Department of Electrical Engineering, University of California, Los Angeles, CA

Transcription

1 Theoy of citig M.V. Simki ad V.P. Roychowdhuy Depatmet of Electical Egieeig, Uivesity of Califoia, Los Ageles, CA We peset empiical data o mispits i citatios to twelve high-pofile papes. The geat majoity of mispits ae idetical to mispits i aticles that ealie cited the same pape. The distibutio of the umbes of mispit epetitios follows a powe law. We develop a stochastic model of the citatio pocess, which explais these fidigs ad shows that about 7-9% of scietific citatios ae copied fom the lists of efeeces used i othe papes. Citatio copyig ca explai ot oly why some mispits become popula, but also why some papes become highly cited. We show that a model whee a scietist picks few adom papes, cites them, ad copies a factio of thei efeeces accouts quatitatively fo empiically obseved distibutio of citatios. I. Statistics of mispits i citatios ow let us come to those efeeces to authos, which othe books have, ad you wat fo yous. The emedy fo this is vey simple: You have oly to look out fo some book that quotes them all, fom A to Z, ad the iset the vey same alphabet i you book, ad though the impositio may be plai to see, because you have so little eed to boow fom them, that is o matte; thee will pobably be some simple eough to believe that you have made use of them all i this plai, atless stoy of yous. At ay ate, if it aswes o othe pupose, this log catalogue of authos will seve to give a supisig look of authoity to you book. Besides, o oe will touble himself to veify whethe you have followed them o whethe you have ot, beig o way coceed i it Miguel de Cevates, Do Quixote Whe scietists ae witig thei scietific aticles, they ofte use the method descibed i the above quote. They ca do this ad get away with it util oe day they copy a citatio, which caies i it a DA of someoe else s mispit. I such case, they ca be idetified ad bought to justice, simila to how biological DA evidece helps to covict cimials, who committed moe seious offeces tha that.

2 2 Ou iitial epot [] led to a lively discussio o whethe copyig a citatio is a poof of ot eadig the oigial pape. Alteative explaatios ae woth exploig; howeve, such hypotheses should be suppoted by data ad ot by aecdotal claims. It is ideed most atual to assume that a copyig cite also failed to ead the pape i questio (albeit this caot be igoously poved). Etities must ot be multiplied beyod ecessity. Havig thus shaved the citique with Occam s azo, we will poceed to use the tem o-eade to descibe a cite who copies. As mispits i citatios ae ot too fequet, oly celebated papes povide eough statistics to wok with. Let us have a look at the distibutio of mispits i citatios to oe eowed pape (o. 5 i Table I.), which at the time of ou iitial iquiy [], that is i late 22, had accumulated 4,3 citatios. Out of these citatios 96 cotaied mispits, out of which oly 45 wee distict. The most popula mispit i a page umbe appeaed 78 times. Table I.. Papes, mispits i citig which we studied. o Refeece K. G. Wilso, Phys. Rev. 79, 499 (969) 2 K. G. Wilso, Phys. Rev. B 4, 374 (97) 3 K. G. Wilso, Phys. Rev. B 4, 384 (97) 4 K. G. Wilso, Phys. Rev. D, 2445 (974) 5 J.M. Kostelitz ad D.J. Thouless, J. Phys. C 6, 8 (973) 6 J.M. Kostelitz, J. Phys. C 7, 46 (974) 7 M.J. Feigebaum, J. Stat. Phys. 9, 25 (978) 8 M.J. Feigebaum, J. Stat. Phys. 2, 669 (979) 9 P. Bak, J. vo Boehm, Phys. Rev. B 2, 5297 (98) P. Bak, C. Tag, ad K. Wiesefeld, Phys. Rev. Lett. 59, 38 (987) P. Bak, C. Tag, ad K. Wiesefeld, Phys. Rev. A 38, 364 (988) 2 P. Bak ad C. Tag, J. Geophys. Res. B 94, 5635 (989) As a pelimiay attempt, oe ca estimate the atio of the umbe of eades to the umbe of cites, R, as the atio of the umbe of distict mispits, D, to the total umbe of mispits, T. Clealy, amog T cites, T D copied, See, fo example, the discussio Scietists Do't Read the Papes They Cite o Slashdot: 34

3 3 because they epeated someoe else s mispit. Fo the D othes, with the ifomatio at had, we have o evidece that they did ot ead, so accodig to the pesumed iocet piciple, we assume that they did. The i ou sample, we have D eades ad T cites, which lead to: R D / T. (I.) Substitutig D = 45 ad T = 96 i Eq.(I.), we obtai that R. 23. The values of R fo the est of the doze studied papes ae give i Table I.2. As we poited out i Ref. [2] the above easoig would be covicig if the people who itoduced oigial mispits had always ead the oigial pape. It is moe easoable to assume that the pobability of itoducig a ew mispit i a citatio does ot deped o whethe the autho had ead the oigial pape. The, if the factio of ead citatios is R, the umbe of eades i ou sample is RD, ad the atio of the umbe of eades to the umbe of cites i the sample is RD/T. What happes to ou estimate, Eq. (I.)? It is coect, just the sample is ot epesetative: the factio of ead citatios amog the mispited citatios is less tha i geeal citatio populatio. Ca we still detemie R fom ou data? Yes. Fom the mispit statistics we ca detemie the aveage umbe of times, p, a typical mispit popagates: p T D D =. (I.2) The umbe of times a mispit had popagated is the umbe of times the citatio was copied fom eithe the pape which itoduced the oigial mispit, o fom oe of subsequet papes, which copied (o copied fom copied etc) fom it. A mispited citatio should ot diffe fom a coect citatio as fa as copyig is coceed. This meas that a selected at adom citatio, o aveage, is copied (icludig copied fom copied etc) p times. The ead citatios ae o diffeet fom uead citatios as fa as copyig goes. Theefoe, evey ead citatio, o aveage, was copied p times. The factio of ead citatios is thus R = + p. (I.3) Afte substitutig Eq. (I.2) ito Eq. (I.3), we ecove Eq. (I.). ote, howeve, that the aveage umbe of times a mispit popagates is ot equal to the umbe of times the citatio was copied, but to the umbe of times it was copied coectly. Let us deote the aveage umbe of citatios copied (icludig copied fom copied etc) fom a paticula citatio as c. It ca be de-

4 4 p temied fom p the followig way. The c cosists of two pats: p (the coectly copied citatios) ad mispited citatios. If the pobability of makig a mispit is M ad the umbe of coectly copied citatios is p the the total umbe of copied citatios is ad the umbe of mispited citatios is M p. As each mispited citatio was itself copied M M c times, we have the followig self-cosistecy equatio fo c : M = + + c p p It has the solutio ( ) c M (I.4) c = p M M (I.5) p c Afte substitutig Eq. (I.2) ito Eq. (I.5) we get: T D D MT =. (I.6) Fom this, we get: R = + c = D T ( MT ) M D (I.7) D The pobability of makig a mispit we ca estimate as M =, whee is the total umbe of citatios. Afte substitutig this ito Eq. (I.7) we get: R D T T D =. (I.8) Substitutig D = 45, T = 96, ad = 43 i Equatio (I.8), we get R. 22, which is vey close to the iitial estimate, obtaied usig Eq.(I.).

