A note on different Bradford multipliers

A Note on Different Bradford Multipliers

L. Egghe LUC, Universitaire Campus, B-3670 Diepenbeek, Belgium* and U/A, Universiteitsplein I, B-2670 Wi/rijk, Belgium

In this note we show that the multiplier k that appears in the law of Bradford is not the average production of articles per author nor the average number p of articles per journal, contradicting some earlier statements of Goffman and Warren and of Yablonsky. We remark however that the Bradford multiplier might be close to p in a lot of cases, being merely a coincidence of the special functional relation between ~1 and k which we develop in full detail. We finally show that K = kplA (p = number of Bradford groups, A = total numbers of articles) is a universal constant for the bibliography. Furthermore, K

is the Bradford multiplier of a group free formulation of Bradford’s law, introduced in an earlier article of the author.

Introduction

The Bradford law is a classic bibliometric law formulated in Bradford (1985): Suppose we have a bibliography con- sisting of articles (and hence journals containing these articles) on a certain topic. Suppose we arrange the journals in decreasing order of the number of articles they contain. Then we can form a group of the r,, most important journals, yielding in total y0 articles (this is no law: we just fix the notation r, and y0 here). If we take then the next journals in the above mentioned ordering (starting with the journal on rank r, + l), then, in order to have again y,, articles in the second group, we need more journals, evi- dently, say r,k with k > 1. Then the Bradford law says that, in order to form a third group of y0 articles, we need rok2 journals and so on: the ith group of y,, articles has r,,k-’ journals and so on until all journals are used: until group p: here we have rOkpm’ journals yielding y, articles. k is called Bradford’s multiplier (w.r.t. the number p).

In Goffman and Warren (1969) as well as in Yablonsky (1980) the relationship between k and the average production is considered. We mean by average production: the average number or. of articles per journal. In Goffman and Warren (1969) however the average production g of articles per author is used and said to approximate k (k related

*Permanent address.

Received March 9, 1989; revised June 12, 1989; accepted July 17, 1989.

0 1990 by John Wiley & Sons, Inc.

to the bibliography of journals and articles). This is misin- terpreted in Yablonsky when he considers a bibliography of authors and articles and uses the average production g as above but with a Bradford multiplier related to the bibliography of journals and articles.

We show that k cannot be put equal to any average production but when chasing p properly, k can be close to p and practically always k I p, a fact that is supported by practical data.

We base our arguments on the following formula for k that has been shown in Egghe (1986), using a joumal- article duality argument:

k = (eYy,)“p (1)

where y = 0.5772 (Euler’s number), hence ey = 1.781, p = the number of Bradford groups, and y,,, is the number of articles in the journal of rank 1 (the most productive journal). From this we have immediately the following trivial result:

Result

k f an average production of articles per journal, nor of articles per author, or whatever. Indeed, according to equation (I), k is not a constant of the bibliography but depends on p, while average production is a constant of the bibliography.

From equation (1) we see that kP is a constant of the bibliography but also this constant is not equal to p. In- deed, using formula (1) we have:

kP = eYy, > y, > p

Hence kP # p. In the next section we will investigate the real relationship

between k and I-L. In the last section we will show that a new group-free formulation of Bradford’s law (introduced in Egghe (1989) and (199Ob)), has a group-free Bradford multiplier K = kp’* where A denotes the total number of articles.

The Relation Between p and k

Consider a bibliography of journals and articles with p as the average number of articles per journal in this bibliography. We suppose the bibliography satisfies Bradford’s

JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE. 41(3):204-209, 1990 CCC 0002-8231/90/030204-06$04.00

law. As shown in Egghe (1985, 1989), this law is equivalent with the law of Lotka:

where f(y) denotes the number of journals with y articles and where A is a constant. Hence, in this formulation,

Ym

c Yf(Y)

EL = y=’ I5 f(u) ’

(3)

y=l

where y,,, denotes the number of articles in the most productive journal. Hence

“m -dL

6 = -$logYm + Y) (4)

Combining formulae (1) and (4) yields:

p = 6 log kP 7r2

(5)

. 1

:p(k) , I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

or

p = 6p log k 7T2

(f-5)

In investigating the relation between p and k it is handy to use the function

f,(k) = ;

or

f,(k) = ?r”k 6p log k

Elementary calculus shows that, for all values of p, f, obtains a minimum in e, has a deflection in e2 and has a vertical asymptote in the abscissa 1. Furthermore,

;Tf,‘k’ = 0, lim f,(k) = +m + k++m >

F:&,(k) = -33, F?&(k) = +m >

but only the<cases k > 1 are of interest in the relation with Bradford’s law. Note also that for all p:

,p=3

p=4

p=5 p=6 -7 ;,s

JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE-April 1990 205

See the next figure for the shape off, (for p = 3,4,5, 6,7,8; cases that are encountered most often in practice).

