Relations between the continuous and the discrete Lotka power function

Transcription

1 Relatios betwee the cotiuous ad the discrete Lotka power fuctio by L. Egghe Limburgs Uiversitair Cetrum (LUC), Uiversitaire Campus, B-3590 Diepebeek, Belgium ad Uiversiteit Atwerpe (UA), Campus Drie Eike, Uiversiteitsplei, B-260 Wilrijk, Belgium ABSTRACT The discrete Lotka power fuctio describes the umber of sources (e.g. authors) with =,2,3,... items (e.g. publicatios). As i ecoometrics, iformetrics theory requires fuctios of a cotiuous variable j, replacig the discrete variable. Now j represets item desities istead of umber of items. The cotiuous Lotka power fuctio describes the desity of sources with item desity j. The discrete Lotka fuctio is the oe that oe obtais from data, obtaied empirically; the cotiuous Lotka fuctio is the oe eeded whe oe wats to apply Lotkaia iformetrics, i.e. to determie properties that ca be derived from the (cotiuous) model. It is, hece, importat to kow the relatios betwee the two models. We show that the expoets of the discrete Lotka fuctio (if ot too high, i.e. withi limits ecoutered i Permaet address: Limburgs Uiversitair Cetrum, Uiversitaire Campus, B-3590 Diepebeek, Belgium. Ackowledgemet: The author is idebted to Prof. Dr. R. Rousseau for iterestig discussios o the topic of this paper. Key words ad phrases: Discrete Lotka fuctio, cotiuous Lotka fuctio, power fuctio.

2 2 practise) ad of the cotiuous Lotka fuctio are approximately the same. This is importat to kow i applyig theoretical results (from the cotiuous model), derived from pratical data. I. Itroductio Lotka s law i its historical formulatio i Lotka (926) is formulated as follows: i a fixed group of authors (or i a bibliography), the umber f( ) of authors with ( =,2,3... ) publicatios is give by the power law: K = () ( ) a f where K ad a are positive costats. The most classical value for the expoet is eve Lotka himself poited out i Lotka (926) that other values of a might occur, but usually a>, makig () a (fastly) decreasig fuctio. a= 2 but Lotka s law is foud to be valid i may applicatios i or eve outside the iformetrics field (see e.g. Wilso (999), Egghe (2004) ad may refereces i both works). I geeral we ca talk about sources (geeralizig authors above) havig (or producig) items (geeralizig publicatios above) ad i this framework, f( ) i () deotes the umber of sources with items. As i other -metrics theories (such as ecoometrics), istead of (), oe uses fuctios of a cotiuous variable j, replacig the discrete variable above. This is doe for mathematical reasos: models ad their properties ca better be uderstood whe oe ca use the formalism of calculus (i.e. mathematical aalysis). This is so because it is easier to evaluate derivatives ad itegrals tha discrete differeces or sums. Hece, i the theory of Lotkaia iformetrics (see e.g. Egghe (2004a)) oe uses the cotiuous variat of (), deoted by ϕ.

3 3 The cotiuous Lotka fuctio ϕ is also give by a power fuctio but of a cotiuous variable j : C ϕ = (2) () j j α where agai C ad α are positive costats. I words: ϕ ( j) is the desity of sources with item desity j. As said above: kowig the cotiuous Lotka fuctio (2) is importat sice (2) is the basis for may derived results, that are ot possible to prove with (), usig discrete sums. More fudametally, without a cotiuous model oe has o relatios with other iformetric (eve ecoometric or liguistical) distributios such as the oes of Pareto ad Zipf. The problem with the cotiuous model (2) above is that its parameters C ad α (the most importat oe) caot be determied directly from a cocrete set of data (e.g. a bibliography). Such a set of data (obviously beig discrete) gives iformatio about the discrete fuctio f i (). From the above it is clear that the followig problem is worth studyig. Problem I. : Determie relatios betwee the discrete Lotka fuctio f ad the cotiuous Lotka fuctio ϕ. Especially determie relatios betwee the expoets a ad α. This problem will be studied i the ext sectio. The relatio betwee f ad ϕ will be clarified usig a third fuctio: the discretized versio of the cotiuous fuctio ϕ, to be itroduced i the sequel.