5 5 The values of R fo the est of the papes ae give i Table I.2. They age betwee % ad 58%. Table I.2. Citatio ad mispit statistics togethe with estimates of R fo twelve studied papes. The citatio data wee etieved fom the ISI database i late 22 ad ealy 23. The way we cout mispits is look at the whole sequece of volume, page umbe ad the yea, which amouts to betwee 8 ad digits fo diffeet studied papes. That is, two mispits ae distict if they ae diffeet i ay of the places, ad they ae epeats if they agee o all of the digits. o. Mispits R % ak Citatios dis- M total (%) Eq. Eq. Eq. fo MC tict (I.) (I.8) (II.4) R = % % % % % % % % % % % % I the ext sectio we itoduce ad solve the stochastic model of mispit popagatio. The model explais the powe law of mispit epetitios (see Fig.II.). If you do ot have time to ead the whole chapte, you ca poceed afte Sectio II. ight to Sectio III.. Thee we fomulate ad solve the model of adom citig scietists (RCS). The model is as follows: whe scietist is witig a mauscipt he picks up seveal adom papes cites them ad copies a factio of thei efeeces. The model ca explai why some papes ae fa moe cited tha othes. Afte that, you ca diectly poceed to discussio i Sectio V. If you have questios, you ca fid aswes to some of them i othe sectios. The esults of Sectio I. ae exact i the limit of ifiite umbe of citatios. Sice this umbe is obviously fiite, we eed to study fiite size effects, which affect ou estimate of R. This is doe i Sectio II.2 usig complicated mathematical methods ad i Sectio II.3 usig Mote Calo simulatios. The limitatios of the simple model aisig fom the istaces like, fo example, the same autho epeats the same mispit, ae discussed i Sectio II.4. I Sectio II.5, we eview the pevious wok o idetical mispits. I shot: some people did otice epeat mispits ad at-

6 6 tibuted them to citatio copyig, but obody deived Eq. (I.) befoe us. The RCS model of Sectio III ca explai a powe law i oveall citatio distibutio, but caot explai a powe law distibutio i citatios to the papes of the same age. Sectio IV. itoduces the modified model of adom-citig scietist (MMRCS), which solves the poblem. The model is as follows: whe a scietist wites a mauscipt, he picks up seveal adom ecet papes, cites them, ad also copies some of thei efeeces. The diffeece with the oigial model is the wod ecet. I Sectio IV.2 the MMRCS is solved usig theoy of bachig pocesses ad the powe law distibutio of citatios to the papes of the same age is deived. Sectio IV.3 cosides the model whee papes ae ot ceated equal but have Dawiia fitess that affects thei citability. Sectio IV.4 studies effects of liteatue gowth (yealy icease of the umbe of published papes) o citatio distibutio. Sectio IV.5 descibes umeical simulatios of MMRCS, which pefectly match eal citatio data. Sectio IV.6 shows that MMRCS ca explai the pheomeo of liteatue agig that is why papes become less cited as they get olde. Sectio IV.7 shows that MMRCS ca explai the mysteious pheomeo of sleepig beauties i sciece (papes that ae at fist hadly oticed suddely awake ad get a lot of citatios). Sectio IV.8 descibes the coectio of MMRCS to the Sciece of Self-Ogaized Citicality. II. Stochastic modelig of mispits i citatios. Mispit popagatio model Ou mispit popagatio model (MPM) [], [3] which was stimulated by Simo s [4] explaatio of Zipf Law ad Kapivsky-Rede [5] idea of lik ediectio, is as follows. Each ew cite fids the efeece to the oigial i ay of the papes that aleady cite it (o it ca be the oigial pape itself). With pobability R he gets the citatio ifomatio fom the oigial. With pobability -R he copies the citatio to the oigial fom the pape he foud the citatio i. I eithe case, the cite itoduces a ew mispit with pobability M. Let us deive the evolutio equatios fo the mispit distibutio. The oly way to icease the umbe of mispits that appeaed oly oce,, is to itoduce a ew mispit. So, with each ew citatio iceases by with pobability M. The oly way to decease, is to copy coectly oe of mispits that appeaed oly oce, this happes with pobability α, whee

7 7 α = ( R) ( M) (II.) is the pobability that a ew cite copies the citatio without itoducig a ew eo, ad is the total umbe of citatios. Fo the expectatio value, we thus have: d = M α. (II.2) d, (whee The umbe of mispits that appeaed K times, K K > ) ca be iceased oly by copyig coectly a mispit which appeaed K times. It ca oly be deceased by copyig (agai coectly) a mispit which appeaed K times. Fo the expectatio values, we thus have: d K d = α (K ) K K K ( K > ). (II.3) Assumig that the distibutio of mispits has eached its statioay state, we ca eplace the deivatives ( d K d ) by atios ( K ) to get: = ; +α M K + K K = + α + K (K >). (II.4) ote that fo lage K: ca be ewitte as: K K + d dk + K, theefoe Equatio (II.4) d K dk + α + α + K + α k K k. Fom this follows that the mispits fequecies ae distibuted accodig to a powe law: K ~ /K γ, (II.5) whee γ =+ α = + ( R) ( M). (II.6)

8 8 Relatioship betwee γ ad α i Eq.(II.6) is the same as the oe betwee expoets of umbe-fequecy ad ak-fequecy distibutios 2. Theefoe the paameteα, which was defied i Eq.(II.), tued out to be the Zipf law expoet. A exact fomula fo k ca also be obtaied by iteatio of Eq.(II.4) to get: K) Γ( γ ) ( K + γ ) Γ( M M K = = Β, Γ α ( K γ ) α (II.7) Hee Γ ad Β ae Eule s Gamma ad Beta fuctios. Usig the asymptotic fo costat γ ad lage K Γ Γ( γ ) ( K + γ ) dt d γ ~ K (II.8) we ecove Eq.(II.5). The ate equatio fo the total umbe of mispits is: T = M + α. (II.9) The statioay solutio of Eq. (II.9) is: T M = = α M R + M RM. (II.) 2 Suppose that the umbe of occueces of a mispit (K), as a fuctio of the ak ( ), whe the ak is detemied by the above fequecy of occuece (so that the most popula mispit has ak, secod most fequet mispit has ak C 2 ad so o), follows a Zipf law: K ( ) =. We wat to fid the umbefequecy distibutio, i.e. how may mispits appeaed times. The umbe of α mispits that appeaed betwee K ad K 2 times is obviously 2 -, whee α α K = C ad K = C. Theefoe, the umbe of mispits that appeaed K times, k, satisfies 2 2 dk K α = d ad hece, = d dk ~ K K.

