verlag .berlin 2005 .singh p , and chandavarkar b. r , and arora s , and agrawal n. performance comparison of executing fast transactions in bitcoin network using verifiable code execution .second international conference on advanced computing , networking and security .2013 .odwyer karl j , and david malone .bitcoin mining and its energy footprint .irish signals and systems conference 2014 and 2014 china - ireland international conference on information and communications technologies .

Abstract: in january 3 , 2009 , satoshi nakamoto gave rise to the `` bitcoin block chain '' creating the first block of the chain hashing on his computer s central processing unit ( cpu ) . since then , the hash calculations to mine bitcoin have been getting more and more complex , and consequently the mining hardware evolved to adapt to this increasing difficulty . three generations of mining hardware have followed the cpu s generation . they are gpu s , fpga s and asic s generations .   this work presents an agent based artificial market model of the bitcoin mining process and of the bitcoin transactions . the goal of this work is to model the economy of the mining process , starting from gpu s generation , the first with economic significance .   the model reproduces some `` stylized facts '' found in real time price series and some core aspects of the mining business . in particular , the computational experiments performed are able to reproduce the unit root property , the fat tail phenomenon and the volatility clustering of bitcoin price series . in addition , under proper assumptions , they are able to reproduce the price peak at the end of november 2013 , its next fall in april 2014 , the generation of bitcoins , the hashing capability , the power consumption , and the mining hardware and electrical energy expenses of the bitcoin network .  * keywords * : artificial financial market , bitcoin , heterogeneous agents , market simulation


quantum computing promises major increases in computing power but poses many experimental hurdles to its implementation .finding more efficient ways to implement quantum gates may allow small scale quantum computing tasks to be demonstrated on a shorter time - scale .a key quantum gate is the toffoli gate .the toffoli gate acts on 3 qubits and in conjunction with the hadamard gate forms perhaps the simplest universal gate set in quantum computing .  in this paperwe show that the number of two - qubit gates required to implement a toffoli gate can be reduced by making one of the qubits in the circuit a qutrit ( or , in the general case , a qudit ) .the qutrit nature will only manifest during the gate - after the gate only the qubit levels will be occupied .remarkably , the additional space afforded by the extra level on one qubit provides a significant reduction in the resource requirements .this reduction is particularly dramatic for optical implementations where systems for applying the envisaged manipulations exist quite naturally .the paper is arranged in the following way . in the next sectionwe introduce the scheme in an abstract , implementation independent , way and demonstrate its increased efficiency . in section iiiwe consider various optical realizations and then summarize and conclude in our final section .we begin by showing how a toffoli gate can be implemented using only three controlled sign ( c - s ) gates plus single qubit unitaries by allowing one of the qubits to be a qutrit . to our knowledgethe most efficient implementation that has been described using only qubits involves five 2-qubit gates , .if we restrict ourselves to using only controlled sign ( c - s ) gates , then six c - s gates plus various single qubit gates are needed .fig.[fig1 ] shows the arrangement to implement a toffoli - sign ( t - s ) gate , i.e. a three qubit gate that applies a sign change on one and only one of the state components and the identity is implemented otherwise .it is of no consequence which state component is sign flipped and the flipped component will vary between our various implementations .all such gates are locally equivalent and can be inter - converted with straightforward single - qubit bit flips . the t - s gate becomes a toffoli gate by placing hadamards before and after the gate on one of the qubits , which then becomes the target qubit .the two control qubits are labelled as usual with logical states and ( where we reserve bold print for indicating logical values ) .the target is a qutrit for which we label the additional state .we assume c - s gates and hadamard gates are available .we note that a cnot gate can be constructed from a c - s gate using a pair of hadamard gates .we require one additional gate that we label , which enables transitions between the and states . the c - s ( and cnot ) gate act on the qubit levels in the usual way .if they act on a qutrit in the state then both gates implement the identity regardless of the value of the control qubit .  ) .similiarly for the c - s gate ( middle two - qubit gate ) .the gate flips the qutrit between the states and .the sign change occurs on the component.,width=302 ]  the first operation we perform is to apply an gate to the target qutrit .the gate is defined by the following actions on the basis states : , , .consider an arbitrary three qubit state where and the ket subscripts label the different qubits according to fig.1 .after the application of the gate on the qutrit ( ) we have we now apply a cnot gate between and to obtain next a c - s gate is applied between and the resulting in now a cnot gate is again applied between and the and finally the gate is again applied to the qutrit .the state is then as expected for a t - s gate with the sign change implemented on the component .this technique can be generalized straightforwardly to higher order n - toffoli gates ( where n is the number of control qubits ) by introducing an n+1 level qudit as the target .fig.[fig2 ] shows an explicit construction of the next level up , the n=3 , t - s gate . in generalthe number of two - qubit gates required for this method is .this seems a significant improvement on previous estimates of optimal gate numbers for higher order toffoli gates .for example reference finds 64 2-qubit gates are required for a 5-toffoli whilst our qudit construction requires only 9 .   