Conclusions

From the above calculations and figures we can conclude that k # )(L, nor is k a linear function of p. The minimum of all functions& is, however, stable in the sense that f, changes very little in a wide neighborhood of the abscissa e, where f, attains its minimum. Furthermore for p = 4 and p = 5, this minimum is close to 1. Both re- marks above yield that if p = 4 or 5, k is in the neighborhood of p for a wide variety of bibliographies (i.e., k’s and p’s). If p increases, k decreases. From p = 6 onwards k < p for a wide variety of k’s and p’s. Only for p = 3 is k > p since the minimal value off,(k) isf,(e) = 1.49 > 1.

The above conclusions conform very well with practical data. In Table 1 we summarize these verifications. The number p of groups has been chosen as they have been in the indicated references. Of course, other values of p were perfectly possible.

In Goffman and Warren (1969) and Yablonsky (1980), one considers the so-called “minimal nucleus” form of Bradford’s law. This is chasing the maximal number of groups, taking into account that the second to last Bradford group must at least contain a journal with two (or more) articles (if not, then the last and the second to last Brad- ford group have the same number of journals, contradicting k > 1). In Egghe (1986) it is shown that the above condition is equivalent with the requirement.

log k > ; (8) L

Hence k > 1.6487. From formula (1) one finds an expression for the number of Bradford groups as a function of k:

TABLE I. Experimental verification.”

Bibliography P k CL Confirmation

Applied Geophysics (Bradford, 1985; Egghe, in press)

Lubrication (Bradford, 1985; Egghe, in press)

ORSA (Kendall, 1960; Egghe, in press)

France data (Aiyepeku, 1977)

Applied Geophysics (Bradford, 1985; Egghe, in press)

USA data (Aiyepeku, 1977)

Germany data (Aiyepeku, 1977)

USA, UK, France, Germany data (Aiyepeku, 1977)

Lubrication (Bradford, 1985; Egghe, in press)

Geography (Aiyepeku, 1977)

UK data (Aiyepeku, 1977)

Circulation data (Goffman & Morris, 1970)

User’s data (Goffman & Morris, 1970)

Transplantation-Immunology (Goffman & Morris, 1970)

Schistosomiasis (Warren & Newill, 1967; Egghe, in press) Mast Cell

(Seley, 1968; Egghe, in press)

3

3

5.49 4.09 Y:k>p

3.40 2.41 Y:k> p

4 4.56 4.76 Y:k=p

2.8 4.48

2.78 4.09

2.3 4.05

1.9 3.20

Y:k < p but k =!= /.L Y:k<p but k + p, Y:k<.p but k + /.I, Y:k < /.L

1.8 5.50

1.69 2.41

1.7 13.27

2.4 4.23

1.4 2.36

1.4 2.22

1.8 4.12

2.03

1.44

5.70

4.05

Y:k< k

Y:k< /.L

Y:k< /L

Y:k<p

Y:k < /.L

Y:k<p

Y:k < p

Y:k < /.L

Y:k< p

a We did not use the Bradford data of Goffman-Warren on Mast Cell and Schistoso- miasis (1969, Table 3), since, as remarked in Egghe (in press), they are not correct. In- stead we used Egghe (in press) for our Bradford-data on Mast Cell and Schistosomiasis in the above table.

206 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE-April 1990

log k = + log(1.78lyJ

log( 1.78 ly,) P= log k

(9)

Hence, from (8)

p < 2 log(1.781y,) (10)

The formula can be calculated right from the “raw” data since y,,, is known, estimating immediately the maximum possible number of Bradford groups that can exist.

For k = p in the situation of log k = f one requires (using (6)):

6~ ,logk=k=e”*=& 7T

Using (1) one has log kP = y + log y,. Hence

f (Y + log y,) = vi Hence

Y, = e ~~4l6-v = 8.46 (12)

It is doubtful that the journal with the highest production has only eight or nine articles in the bibliography. Hence also in this case is k # p.

Interpretation of the Bibliography Constant kP

From equation (1) we have kp = e”y,, a constant for the bibliography. Since this constant contains the p-dependent Bradford multiplier k it is natural to ask: can kP be in- terpreted as a “group-free Bradford multiplier” of a new “Bradford law,” and, if yes, what will then be the relation between this new law of Bradford and the old, historical one (Bradford, 1985).

Incidentally, in Egghe (1989) and (1990b), we introduced already a group-free law of Bradford in the context of duality theory of IPP’s (Information Production Pro- cesses). IPP’s are continuous models of bibliographies (but other examples exist, see Egghe (1989) and (1990a)). We briefly repeat the definitions.