9 9 The expectatio value fo the umbe of distict mispits is obviously D M =. (II.) Fom Equatios (II.) ad (II.) we obtai: D T R =, (II.2) T D which is idetical to Eq.(I.8). Oe ca ask why we did ot choose to extact R usig Equatios (II.) o (II.6). This is because α ad γ ae ot vey sesitive to R whe it is small (i fact Eq. (II.) gives egative values of R fo some of the fittigs i Figue II.). I cotast, T scales as / R. We ca slightly modify ou model ad assume that oigial mispits ae oly itoduced whe the efeece is deived fom the oigial pape, while those who copy efeeces do ot itoduce ew mispits (e.g. they do cut ad paste). I this case oe ca show that T = M ad D = M R. As a cosequece Eq.(I.) becomes exact (i tems of expectatio values, of couse).

10 Figue II.. Rak-fequecy distibutios of mispits i efeecig fou highpofile papes (hee the ak is detemied by the fequecy so that the most popula mispit has ak, secod most fequet mispit has ak 2 ad so o). Figues a), b), c), ad d) ae fo Papes 2, 5, 7, ad of Table I.. Solid lies ae fits to Zipf Law with expoets a).2; b).5; c).66; d) Fiite-size coectios Pecedig aalysis assumes that the mispit distibutio had eached its statioay state. Is this easoable? Eq.(II.9) ca be ewitte as:

11 d( T / ) = d l. (II.3) M ( T / ) ( α) atually the fist citatio is coect (it is the pape itself). The the iitial coditio is = ; T =. Eq.(II.3) ca be solved to get: T M M = α + R+ M α R M M R = M R (II.4) This should be solved umeically fo R. The values obtaied usig Eq.(II.4) ae give i Table I.2. They age betwee 7% ad 57%. ote that fo oe pape (o.4) o solutio to Eq. (II.4) was foud 3. As is ot a cotiuous vaiable, itegatio of Equatio (II.9) is ot pefectly justified, paticulaly whe is small. Theefoe, we eexamie the poblem usig a igoous discete appoach due to Kapivsky ad Rede [6]. The total umbe of mispits, T, is a adom vaiable that chages accodig to T ( + ) ( ) T T ( ) with pobability M α = (II.5) T ( ) ( ) T + with pobability M + α afte each ew citatio. Theefoe, the expectatio values of T obey the followig ecusio elatios: ( ) T T ( + ) = T( ) + α + M (II.6) To solve Eq. (II.6) we defie a geeatig fuctio: 3 Why did this happe? Obviously, T eaches maximum whe R equals zeo. M T =. Fo pape Substitutig R = i Eq.(II.4) we get: ( ) o.4 we have = 2,578, M = D/ = 32/2,578. Substitutig this ito the pecedig equatio, we gett MAX = 239. The obseved value T = 263 is theefoe highe tha a expectatio value of T fo ay R. This does ot immediately suggest discepacy betwee the model ad expeimet but a stog fluctuatio. I fact out of,, us of Mote-Calo simulatio of MPM with the paametes of the metioed pape ad R=.2 exactly 49,72 us (almost 5%) poduced T 263. MAX

12 2 ( ω) = T( ) χ ω (II.7) = ω ad summig ove Afte multiplyig Eq. (II.6) by the ecusio elatio is coveted ito the diffeetial equatio fo the geeatig fuctio ( ω) dχ dω = + α ( )χ + M ω ( ) 2 (II.8) Solvig Eq.(II.8) subject to the iitial coditio ( ) = T( ) = χ gives ( ω) M ( ) ( ) = 2 +α α ω ω χ (II.9) Fially we expad the ight had side of Eq. (II.9) i Taylo seies i ω ad equatig coefficiets of ω obtai: T ( ) M Γ( + α) ( ) ( ) = α Γ + α Γ + (II.2) T Usig Eq. (II.8) we obtai that fo lage ( ) M α Γ = α ( + α) This is idetical to Eq. (II.4) except fo the pe-facto Γ( +α) (II.2). Paameteα (it is defied i Eq. (II.)) ages betwee ad. Theefoe, the agumet of Gamma-fuctio ages betwee ad 2. Because Γ ( ) = Γ( 2) = ad betwee ad 2 Gamma fuctio has just oe extemum Γ (.466 K) =.8856K, the cotiuum appoximatio (Eq.(II.4)) is easoably accuate.

13 3 3. Mote-Calo simulatios I the pecedig sectio, we calculated the expectatio value of T. Howeve, it does ot always coicide with the most likely value whe the pobability distibutio is ot Gaussia. To get a bette idea of the model s behavio fo small ad a bette estimate of R we did umeical simulatios. To simplify compaiso with actual data the simulatios wee pefomed i a mico-caoical esemble, i.e. with a fixed umbe of distict mispits. Each pape is chaacteized by the total umbe of citatios,, ad the umbe of distict mispits, D. At the begiig of a simulatio, D mispits ae adomly distibuted betwee citatios ad choological umbes of the citatios with mispits ae ecoded i a list. I the ext stage of the simulatio fo each ew citatio, istead of itoducig a mispit with pobability M, we itoduce a mispit oly if its choological umbe is icluded i the list ceated at the outset. This way oe ca esue that the umbe of distict mispits i evey u of a simulatio is equal to the actual umbe of distict mispits fo the pape i questio. A typical outcome of such simulatio fo Pape o.5 is show i Fig. II.2. adom-citig model actual data fequecy ak Figue II.2. A typical outcome of a sigle simulatio of the MPM (with R=.2) compaed to the actual data fo Pape o.5 i Table I..

14 4 Figue II.3. The outcome of,, us of the MPM with =43, D=45 (paametes of pape o.5 fom Table I.) fo fou diffeet values of R (.9,.5,.2, fom left to ight). To estimate the value of R,,, us of the adom-citig model with R =,.,.2.9 wee doe. A outcome of such simulatios fo oe pape is show i Fig. II.3. The umbe of times, R, whe the simulatio poduced a total umbe of mispits equal to the oe actually obseved fo the pape i questio was ecoded fo each R. Bayesia ifeece was used to estimate the pobability of R: ( R) R P (II.22) = R R Estimated pobability distibutios of R, computed usig Eq. (II.22) fo fou sample papes ae show i Figue II.4. The media values ae give i Table I.2 (see the MC colum). They age betwee % ad 49%.

