On the influence of growth on obsolescence

Scientometrics, VoL 2Z No. 2 (1993) 195-214

ON THE I N F L U E N C E OF G R O W T H ON OBSOLESCENCE

L. EGGHE

LUC, Universitaire Campus, B-3590 Diepenbeek (Belgium) + and

UIA, Universiteitsplein 1, B-2610 Wih'ijk (Belgium)

(Received December 14, 1992)

In many papers, the influence of growth on obsolescence is studied but a formal model for such an influence has not been constructed. In this paper, we develop such a model and find different results for the synchronous and for the diachronous study. We prove that, in the synchronous case, an increase of growth implies an increase of the obsolescence, while, in the diachronous case, exactly the opposite mechanism is found. Exact proofs are given, based on the exponential models for growth as well as obsolescence. We leave open a more general theory.

I.- INTRODUCTION

The influence of growth on the obsolescence of a research subject, as

expressed by the ci tat ion/referencing behavior in its literature, has been the

subject of many research in the past. More than twenty years ago already, it was

M.B. Line and B.C. Brookes, who remarked the problem of measuring obsolescen-

ce for a growing literature {see [1], [2]). Even earlier occasional remarks on this

can be found in the references therein. From these early publications it is clear that

one must define unambiguous ly what is meant by obsolescence, growth and even

"on what" these events take place (i.e. the "literature'}.

In general one defines a fixed research subject that is examined when time

passes : into the past for a synchronous study, when studying reference lists in

existing publications; into the future for a diachronous study, when studying the

+ Permanent address.

Scientometrics 27 (1993) Elsevier, Amsterdam - Oxford-New York - Tokyo

Akad~miai Kiad6

L. EGGHE: INFLUENCE OF GROWTH ON OBSOLESCENCE

future use of a publication. The problem hereby is that a subject is not constant

but is in evolution, certainly when the considered time per iods axe long (e.g. ten

years or more, which is often the case). This problem is not considered here : w e

assume clari ty on this topic or - more correctly - we only assume that we have a

set of publications, g rowing in time. This set of publications will be called the

"literature' .

Growing in t ime means that for increasing t {t = time} the number of

publications at t ime t increases. In this context, one can consider "pure" growth

(i.e. for every t, the number of articles published at time t is larger than the

number of articles publ ished at t ime t-l) or cumulative growth (which is, in fact,

true for any piece of literature!). Saying that there is an influence of growth on

obsolescence means - in the lat ter case - that the obsolescence of a "literature" is

dependent on the t ime per iod that one considers and hence that obsolescence of a

subject in itself is an undef ined thing! As stated by Vickery in a supplement to [1],

"obsolescence" is a function of growth and obsolescence (note the " " here)! In [3]

one studies the relat ion be tween growth and obsolescence in the literature of

thcrmoluminescent dosimctry.

Another ambiguous aspect of growth is : do we consider g rowth of the

l i terature (measured by the number of publications) or g rowth of the number of

au thors {or of researchers} or both {d. the papers [2], [4]}. In this paper , we restrict

ourselves to the s tudy of the growth of the literature.

196 Scientometrics 27 (1993)


Finally obsolescence must be defined in a clear way. Do we s tudy the

obsolescence in a synchronous or a diachronous way? In the former case one fixes

the ci t ing l i terature and studies the age distr ibution of the references therein; in

the lat ter case, one fixes the cited literature and s tudies the use of this l i terature

after its publication. Both mechanisms are alike, but different to handle. Papers

referr ing to this difference are e.g. [5], the letter of Sandison [6] and, more

recently, [7] and [8]. In this paper we will be discussing b o t h approaches of

obsolescence.

In the recent paper [9] of Rao and Meera one invest igates the concrete

inf luence of the growth rate of the literature on the rate that this piece of l i terature

becomes obsolete. It must be stressed that their s tudy deals only wi th synchronous-

obsolescence. From an heuristic-philosophical point of view, it is not at all d e a r

that there should be an influence and we will not deal wi th this p roblem here.

