{"id":54,"date":"2018-07-09T04:36:14","date_gmt":"2018-07-09T04:36:14","guid":{"rendered":"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=54"},"modified":"2018-12-07T05:40:10","modified_gmt":"2018-12-07T05:40:10","slug":"bradford-distributions-an-overview","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/chapter\/bradford-distributions-an-overview\/","title":{"rendered":"Bradford Distributions: An Overview"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nI. Objectives<\/strong>\r\n\r\n \r\n\r\nAfter going through this case study you will come to know about the following:\r\n\r\n \r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 To study and understand Derivation of equations for Bradford distribution by various bibliometricians.\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 To discuss viewpoints of some bibliometricians on the law.\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 To study the Ambiguity between verbal and graphical representations of Bradford distribution.\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 To discuss Bradford-Zipf distribution.\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 To study the Characteristics of bibliometric distribution, etc.\r\n\r\n \r\n\r\nII. Learning Outcome<\/strong>\r\n\r\n \r\n

After completion of this module, you will be certainly knowledgeable with regard to Bradford distribution and related work. At the end of this module, you gained knowledge on various aspects of Bradford's law -- Bradford-Zipf distribution, ambiguity between verbal and graphical interpretation of Bradford's law, Leimkulher distribution; computational aspects of baradford's law.<\/p>\r\n \r\n\r\nIII. Module Structure<\/strong>\r\n\r\n \r\n\r\n1 Introduction\r\n\r\n2.\u00a0 Cole's Formulation\r\n\r\n3.\u00a0 Leimkuhler\u2019s Formulation\r\n\r\n4.\u00a0 Brookes Formulation\r\n\r\n5.\u00a0 Naranan\u2019s Viewpoint\r\n\r\n6.\u00a0 Bookstein\u2019s Viewpoint\r\n\r\n7.\u00a0 Bradford Multiplier\r\n\r\n8.\u00a0 Ambiguity between Verbal and Graphical Statements\r\n\r\n9.\u00a0 Bradford-Zipf Distribution\r\n\r\n10.\u00a0 Characteristics of Bibliometric Distribution\r\n\r\n11.\u00a0 References\r\n\r\n \r\n\r\n1. Introduction<\/strong>\r\n\r\n \r\n

The law of scattering was propounded by Samuel Clement Bradford (1878- 1948), a British librarian, mathematician and document artists at the Science Museum in London after a laborious study of scientific literature in mid-1930s. After examining the distribution of scientific literature in periodicals and their coverage in abstracting and indexing periodicals he realized that the distribution of literature follow a particular pattern [1, 2]. He opined that \u2018the nucleus of periodicals devoted to the given subject must contain, individually, more articles on that subject than periodicals dealing with related subjects\u2019 [3]. \u2018In consequence, it is possible to arrange periodicals in zones of decreasing productivity, in regard to papers on a given subject, and the numbers of periodicals in each zone will increase as their productivity decreases\u2019 [3]. He described a scattering pattern of journals in the area of applied geophysics and lubrication. He plotted the partial sums of references against the natural logarithm of the partial sum of numbers of journals, and he noticed that the resulting graph is a straight line. On the basis of this observation, he suggested the following linear relation to describe a scattering phenomenon [2]<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n

F(x) = a + b log x.<\/p>\r\n \r\n

F(x) is the cumulative number of references contained in the first x most productive journal; a and b are constants. The following figure is a hypothetical, but typical, log-linear curve (as described by Bradford) showing aggregates of articles on a given subject corresponding to the number of journals.<\/p>\r\n\"\"\r\n\r\n \r\n

This type of a curve is usually called a Bradford curve; In X-axis: Partial sum of Journals (in log scale). In Y-axis: Partial sum of articles contained in X top most journals (in linear scale)\u00a0P1 in the figure is the point at which the straight line part of the curve begins. Draw Y1P1, Y2P2, and Y3P3 such that they are parallel to the X-axis and OY1 = Y1 Y2 = Y2 Y3. Draw P1X 1, P2X 2, and P3 X3 such that they are parallel to Y-axis. Since P1P3 is a straight line and since Y 1Y2 = Y 2Y 3, X1X2 and X2X3 are equal, say r units. Let the distance between O and X is s units. Thus, if \u03b1, \u03b2 and \u03b3 are the positive real numbers corresponding respectively to the logarithmic abscissa OX1, OX2 and OX3, we have, log \u03b1 = s, log \u03b2 = s+r, and log \u03b3 = s+2r.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\nThat is, \u03b1 = 10s, \u03b2 = 10r+s = 10s.10r , and \u03b3 = 10s+2r = 10s.102r\r\n\r\n \r\n

Substituting n = 10r, we see that the natural numbers \u03b1, \u03b2, and \u03b3 are related to each other as 1:n:n2. On the basis of this relationship and also since OX1 represents a number of periodicals in a subject area Bradford stated his law as<\/p>\r\n \r\n