15 5 ow let us assume R to be the same fo all twelve papes ad compute Bayesia ifeece: ( R ) i= P = 2 R 2 i= i R i R (II.23) The esult is show i Fig. II.5. P(R) is shaply peaked aoud R=.2. The media value of R is 8% ad with 95% pobability R is less tha 34%. But is the assumptio that R is the same fo all twelve papes easoable? The estimates fo sepaate papes vay betwee te ad fifty pecet! To aswe this questio we did the followig aalysis. Let us defie fo each pape a pecetile ak. This is the factio of the simulatios of the MPM (with R =.2) that poduced T, which was less tha actually obseved T. Actual values of these pecetile aks fo each pape ae give i Table I.2 ad thei cumulative distibutio is show i Fig. II.6. ow if we claim that MPM with same R =.2 fo all papes ideed descibes the eality the the distibutio of these pecetile aks must be uifom. Whethe o ot the data is cosistet with this, we ca check usig Kolmogoov-Smiov test [7]. The maximum value of the absolute diffeece betwee two cumulative distibutio fuctios (D-statistics) i ou case is D =.5. The pobability fo D to be moe tha that is 9%. This meas that the data is pefectly cosistet with the assumptio of R =.2 fo all papes.

16 6 Figue II.4.Bayesia ifeece fo the pobability desity of the eades/cites atio, R, computed usig Eq. (II.22). Figues a), b), c), ad d) ae fo Papes o. 2, 5, 7, ad (Table I.).

17 7 6 5 pobability desity R Figue II.5. Bayesia ifeece fo the eades/cites atio, R, based o twelve studied papes computed usig Eq.(II.23).

18 8 cumulative pobability distibutio % 2% 4% 6% 8% % pecetile ak Figue II.6. Cumulative distibutio of the pecetile aks of the obseved values of T with egad to the outcomes of the simulatios of the MPM with R =.2 (diamods). Fo compaiso the cumulative fuctio of the uifom distibutio is give (a lie). Oe ca otice that the estimates of M (computed as M=D/) fo diffeet papes (see Table I.2) ae also diffeet. Oe may ask if it is possible that M is the same fo all papes ad diffeet values of D/ ae esults of fluctuatios. The aswe is that the data is totally icosistet with sigle M fo all papes. This is ot uexpected, because some efeeces ca be moe eo-poe, fo example, because they ae loge. Ideed, the most-mispited pape (o.2) has two-digit volume umbe ad five-digit page umbe. 4. Opeatioal limitatios of the model Scietists copy citatios because they ae ot pefect. Ou aalysis is impefect as well. Thee ae occasioal epeat idetical mispits i papes, which shae idividuals i thei autho lists. To estimate the magitude of this effect we took a close look at all 96 mispited citatios to pape 5 of Table I.. It tued out that such evets costitute a mioity of epeat mispits. It is ot obvious what to do with such cases whe the autho lists ae ot idetical: should the set of citatios

19 9 be couted as a sigle occuece (ude the pemise that the commo co-autho is the oly souce of the mispit) o as multiple epetitios. Coutig all such epetitios as oly a sigle mispit occuece esults i elimiatio of 39 epeat mispits. The umbe of total mispits, T, dops fom 96 to 57, bigig the uppe boud fo R (Eq.(I.)) fom 45 23% 45 up to 29 %. A alteative appoach is to subtact all the epetitios of each mispit by the oigiatos of that mispit fom o-eades ad add it to the umbe of eades. Thee wee such epetitios, which iceases D fom 45 up to 56 ad the uppe boud fo R (Eq.(I.)) ises to 56 29%, which is the same value as the pecedig estimate. 96 It would be desiable to edo the estimate usig Equatios (II.2) ad (II.4), but the mispit popagatio model would have to be modified to accout fo epeat citatios by same autho ad multiple authoships of a pape. This may be a subject of futue ivestigatios. Aothe issue bought up by the citics [8] is that because some mispits ae moe likely tha othes, it is possible to epeat someoe else s mispit puely by chace. By examiig the actual data, oe fids that about two thid of distict mispits fall i to the followig categoies: a) Oe mispited digit i volume, page umbe, o i the yea. b) Oe missig o added digit i volume o page umbe. c) Two adjacet digits i a page umbe ae itechaged. The majoity of the emaiig mispits ae combiatios of a), b), c), fo example, oe digit i page umbe omitted ad oe digit i yea mispited 4. Fo a typical efeece, thee ae ove fifty afoemetioed likely mispits. Howeve, eve if pobability of cetai mispit is ot egligibly small but oe i fifty, ou aalysis still applies. Fo example, fo pape o.5 (Table I.) the most popula eo appeaed 78 times, while thee wee 96 mispits i total. Theefoe, if pobability of cetai mispit is /5, thee should be about such mispits, ot 78. I ode to explai epeat mispits distibutio by highe pobability of cetai mispit this pobability should be as big as This is extemely ulikely. Howeve, fidig elative popesities of diffeet mispits deseves futhe ivestigatio. Smith oticed [9] that some mispits ae i fact itoduced by the ISI. To estimate the impotace of this effect we explicitly veified 88 mispited (accodig to ISI) citatios i the oigial aticles. 72 of them wee exactly as i the ISI database, but 6 wee i fact coect citatios. To be pecise some of them had 4 Thee ae also mispits whee autho, joual, volume ad yea ae pefectly coect, but the page umbe is totally diffeet. Pobably, i such case the cite mistakely took the page umbe fom a eighboig pape i the efeece list he was liftig the citatio fom.

20 2 mio iaccuacies, like secod iitial of the autho was missig, while page umbe, volume ad yea whee coect. Appaetly, they wee victims of a eoeous eo coectio [9]. It is ot clea how to cosistetly take ito accout these effects, specifically because thee is o way to estimate how may wog citatios have bee coectly coected by ISI []. But give the elatively small pecetage of the discepacy betwee ISI database ad actual aticles ( % ) this ca be take as a oise with which we ca live. atio of eades to cites umbe of citatios Figue II.7. Ratio of eades to cites as a fuctio of total amout of citatios fo R =.2, computed usig Eq.(II.24). It is impotat to ote that withi the famewok of the MPM R is ot the atio of eades to cites, but the pobability that a cite cosults the oigial pape, povided that he ecouteed it though aothe pape s efeece list. Howeve, he could ecoute the pape diectly. This has egligible effect fo highly-cited papes, but is impotat fo low-cited papes. Withi the MPM famewok the pobability of such a evet fo each ew citatio is obviously, whee is the cuet total umbe of citatios. The expectatio value of the tue atio of eades to cites is theefoe: R l + = ( ) = R + ( R) R + ( R) ( 2 ). (II.24)