or ) .similiarly for the c - s gate .the gate flips the ququit between the states and .the gate flips the ququit between the states and .the sign change occurs on the component.,width=302 ]  these results suggest that the computational depth of quantum processing circuits might be significantly reduced by employing these techniques . in the next sectionwe discuss various ways in which the required manipulations can be realized in optics .we now turn specifically to an optical encoding and show that for this encoding the manipulations discussed in the previous section have natural physical realizations .first consider dual - rail encoding in which the logical qubit states are given as and where is the fock state of the optical mode ( with plain text reserved for fock state occupation number ) .the optical modes are orthogonal and may represent , for example , different polarization or spatial modes . to create a qutrit we simply add an extra mode such that , in the language of the previous section , , and , where we have simplified our nomenclature such that , where labels the new mode . in the following we will show that linear optical elements plus proposed two qubit optical gates are sufficient to implement the operations used in the previous section .we begin by discussing deterministic two qubit gates based on strong non - linearities .we then consider heralded non - deterministic gates based on measurement induced non - linearities .finally we consider demonstration gates based on post - selected measurement induced non - linearities . in each casewe find considerable advantages to our qudit implementation .it has long been known that a strong cross - kerr non - linearity enables the implementation of universal optical quantum gates on the encoding we have just introduced . in particular a non - linear medium can be used to induce a cross - kerr effect between the relevant photon modes .ideally the cross - kerr effect will produce the unitary evolution $ ] , where is the annihilation operator for one optical mode and another .consider the schematic set - up of fig.[fig3 ] .two polarization encoded qubits are converted into spatial dual rail qubits using polarizing beamsplitters .one mode from each of the qubits is sent through the cross - kerr material .the operation of this device on an arbitrary two qubit input state is given by the following evolution :  .pbs are polarizing beamsplitters.,width=302 ]    only when both modes entering the kerr material are occupied is a phase shift induced . if we now choose the strength of the non - linearity such that , the effect is to flip the sign of one element of the superposition producing a c - s gate .we can introduce a third spatial mode by using polarization rotation and polarizing beamsplitters and hence implement the control gate sequence required in section 1 as shown in fig.[fig4 ] .notice , from eq.[c - k ] , that this c - s gate implements the identity if both polarisation modes are unoccupied as required .only 3 interactions are required compared with the 5 or 6 that would be needed to implement the toffoli by the usual qubit gates .thus the same saving is made as discussed for the abstract case .gates by accessing an additional spatial mode .the half - wave plates ( ) are set at 22.5 degrees so as to act as hadamard gates and the c - s gates are implemented as per fig.[fig3].,width=302 ]  unfortunately kerr materials with the required strength of non - linearity are not presently available . as a result we now consider non - deterministic implementations based on measurement induced non - linearites .knill , laflamme and milburn have shown that scalable quantum gates can be implemented on optical qubits using linear elements , photon counters and photon sources .in particular , using linear optical elements , photon counters and an entangled photon pair as resources , it is possible to implement the optical c - s gate of the previous section non - deterministically with a probability of success of 1/4 .gate success is heralded by the photon counter signature obtained from the entangled pair after interaction with the qubits .if we used the direct method of implementing a toffoli via a sequence of c - s gates and single qubit unitaries we would require 6 entangled pairs and the probability of success would drop to .recently fiur suggested a dedicated scheme requiring only 3 entangled pairs with a probability of success of .his method offered significant reduction in the number of entangled pairs required , but only a modest increase in the success probability .our method further reduces the number of additional entangled pairs and dramatically improves the success probability . in the followingwe will adapt our qutrit techniques to linear optics and obtain a heralded toffili gate requiring only 2 entangled pairs and working with a probability of success of .gates by accessing an additional spatial mode .the half - wave plates ( ) act as hadamard gates and the c - s gates are implemented using heralded non - deterministic gates .the sign change occurs on the component.,width=302 ]  the proposed set - up is shown in fig.[fig5 ] .it is similar to the deterministic gate set - up except that the c - s gates are assumed non - deterministic and the final two - qubit gate is replaced with a passive quantum filter .notice that , because we are also exploiting the polarization degree of freedom on the second spatial mode , the target is now effectively a ququit ( a 4-level quantum system ) .each 2 qubit gate works with a probability of success and requires an entangled pair for their operation .the filter succeeds with probability and does not require any ancilla qubits for its operation .the gate works in the following way : the state after the second c - s gate ( see fig.[fig5 ] ) is conditional on the correct photon counting outcomes from the previous two c - s gates ( probability of success ) . here and refer to horizontally and vertically polarized single photon states respectively and refers to the vacuum state ( both polarization modes unoccupied ) , i.e , and .the initial state is as defined in eq.[3qubits ] and the ordering of the kets corresponds to top to bottom in fig.