\u201cBradford stated that \u201cIf scientific journals are arranged in order of decreasing productivity of articles on a given subject, they may be divided into a nucleus of periodicals more particularly devoted to subject, they and several groups of zones containing the same number of articles as the nucleus, when the zones will be 1:n:n2 \u2026.\u201d<\/p>\r\n \r\n\r\nThis is called Law of Scattering or Bradford's law.\r\n\r\n \r\n

Since then a great deal of work has been done with this law [17,9,10, 13]. An attempt is made in this module to highlight some of those works.<\/p>\r\n \r\n\r\n2. Cole's Formulation<\/strong>\r\n\r\n \r\n

Cole [8] also experimented with the law and named the slope of the curve as the reference scattering co -efficient and concluded that the coefficient might be the characteristic of the subject field. For petroleum literature, Cole obtained the relationship<\/p>\r\n \r\n\r\nF(x) = 1+ b log10 x (x > c)\r\n\r\n \r\n

where F(x) stands for the cumulative number of papers contained in the x number of most productive journals, and c represents the number of journals figuring in the nuclear zone. For petroleum literature Cole found the value of b as 0.43.<\/p>\r\n \r\n\r\n3. Leimkuhler\u2019s Formulation<\/strong>\r\n\r\n \r\n

Ferdinand F. Leimkuhler [13]Professor of Industrial Engineering of the Purdue University of the United States analyzed all the published data on Bradford distribution and derived the following equation applying statistical techniques.<\/p>\r\n \r\n

ln(1+\u03b2x)<\/p>\r\n

F(x) =-----------------, (0\u2264 x \u2264 1)<\/p>\r\n

ln (1 + \u03b2)<\/p>\r\n \r\n

In the equation, F(x) stands for the cumulative fraction of the references, x for the corresponding fraction of the most productive journals, and \u03b2 is a constant related to the document collection. Brookes [5] opined that the equation is but a compromise since Leimkuhler accepted the empirical data as both complete and exact. Brookes further observed that \u201cUnfortunately, though Leimkuhler\u2019s formulation can be used theoretically without difficulty, it has some disadvantages for the practical document a list. The numerical evaluation of the key parameter \u03b2 requires tedious statistical computation and the solving of an implicit equation by approximation methods....In fact it was the exasperation evoked by an attempted practical application of Leimkuhler\u2019s formulae that led the author of this paper to seek a simpler formulation of the Bradford distribution\u201d [5: p249]<\/span><\/p>\r\n \r\n

4. Brookes Formulation<\/strong><\/p>\r\n \r\n

Brookes [5, 6] formulation of the Bradford distribution follows. Suppose R(n) is the cumulative total of relevant papers found in the first n journals when all the journals are ranked in order of decreasing productivity, the Bradford\u2019s law requires that<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

\"\"The only function that fully satisfies this condition is<\/span><\/strong><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0R(n) = k log n, where k is a constant\u00a0 \u00a0 \u00a0 Eq.4<\/strong>\r\n\r\n<\/div>\r\n

Brookes has provided another model<\/span><\/strong><\/p>\r\n\r\n

\r\n

<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

The values we get are a = 188, and b = .382<\/p>\r\n

With the values of a and b we can now determine the value of the cumulative total of references for the periodical of any rank.<\/p>\r\n \r\n

For testing, let us take 45th rank.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\nWe have\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 R(n) = anb<\/sup>\r\n\r\nPutting the values , we get R(n) = 188 x 45.382\r\n\r\n=\u00a0 188 x 4.276\r\n\r\n=\u00a0 803.888\r\n\r\n=804, which is quite close to the observed value of 802.\r\n\r\n \r\n

Bradford considered the Bibliograph to be a straight line which has resulted in two different formulations, one is verbal and the other is graphical. The algebraic expressions for the two<\/p>\r\n \r\n\r\nformulations given by Brookes are:\r\n\r\nR(n) = j log (n\/t + 1) for the verbal formulation, and\r\n\r\nR(n) = k log n\/s for the graphical formulation\r\n\r\n \r\n\r\n5. Naranan\u2019s Viewpoint<\/strong>\r\n\r\n \r\n

In 1970 Naranan [14] opined that (i) Bradford\u2019s law of bibliography of scientific literature is explainable in terms of an underlying power law distribution of the number of articles in scientific journals; (ii) the law emerges as a natural consequence of exponential growth of scientific literature and journals at comparable rates; and (iii) a model like this predicts a strong correlation between the age of a journal and the number of articles it carries. The author was hopeful that the proposed mechanism might find wider application in many other fields of science.<\/p>\r\n \r\n