21 2 The values of R * fo papes with diffeet total umbes of citatios, computed usig Eq.(II.24), ae show i Fig.II.7. Fo example, o aveage, about fou people have ead a pape which was cited te times. Oe ca use Eq.(II.24) ad empiical citatio distibutio to estimate a aveage value of R fo the scietific liteatue i geeal. The fomula is: ( ) R i i R = (II.25) i Hee the summatio is ove all of the papes i the sample ad i is the umbe of citatios that ith pape had eceived. The estimate, computed usig citatio data fo Physical Review D [] ad Eqs (II.24) ad (II.25) (assumig R =.2), is R Compaiso with the pevious wok The bulk of pevious liteatue o citatios was coceed with thei coutig. Afte extesive liteatue seach we foud oly a hadful of papes which aalyzed mispits i citatios (the pape by Steel [2], titled idetically to ou fist mispit pape, i.e. Read befoe you cite, tued out to use the aalysis of the cotet of the papes, ot of the popagatio of mispits i efeeces). Boadus [3] looked though 48 papes, which cited both the eowed book, which misquoted the title of oe of its efeeces, ad that pape, the title of which was misquoted i the book. He foud that 34 o 23% of citig papes made the same eo as was i the book. Moed ad Vies [4] (appaetly idepedet of Boadus, as they do t efe to his wok), foud idetical mispits i scietific citatios ad attibuted them to citatio copyig. Hoema ad owicke [5] looked though a umbe of papes, which deal with the so-called Otega Hypothesis of Cole ad Cole. Whe Cole ad Cole quoted a passage fom the book by Otega they itoduced thee distotios. Hoema ad owicke foud seve papes which cite Cole ad Cole ad also quote that passage fom Otega. I six out of these seve papes all of the distotios made by Cole ad Cole wee epeated. Accodig to [5] i this pocess eve the oigial meaig of the quotatio was alteed. I fact, ifomatio is sometimes defied by its popety to deteioate i chais [6]. While the factio of copied citatios foud by Hoema ad owicke [5], % agees with ou estimate, Boadus umbe, 23%, seems to disagee with it. ote, howeve, that Boadus [3] assumes that citatio, if copied - was copied fom the book (because the book was eowed). Ou aalysis idicates that majoity of citatios to eowed papes ae copied. Similaly, we su-

22 22 mise, i the Boadus case citatios to both the book ad the pape wee ofte copied fom a thid souce. III. Copied citatios ceate eowed papes?.the model of adom-citig scietists Duig the Mahatta poject (the makig of uclea bomb), Femi asked Ge. Goves, the head of the poject, what is the defiitio of a geat geeal [7]. Goves eplied that ay geeal who had wo five battles i a ow might safely be called geat. Femi the asked how may geeals ae geat. Goves said about thee out of evey huded. Femi cojectued that cosideig that opposig foces fo most battles ae oughly equal i stegth, the chace of wiig oe battle is ½ ad the chace of wiig five battles i a ow is 2 5 = 32. So you ae ight, Geeal, about thee out of evey huded. Mathematical pobability, ot geius. The existece of militay geius also questioed Lev Tolstoy i his book Wa ad Peace. A commoly accepted measue of geatess fo scietists is the umbe of citatios to thei papes [8]. Fo example, SPIRES, the High-Eegy Physics liteatue database, divides papes ito six categoies accodig to the umbe of citatios they eceive. The top categoy, Reowed papes ae those with 5 o moe citatios. Let us have a look at the citatios to oughly eightee ad a half thousad papes 5, published i Physical Review D i []. As of 997 thee whee about 33 thousads of such citatios: eightee pe published pape o aveage. Howeve, foty-fou papes wee cited five huded times o moe. Could this happe if all papes ae ceated equal? If they ideed ae the the chace to wi a citatio is oe i 8,5. What is the chace to wi 5 cites out of 33,? The calculatio is slightly moe complex tha i the militaistic 5 I ou iitial epot [22] we metioed ove 24 thousad papes. This umbe is icoect ad the eade suely udestads the easo: mispits. I fact, out of 24,295 papes i that dataset oly 8,56 tued out to be eal papes ad 5,735 papes tued out to be mispited citatios. These papes got 7,382 out of 35,868 citatios. That is evey distict mispit o aveage appeaed thee times. As oe could expect, cleaig out mispits lead to much bette ageemet betwee expeimet ad theoy: compae Fig. III. ad Fig. of Ref [22].

23 23 case 6 5, but the aswe is i, o, i othe wods, it is zeo. Oe is tempted to coclude that those foty-fou papes, which achieved the impossible, ae geat. I the pecedig sectios, we demostated that copyig fom the lists of efeeces used i othe papes is a majo compoet of the citatio dyamics i scietific publicatio. This way a pape that aleady was cited is likely to be cited agai, ad afte it is cited agai it is eve moe likely to be cited i the futue. I othe wods, uto evey oe which hath shall be give [Luke 9:26]. This pheomeo is kow as Matthew effect, 7 cumulative advatage [2], o pefeetial attachmet [2]. The effect of citatio copyig o the pobability distibutio of citatios ca be quatitatively udestood withi the famewok of the model of adomcitig scietists (RCS) [22] 8, which is as follows. Whe a scietist is witig a mauscipt he picks up m adom aticles 9, cites them, ad also copies some of thei efeeces, each with pobability p. 6 If oe assumes that all papes ae ceated equal the the pobability to wi m out of possible citatios whe the total umbe of cited papes is is give by m ( ) the Poisso distibutio: p = e. Usig Stilig s fomula oe ca m! e ewite this as: l( p) ml. Afte substitutig = 33,, m m = 5 ad = 85 ito the above equatio we get: l( p ), 8, o p Sociologist of sciece Robet Meto obseved [9] that whe a scietist gets ecogitio ealy i his caee he is likely to get moe ad moe ecogitio. He called it Matthew Effect because i Gospel accodig to Mathew (25:29) appea the wods: uto evey oe that hath shall be give." The attibutio of a special ole to St. Matthew is ufai. The quoted wods belog to Jesus ad also appea i Luke ad Mak s gospels. evetheless, thousads of people who did ot ead The Bible copied the ame Matthew Effect. 8 Fom the mathematical pespective, almost idetical to RCS model (the oly diffeece was that they cosideed a udiected gaph, while citatio gaph is diected) was ealie poposed i Ref. [27]. 9 The aalysis peseted hee also applies to a moe geeal case whe m is ot a costat, but a adom vaiable. I that case m i all of the equatios that follow should be itepeted as the mea value of this vaiable.

24 24 The evolutio of the citatio distibutio (hee K deotes the umbe of papes that wee cited K times, ad is the total umbe of papes) is descibed by the followig ate equatios: d = m, (III.) d d d = m ( + p( K ) ) ( pk ) K K + K, which have the followig statioay solutio: = ; m + ( K ) + p K = K. (III.2) + m + pk Fo lage K it follows fom (III.2) that: K / γ ~ K ; γ = +. (III.3) m p Citatio distibutio follows a powe law, empiically obseved i [23], [24], [25]. A good ageemet betwee the RCS model ad actual citatio data [] is achieved with iput paametes m = 5 ad p =. 4 (see Figue III.). ow what is the pobability fo a abitay pape to become eowed, i.e. eceive moe tha five huded citatios? Iteatio of Eq. III.2 (with m = 5 ad p =. 4 ) shows that this pobability is i 42. This meas that about 44 out of 8,5 papes should be eowed. Mathematical pobability, ot geius. O oe icidet [26] apoleo (icidetally, he was the militay commade, whose geius was questioed by Tolstoy) said to Laplace They tell me you have witte this lage book o the system of the uivese, ad have eve eve metioed its Ceato. The eply was I have o eed fo this hypothesis. It is wothwhile to ote that Laplace was ot agaist God. He simply did ot eed to postulate His existece i ode to explai existig astoomical data. Similaly, the peset wok is ot blasphemy. Of couse, i some spiitual sese, geat scietists do exist. It is just that eve if they would ot exist, citatio data would look the same.