Wha t w e can investigate is the question : "Is there an influence, yes or no and of

wha t type?

Experimental evidence is given in [9] that, in the synchronous case, the

faster the l i terature grows, the faster it becomes obsolete. Wal lace in [10] studies

the same problem for the diachronous way but does not get clear results in this

case.

For acquisi t ion or archival purposes it would be "ideal" to have that the

most impor tan t journals have the smallest obsolescence rate [synchronous as well

as diacl~_ronous}. This is however contrary to the f indings in [9]. No need to say

that a theoretical confirmation of this vein complicates acquisi t ion or archival

Scientometrics 27 (1993) 197


policies. Apar t from [2] {assuming constant "utili ty') I do not know of any model-

theoretic paper that deals with this problem.

Therefore the theoretical investigation of the influence of growth on

obsolescence of li terature is very important . The s tudy is not simple, however, and

therefore we will restrict ourselves to the "easiest" case of exponential growth as

well as exponential decay. In the first case we suppose a function :

g(t) =gat [I)

where a > 1 and where g(t} denotes the number of publications at time t.

In the second case we suppose a function :

c (t) = C b t (2)

where 0 < b < 1 and where, in the synchronous case, c{t) describes the density of

the n ~ of references in a publ icat ion that are to publications of t years ago t

(i.e. ~ c(t'}dt' denotes the number of references that are to publications that are t U

years younger}. In the d iachronous case, c(t} denotes the number of dtat ions to a

publication that is t years old. Note that the growth function measures "number

of" while the obsolescence function measures "densities of numbers of". This

difference is dassical in the l i terature on growth o r obsolescence.

The author is well aware of the s impli fying aspect of these assumptions. In

fact in [11], an extended s tudy of g rowth models was made, proving that Gom-

pertz curve (a kind of logistic curve} and the power law are well-fitting models. In

[12] one proves that the lognormal dis t r ibut ion fits the synchronous obsolescence

curves very well. The point is that the present study, using only exponential

functions, is a lready complicated and that it is not clear to extend these results to

the above mentioned more general laws. Nevertheless, as one knows, the exponen-

tial models are the basis of g rowth and obsolescence and are the "building blocks"

of more refined models. It can fur thermore be assumed that the results of this

s tudy can be used in other, more general situations.



Our conclusions are as follows : in the synchronous case we confirm the

findings of e.g. Rao and Meera, i.e. that the h igher the growth rate of the literature

is, the faster it becomes obsolete. In the d iachronous case, however, we find the

opposi te effect : the faster the l i terature g rows the s lowest it becomes obsolete.

Note, however, that for "pure" obsolescence in the sense of Vickery such a

different result would imply a contradiction. For "obsolescence" {as used by

Vickery with the " "} such different results can occur b y the fact that here we do

not measure pure obsolescence but obsolescence be ing influenced by the size (and

publication dates) of the literature.

II. THE MODELS

II.1. Synchronous case

�9 0 t T

Fig.1 : Time axis : synchronous case

Let T > 0 be the time abscis represent ing the present time, indicating that

we s tudy a growth process of a piece of l i terature that was "born" at t ime 0 (hence

T units (e.g. years} ago). At every t ime t e [0,T] we will s tudy the references in

each publicat ion that was publ ished at t ime t. We adop t a further simplification :

we assume that the aging factor b is fixed for all publicat ions at every time t

[O,X].

Fix t ~ [0,T] arbitrarily. The densi ty function of the references in every

publication, published at time t is :

C(t') : C b t' {31



t' E [0,oo[, to be counted from t on (backwards, i.e. to the past). For ease of

reasoning we will reduce all these formulae to the s tar t ing point T (i.e. the

present). Then (3) reduces to :

c(tll) I = C b tu-tr-t) , t tl 6 [T-t,**[ (4)

[ = 0 , t II e [0,T-t[

Hence, denoting XA for the characteristic function of a set A (i.e. XA(x) = ] if x

A and 0 if x �9 A}, we find that, w.r.t the present :

c(t II) = c b tu-(r-t) Xtr-t,-[ (tl/) (5)

for every t" ~ [0,~[ {note that t" is now independent from t}.