[fig5 ] .instead of using a third c - s gate as in the deterministic scheme , half - wave plates oriented at 22.5 degrees are applied to both target modes leading to the state where and .the third and fourth modes are then recombined on a polarizing beam - splitter and the output mode is conditioned on a zero detection at the second output port of the polarizing beamsplitter .the probability of the zero detection is producing the conditional output state hence we have produced a t - s gate with the phase flip occurring on the component .the probability of success is .the proposed heralded circuit represents a five - photon experiment .such experiments are feasible , but difficult . in the following section we propose an in principle demonstration requiring only three photons .we now consider the construction of a post - selected gate . by thiswe mean a gate in which the photons act as their own ancilla , with success heralded by successful detection of a photon for each qubit , so - called coincidence detection . in this way a t - s gate requiring only the 3 photons needed to represent the 3 qubits could be constructed .for example , we could substitute post - selected c - s gates into the circuit of fig.[fig5 ] .these gates operate with a probability of success of so this would produce a post - selected t - s gate requiring only three photons and working with a probability of success of .however , it turns out the gate can be optimized for maximum success probability by using the techniques for chaining post - selected c - s gates described in ref. and in this way achieves a success probability of as we now describe .fiur also considered this problem and proposed a different architecture with a success probability of about  .  .the figure is represented , for clarity , as if all modes are spatial . in an experimental realization polarization modeswould be utilized where ever possible.,width=340 ]  our proposed set - up is shown in fig.[fig6 ] .the important part of the circuit is the central string of two interferometers created from the top target mode ( ) .firstly , if this mode is unoccupied , photons can only emerge in all three output qubits if they remain in their respective modes .in this case no phase flip occurs .if the top target mode is occupied we must consider passage of the photon through the central interferometers .the first interferometer is anti - balanced if the bottom mode of the first control mode , , is unoccupied . as a result the photon couples completely into the bottom mode of the second interferometer with a phase flip .on the other hand , if the bottom mode of the first control is occupied there will be no phase change at the one third beamsplitter in the first interferometer due to two photon interference ( assuming a single photon exits from each port of the beamsplitter ) . in this casethe first interferometer is balanced and the photon will couple completely into the top mode of the second interferometer .now we turn to the second interferometer .if the top mode of the second control , , is unoccupied then passage through the bottom mode of the interferometer induces a phase flip . but recall this will only occur if the photon has undergone a phase flip in the first interferometer . thus if the photon subsequently makes it into the output target mode it will not have undergone a phase flip .similarly if the target photon couples into the top mode of the second interferometer then it will not have undergone a phase flip in the first interferometer and will not undergo any further phase flip if it subsequently makes it to the output target mode , regardless of the state of .finally , if the photon couples into the bottom mode of the second interferometer ( meaning that had its bottom mode unoccupied , and the photon has picked up a phase flip ) , and has its top mode occupied , two - photon interference at the one third beamsplitter in the second interferometer then leads to no phase flip . thus if the target photon subsequently makes it to the output , this state component will carry a phase flip . summarizing , we find that the only state component that will carry a phase flip will be the one in which every qubit was in its logical zero state , where we take occupation of the top mode to be logical zero .the beamsplitter ratios are picked such that all state components have equal probability to lead to an event with a photon appearing in all the output qubits .that probability turns out to be .we have discussed the implementation of toffoli gates in circuits where one of the qubits is allowed to be a qutrit , or more generally a qudit and shown that increased gate efficiency can be achieved . in particular we showed that instead of 6 c - s gates , as required for a qubit only circuit , introducing a qutrit allows this to be reduced to 3 c - s gates . for an -toffolithe required gate number is if a level qudit is available .we showed that for deterministic optical quantum processing a natural implementation method could be identified . for non - deterministic optical approaches further simplificationscould be identified leading to a heralded toffoli gate with a probability of success of and a post - selected gate with a probability of success of .these latter results open the door to experimental optical demonstrations of toffoli gates in the near future and make small scale applications appear more feasible .it is interesting to note that a non - universal set of qubit gates , c - s plus hadamard ( an additional gate is required to make this a universal set ) , becomes universal , toffoli plus hadamard , when a single additional quantum level is introduced .we thank j. l. dodd , g. j. pryde and j. l. obrien for useful discussions .this work was supported by the australian research council and the dto - funded u.s.army research office contract no .w911nf-05 - 0397 .we note that a 3 two - qubit gate construction does exist for the so - called `` simplified toffoli gate '' that differs from the true toffoli gate by a non - trivial phase factor : n. margolus , in _ collision based computation _ , editor a. adamatzky , page 107 ( springer - verlag , 2002 ) .

Abstract: the simplest decomposition of a toffoli gate acting on three qubits requires _ five _ 2-qubit gates . if we restrict ourselves to controlled - sign ( or controlled - not ) gates this number climbs to six . we show that the number of controlled - sign gates required to implement a toffoli gate can be reduced to just _ three _ if one of the three quantum systems has a third state that is accessible during the computation , i.e. is actually a qutrit . such a requirement is not unreasonable or even atypical since we often artificially enforce a qubit structure on multilevel quantums systems ( eg . atoms , photonic polarization and spatial modes ) . we explore the implementation of these techniques in optical quantum processing and show that linear optical circuits could operate with much higher probabilities of success .