25 25 umbe of citatios. pobability eal citatio data adom-citig model Figue III.. Outcome of the model of adom-citig scietists (with m = 5 ad p =.4) compaed to actual citatio data. Mathematical pobability athe tha geius ca explai why some papes ae cited a lot moe tha the othes. 2.Relatio to pevious wok Ou oigial pape o the subject [22] was stimulated by the model itoduced by Vazquez [26]. It is as follows. Whe scietist is witig a mauscipt, he picks up a pape, cites it, follows its efeeces, ad cites a factio p of them. Aftewad he epeats this pocedue with each of the papes that he cited. Ad so o. I two limitig cases ( p = ad p = ) the Vazquez model is exactly solvable [26]. Also i these cases it is idetical to the RCS model (m = case), which i cotast ca be solved fo ay p. Though theoetically iteestig, the Vazquez model caot be a ealistic desciptio of the citatio pocess. I fact, the esults peseted i two pecedig sectios idicate that thee is essetially just oe ecusio, that is, efeeces ae copied fom the pape at had, but hadly followed. To be pecise, esults of two pecedig sectios could suppot a geealized Vazquez model, i which the efeeces of the pape at had ae copied with pobability p,

26 26 ad aftewads the copied efeeces ae followed with pobability R. Howeve, give the low value of this pobability ( R. 2 ), it is clea that the effect of secoday ecusios o the citatio distibutio is small. The book of Ecclesiastes says: Is thee ay thig wheeof it may be said, See, this is ew? It hath bee aleady of old time, which was befoe us. The discovey epoted i this sectio is o exceptio. Log ago Pice [2], by postulatig that the pobability of pape beig cited is somehow popotioal to the amout of citatios it had aleady eceived, explaied the powe law i citatio fequecies, which he had ealie obseved [22]. Howeve, Pice did ot popose ay mechaism fo that. Vasquez did popose a mechaism, but it was oly a hypothesis. I cotast, ou pape is ooted i facts. IV. Mathematical theoy of citig. Modified model of adom-citig scietists "... citatios ot oly vouch fo the authoity ad elevace of the statemets they ae called upo to suppot; they embed the whole wok i cotext of pevious achievemets ad cuet aspiatios. It is vey ae to fid a eputable pape that cotais o efeece to othe eseach. Ideed, oe elies o the citatios to show its place i the whole scietific stuctue just as oe elies o a ma's kiship affiliatios to show his place i his tibe." Joh M. Zima, FRS (Ref. [28]) I spite of its simplicity, the model of adom citig scietists appeaed to accout fo the majo popeties of empiically obseved distibutios of citatios. A moe detailed aalysis, howeve, eveals that some featues of the citatio distibutio ae ot accouted fo by the model. The cumulative advatage pocess would lead to oldest papes beig most highly cited [5], [2], [29]. I eality, the aveage citatio ate deceases as the pape i questio gets olde [23], [3], [3], [32]. The cumulative advatage pocess would also lead to a expoetial distibutio of citatios to papes of the same age [5], [29]. I eality citatios to papes published duig the same yea ae distibuted accodig to a powe-law (see the ISI dataset i Fig.(a) i Ref. [25]). Some of these efeeces do ot deal with citig, but with othe social pocesses, which ae modeled usig the same mathematical tools. Hee we ephase the esults of such papes i tems of citatios fo simplicity.

27 27 I this sectio, we study the modified model of adom-citig scietists [33]: whe a scietist wites a mauscipt, he picks up seveal adom ecet papes, cites them ad also copies some of thei efeeces. The diffeece with the oigial model is the wod ecet. We solve this model usig methods of the theoy of bachig pocesses [34] (we eview its elevat elemets i Appedix A), ad show that it explais both the powe-law distibutio of citatios to papes published duig the same yea ad liteatue agig. A simila model was ealie poposed by Betley, Hah & Shea [35] i the cotext of patets citatios. Howeve they just used it to explai a powe law i citatio distibutio (fo what the usual cumulative advatage model will do) ad did ot addess the topics we just metioed. While wokig o a pape, a scietist eads cuet issues of scietific jouals ad selects fom them the efeeces to be cited i it. These efeeces ae of two sots: Fesh papes he had just ead to embed his wok i the cotext of cuet aspiatios. Olde papes that ae cited i the fesh papes he had just ead to place his wok i the cotext of pevious achievemets. It is ot a ecessay coditio fo the validity of ou model that the citatios to old papes ae copied, but the pape itself emais uead (although such opiio is suppoted by the studies of mispit popagatio). The ecessay coditios ae as follows: Olde papes ae cosideed fo possible citig oly if they wee ecetly cited. If a citatio to a old pape is followed ad the pape is fomally ead the scietific qualities of that pape do ot ifluece its chace of beig cited. A easoable estimate fo the legth of time a scietist woks o a paticula pape is oe yea. We will thus assume that ecet i the model of adomcitig scietists meas the pecedig yea. To make the model mathematically tactable we efoce time-discetizatio with a uit of oe yea. The pecise model to be studied is as follows. Evey yea papes ae published. Thee is, o aveage, efeeces i a published pape (the actual value is somewhee betwee ef 2 ad 4). Each yea, a factio α of efeeces goes to adomly selected pecedig yea papes (the estimate fom actual citatio data is α. (see Fig. 4 i Ref. [23]) o α. 5 (see Fig. 6 i Ref. [36]). The emaiig citatios ae adomly copied fom the lists of efeeces used i the pecedig yea papes. The ucetaity i the value of α depeds ot oly o the accuacy of the estimate of the factio of citatios which goes to pevious yea papes. We also abitaily defied ecet pape (i the sese of ou model), as the oe published withi a yea. Of couse, this is by ode of magitude coect but the tue value ca be aywhee betwee half a yea ad two yeas.

