Hierarchical distributions and Bradford's law

Hierarchical Distributions and Bradford’s Law Aparna Basu National Institute of Science, Technology & Development A probabilistic model of random fragmentation of the unit line provides the formal underpinning for deriving a distribution of the articles published in any field, over journals ranked in decreasing order of productivity. No assumptions need to be made about the causal mechanism that brings about such a distribution. Interestingly, the proportion of articles, p, that may be obtained from some given proportion, 9, of the most productive journals, is found to be greater than 9 by a factor -9 In 9. This may be interpreted as the additional “information” retrieved over the unranked case, and is a direct consequence of the procedure of ranking the journals. While the distribution obtained reproduces the general shape of a cumulative frequency log-rank graph of publications data, to ensure good fit to data, a parameter has to be introduced. This parameter may be considered to incorporate the effects of possible deviation from randomness, and is suggested as an indirect measure of concentration. 0 1992 John Wiley 81Sons, Inc. Bradford’s Law Bradford’s “law” is the name given to an empirical relationship, first reported by S. C. Bradford (1934), Librarian, Science Museum Library, London, that describes the distribution of scholarly articles in any particular discipline in relevant journals. The law gained wide attention after the publication of Bradford’s book, Documentation (Bradford, 1948). Bradford found that a small core of journals publish the bulk of articles related to any particular discipline. By ranking the journals in decreasing order of “productivity,” he was able to divide the articles into equal zones and show that the zones contained journals in the The law was first confirmed by ratio of l:n:n’.... Bernal (1948), and has since been verified for a number of disciplines (Aiyepeku, 1977; Bulick, 1978; Kendall, An earlier version of this article was presented at the Third International Conference on Informetrics, Bangalore, August 1991, under the title, “On the Theoretical Foundations of Bradford’s Law.” Received November 0 1992 by John 18, 1991; revised Wiley & Sons, Inc. February 14, 1992. Studies, Hillside Road, New Delhi 110 012, India 1960; Lawani, 1973). Subsequently, Leimkuhler (1967) gave a mathematical distribution for the same phenomenon. This was simplified by Brookes (1968) to suggest a linear relationship between the logarithm of the rank of a journal and the number of articles appearing in journals of that rank or better. For convenience, these formulations will hereafter be alluded to under the generic name “Bradford’s law.” In practical terms, the law may provide a heuristic tool by which libraries and information services can decide the extent of journal coverage they wish to incorporate into their services in a costefficient manner (Tague, 1988). Although Bradford’s law is well known and utilized in the field of documentation, the philosophical content of the law ranges beyond a mere description of the scatter of literature in journals. It appears to hold for diverse data such as batting totals in cricket, and questions asked by members after a conference presentation (Brookes, 1977). This points not only to its range of applicability, but even more so to its fundamental character. In common with several other “informetric” laws like the ZipfPareto laws of word frequencies and the distribution of income (Egghe, 1991; Pareto, 1897; Zipf, 1972), and Lotka’s law of scientific productivity (Chen & Leimkuhler, 1986; Lotka, 1926), Bradford’s law tends to demonstrate that certain systems, left to themselves, produce highly unequal distributions where most of the “information,” be it publications or wealth, is concentrated in a small population of “sources,” while the remaining “information” is thinly spread out over the rest of the population. Attempts have been made to find a theoretical foundation for all these laws. Price (1976) has proposed a unifying theory based on a model of “cumulative advantage processes.” Brookes (1977) explains it as “an empirical law of social behaviour that pervades all social activities,” while Zipf (1972) has formulated his “principle of least effort in human endeavour” as an underlying mechanism for such regularities. In spite of a large body of literature on the subject, there has been a lingering feeling among the practitioners of the art that the theoretical basis of Bradford’s law remains to be fully explained. J. E. Kansay (1971) writes, in Kent5 Encyclopaedia ofLibrary and Informa- JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATIONSCIENCE. 