paper citation count itself and various quantities derived from it are used as influential indicators of research impact . at the same time , it is well known that the cumulative number of citations received by academic publications strongly depend on paper age and field  .old papers have had more time to acquire citations than recent ones , and their advantage is further enhanced by the preferential attachment mechanism . on the other hand ,different academic fields adopt very different citation practices ( see for a review on the topic ) , which results in a strong dependence of the mean number of citations on academic field , as shown in several works ( , among others ) .a natural question arises : how can we `` fairly '' use citation - based indicators to compare papers from different fields and of different age ?the problem of comparing papers from different fields is usually referred to as the _ field - normalization _ problem .several approaches to address this question have been proposed in the literature ( see for a recent review ) .a particularly simple approach is to divide each paper s citation count by the mean number of citations for papers of the same field published in the same year .the results by suggested that this indicator , called _ relative citation count _, produces a ranking that is statistically consistent with the hypothesis of a ranking that is not biased by field and age .this finding has been challenged by subsequent works by and , which leaves the debate on age- and field- normalization procedures still open .  in this article, we analyze a large dataset from microsoft academic graph to show that existing indicators of impact , including the relative citation count , fail to produce rankings that are not biased by age and field . to simultaneously assess these biases , we present a new procedure based on the mahalanobis distance  .this permits to compare the ranking by a given indicator with those obtained with a simulated unbiased sampling , and hence to quantify the overall ranking bias .an analytic result derived in this paper allows us to assess the contribution of each field to the overall ranking bias .it is worth noticing that while we focus on the biases by age and field , our bias assessment procedure can be easily extended to detect any other kind of information bias .we also present the first systematic study of the possible bias by field of pagerank score and of its age - rescaled version introduced by . the motivation to analyze these network - based indicators comes from the finding that they outperform other metrics in identifying expert - selected milestone papers .however , the application of pagerank and its variants to academic citation networks focused on datasets composed of papers from a single field  . to our best knowledge ,the metric s possible bias by academic field is still unexplored and we are the first authors to address it .we introduce two novel indicators of impact motivated by the -score : age- and field - rescaled citation count and age- and field - rescaled pagerank .we find that the novel indicators produce paper rankings that are much less biased by age and field than the rankings produced by the other analyzed indicators . nevertheless ,also the mahalanobis distance observed for the new indicators is not statistically consistent with ones obtained for a simulated unbiased process .this indicates that the problem of achieving an ideal unbiased ranking of the publications remains open .the rest of our article is organized as follows : section 2 describes the analyzed dataset of publications obtained from the microsoft academic graph .section 3 presents existing paper - level impact indicators and reports their bias by scientific field . in section 4, we introduce a rescaling procedure for citation count and pagerank scores motivated by the -score . in section 5, we introduce a general procedure to test for any kind of ranking bias , and present its application to assess the field and age bias of the rankings by the indicators studied here . in section 6 ,we conclude by discussing possible limitations of our analysis and future research directions .we analyze a bibliographic dataset which was provided for the kdd cup 2016 .this data is a dump of the _ microsoft academic graph _ ( mag ) and contains more than 126 millions of publications and more than 467 millions citations .each publication is also endowed with various properties such as unique i d , publication date , title , journal i d , etc .we pre - processed the data ( details are provided in appendix [ app : kdd ] ) to remove from the analysis papers with incomplete information , ending up with unique publications and citations .the mag has a field classification at paper level . in the kdd cup dump , there are main fields and numerous subfields up to hierarchical levels of subsubfields .however , all the different subfields can belong to several main fields , meaning that each publication can belong to more than one main field .we use here the field classification at the highest hierarchical level , i.e. , we only consider the main fields . when calculating the citation count and pagerank score of papers ( see sec .[ sec : short - ex - metric ] ) , we consider the publications that belong to more than one field only once . in this way, we do not modify the number of citations that each paper receives and gives , and we do not change the topology of the network on which the pagerank scores are calculated . on the other hand , in agreement with , to compute the fields size( see tab .[ tab : fields ] in appendix [ app : kdd ] ) and the field - rescaled metrics ( see sec . [sec : short - ex - metric ] and sec .[ sec : def - new - metrics ] ) , each publication can be considered multiple times in the analysis , once for each field the publication belongs to . with this , each field is represented by all its publications even if some of these are shared with other fields .we now define the four existing metrics analyzed in this work : citation count , relative citation count , pagerank score , age - rescaled pagerank score ( sec .[ sec : definition ] ) .furthermore , we show that these metrics are severely biased by scientific field ( sec . [sec : field_bias ] ) .[ [ citation - count - c ] ] citation count , + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the citation count of node is simply the number of citations received by paper . in terms of the citation network s adjacency matrix ( in a directed network , if node points to node , otherwise ) , we can express the citation count as .