Since there are c(t"}dt" of these references referring to publ icat ions of t ime t" and

g ' ( t )dt publicat ions published at time t (each wi th the same function c(.) for the

references) w e have :

(in a) g a t c b t#-(T-t) XtT-t.-[ (t/l) dt dt" {6)

rderences that are for publications of t" (years} ago in the publications that were

published t (years] ago. So

h(t It) =(f (Ina)ge t eb t'-r XtT-t,-[ (tt/)dt] dt" (71 L% )

denotes the total number of references to publicat ions of t" (years) ago. We

calculate (7)

h(t II) = g c b t"-r ina li (ab) t X[T-t,-[ (tl/) dt} dt II {8)

Now, by definit ion of X we have :

XIT-t.-[ ( t l l ) = Xtr-t'I.,,[ ( t ) (9)



for all t,t" E [0,oo[. Subst i tu t ing [9) in [8) yields :

h{t #) = g c b t'-r in a I[ (ab)t Xtr-t".-t (t) dt] ) dt u

(lo)

h(t u) _ g c b t"-r l-l~b" ((a~) T - (~b)r-t~ dt tt (if ab # I)

h(ttt) = ~ e a r bt- (I - (ab)-tu) dt tt in b

Hence, going back to the variable t we see that :

fo r all t ~ [0,**[.

If a b = 1, t h e n w e h a v e [from [1011 :

h(t) = g c b t-r (in a) t dt 112}

In both cases hit) denotes the number of references to publications of t

years ago, hence :

u = h(t) [13} dt

is the corresponding density function.

The function [13} represents the synchronous aging function of the whole

piece of literature up to the present time. Note that the parameter a figures in



these functions. We hence have proved that synchronous obsolescence is indeed

dependen t on the growth of the literature - a non trivial fact, at least to m y mind!

11.2. D i a c h r o n o u s case

0 t T - 4 - . . . . . . . .

T O

Fig.2 : Time axis : diachronous case

N o w w e have a t ime axis as in Fig.2 : at the present t ime T O we s tudy the

li terature, publ i shed in the per iod [0,T] and is diachronously followed until "now' ,

i.e. TO, T O z T. Note that the period [T,T0] is a "non growth" per iod {i.e. that we

have not inc luded the publications in this period} which is more general than

requi r ing that T = T O {studying growth until "now'} lin II.1 this distinction would

have been meaningless and hence not included}.

Fix t G [0,T0] arbitrarily. T h e density function for the citations a t t ime t '

for every publ icat ion at t ime t E [0,T] is

C (t I) = C b t'

t ' �9 [0,T0-t], counted from t on (forward, i.e. to the future). Making t = 0 as our

reference po in t we have "

C (t It) = 0 , t tl 6 [0, t]

= C b t/'-t , t II 6 [t,T o]

(14}

for t" �9 [0,T0].

Hence

c ( t #) = c b e"-t Xtc.r0] (tlz) {15}

for t" e [0,T0]. So, as in the previous section,



h(t#) = (To]" (in a) g a t c b t'-t ~([t,T.] (t/I) dt} dt II (16)

is the total number of citations, given at time t" c [O,To]. Since t" s T O (since T O is

the present time), we have that

XttoTo] (t//) = Xto.t"] (t)

for all t ~ [0,T].

Hence

h(t it) = (in a) g a t C b tumt dt dt tl,

117)

ttt~T

(181

=li (ina)ga t cbt"-tdtldt/1, T < tt1<T o

yielding

h ( t//) g c b tu t" In a dt// t" ~ T = ((~) - I) in a - in b '

(19)

gcbt,,((b) r ) ina d r " T x t'zx T o = - 1 ina - inb '

for the diachronous case. Note again the influence of a on h as in the synchronous

case. Again,

u = h(t) (13) dt

by definition.



How obsolescence is dependen t of a will be investigated in the next

section.

III. THE INFLUENCE OF a {GROWTH} ON 7 (OBSOLESCENCE}

There are many ways to calculate the "obsolescence rate" (or "aging rate").