43(7):494-500, 1992 ccc 0002~8231/92/070494-07$04.00 tion Sciences, “until an acceptable theoretical proof of its empirical stability is found, the Law of Scattering is not likely to be accepted as a fundamental law, but will continue to be regarded as a statistical curiosity.” B. C. Brookes (1977) writes, “in some undefinable way the Bradford law seems to stand apart from other mathematical regularities. . . to be unrelatable to conventional mathematical or statistical ideas,” and again (Brookes, 1979), “. . .We may.. . be mistaken in continuing to search for that single formulation embracing all Bradford phenomena which has eluded capture for more than forty years.” [For an account of Bradford’s law and related work one may see Garfield: (1980).] The relationship of Bradford’s law with other informetric laws which find application in areas as far removed from each other as linguistics, economics, and scientometrics, all suggest that the underlying mechanism is not peculiar to bibliometrics alone, but may be a manifestation of a more general “law of large numbers” for hierarchical distributions. As argued by Ijiri and Simon (1977), “if the very same regularity appears among diverse phenomena having no obvious common mechanism, then chance operating through the laws of probability becomes a plausible candidate for explaining that regularity.” Therefore, we adopt a probabilistic model based on the random and unequal partitioning of articles among journals. From this model we obtain an expression which asymptotically behaves like Bradford’s law. The model, however, is not causal in that it does not seek to explain the underlying causes of the detailed, time-dependent process by which the Bradford relation establishes itself. We review briefly, though not exhaustively, the existing literature on Bradford’s law in the next section, in particular, the theoretical developments and issues discussed by various authors in the last nearly 50 years. [For a comprehensive review of empirical models of the Bradford law one may see Qiu (1990).] In a later section, we explain our model and derive a mathematical formulation for a law of scattering. Its behavior at low, intermediate, and large values of In r is discussed in the Appendix, where we show that, at intermediate ranks, the behavior is in accordance with the simplified version of Leimkuhler’s law. In the concluding section we discuss some of the implications and limitations of our model and indicate how the formulation can be modified to take into account different concentrations in the data. Theoretical Developments The first theoretical expression for the scatter of articles over a ranked set of journals was given by Leimkuhler (1967). It was subsequently simplified by Brookes (1968) to the form B(r) = a + k log r (1) where B(r) represents the total number of articles published in journals up to rank r, and a and k are constants. This is a widely used formulation and essentially describes only the central linear portion of the curve of B(r) versus log r. The initial or nuclear zone of highly productive journals is anomalous in that it deviates from linearity expressed in eq. (1). Brookes has put forward a mixed Poisson model to explain the nonlinear parts of the scattering curve which he calls the hybrid Bradford curve, and uses two sets of equations, one for the nuclear zone and one outside it (Brookes, 1977). He has also suggested sociological factors for the nonlinearity in the core region referred to as nuclear “restraint” or “enhancement .” The journals of low productivity (large r) lie below the line defined by eq. (1) on a curve called the Groos droop (Groos, 1967). The Groos droop has been a point of controversy. Brookes (1968) adopted the view that it reflects the essential incompleteness of a bibliographic search, and more exhaustive searches are likely to restore the curve to linearity. However, after painstaking searches, O’Neill (1970) and others have concluded that the Groos droop is not due to incomplete search, but is an integral part of the scatter process. An explanation offered by Egghe and Rousseau (1988) relates the droop to combining bibliographies in different fields. Rousseau (1988) has been able to reproduce curves both with and without the droop for Lotka’s law. In a recent study by Qiu (1990), all existing models of Bradford’s law were tested for goodness-of-fit with 19 selected data sets from the Bradford literature using the x2 and KolmogorovSmirnov tests. Qiu concludes from this that most of the cumulative rank-frequency models for Bradford’s law fail to reproduce the Groos droop. Different forms of the core and droop regions have been lucidly explained by Chen and Leimkuhler (1987) using an index, and also discussed by Eto (1988). Explanatory models of the Bradford phenomenon have been offered by Naranan (1970), Price (1976), and Karmeshu (1984). These are based on models of radioactive decay, Shockley’s ideas of scientific productivity (1957), and the “cumulative advantage process.” Other theoretical contributions with a different approach were done by Yablonsky (1985) using stable non-Gaussian distributions, and Sen (1989) using Bose-Einstein statistics. Earlier, unequal or skew distributions, which describe all Bradford-like phenomena, have been discussed by Simon (1948, 1977), and Fairthorne (1969). Before we begin elaborating our model, it may not be out of place to make a small digression of the nature of various categories of models. Models may, in general, be empirical, theoretical, or formal. Empirical formulations are usually derived from regularities in the data, and provide a description of an observed result without offering causal or explanatory hypotheses for the process leading to the observed effect. Theoretical models JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATIONSCIENCE-August 1992 495 typically try to explain the