28 28 2. Bachig citatios Whe is lage, this model leads to the fist-yea citatios beig Poissodistibuted. The pobability to get citatios is p λ! λ ( ) = e, (IV.) whee λ is the aveage expected umbe of citatios λ = α ef. (IV.2) The umbe of the secod-yea citatios, geeated by each fist yea citatio (as well as, thid-yea citatios geeated by each secod yea citatio ad so o), agai follows a Poisso distibutio, this time with the mea λ = ( α ). (IV.3) Withi this model, citatio pocess is a bachig pocess (see Appedix A) with the fist yea citatios equivalet to childe, the secod-yea citatios to gad childe, ad so o. As λ <, this bachig pocess is subcitical. Figue IV. shows a gaphical illustatio of the bachig citatio pocess. Substitutig Eq.(IV.) ito Eq.(A) we obtai the geeatig fuctio fo the fist yea citatios: ( ) λ ( z) = e z f. (IV.4) Similaly, the geeatig fuctio fo the late-yeas citatios is: ( )λ ( z) = e z f. (IV.5) The pocess is easie to aalyze whe λ = λ, o = λ α = ef, as the we have a simple bachig pocess, whee all geeatios ae goveed by the same offspig pobabilities. The case whe λ α λ λ we study i Appedix B.

29 29 Figue IV.. A illustatio of the bachig citatio pocess, geeated by the modified model of adom-citig scietists. Duig the fist yea afte publicatio, the pape was cited i thee othe papes witte by the scietists who have ead it. Duig the secod yea oe of those citatios was copied i two papes, oe i a sigle pape ad oe was eve copied. This esulted i thee secod yea citatios. Duig the thid yea, two of these citatios wee eve copied, ad oe was copied i thee papes. A. Distibutio of citatios to papes published duig the same yea Theoy of bachig pocesses allows us to aalytically compute the pobability distibutio, P ( ), of the total umbe of citatios the pape eceives befoe it is fogotte. This should appoximate the distibutio of citatios to old papes. Substitutig Eq.(IV.5) ito Eq.(A8) we get: P d! dω ( ω ) λ ( ) ( ) λ = e = e ω= λ! (IV.6) Applyig Stilig s fomula to Eq.(IV.6), we obtai the lage asymptotic of the distibutio of citatios:

30 3 P 3 2 (IV.7) ( λ l λ ) ( ) e 2π λ Whe λ << we ca appoximate the facto i the expoet as: 2 ( λ ) 2 λ l λ. (IV.8) As λ <<, the above umbe is small. This meas that fo << 2 λ the expoet i Eq.(IV.7) is appoximately equal to ad the ( ) 2 behavio of P ( ) is domiated by the >> 2 ( λ) 2 the behavio of ( ) facto. I cotast, whe 3 2 P is domiated by the expoetial facto. Thus citatio distibutio chages fom a powe law to a expoetial (suffes a expoetial cut-off) at about 2 c = (IV.9) λ l λ ( λ) 2 citatios. Fo example, wheα =. Eq.(IV.3) gives λ =. 9 ad fom Eq.(IV.9) we get that the expoetial cut-off happes at about 2 citatios. We see that, ulike the cumulative advatage model, ou model is capable of qualitative explaatio of the powe law distibutio of citatios to papes of the same age. The expoetial cut-off at 2, howeve, happes too soo, as the actual citatio distibutio obeys a powe law well ito thousads of citatios. I the followig sectios we show that takig ito accout the effects of liteatue gowth ad of vaiatios i papes Dawiia fitess ca fix this. I the cumulative advatage (AKA pefeetial attachmet) model, a powe law distibutio of citatios is oly achieved because papes have diffeet ages. This is ot immediately obvious fom the ealy teatmets of the poblem [4], [2], but is explicit i late studies [5], [2], [29]. I that model, the oldest papes ae the most cited oes. The umbe of citatios is maily detemied by pape s age. At the same time, distibutio of citatios to papes of the same age is expoetial [5], [29]. The key diffeece betwee that model ad ous is as follows. I the cumulative advatage model, the ate of citatio is popotioal to the umbe of citatios the pape had accumulated sice its publicatio. I ou model, the ate of citatio is popotioal to the umbe of citatios the pape eceived duig pecedig yea. This meas that if a ulucky pape was ot cited duig pevious yea it will eve be cited i the futue. This meas that its ate of citatio will be less tha that i the cumulative advatage model. O the othe had,

31 3 the lucky papes, which wee cited duig the pevious yea, will get all the citatio shae of the ulucky papes. Thei citatio ates will be highe tha i the cumulative advatage model. Thee is thus moe statificatio i ou model tha i the cumulative advatage model. Cosequetly, the esultig citatio distibutio is fa moe skewed. B. Distibutio of citatios to papes cited duig the same yea We deote as ( ) equilibium distibutio of ( ) equatio: the umbe of papes cited times duig give yea. The should satisfy the followig self-cosistecy λm λ = e m=!! ( ) ( ) ( ) ( ) λm λ m e + (IV.) Hee the fist tem comes fom citatio copyig ad the secod fom citig pevious yea papes. I the limit of lage the secod tem ca be eglected ad the sum ca be eplaced with itegal to get: λ e m ( ) = dm( m)( m)! λ (IV.) I the case λ = oe solutio of Eq.(IV.) is ( m) = C, whee C is a abitay costat. Clealy, the itegal becomes a gamma-fuctio ad the factoial i the deomiato cacels out. Howeve, this solutio is, meaigless sice the total umbe of citatios pe yea, which is give by cite = = m ( m) m (IV.2) diveges. I the case λ <, ( m) = C is o loge a solutio sice the itegal gives C λ. Howeve ( m) = C m is a solutio. This solutio is agai meaigless because the total umbe of yealy citatios give by (IV.2) agai diveges. Oe ca look fo a solutio of the fom C ( m) = exp ( µ m). (IV.3) m

32 32 Afte substitutig Eq. (IV.3) ito (IV.) we get that ( ) same fuctio but istead of µ with ( λ) is give by the µ = l + µ. (IV.4) The self cosistecy equatio fo µ is thus ( λ) µ = l + µ. (IV.5) The obvious solutio is µ = which gives us the peviously ejected solutio ( m) = C m. It is also easy to see that this statioay solutio is ustable. If µ slightly deviates fom zeo Eq. (IV.4) gives us µ = µ λ. Sice λ < the deviatio fom statioay shape will icease the ext yea. Aothe solutio of Eq. (IV.5) ca be foud by expasio of the logaithm up to the secod ode i µ 2 λ. Oe ca show that this solutio is stable. Thus, we get: µ. It is ( ) C exp. (IV.6) m ( m) ( 2( λ) m) Afte substitutig this ito Eq. (IV.2) we get ( λ) cite C 2. (IV.7) The solutio which we just peseted was stimulated by that obtaied by Wight [37], who studied the distibutio of alleles (alteative foms of a gee) i a populatio. I Wight s model, the gee pool at ay geeatio has costat size g. To fom the ext geeatio we g times select a adom gee fom cuet geeatio pool ad copy it to ext geeatio pool. With some pobability, a gee ca mutate duig the pocess of copyig. The poblem is idetical to ous with a allele eplaced with a pape ad mutatio with a ew pape. Ou solutio follows that of Wight but is a lot simple. Wight cosideed fiite g. ad as a cosequece got Biomial distibutio ad a Beta fuctio i his aalog of Eq.(IV.). The simplificatio was possible because i the limit of lage g Biomial distibutio becomes Poissoia. Alteative deivatios of Eq.(IV.6) ca be foud i Refs [33] ad [38].