In the case of function (2) a natural way to express the aging rate is as follows :

calculate :

c(t+l) _ b {20) c(t)

We then refind b as a measure of obsolescence {i.e. the aging rate is large if b is

small and vice-versa].

This method is common in many publicat ions {see e.g. [1211 and the

references therein} and will therefore also be used here for 7- We note however

that the calculation of T ( t + 1) "r ( t ) mixes two approaches : the continuous

approach with the discrete one. Indeed, the function 7 is expressed in the continu-

ous setting {i.e. t , [0,*-*[) while the compar ison of u247 versus 7{t} is a conse-

quence of the habit of compar ing next year with the previous one, i.e. a discrete

approach.

In general, we could use many measure for the aging rate of the form :

7 (~ ( r {21) y ( t )

where r is any function such that (p{t) > 1 for t > 0. An interesting example is

cp{t} = 2t where we investigate the aging after a "double" periode. Also this

approach will be used {see section IV}.



III.1. Synchronous case

For

a(t) = Y (t+l) ~(t)

we have the following result :

(22]

Theorem III.1 :

Oa - ~ ( t , a , b ) < 0

for every t e [0,o*[, i .e the larger a (the faster growth}, the smaller a {the faster the

obsolescence works).

ProOf :

We prove the theore~n orgy for t ~ IN.

{1} Let ab ~ 1

Then aq9 ~ 1 for all a ' in a ne ighborhood of a. Hence we have formula {23}

{by definition [22} and formula (11))

at+i a(t,a,b) - {23}

b~____ 1 at

Hence [24) is valid :

_ _ a t * 2 at-.1 at.1 aa (t,a,b)= Oa < 0 {24)

if and only if



b t*l 1 a t+l

bt _ _!_ 1 a t

I > --

< --

t+l for b > ta a

t+l for b < ! ta a

(25)

{i} L e t b > 1/a

W e h e n c e h a v e to p r o v e :

(ab) t.1 _ 1 > t+___!l

(ab) t _ i t {26)

This is equivalent to

tCab) t (ab-l) > (ab) t- 1

But :

(ab) t - I = (ab-l) ((ab) t-1 + (ab) t-2 + .

since ab > 1.

. . + I) < (ab-l) t (ab) t

{ii} L e t b < l / a

N o w w e h a v e to s h o w

1 -- (ab) t+l <

1 - (ab) t

t*l

t {27)

or, equivalently :

t(ab) t (l-ab) < 1 - (ab) t

But

1 - (ab) t = (l-ab) (l+ab .... +(ab) e-l) > (l-ab) t (ab) t

since ab < 1.



(2) L e t a b = l

In this case w e will verify the result directly, w i t h o u t us ing der iva t ives .

N o w (by {22} a n d {12)} :

a (t) = y (t+l) _ (t+l) b t§

y (t) t b t-T

a(t) = b t+l t

(28)

(i) Le t a = 1 / b < a '

W e hence m u s t ver i fy {using (23} and (28}} :

t*l b > t

b t.~ _ 1 a'tLX

bt _ 1 a,t

1 Since b > ~ , w e h a v e the condi t ion a '

(a~))t*1 _ t ba t-b a l> -t

H e n c e the c o n d i t i o n

(a%)) ( ( a ~ ) ) t _ i ) > t ( a ~ - i )

But :

(a~o) t _ 1 = (a~ - I) ((a~))t-1 + (a%))t-2 +

since a ' b > 1. H e n c e (29) is verified, again since a 'b > 1.