4 4 II. Comparig f ad ϕ : itroductio of the discretized versio of ϕ It is clear that, if α > C C T j dj dj j = = = α ϕ() (3) α ad, if α > 2 : C C A j j dj dj j 2 = ϕ() = α = α (4) A Formulae (3) ad (4) imply µ = T 2A T 2µ α = = A T µ (5) AT A C = = A T µ (6) I the same way, for the discrete Lotka fuctio f, we have f( ). (7) a T= = = = K Hece T K = ζ (8) ( a)

5 5 Also sice f( ) (9) a A= = = = K we have ( a ) ζ( a) A ζ µ = = (0) T for a 2, where ζ > ( ) a a = deotes Riema s Zeta fuctio. So the kowledge of T (the = total umber of sources) ad of A (the total umber of items) determie (withi the rage α, a> 2) α ad C (the two parameters of the cotiuous Lotka fuctio) via (5) ad (6) ad a ad K via (8) ad (0). The latter determiatio goes as follows: sice we kow A ad T, we kow A µ =, the average umber of items per source. The we ca determie a umerically T if we have a table of a versus ζ ( a ) ζ( a) (which is provided i this article see Appedix from a= 2. o (i 0.0 icremets)). The, sice T ad a are kow, K follows from (8), usig a table of a versus (2004a). ζ( a) which ca be foud i Egghe ad Rousseau (990) ad Egghe Note that K f( ) C ϕ( ) = =. I fact eve more is true: if α > 2, the ( ) ϕ( ) K= f < T< C=, () showig that f ad ϕ are differet fuctios. The first iequality is obvious sice f( ) is the umber of sources with oe item ad sice T is the total umber of sources. The secod iequality trivially follows from (3). That ϕ ( ) > T

6 6 is ot couter-ituitive sice every j (hece also j = ) deotes a item desity (compare with a Gaussia desity which has a total area of but which ca have a peak (at 0) as high as we wish). The above yields f ad ϕ, give A ad T, so theoretically the relatio betwee f ad ϕ is established. However, e.g. due to the occurrece of the cumbersome fuctio ζ i (8) ad (0), we will cotiue our search for more practical relatios betwee the fuctios f ad ϕ. This is obtaied by discretizatio of the fuctio ϕ, which is defied ow. Defiitio II. : Let ϕ be as i (2). The discretized versio of ϕ, deoted I( ϕ ), is defied o as follows: for every =,2,3, ( ϕ)( ) ϕ( ) I = j dj= C dj (2) j α For α we hece have I ( ϕ)( ) C = α α ( ) α + (3) It is clear that, i exact mathematical terms, I( ϕ ) is ot a power fuctio, hece I( ϕ ) is ever equal to f. But both fuctios represet discrete size-frequecy fuctios of the same iformetric system. So f ad I( ϕ ) should have the same shape. Note that I ( ϕ)( ) C = α ( ) α α + I C d α d α ( ϕ)( )

7 7 C I ϕ (4) ( )( ) α So I( ϕ ) ad ϕ have the same shape (heuristic argumet). For I( ϕ ) ad f to have the same shape we hece require C α ad K a to have the same shape. This implies a α (5) for values of a ad α ot too high (if these values are high the, eve if they are very differet, C K a 0 α for 2 ). Whether (5) is true for values of a ad α ot too high ca be proved i a exact (umerical) way as we will do i the sequel. Note first that it follows from the exhaustive list of examples i Egghe (2004a) (Chapter I) that most expoets are below 4 (ad most commoly aroud 2). For these values we ca ideed prove (5) as follows. Use (5) ad (0) (ad hece also the table i the Appedix) to obtai the followig table of α - ad a-values i fuctio of A µ = T (ote that (5) ad (0) imply that both a ad α oly deped o µ ad ot o the two values of T ad A themselves!).

8 8 Table. Compariso of values of α ad a for several values of A µ = T µ α a We see that α ad a are ideed comparable ad closer to each other the closer they are to 2, the most commo value. We also ote from (4), (5) ad (0) that α > 2 if ad oly if a> 2 (hece also α 2 if ad oly if a 2 ), eve if a ad α are very differet (for high values). This is a importat coclusio because may, from Lotka s fuctio ϕ derived properties (e.g. shape of the cumulative first-citatio distributio, existece of the Groos droop, cocetratio ad fractal properties, - see Egghe (2004a)), are commo for differet above 2 ad for differet αs below 2 but are differet for values of α below ad above 2. αs Ideed, data show a Groos droop (see Groos (967)) if ad oly if α < 2 (see Egghe (985, 2004a) or Egghe ad Rousseau (990)). The cumulative first-citatio distributio is S-shaped if ad oly if results, the value α > 2 ad is cocave if ad oly if α 2 (see Egghe (2000)). I all these α = 2 is a turig poit. So, from the above, the practical calculatio of the expoet a (based o the data) yields a estimate of C α which determies ϕ () j =. j α Note also that (3) ad (3) imply that

9 9 I( ϕ)( ) = T < T (6) 2 α, a property that is shared with the discrete fuctio f (ad ot with the cotiuous fuctio ϕ itself). I fact it is easy to see that I( )( ) improved by the followig propositio. ϕ < T for every. Iequality (6) ca eve be Propositio II.2 : If α = a> 2, the I( ϕ )( ) < K= f( ) < T< C=ϕ ( ) (7) Proof : We oly have to show that (usig ()) I( ϕ )( ) < K. Usig (6) ad (8) we hece have to show (usig that a = α ) that < ( ) 2 α ζ α or < α α 2 k= k Hece we must prove that < α α α α α or < (8) α α α α α But the left-had side of (8) equals