process through plausible assumptions regarding the dynamics of the situation so as to reproduce the observed effect: “. . . assumptions must be chosen that will yield an equilibrium resembling the observed one. . . underlying assumptions should provide a plausible explanatory mechanism for the phenomena” (Simon, 1977). A formal model tries to capture the result through certain general principles such as the central limit theorem, conservation laws, or the maximum entropy principle, without reference to causal or dynamical factors. The model proposed in the next section may be considered to fall into this last category. In this connection, one may compare the views of Simon (1977), who stresses the explanatory character of models, with those of Bookstein (1990), who has cautioned that several alternative models based on different dynamical assumptions may yield the same equilibrium solution, thus making it impossible to decide between them. Bookstein has suggested further investigation of formal models. Random Partition of the Unit Line Our model for the distribution of articles published in journals is based only on probabilistic considerations, namely, the expected sizes of groups that arise when a set of articles is randomly subdivided into a number of groups equal to the number of journals. The sizes of the groups are assumed to be unequal. This inequality constraint ensures the generation of a hierarchy of journals. Even though the generation of the hierarchy is time dependent, Bradford’s law itself does not explicitly incorporate time. In other words, what is observed as the Bradford law is an equilibrium or stationary state of a dynamic process. Our aim is to obtain the hierarchic content of the equilibrium state, without making assumptions about the causal or dynamical nature of the process. Consider the subdivision of a set of articles into groups corresponding to different journals. If the total number of articles and journals is very large, our problem involving discrete variables, viz. articles and journals, may be considered in terms of an equivalent continuous problem of random, unequal subdivisions of a line. The subdivision of a line of unit length into two parts involves dropping a point at random anywhere along the line. In general, the distribution of such points will follow a Gaussian distribution with a peak at the mid-point. For unequal division, the mid-point is excluded. Now the random point may fall on either side of the mid-point with equal probability. If it falls in the left half, then, on an average, the most probable point on which it will fall is the mid-point of the left half, i.e., at a distance of a from the left end and i from the right end. Similar arguments apply if the point falls in the right half. If an experiment is done several times, then the expected values of the smaller piece will be + and the larger $, respectively [see, e.g., Laing (1986)]. 496 More generally, if the line is to be subdivided into N unequal parts, the sizes will be as follows: Size Rank N N-l (;x;)+($X&)+...+($+) N-x ($x;)+(;x&)+...+; 1 where ranks have been assigned in decreasing order of length. The expected length of the piece of ith rank is: To obtain the cumulative number of articles, CE,, published in journals up to rank i, we take the sum of expectation values of ranks 1 to i, or El, EZ, E3, . . . Ei, where E,=$ [ E2 = ; 1 +L+ i I+++$+... “‘X 1 . . . +l+ i ++++ 1 “‘N I 1 .,.$ f+ ... +i+ i Ej =; E; = $ 1 L+ i “‘Jv I If we add terms vertically, we find that the first i items are equal to unity. Thus, i temn -i i+l 1 + 1 + 1 + . . . + -+- CE’;=iEj=$ j=l i i+2 1 2,’1+CJi+j] i “‘E N-i j=l 1 (3) This is an exact result which can be further simplified when the number of pieces becomes large. In the limit of large ZV,we have the relation (Abramowitz & Stegun, 1972): N*~[+ ] ) lim 2 & - In N JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATIONSCIENCE-August = y Euler’s constant = 0.57 1992 (4) Similarly, if i is large, then eq. (3) reduces to: I Eq. (5) gives the total fractional length contained in all pieces up to rank i, and is the main result on which the rest of our study will be based. In order to use this “rank-size” relationship for bibliometric applications, we write: 1 (6) 1 + In N - In r for the fraction F(r) of papers published in all journals up to rank r, among a total of N journals which have published at least one paper in the field. The total number of articles published in journals up to rank r is: B(N) B(r) = - N r[l + In N - In r] (7) where B(N) is the total number of articles published in the field. In terms of the average productivity p = B(N)/N we may write: B(r) = r[l + In N] - r In r CL In this form, the ratio of the productivity of journals of rank r and the average productivity, p, depends only on r, except for the term (1 + In N) that depends on the size N of the journal collection. This makes the c o mparison between bibliographies of different sizes or average productivities relatively straightforward. In Figure 1, the curve represented by eq. (7) has been plotted for the total number of journals and articles corresponding to the bibliography of agriculture (Lawani, 1973). We see that it reproduces the general features of PLOTOFB(r)vs.Ln(r) 2800I 21002 iii 1400- the Bradford curve, namely the