{29)

. . . + I ) > ( a ~ - I ) t

(ii) Let a ' < a = 1 /o

N o w w e m u s t p r o v e



b t.1 i a,t+l

1 b t - a,t

> t+l b t

1 Since b < --~ , we have that this is equivalent with a '

or

But

t > t b a + ba - (ab) t'1

t (l-ba) > ba (I- (ab) t) {30)

I - (ab) t = (l-ab)(l+ab+...+(ab) t-l) < (l-ab) t

since ab < 1. A g a i n since ab < 1, condi t ion is proved. This concludes the proof of

the whole theorem. []

Note : In the l im i t i ng case (t -o ~) we have (for ab * 1} :

lira ~(t,a,b) = Lira i (ab) t'1 - i t-- t-- a (ab) t- I

= ! , ab<1 a

= b, ab>1

Hence, a decreases wi th a indef ini te ly R--ab < 1 or becomes (for high t} re la t ively

u n i n f l u e n c e d b y the g r o w i n g rate (if ab > 1).

208 Sciemometrics 27 (1993)

L. E GGHE : INF L UENCE OF G R O W T H ON OBSOLESCENCE

111.2. Diachronous case

Based on {19} and {13} we now have :

~{t) = g c (a t - b t) in a In a - in b

in a = g c b t - 1 ina - Inb

Now

(t) -- YCt+1)

implies that

a(t) at§ - bt'X t+l ~ T a t - b t

, t~T

, T ~ t S T o

{31}

= b , t aT {32}

= b , t < T AND t+l > T

We have the following result.

T h e o r e m 111.2 :

c3...~a ( t , a , b ) > 0 0a

for every t e [0,T[, i.e. the larger a {the faster growth}, the larger a {the slower the

obsolescence works}, and a is independent of a for t z T.

Scientometrics 27 (1993) , 209


P r o o f :

W e p r o v e th is r e s u l t fo r t E IN.

T h e last a s s e r t i o n is t r ivia l , s o w e r e s t r i c t o u r s e l v e s to t h e case t �9 [0,T[.

[1) t § x T

In th i s case :

O~a = (at-bt)(t+l) a t - (a t§ - b t`x) ta t-~

/ga (at _ bt)2 > 0

if a n d o n l y if

a (a t - b t) > t b ~ ( a - b )

But

a t - b t = (a-b)(a ~-I + a t-a b + ... + b t-l) > (a-b) t b t

s i n c e a > b {in fact a > 1 > b).

(2} t < T, t § - T

W e n o w p r o v e d i r e c t l y t h a t a i n c r e a s e s w i t h a. H e n c e , if a < a ' w e h a v e to

p r o v e tha t

i -i

<

I

[33)

Thi s is e q u i v a l e n t w i t h

+ 1 < a t a T

b t+T §

P u t t i n g



x

- -

w e h a v e the c o n d i t i o n :

(xT-1)6,~-!) < (xt-1)CyT-1)

E q u i v a l e n t l y :

{34}

Thi s is i n t u i t v e l y d e a r s ince 1 < x < y; a n exac t p r o o f fo l lows . T h e left h a n d s ide

of (34) e q u a l s

x t + xty+ ...+ xty t-I

... + xt+ly t-1 + X t § + x t + l y +

+ . . .

+ x T-1 + x T - l y + ... + x T - l y t-1

T h e r i gh t h a n d s ide of {34} equa l s

y t + y t x + ... + ytxt-1

+ y t + l + y t + l x + ... + y t+ lx t -1

+ . . .

+ yT-1 + yT-1 x + ... + yT- lx t -1

If w e c o m p a r e the respec t ive c o l u m n s in the a b o v e b locks {say the i th

co lumn} , we f ind :



first b lock :

xty i-1 + x t + l y i-1 + ... + xT- ly i-1 {35)

second block :

ytxi-1 t+l i-1 yT-lxi-1 +Y X + ' " + (36)

N o w [35} equals

x i - ly i-1 {x t - i+l + x t-i+2 + ... + x T-i) {37}

a n d {36} equals

yi- lxi-1 [ y t - i + l . yt-i+2 + ... + yT-i} {38)

N o w (37) < (38} since x < y since a > b and t < T. This p roves [33] a n d hence also

(34) a n d the whole theorem. []

IV. O T H E R O B S O L E S C E N C E RATES

Ins tead of evaluat ing obsolescence th rough the func t ion a, we can also

invest igate funct ions as :

a" = 3' (2u) [39} ~(t)