[ [ relative - citation - count - cf ] ] relative citation count , + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  to overcome citation count s bias by paper age and academic field , defined the _ relative citation count _ of paper as , where denotes the mean citation count for papers published in the same field and year as paper . throughout this paper, we always refer to the main fields provided in the mag dataset .[ [ page - rank - p ] ] page rank , + + + + + + + + + + + + + + + + + + + + + + + + + +  originally devised by to rank webpages in the world wide web , pagerank algorithm has attracted considerable interest of the scientometrics community .the rationale behind the application of the metric to citation networks is that not all citations should be counted the same : citations coming from influential papers should count more than citations from obscure articles .the pagerank scores of papers are usually written in a vector defined by the following equation where is a constant , is the random - walk transition matrix with elements , is the number of references in paper , and is a uniform teleportation vector with elements for all papers ..  can be interpreted as the stationary equation of a stochastic process on the citation network . in this process, a random walker is placed on each paper and he / she either follows a citation edge with probability , or jumps to a randomly chosen paper with probability .when the number of walkers on each paper reaches a stationary value , we obtain the pagerank score of a paper by calculating the fraction of walker on this paper . in our analysis, we set which is the usual choice in citation networks .pagerank is based on a static , time - aggregated perspective of the considered network . in general ,such perspective has been shown to be limiting for the analysis of evolving networks  . while the resulting metric s bias towards old papers has already been studied in the literature , its possible bias by academic field is still unexplored and we address it in section [ sec : field_bias ] .  [[ age - rescaled - page - rank - rap ] ] age - rescaled page rank , + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  to suppress the age bias of pagerank , proposed to rescale the pagerank score by comparing each paper s score with the scores of papers of similar age . assuming that the papers are ordered by older to younger, one computes the mean value and the standard deviation of pagerank scores over papers around , i.e. $ ] .consequently , the rescaled pagerank score of paper is defined as  applied rescaled pagerank to the network of physics papers published by the american physical society journals to show that the resulting ranking is not biased by paper age and , as a result , it allows us to identify seminal publication much earlier than rankings by metrics that are biased against recent papers . in the following, we set as in .after having described a set of existing metrics , we now apply them to the mag dataset to show that the rankings that they produce are biased by scientific field . for a ranking that is not biased by scientific field, the number of top - ranked publications from each field should be proportional to the total number of publications from that field .in other words , for an unbiased ranking , we expect papers from field among the top papers in the ranking , where is the total number of publications from field  . in the following ,we denote by the number of publications from field in the top- of the ranking by metric .we restrict our analysis to ; results for other values of are available upon request from the authors .    in the top panels of fig .[ fig : rescaled_cit_bad ] , we illustrate the field bias of citation count , , and relative citation count , .the presence of strong biases is evident for both metrics because there are fields whose ratio is far away from the expected value . in particular ,environmental science is extremely over - represented in the top of the ranking by citation count .we argue that this bias comes from the fact that publications from this field have a mean citation count almost twice as big compared to publications belonging to other fields ( see table [ tab : fields ] ) . for relative citation count, we find a better agreement with what we would expect from an unbiased indicator .however , relatively large deviations are still evident , especially for the field of political science .in the bottom panels of fig . [fig : rescaled_cit_bad ] , we report the distributions of and for each field .these panels show that the bias by field is not limited to the top papers in the ranking , but it arises from systematic differences between the score distributions across different fields . for example , when looking at the distribution of , papers in the field of political science have consistently smaller probability to have more than one citation compared to other fields . for a detailed discussion about the bias of the ranking by , we refer to appendix [ appendix : radicchi ] .[ fig : rescaled_pg_bad ] reports the same analysis for pagerank scores , , and age - rescaled pagerank scores , .this figure provides the first study of the dependence of pagerank score on academic field .the top panels of fig .[ fig : rescaled_pg_bad ] show that the top positions of both rankings are biased by field , and both rankings overestimate the impact of publications in the field of environmental science .again , we argue that this happens because the mean indegree of publications from environmental science is approximately twice as big compared to publications that belong to other fields ( see table [ tab : fields ] ) . from the bottom - left panel of fig . [fig : rescaled_pg_bad ] , we find that the full distribution of scores of page rank have a similar shape , but different broadness . these differences are slightly smaller for the age - rescaled pagerank , with the exception of the field of environmental science ( see bottom right panel of fig . [fig : rescaled_pg_bad ] ) .in this section , we introduce two novel indicators of paper impact : the age- and field - rescaled citation count , , and the age- and field - rescaled pagerank , .the two indicators , and , are obtained from citation count and pagerank score , respectively , through a rescaling procedure .this procedure is based on the -score and is aimed at suppressing age and field bias .it is conceptually close to the one presented by , and is an extension of those introduced by and .   to calculate the age- and field - rescaled citation count of a paper belonging to a field , we first compute the mean and the standard deviation of the citation count of papers of the _ same field _ and of _ similar age _ as paper .in particular , and are computed over the papers that belong to the same field as paper and that are among the closest papers to as measured by the distance between their rank by age .then , the age- and field - rescaled citation count score is defined as the averaging window size is a parameter of the method , which we set to .  