core, droop, and the approximately linear portion at intermediate ranks. Unlike empirical models, the functional form arises naturally, and depends only on the total number of articles and journals. There are no free parameters to be obtained from best fit to data. Moreover, unlike causal models there has been no necessity to assume any particular type of behavior that determines the dynamics. Model Interpretation Bradford’s Law and Relation to The model [eq. (7)] can be shown to reduce to the form B(r) = a + k log r in the neighborhood of the point of inflection at r = N/e. Hence, Bradford’s law is an approximate form of eq. (7) in this region. The behavior in the core and droop regions is also obtained (Appendix I). In the fractional form, the relation reduces to the fairly simple equation P = 4(1 - 1% 4) (9) wherep is the fraction of articles that appear in a fraction q of the most productive journals. This can be interpreted in the following way. The ranking of the journals provides an additional amount of information represented by -q log q, over the nonhierarchical case for whichp = q. In other words, in the unranked case, if a given fraction of the journals is randomly selected, one will come across a number of articles which, on an average, will constitute the same fraction of the total number of articles. If, on the other hand, the same fraction of journals is taken from the most productive journals, then the fraction of articles they contain will be greater by -q log q. As explained earlier, the random partitioning model, as developed, does not distinguish between different data sets, since there are no free parameters in the model. Clearly data can be more or less concentrated (Pratt, 1977): the two limiting cases being the “monopoly” situation when all the articles are concentrated in one journal, and its antithesis, a situation of perfect equality when all journals publish the same number of articles. In order to account for this variation we modify our basic formula [eq. (9)] to introduce a single parameter, (Y,which will be an indirect measure of concentration in the data: P = [q(l - 1% 700 - 0 1 2 3 I 4 5 6 7 lo g ra nk FIG. 1. The form of B(r) calculated as a function of In r for r= 1,2 , . N (eq. (7)]. The total number of articles, B(N), and journals, N, have been taken as 2284 and 374, corresponding to the data on agriculture (Lawani, 1973). The thickness of the line indicates the density of points at higher values of In r. 4)la (10) where (Ymay also be assumed to incorporate deviation from randomness. The modification is consistent with the boundary conditions (q = 0,~ = 0) and (q = 1, p = 1). Detailed comparisons with data will be made in a subsequent publication where we shall discuss the properties of the parameter a, and show that it may be linked to a concentration measure. For an overview of existing measures of concentration one may see Egghe and Rousseau (1990). For the single data set considered JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATIONSCIENCE-August 1992 497 in this study the model was fit using regression. The agreement as indicated in Figure 2 is excellent at LY= 0.78 (see also Table 1). Our model generates a strict hierarchy of expectation values with b(1) > b(2) > . . .b(r). . . > b(N), where b(r) is the number of articles in the rank level 1. In other words, no two rank levels may be expected to have the same number of articles. This need not be a limitation of the model, even though it is always possible that more than one journal will carry the same number of articles. In the latter case the maximal rank is taken. On the other hand, in the event that a strict hierarchy is actually generated in the literature (and there is no reason why it should not be), frequency relations which express the number of journals, J(p), carryingp articles as a power law of the type J(p) = pek (as in Lotka’s law) cease to meaningful. However, as we have shown, Bradford’s law holds equally in this case. It appears, therefore, that the content of asymptotic power laws and Bradford’s law is not the same. Therefore, the remarks by Brookes (1970) regarding power laws seem to be in order. Particular cases where our formulation is not likely to fit the data are for small collections, high ranks (i.e., r = 1,2,3.. .), and when the Bradford curve has a rising tail. For high ranks the exact formula [eq. (3)] should be used. Conclusions In this study, we have derived a theoretical expression for the distribution of articles in journals, based on a model of the random and unequal partitioning of any entity-in this case the total number of articles. While we do not wish to suggest that random partitioning is the mechanism or process by which articles gradually accrete in the space of journals, a random hierarchical Model Line and Data for Bibliography of Agriculture 1.2r 4 FIG. 2. Data on publications in agriculture (Lawani, 1973) and model line [eq. (lo)]. (The model was fit to the data using the regression algorithm in the NONLIN module of the statistical package SYSTAT. The parameters of fit are: f - value = 0.002/8.886, a = 0.777, ASE = 0.003). 