33 33 3. Scietific Dawiism ow we poceed to ivestigate the model, whee papes ae ot ceated equal, but each has a specific Dawiia fitess, which is a bibliometic measue of scietific fags ad claws that help a pape to fight fo citatios with its competitos. While this paamete ca deped o factos othe tha the itisic quality of the pape, the fitess is the oly chael though which the quality ca ete ou model. The fitess may have the followig itepetatio. Whe a scietist wites a mauscipt he eeds to iclude i it a cetai umbe of efeeces (typically betwee 2 ad 4, depedig o implicit ules adopted by a joual whee the pape is to be submitted). He cosides adom scietific papes oe by oe fo citatio, ad whe he has collected the equied umbe of citatios, he stops. Evey pape has specific pobability to be selected fo citig, oce it was cosideed. We will call this pobability a Dawiia fitess of the pape. Defied i such way, fitess is bouded betwee ad. I this model a pape with fitess φ will o aveage have ( ϕ) α ef ϕ ϕ p λ = (IV.8) fist-yea citatios. Hee we have omalized the citig ate by the aveage fitess of published papes, ϕ, to isue that the factio of citatios goig to p pevious yea papes emaied α. The fitess distibutio of efeeces is diffeet fom the fitess distibutio of published papes, as papes with highe fitess ae cited moe ofte. This distibutio assumes a asymptotic fom p ( ϕ), which de- p ϕ, ad othe pa- peds o the distibutio of the fitess of published papes, ( ) ametes of the model. Duig late yeas thee will be o aveage λ ( ϕ) ( α ) ϕ ϕ ϕ = (IV.9) ext yea citatios pe oe cuet yea citatio fo a pape with fitess φ. Hee, is the aveage fitess of a efeece. p A. Distibutio of citatios to papes published duig the same yea Let us stat with the self-cosistecy equatio fo p ( ϕ), the equilibium fitess distibutio of efeeces:

34 34 p ( ϕ) ϕ p α ϕ ( ϕ) ( α ) ( ϕ) p = (IV.2) solutio of which is: p + ϕ p ϕ p ( ϕ) α ϕ = p p ( ϕ) ϕ ( α ) ϕ ϕ p (IV.2) Oe obvious self-cosistecy coditio is that ( ϕ) dϕ =. p (IV.22) Aothe is: ( ϕ) dϕ ϕ. ϕ p = Howeve, whe the coditio of Eq. (IV.22) is satisfied the above equatio follows fom Eq. (IV.2). p, Let us coside the simplest case whe the fitess distibutio, ( ϕ) is uifom betwee ad. This choice is abitay, but we will see that the esultig distibutio of citatios is close to the empiically obseved oe. I this case, the aveage fitess of a published pape is ϕ =. 5. Afte substitutig this ito Eq. (IV.2), the esult ito Eq. (IV.22), ad pefomig itegatio we get: α = l ( α ) ϕ ) ( α ) ϕ + ( ) ( α ) ϕ Sice α is close to, 2 2 ϕ must be vey close to α with the latte eveywhee but i the logaithm to get: p p (IV.23), ad we ca eplace it α 2α ϕ = e (IV.24)

35 35 Fo papes of fitess ϕ, citatio distibutio is give by Eq. (IV.6) o Eq. (IV.7) with λ eplaced with λ ( ϕ) P (, ϕ) ( ϕ), give by Eq.(IV.9): ( λ( ϕ ) l λ( ϕ )) e. (IV.25) Whe =. 2π λ 3 2 α 2 3 α, Eq. (IV.24) gives ( ) (IV.9) it follows that ( ) ( ) ϕ. Fom Eq. λ = α ϕ. Substitutig this ito Eq. (IV.9) we get that the expoetial cutoff fo the fittest papes ( ϕ = ) stats at about 3, citatios. I cotast, fo the ufit papes the cut-off is eve stoge tha i the model without fitess. Fo example, fo papes with fitess ϕ =. we get λ (.).( α ) ϕ. λ = λ ad the decay facto i the expoet becomes (.) l (.) This cut-off is so stog tha ot eve a tace of a powe law distibutio emais fo such papes. To compute the oveall pobability distibutio of citatios we eed to aveage Eq.(IV.25) ove fitess: P ( ) dϕ 2π 3 2 λ ( ϕ) e ( λ( ϕ ) l λ( ϕ )) (IV.26) We will cocetate o the lage asymptotic. The oly highest-fitess papes, which have λ ( ϕ) close to, ae impotat ad we ca appoximate the itegal i Eq. (IV.26), usig Eq. (IV.8), as: α dϕ exp ϕ ϕ 2 ϕ = 2 α 2 2 dze α ϕ 2 2 z We ca eplace the uppe limit i the above itegal with ifiity whe is lage. The lowe limit ca be eplaced with zeo whe << c, whee c = 2 α ϕ 2. (IV.27)

36 36 P I that case the itegal is equal to π 2, ad Eq.(IV.26) gives: ( ) ϕ 2. (IV.28) 2 ( α ) I the opposite case, >> c, we get: P ( ) ϕ c 2. 5 ( ) 4 α e c (IV.29) Whe α =., 5 c = 3. Compaed to the model without fitess, we have a modified powe-law expoet (2 istead of 3/2) ad a much elaxed cut-off of this powe law. This is cosistet with the actual citatio data show i the Fig. IV.2. As was aleady metioed, because of the ucetaity of the defiitio of ecet papes, the exact value of α is ot kow. Theefoe, we give c fo a age of values of α i Table IV.. As log as α. 5 the value of c does ot cotadict existig citatio data. Table IV.. The oset of expoetial cut-off i the distibutio of citatios, c, as a fuctio of α, computed usig Eqs.(IV.27) ad (IV24). α c E+5 7.2E+9 The majo esults, obtaied fo the uifom distibutio of fitess, also hold fo a o-uifom distibutio, which appoaches some fiite value at its uppe exteme p p ( ϕ = ) = a >. I Ref. [33] we show that i this case ( α ) ϕ is vey close to uity whe α is small. Thus we ca teat Eq.(IV.26) the same way as i the case of the uifom distibutio of fitess. The oly chage is that Eqs. (IV.28) ad (IV.29) acquie a pe-facto of a. Thigs tu out a bit diffeet whe p ( ) =. I Appedix C we coside the fitess distibutio, which vaishes at = p ϕ as a powe law: ( ϕ) = ( θ + )( ϕ) θ p. p