differently from , for the computation of the -score , we use temporal windows with the same number of publications , which in general corresponds to real - time intervals of different duration .this choice is supported by recent findings ( ) that indicate that in citation networks , time is better defined by number of publications than by real time . our analysis ( not shown )confirms that when computation temporal windows are of a fixed temporal length , the bias removal is inferior to that achieved with temporal windows containing a fixed number of publications .differently from the relative citation count , is expected to have not only uniform mean value across different publication dates and fields , but also uniform standard deviation .this should lead to a more balanced ranking of the papers .we show in the following that this is indeed the case .previous works have shown that pagerank is biased towards old papers in scientific citation networks . moreover ,we have shown in sec .[ sec : short - ex - metric ] that pagerank score is biased by scientific domain . to simultaneously suppress these two biases, we propose the age- and field - rescaled pagerank score . is defined similarly as : we compute the mean value and the standard deviation of the pagerank scores of the papers that belong to the same field as paper and that are among the closest papers to as measured by the distance between their rank by age .the age- and field - rescaled pagerank score is then defined as in the following , we set .  in the top panels of fig .[ fig : rescaled_good ] , we show that in the top-1% of the rankings by and each field appears well represented .in fact , the deviations from the expected value are very small especially if compared to the deviations of the other rankings ( see top panels in figs .[ fig : rescaled_cit_bad ] and [ fig : rescaled_pg_bad ] ) . in the bottom panels of fig .[ fig : rescaled_good ] , we report that the the full score distributions for papers from different fields collapse extremely well top of each other thanks to the rescaling procedure .we begin this section by introducing a new methodology to assess a ranking s bias based on the mahalanobis distance ( paragraph [ sec : mahala ] ) .then , we use this to quantify the bias by field ( paragraph [ sec : field_bias2 ] ) and the bias by age and field ( paragraph [ sec : field_age_bias ] ) .while figs . [ fig : rescaled_cit_bad ] and [ fig : rescaled_pg_bad ] illustrate the substantial field bias of the existing metrics , the bias is much weaker ( if any ) for the new age- and field - rescaled metrics in figure [ fig : rescaled_good ] .now we quantify this improvement by extending the statistical tests of bias suppression presented by . similarly to these works , we assume that a ranking is unbiased if its properties are consistent with those of an unbiased selection process .let us first analyze the problem of assessing the bias by field . consider an urn which contains marbles , each of them corresponding to one of the publications present in our dataset .an _ unbiased selection process _ then corresponds to sampling from this urn at random without replacement a fixed number of publications . from the extracted sample ,we count the number of publications that belong to each field , , and record these numbers in the vector ; here denotes the number of fields .the probability to observe a certain vector , , is given by the _ multivariate hypergeomentric distribution _( mhd ) where is the total number of publications in field . following this selection process , among the extracted publications ,the expected number of publications for field is .assume that the actual ranking by a given metric features publications from field in the top of its ranking .in general , the observed deviates from its expected value .a simple approach to quantify this deviation would consist in computing the -score , defined as , where is the expected standard deviation for field according to the mhd specified by eq . .there are however two shortcomings of the -score .first , the -score only gives partial information for a mhd how far from the expected values we are in units of standard deviations but it does not provide information on how statistically significant the deviations are .second , to quantify the overall bias of a given indicator , we would need to aggregate the -scores from the different fields .for example , we could take the average -score , but this would neglect the correlation between the different fields coming from the constraint .  to overcome these two problems ,we follow a different approach .we first run various numerical simulations that reproduce the unbiased selection process .these simulations produce a set of ranking vectors which are distributed according to eq .  around the vector of expected values , .differently from  , we do not estimate the confidence interval for the different fields separately .we calculate instead the mahalanobis distance ( ,  and appendix [ app : md ] ) for each simulated vector from , and construct the distribution of s obtained by the simulated unbiased selection process .the inset of the left panel of fig .[ fig : md_field ] reports the distribution of the for simulations .the distribution is centered around its mean value of and the upper bound for the confidence level is around . for the unbiased sampling processis extremely close to the number of degrees of freedom of our problem .this stems from the fact that the mhd defined by eq.  converges to a multivariate gaussian distribution ( mgd ) as we increase the number of publications while keeping fixed and small .our dataset is large enough for this approximation to be accurate .the of a mgd is distributed as a variable with average equal to the number of degrees of freedom , i.e. 18 since we have 19 distinct fields and one constraint . ] for an _ ideal unbiased ranking _ , we would expect its to fall into the confidence interval of the distribution of the obtained from the simulated unbiased sampling process .the methodology presented above is easily generalized to simultaneously assess a ranking s bias by age and field .to add the temporal dimension to the bias assessment procedure , we split the publications into equally - sized age groups , and repeat the above analysis by using different categories of publications . in sec .[ sec : field_age_bias ] , we set representing the number of fields and as in , and thus we obtain age - field groups of different sizes .     