IPP’s and group-free version of Bradford’s law

An IPP is a triple

(&I, V)

where S is the set of sources and I is the set of items. Both sets form an interval: S = [0, Z’j, I = [0, A], and V is a “device function”

v:s - I,

a strictly increasing differentiable function, stating which items belong to which source. The advantage of such a continuous set-up has already been explained in Egghe (1989, 1990a), but has been used in many other earlier formula- tions. The main reason of introducing continuous models is that one can use infinitesimal calculus. Furthermore real

bibliographies are included in such a model since, if T and A denote the total number of journals and articles respec- tively in a bibliography, then {1,2, . . . , T} C [0, T] and {1,2,, , . ,A} C [0, A]. In this sense, the real bibliography is included in the continuous model and the results apply.

We suppose that we have an IPP. We say that this IPP satisfies Bradford’s law (group-free), if there is a number K > 1 such that, for every i E [O,A], the number of sources needed to have the first i items is given by

S(i) = C iKi’di’ (14)

K is called the group-free Bradford multiplier. Of course, on an IPP we can also formulate Bradford’s

law in the classical sense just as we did in the introduction. We now have the following theorem.

Theorem Let @,I, V) be an IPP. Then the following assertions

are equivalent:

(i) The IPP satisfies Bradford’s law, for every chosen number of groups p E IN (the natural numbers).

(ii) The IPP satisfies the group-free version of Bradford’s law.

In case (i) let us denote (for any p E IN) by k(p) the Bradford multiplier for p groups and, in case (ii) let us denote by K the group-free Bradford multiplier. Then one has

K = k(p)p’A (13

Hence:

K = (eYy,)liA (16)

where ey = 1.781 . . . , y, is the number of items in the most productive source and A is the total number of items. Hence K is a constant for the IPP.

Proof: (i) + (ii). This is the hardest part of the proof. Let, firstly, i E [0, A] be such that there is a 4 and p E IN such that i = qA/p. We then apply (1) with p as above. This yields p groups of resp. (we write ro(p) and k(p) instead of r, and k to show the p dependence)

rob), ro(pMp), . . ., ro(pN(p)P-’

sources, each containing ye(p) = A/p items. Hence

S(i) = robI + rob)k(p) + . . . + ro(pN(p)q-’

= robI k(p)’ - 1 k(p) - 1

= k(-?;” I (k(p)@” - 1)

= ,;;f 1~ log k(p) kj;zk(p;

rob)

I

i/A

= k(p) - 1 P log k(p) (kb)P)’ 4 0

r,(p) = k(p) - 1

P log k(p) ’ (k(pY’A)’ 4 (17)


Furthermore,

ro(p) + ro(pMp) + . . . + ro(p)k(p)P-’ = T,

the total number of sources. Hence

r(p) = WP) - 1) 0

k(P)P - 1

Hence

c = r,(p) k(p) - 1 P log k(p)

c = Tlog (kb)P)

k(pY - 1

c = TY + log Ym e’v, - 1

(by (l)), a constant of the IPP. Also, if we denote

K = k(p)piA

(18)

(19)

cm

then we see, again using (I), that K is a constant of the IPP, being

K = (eyy,)“A (16)

In conclusion:

S(i) = C I

‘K”di’ ( 14) 0

is valid for all i in [O,A] of the form qA/p. Now the set

5 p,q,p E IN is dense in the set [O,A]: We say that a set X is dense in a set Y if every element of Y is the limit of a sequence of ele- ments of X. It is a well known and easy to prove fact that if a function is continuous on Y and known on X, then this function is known on Y: i.e., there exists only one continuous extension from X to Y, if X is dense in Y.

Supposing the function S to be continuous (a quite natural requirement) and noting that the function

i E [O,A] -

is continuous and an extension (to [O,A]) of S, we can conclude that, for all i E [O,A]:

S(i) = cjiK’,dir ) (14) 0

where C and K are constants of the IPP.

(ii) j (i). Let p E IN be arbitrary. Define ye(p) = A/p. Then, according to (ii)

rob> = Go) = CK” di ’

= sK(KYo - 1) (21)

Furthermore,

S(2yo) - S(y,) = jboC.K1’ di’ YO

= CKYO(KYO

log K

Hence we see that

S(So) - S(yo) = robWo

= ro( p)KA@

1) (22)

(23)

In the same way we can show that, for every q = 1,2,. . ,p:

S(qyo) - S((q - 1)~~) = robI (KA’p)g

Hence, if we put

k(p) = KA@

then

(24)

(25)

S(wo) - S((q - 1)~~) = ro(pNb)”

for every q = 1,2, . . . ,p, showing that the IPP satisfies Bradford’s law for p groups.

From the above proofs, point 2 of the assertion is auto- matically proved:

K = k(p)p’A (15)

and, using (1)

K = (eYy,)‘” •i (16)

Note

From an earlier paper of mine (Egghe, 1985) one could conclude (see p. 175-176) that property (i) in the above theorem is equivalent with (reusing “bibliography” instead of IPP):

(i’) The bibliography satisfies Bradford’s law, for a certain number of groups p E IN.