498 TABLE 1. Data on bibliography of agriculture (Lawani, 1973), and values of B(r) computed from eqs. (3), (7), and (10) with ct = 0.777. Rank 1.000 2.000 3.000 4.000 5.000 6.000 8.000 10.000 11.000 13.000 15.000 16.000 17.000 18.000 19.000 20.000 22.000 25.000 28.000 29.000 36.000 39.000 44.000 47.000 50.000 60.000 68.000 79.000 92.000 103.000 121.000 146.000 172.000 212.000 261.000 374.000 Cum. Arts 80.000 150.000 201.000 242.000 275.000 307.000 369.000 429.000 458.000 514.000 568.000 584.000 609.000 633.000 655.000 676.000 716.000 773.000 827.000 844.000 956.000 1001.000 1071.000 1110.000 1146.000 1256.000 1336.000 1435.000 1539.000 1616.000 1724.000 1849.000 1953.000 2073.000 2171.000 2284.000 Eq. (3) Eq. (7) 39.712 73.318 103.870 132.386 159.375 185.144 233.772 279.322 301.147 343.179 383.325 402.773 421.840 440.547 458.915 476.961 512.153 562.885 611.415 627.144 731.716 773.881 841.071 879.692 917.143 1034.373 1120.717 1230.163 1347.504 1437.941 1570.749 1728.453 1864.656 2028.300 2166.380 2284.000 42.286 76.106 106.731 135.280 162.287 188.063 236.696 282.243 304.065 346.087 386.222 405.664 424.725 443.425 461.786 479.826 515.003 565.714 614.222 629.943 734.462 776.603 843.754 882.351 919.778 1036.928 1123.208 1232.565 1349.801 1440.149 1572.811 1730.312 1866.302 2029.621 2167.302 2284.000 W (10) 102.931 162.500 211.334 254.073 292.671 328.189 392.406 449.908 476.710 527.152 574.066 596.396 618.057 639.100 659.569 679.503 717.903 772.249 823.224 839.550 945.905 987.811 1053.559 1090.819 1126.603 1236.590 1315.821 1414.323 1517.780 1596.143 1709.255 1840.822 1952.281 2083.774 2192.798 2284.000 model may nevertheless provide a “snapshot” of the final distribution much in the same way that thermodynamics can estimate several equilibrium parameters without going into the details of molecular dynamics. Our analysis is “information-theoretic” in spirit, since it does not assume anything other than a hierarchy, which is simply a property of ranked sets. Our result is also consistent with information theory since -q log q is the added information provided by ranking. Apart from (Y,the constants appearing in this expression are not free parameters to be determined by comparison with data, but functions of the total number of articles, B(N), and the total number of journals, N. If N is not known, as in the case of an incomplete search, N may be treated as a free parameter to be determined from the best fit to the data. The expression asymptotically reduces to the linear Bradford law in the neighborhood of I = (N/e). It reproduces the general shape of the Bradford curve including the initial nuclear zone, the central approximately linear portion, and the Groos droop. The rela- JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATIONSCIENCE-August 1992 tion of B(r) and In r is statistical and holds when the partitioning takes place between a large number of journals. The same may be true of Bradford’s law, since it fails to hold, for example, when used for an individual’s output (Bonitz, 1980). The probabilistic interpretation of B(r) as an expectation values also permits it to take nonintegral values. To account for varying levels of concentration in the data, our basic expression has been modified through the introduction of a single parameter. While the agreement with the data of publications in agriculture (Lawani) is good, it is not included as a validation of our model. In fact, the relation needs to be evaluated by fitting to other data sets. Further, cases where the relation fails to hold also need to be examined. Other aspects of the Bradford curve which have not been included in this discussion, such as the different shapes of the nuclear and tail regions, mature and immature bibliographies, time span of collections, and size dependence and concentration also need to be considered. Yet another important direction for future research is the application of Bradford’s law for bibliometric prediction (Burrell, 1989). Finally, an overarching law could be sought that incorporates the characteristics of both Bradford’s and Lotka’s laws. Acknowledgments The author is grateful to Dr. Ashok Jain for introducing her to the Bradford law; Mr. K.C. Garg, Mr. S. Arunachalam, and Lalita Sharma for helpful references, and Mr. M. M. L. Chandna for typing the manuscript. Special thanks are due to Dipankar Basu for a critical reading of the manuscript. The author is especially grateful to two anonymous referees for their valuable suggestions. Appendix 1: Asymptotic Bradford’s Law Behavior and Let us examine the behavior of B(r) as a function of In I, by rewriting eq. (7) as: B(r) = &“‘[l + In N - In r] (Al) For small values of In r, the expression in parentheses is dominated by In N, so that B(r) = p In Ne’” r (4 This exponential growth corresponds to the observed core or nuclear zone in Bradford plots of bibliographic data. At r0 = N/e, B(r) has a point of inflection. A Taylor expansion in the neighborhood of this point gives B(r) = BT[(3 - In N) + In F-] 643) In this region, B(r) is approximately linear in relation to In r and expresses Bradford-like behavior of the type B(r) = A + k In r with the following parameters: Slope k = B\e X-intercept: A = In N - 3 Y-intercept = -(B\e) (In N - 3) (A41 Beyond this region, for larger values of In r, the slope of B(r) gradually decreases until it becomes zero at r = N, since dB(r)/d In r = pe’” ‘[In N - In r] (A% vanishes at r = N. 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