the rankings produced by different metrics differ greatly by their ( see fig . [ fig : md_field ] ) . as expected , the metrics that are not field - rescaled ( , , and ) are far from being unbiased . at the same time , relative citation count that is rescaled by field performs only slightly better than pagerank which is ignorant of any field information .the best results by a wide margin are achieved by our metrics , and , obtained using the new rescaling .nevertheless , both these metrics fail to meet the 95% upper bound achieved by simulated unbiased rankings . as disappointing as it may seem , this finding is not entirely surprising as the proposed rescaling procedures focus on equalizing the first two moments of the respective quantities ( and ) whereas the quantities distributions can differ also by higher moments .  to understand which field contributes the most to the resulting values, we derived an alternative analytic expression for the  where we omit the metric superscript from the notation for and for simplicity .we have proven this formula analytically for , and we have numerically tested it for and ( see appendix [ app : md ] ) ; it remains open to prove it in arbitrary dimensions .[ cols= " > , > , > , > , > , > , > " , ]   we first observe that the distributions of both incoming and outgoing edges are broad ( see fig . [ distribution ] ) .both these distribution in fact have long and heavy tails .in addition , we also observe strong variations of the mean citation count across fields ( see table [ tab : fields ] for details ) .for example , publications belonging to the field of environmental science and of political science have mean citation count two times bigger and smaller , respectively , compared to the average citation count across all fields .in agreement with the findings by , the strong variety in mean citation count per publication for the various fields confirms that the corresponding communities exhibit different citation behavior , which calls for field normalization procedures .     here , we address the following question : to which extent is the kdd dump of the mag in accordance with the most updated online version of the mag ? for this comparison , we divided the analyzed paper - field pairs ( one paper - field pair is composed of a paper and the fields that paper belongs to ) into age and field groups , as described in sec .[ sec : quant - rank - bias - by - field - and - age ] . from each group, we randomly choose the of papers . following this procedure, we obtained a representative sample with respect to field and age composed of approximately papers .first , we have matched the paper i d in hexadecimal format present in the data released for the kdd cup to the paper i d in int64 format present in the on - line version of the mag . for papers, we manually verified that the paper ids were exactly the same in the two datasets , albeit represented in different formats .then , using the academic knowledge api , we have downloaded the number of citations for each sampled paper .  for the of the sampled papers , we were not able to find their corresponding paper in the online mag . for unmatched papers , we found out that these were papers with duplicates in the kdd cup data , i.e. these papers are present in the kdd data with two distinct ids , and only one of the two ids is present in the mag .for the matched papers ( of the sample ) , we compared the number of citations reported in the kdd dump of the mag with the number of citations reported on the online version of the mag .since the online mag has a longer time span than the kdd data , in the absence of noise , we would expect the number of citations in the kdd data to be smaller than or equal to the number of citations reported in the online mag .we find that the two citation counts are highly correlated ( fig .[ fig : cit_corr ] ) , and only the of the sampled papers have more citations in the kdd data compared to the online version of the mag .this sets a lower boundary for the error in the percentage of papers with wrong number of citations to about .    to summarize , after our filtering procedure , we find that the data released for kdd cup 2016 has about of papers with duplicates .in addition , about of the matched papers have errors in their citation count .this means that we have correct citation information for about of the analyzed papers .the mahalanobis distance ( ) is an established measure in statistics which generalizes the concept of -score to multivariate distributions by taking into account also possible correlations between the random variables .its definition reads where is the inverse of the covariance matrix , and are two vectors containing the random variables .when the covariance matrix is diagonal , i.e. the random variables are not correlated , than the is equivalent to the square root of the sum of the squares of the -scores .  in sec .[ sec : field_bias2 ] , we have used eq . , an expression for the valid when the covariance matrix comes from a multivariate hypergeometric distribution ( mhd ) , i.e. , when the elements of the matrix are where is the kronecker delta , is the number of papers of category , is the total number of papers , is the number of paper categories , and is the number of sampled papers .it is worthy to remember that even though we have different categories , we only have degrees of freedom .here , we derive eq .for the case of a mhd in three dimensions , i.e. , for . in this case, the covariance matrix is : and the inverse of the covariance matrix is where denotes the determinant of the covariance matrix , .then , let us consider two random column vectors extracted from a -dimensional mhd , and such that where is the number of sampled papers . substituting eq .  in eq ., we write the square of the between and as ^ 2 k_1 k_2 \right\}\end{aligned}\ ] ] where we have used . recalling that , we know that , so we write then , by using the relation , we have : noticing from eq .  that , we obtain finally , if we choose one of the two vectors to contain the expected values , , we re - obtain eq .  since .to be precise , the covariance matrix is not defined for , however the relation holds and therefore also the final result .  using mathematica or similar softwares , it is easy to prove analytically that eq.  holds for small dimensions. we have verified it until 6 dimensions .moreover , we have numerically tested this formula by calculating the s between the ranking vectors of the indicators and the vector of expected values , , with two different alternative methods : ( 1 ) by using eq . , i.e. , by inverting the covariance matrix , and ( 2 ) by using the eigenvalue decomposition of the covariance matrix is symmetric and it has maximal rank because it is the covariance matrix of a multivariate distribution .therefore , we can diagonalize it , where the columns of form an orthonormal basis ; we can also write . with this , we have , where \{ } are the eigenvalues of and \{ } are the coordinates of in the basis which diagonalizes , i.e. where the last equality comes from the orthonormality of which implies . ]. the results of the three methods were all compatible with each other up to 10 decimal digits . the advantage of using eq .is that we can calculate the between two arbitrary vectors without dealing with any ( computationally slow ) matrix inversion or diagonalization , and the number of needed calculations scales linearly with the number of dimensions .importantly , eq .  