The argument there was:

(i’) - Leimkuhler’s law - (i) (26)

and obviously (i) + (i’). Here Leimkuhler’s law means:

R(r) = a log(1 + br)

where R(r) denotes the number of items in the sources of rank r or lower. However, the point is that, more correctly, implications (26) should read:

(i’) @ Leimkuhler’s law, but only for p points r (being the points roCjzoki for i = 0, , . . .,p - 1) (see also Rousseau (1987)).

(i) @ Leimkuhler’s law for all r E [0, T] (assuming a continuous function R).

The proof of these equivalences can be read from the lines of the proof in Egghe (1985), p. 175-176. It is now clear, that (i’) + (i). In fact an easy counterexample is as

208 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE-April 1990

follows. Take p = 3 and the following sources and items (continuously):

Sources are indexed in [0, r] and items are indexed in [O,A]. The sources in [0, $[have together 4 items but they all are equally productive. The sources in [f, y[have together 3 items but they all are equally productive. Finally the sources in [y, r] have together 4 items but they all are equally productive.

It is clear that the above continuous bibliography satisfies Bradford’s law with p = 3 and k = 2 but does not satisfy Bradford’s law for any other p E IN, p > 3.

I have made this counterexample on a continuous bibliography to be in perfect accordance with the above theorem. The practical discrete equivalent of the above counterexample is the following:

source 1 has 12 items source 2 has 6 items source 3 has 6 items source 4 has 3 items source 5 has 3 items source 6 has 3 items source 7 has 3 items

Again this bibliography satisfies Bradford’s law for p = 3 and k = 2 but does not satisfy Bradford’s law for any other p.

Conclusions

We think that this article sheds new light on Bradford’s law and on Bradford’s multiplier. Its relation with the average production p is shown and it is also shown that, usually, the p-dependent multiplier k(p) is smaller than /L.

The meaning of the bibliographic constant k(p)p is explained; one has

K = k(p)plA

where K is the group-free Bradford multiplier, hence being also a constant for the bibliography.

Finally, the group-free version of Bradford’s law has the advantage of looking at Bradford’s law as a function

S(i) = Cii k’dj 0

just as we consider the laws of Lotka, Zipf, Mandelbrot, Leimkuhler, and so on, as functions. So far, Bradford’s law, formulated in its group dependent way could not be regarded as a functional expression (since k(p), ro( p) and yo( p) are p dependent).

References

Aiyepeku, W. 0. (1977). The Bradford distribution theory: the com- pounding of Bradford periodical literatures in geography. 1. of Docu- menration, 33, 210-219.

Bradford, S. C. (1985). Sources of information on specific subjects. En- gineering, 137, 85-88, 1934. Reprinted in J. of Inform&on Sci., 10, 176-180.

Egghe, L. (1985). Consequences of Lotka’s law for the law of Bradford. J. of Documenrution, 41, 173-189.

Egghe, L. (1986). The dual of Bradford’s law. J. of the Amer. Sot. for Information Sci., 37, 246255.

Egghe, L. (In press). Applications of the theory of Bradford’s law to the calculation of Leimkuhler’s law and to the completion of bibliographies. To appear in J. of the Amer. Sot. for Information Sci.

Egghe, L. (1989). The duality of informetric systems wifh applications to the empirical laws. Ph.D. thesis, City University, London, 1989.

Egghe, L. (1990a). The duality of informetric systems with applications to the empirical laws. J. of Inform&on Science (in press).

Egghe, L. (1990b). New Bradfordian laws equivalent with old Lotka laws, evolving from a source-item duality argument. Proceedings of the Second International Conference on Bibliometrics, Scientometrics and Informetrics. London (Canada), L. Egghe and R. Rousseau (eds.), Elsevier, Amsterdam (in press).

Goffman, W. & Morris, T. G. (1970). Bradford’s law and library acquisi- tions. Nature, 226, 922-923.

Goffman, W. & Warren, K. S. (1969). Dispersion of papers among journals based on a mathematical analysis of two diverse medical literatures. Nature, 221, 1205-1207.

Kendall, M. G. (1960). The bibliography of operational research. Opera- tional Research Q, 11, 31-36.

Rousseau, R. (1987). Ben vleugje bibliometrie: de equivalentie tussen de wetten van Bradford en Leimkuhler. Wiskunde en Onderwijs, 13, 71- 78.

Seley, H. (1968). The mast cells. London; Butterworth. Warren, K. S. & Newill, V. A. (1967). Schistosomiasis, a bibliography of

the world’s literature from 1852-1962. Cleveland: Western Reserve univ. Press.

Yablonsky, A. K. (1980). On ftmdamental regularities of the distribution of scientific productivity. Scientometrics, 2, 3-34.


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A note on different Bradford multipliers