10 0... α α α α which is strictly iferior to sice all umbers betwee brackets are positive. ~ It is easy to see that, if α = a is large eough, we have that I( ϕ )( ) > f( ). Ideed, this relatio is satisfied if (use (3), (8) ad (3)) α ( ) + ζα ( ) > + which is always true for = 2,3,... ad α large eough (sice lim ζα ( ) = ). Of course, by defiitio of I( ϕ ), we always have that I( ϕ)( ) ϕ( ) α < for every. We also have the followig propositio, showig the closeess of f ad I( ϕ ) for large α. Propositio II.3 : If α = a, the lim f ( ) I( ϕ)( ) = 0 α (9) for every. Proof : If α = a, the, usig (3), (8) ad (3) T lim f ( ) I( ϕ)( ) lim T α = α α ζα ( ) α ( ) α + (20) sice i the above limit we ca assume that α > 2. If = the (20) equals

11 T lim 0 α α 2 = lim α α α 2 3 while, if, the same value 0 is trivially obtaied. ~ Table 2 illustrates the above propositio for. We use the table of = ζ ( α) versus α that e.g. ca be foud i Egghe ad Rousseau (990). Note that the values ζ ( 2) ad ( 4) ζ are explicitely kow sice ζ ( 2) 2 π = = (2) 2 = 6 ad ζ ( 4) 4 π = = (22) 4 = 90 as e.g. ca be foud i Gradshtey ad Ryzhik (965) or i Abramowitz ad Stegu (972). Table 2. Compariso of I( ϕ )( ) ad K (both divided by T) α =.5 α = 2 ζ ( α) 2 α π = α = 2.5 α = 3 α = 3.5 α = π =

12 2 Fig. illustrates, qualitatively, the relatio betwee f, ϕ ad I( ϕ ). C T K I(ϕ) () ϕ f I(ϕ) Fig. Qualitative illustratio of the relatio betwee f, ϕ ad I( ϕ ) Note that, from (7), the order of the fuctio values i is as depicted. For = 2,3,...,I( ϕ) ca itersect f (if α is large eough) but remais below ϕ for every. Note also, however, that ( ) ϕ( ) f < for all. This is see as follows. Formulae (3) ad (8) imply that ( ) ϕ( ) f < is equivalet with (for α > ) ( α ) ζ( α) > (23) Puttig δ= α > 0, (23) is equivalet with > (24) + = δ δ

13 3 But, : δ + > δ j δ+ δ+ dj as is obvious sice j o, +. So > ] ] + δ δ > dj δ+ δ + = = j δ = dj = j δ + sice δ > 0. This proves (24) ad hece that ( ) ϕ( ) f < (25) for all. III. Coclusios I this paper we discussed the discrete Lotka fuctio f (formula ()), obtaied from practical data, ad the cotiuous Lotka fuctio ϕ (formula (2)), eeded for applyig results from Lotkaia iformetrics theory (see Egghe (2004a)). We showed that the discrete Lotka fuctio f ad the theoretical Lotka fuctio ϕ, although they are differet power laws, ca be calculated from each other i a exact (but umerical) way. We provide the umeric key to calculate oe from the other.

14 4 The simplest ad most importat result is the fact that, for values of the expoets a ad α ot too high (which is usually true i practise) we have that a iformetrics theory ca be applied usig the empirical value of a. α, showig that Lotkaia Refereces M. Abramowitz ad I.A. Stegu (972). Hadbook of mathematical fuctios, with formulas, graphs ad mathematical tables. Dover, New York (USA), 972). L. Egghe (985). Cosequeces of Lotka s law for the law of Bradford. Joural of Documetatio 4(3), 73-89, 985. L. Egghe (2000). A heuristic study of the first-citatio distributio. Scietometrics 48(3), , L. Egghe (2004a). Lotkaia Iformetrics. Elsevier, Oxford, to appear, L. Egghe (2004b). The source-item coverage of the Lotka fuctio. Scietometrics 6(), 03-5, L. Egghe ad R. Rousseau (990). Itroductio to Iformetrics. Quatitative Methods i Library, Documetatio ad Iformatio Sciece. Elsevier, Amsterdam, the Netherlads, 990. I.S. Gradshtey ad I.M. Ryzhik (965). Table of Itegrals, Series ad Products. Academic Press, New York, USA, 965. O.V. Groos (967). Bradford s law ad the Keea-Atherto data. America Documetatio 8, 46, 967. A.J. Lotka (926). The frequecy distributio of scietific productivity. Joural of the Washigto Academy of Sciece 6(2), , 926. C.S. Wilso (999). Iformetrics. Aual Review of Iformatio Sciece ad Techology (ARIST) 34 (M.E. Williams, ed.), , 999.

15 5 Appedix Values of a versus ζ ( a ) ζ( a) a ζ( a- )/ζ( a ) a ζ( a- )/ζ( a ) a ζ( a- )/ζ( a)