allows also to assess the individual contribution of each dimension ( i.e. of each category ) to the . to our best knowledge ,we are the first ones to have derived such explicit formula for when the covariance matrix and the random vectors come from a mhd .according to , the distribution of the indicator is log - normal : ^ 2/2\sigma^2}dc^f\ ] ] where and is fitted from the data . when eq .is verified , then also the distributions of citation count , , for all the individual fields , , are lognormal : ^ 2/2\sigma^2}dc_i\ ] ] where is the mean of . for lognormal distributionsthe variance is proportional to the square of the mean and the constant of proportionality is . from eq . , we see that the citation counts are distributed lognormally with mean and variance . recalling that , we have that the mean is , as it is expected , while the variance becomes .thus , when eq .is verified , the variance of the empirical distribution of the citations for each field has to be proportional to the square of the mean citation count .moreover , the constant of proportionality has to be for every field and year .the analytic result just presented is in line with the eq .( c.1 ) given in appendix c of .there it is shown that a rescaling procedure based on diving the original score by their first moment works if the ratio between standard deviation and mean is constant . in the case of the relative citation ratio, we can calculate analytically such constant using the lognormal distribution and obtain the fitting parameter .obtained by calculating the empirical ratio , , between the variance and the square of the mean citation count of each field and year ., scaledwidth=45.0% ]  in fig .[ fig : rfc_universal ] , we report the distribution of obtained by calculating the empirical ratio between the variance and the square of the mean , , and by inverting the relation for every field and year . if the universality claim was correct , we would expect a narrow distribution of .by contrast , we find that ranges between and across different fields and years .we argue that the broad range of is the reason why the first moment rescaling introduced in does not work in the analyzed dataset .in sec . [ sec : quant - rank - bias - by - field - and - age ] , we have shown that the rankings by citation count and pagerank can be substantially leveled off across different fields and age groups through a rescaling procedure based on the -score .the two resulting indicators , and , are the least biased among the indicators considered in this paper .it is important to notice that while citation count has a direct interpretation and it is widely used in research evaluation ( ) , pagerank score is a more sophisticated quantity which has not yet be turned into a standard tool for research assessment .if the rankings by rescaled pagerank and rescaled citation count brought similar information , rescaled citation count might be preferred due to its simpler interpretation and easier computation .  differently from the citation count, pagerank score uses information on the whole network topology to compute each paper s score . while there exists no universal criterion to decide whether pagerank score leads to a better ranking than the citation count, the results by and suggest that in citation networks , pagerank score improves our ability to find groundbreaking publications in the data . on the other hand ,a node that received many citations is more likely to achieve larger pagerank score showed that pagerank score is on average proportional to citation count for uncorrelated networks .l*4cr metrics & & ( ) & & ( ) + & & & & + & & & & +  we address now the question : _ to what extent the rankings by ( rescaled ) pagerank and ( rescaled ) citation count differ ?_ we focus on two correlations : ( 1 ) the correlation between citation count and pagerank , which has been of interest for previous studies ( ) due to the essential role played by pagerank algorithm in determining the success of google s web search engine ; ( 2 ) the correlation between the two rescaled indicators and .the measured correlations are all positive , significantly larger than zero , yet significantly smaller than one ( see table [ tab : corr ] ) .this is interesting as it may point out that , in analogy with the findings by and , network topology brings useful information that is neglected by citation count . whether the additional information used by pagerank metric can be used to identify groups of significant papers will be the subject of future research .differently than the rankings by citation count and pagerank score ( not shown here ) , the top papers identified by and come from diverse historical period and diverse fields . due to the lack of time bias ,also very recent papers can reach the top of the rankings by and .it would be instructive to look at the top papers as ranked by rescaled citation count and rescaled pagerank .however , the original mag datasets presents noisy entries , as reported by the kdd cup 2016 ( see appendix [ app : kdd ] for more details ) , and it causes the scores of some recent papers to be over - estimated , which makes some of the entries in the top- of the rankings by the rescaled scores unreliable .for this reason , we do not show the rankings here . at the same, this problem does not affect the statistical results presented in the previous sections .

Abstract: it is widely recognized that citation counts for papers from different fields can not be directly compared because different scientific fields adopt different citation practices . citation counts are also strongly biased by paper age since older papers had more time to attract citations . various procedures aim at suppressing these biases and give rise to new normalized indicators , such as the relative citation count . we use a large citation dataset from microsoft academic graph and a new statistical framework based on the mahalanobis distance to show that the rankings by well known indicators , including the relative citation count and google s pagerank score , are significantly biased by paper field and age . we propose a general normalization procedure motivated by the -score which produces much less biased rankings when applied to citation count and pagerank score .