~.. = ~ (3t) [40)

a n d so on. So is a* measur ing the decline in use after a "doubled" period. In the

s y n c h r o n o u s case, we have (for ab * 1} �9

b2t _ 1

m'(t,a,b) = a2t -b t + -!-I 1 a t

b t - __ a t



A g a i n a* d e c r e a s e s w i t h a. In general , for a *{n)

(Z "(n) (t,a,b) -

b nt _ 1 a nt

bt _ _!_ 1 a t

w e h a v e :

~.{n} (t,a,b) b {n-t~t + b(n-z)t b{n-3)t 1 = - - + - - + . �9 . + - - a t a 2t a (n-l) t

w h i c h is d e c r e a s i n g w i t h a for every n ~ ~ . This r e in fo rces t he p r e v i o u s d i s cus s i -

o n o n t he i n f l u e n c e of g r o w t h on s y n c h r o n o u s obso lescence . In t he d i a c h r o n o u s

case, for t + l x T {the o n l y i m p o r t a n t case} we h a v e :

~.{n> (t,a,b) - ant - bnt

a t _ b t

H e n c e ,

a *(n) (t,a,b) = a (n-x)t + aln-2)tb t + a(n-3)tb 2t + �9 �9 . + b (n-l)t

i n c r e a s i n g in a, for e v e r y n ~ IN. This aga in re inforces the p r e v i o u s d i s c u s s i o n o n

t h e i n f l u e n c e of g r o w t h o n d i a c h r o n o u s obsolescence.

The author is grateful to Prof. Dr. L K. Ravichandra Rao for interesting discussions on this paper on the occasion of the third conference on informetrics in Bangalore (India) in August 1991. The author is grateful to Prof. Q. L. Burrell for his written communication on this paper, including the completion of some proofs. The author thanks anonymous referees for their valuable remarks which lead to the result on the diachronous study of obsolescence.



References

I. M . B . LINE, The 'half-life' of periodical literature: apparent and real obsolescence, Journal of Documentation, 26 (1970) 46-54.

2. B .C . BROOKES, The growth, utility, and obsolescence of scientific periodical literature, Journal of Documentation, 26 (1970) No. 4, 283-294.

3. GILDA GAMA DE QUEIROZ, F. W. LANCASTER, Growth, dispersion and obsolescence of the literature: a case in thermoluminescent dosimetry, Journal of Research Communication Studies, 2 (1979-1981) 203- 217.

4. M.R. OLIVER, The effect of growth on the obsolescence of semiconductor physics literature, Journal of Documentation, 27 (1971) No. 1, 11 - 17.

5. M.B. LINE, A. SAND1SON, 'Obsolescence' and changes in the use of literature with time, Journal of Documentation, 30 (1974) No. 3, 283- 350.

6. A. SANDISON, References/citations in the study of knowledge, Journal of Documentation, 31 (1975) No. 3, 195-198.

7. E .R . STINSON, Diachronous vs. Synchronous Study of Obsolescence, Ph. D. Thesis, 1981. 8. E . R . STINSON, F. W. LANCASa~R, Synchronous versus diachronous methods in the measurement of

obsolescence by citation studies, Journal of Information Science, 13 (1987) No. 2, 65 - 74. 9. I .K. RAVICHANDRA RAO, B. M. MEERO, Growth and obsolescence of literature: an empirical study,

Proceech'ngs of the Third International Conference on lnformetrics, I. IC R. RAo, (Ed.), Bangalore (India), August 1991, pp. 377-394, 1992.

10. D. P. WALt.aCE, The relationship between journal productivity and obsolescence, Journal of the American Society for Information Science, 37 (1986) No. 3, 136-145.

11. L. EC, GHE, I. K. RAV1CHA.NDRA RAt, Classification of growth models based on growth rates and its applications, Scientometrics, 5 (1992) No. 1, 5-46.

12. L. EGGHE, I. K. RAVICHANDRA RAO, Citation age data and the obsolescence function: fit and explanations, Information Processing and Management, 28 (1992) No. 2, 201 -217.


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On the influence of growth on obsolescence