Brookes [7] pointed out that Naranan\u2019s analysis was not valid for Bradford. However, his paper provided a plausible model of Lotka\u2019s law with suitable verbal amendments. The comments of Brookes as to the paper are being reproduced verbatim. \u201cThe inverse square law of scientific authorship has hitherto been regarded as an inexplicable and useless scientific oddity. Naranan\u2019s model of it is therefore welcome. And, together with other measures of scientific productivity, Lotka\u2019s law has recently been applied by Dobrov and Korennoi in determining the optimum size of research institutes in USSR\u201d.<\/p>\r\n \r\n

Hubert [ 11] was of the view that Naranan interpretation of the original form of Bradford\u2019s law does not follow a stochastic argument based on his assumptions.<\/p>\r\n \r\n\r\n6. Bookstein\u2019s Viewpoint<\/strong>\r\n\r\n \r\n

In his paper published in 1976, Bookstein [4] analyzed the distributions of Lotka, Zipf, Bradford and Leimkuhler and adopted a point of view that allows us to understand that these distributions are in fact the different versions of a single theoretic distribution. He generalized these distributions with the following words. \u201cAll of these distributions are almost equivalent . . . In each case we have a set of entities (for example, chemists, words) producing events (publications, occurrences) over some dimension of extension (time, length of text) and in each case the distribution describes the number of occurrences of events over a fixed interval of that dimension. Under these conditions it is possible to describe the same distribution in at least four\u00a0distinct ways; these modes of description are represented above by the distributions of Lotka, Zipf, Bradford, and Leimkuhler\u201d.<\/span><\/p>\r\n \r\n

Physicists all over the world have tried to unify four natural forces, i.e. electromagnetic force, gravitational force, weak nuclear force, and strong nuclear force for the last hundred years or so. Bookstein has done the same thing for bibliometric distributions. He has shown that basically all the four bibliometric distributions are different versions of the same distribution.<\/span><\/p>\r\n \r\n

7. Bradford Multiplier<\/strong><\/p>\r\n \r\n

The number of periodicals in the three zones of Bradford distribution generally follows the ratio 1:n:n2, where n is the Bradford multiplier. Ravichandra Rao [16] analyzed the Bradford multiplier with a small sample of 12 datasets using t test. An attempt has also been made to identify a suitable model to explain the law of scattering. Among the various methods tried log normal fits much better than many models including the log linear model.<\/span><\/p>\r\n \r\n

8. Ambiguity between Verbal and Graphical Statements<\/strong><\/p>\r\n \r\n

Bradford's law can be looked into two different ways -- graphically and verbally. This was first observed by Vickery [18].Bradford\u2019s verbal formulation of the law is recorded as \u201cIf scientific journals are arranged in order of decreasing productivity of articles on a given subject, they may be divided into a nucleus of periodicals more particularly devoted to the subject and several groups or zones containing the same number of articles as the nucleus, when the number of periodicals in the nucleus and succeeding zones will be as 1: n: n2\u201d.[2: p. 154].In 1948, Brian C. Vickery [18] contributed an important paper on Bradford\u2019s law. He analyzed about 1600 journal references and compared his results with Bradford\u2019s and found an inconsistency. He remarked \u2013 \u201cWe can \u2026 regard the theoretical distribution of papers on a given subject in scientific periodicals as derived by Bradford, as fully corroborated by the distributions observed in the sample investigations. The rectilinear relation . . . incorrectly assumed by Bradford to be identical with his theoretically derived relation, fits only the upper portion of the observed curve (Figs. 2 and 3). The theoretical relation itself , however, enables us to predict the whole curve\u201d.Vickery showed that if n mjournals contribute a cumulative m papers, and nm is greater than the nucleus, then the verbal formulation is equivalent to the expression nm : n 2m - nm : n3m - n2m: . . . :: 1: am: am2: . . . The graphical formulation is equivalent to the expression nm : n2m : n3m:\u00a0 . . . :: 1: bm: bm2: . . . This apart, when the graph is plotted with the data of verbal formulation it takes a different shape compared to the shape of the graph with complete data set. In the verbal expression, the data in Table 1 will take the following shape and generate a curve as given in Fig. 2.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nTable 2 \u2013 Distribution of Articles according to Zones<\/strong>\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n\"\"Comparing the two bibliographs we find the following:<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n

i. In the verbal formulation, the entire data is not available. What we get is practically a summary of the entire data set.<\/p>\r\n

ii. The Bibliograph in Fig 2 is incomplete, inasmuch as it does not indicate the starting point of the curve.<\/p>\r\niii. The last portion of the graph in both Figs. 2 and 3 is a straight line.\r\n

iv. In the graphical presentation of some Bradford distributions, a droop is observed at the end of the graph, which is not seen with the data of verbal formulation.<\/p>\r\n \r\n

With these, the distinction between verbal formulation and the graphical presentation becomes quite clear and the shortcomings of the verbal formulation apparent. Brookes have provided equations both for verbal formulation as well as the graphical formulation. The equations are given under Brookes\u2019 formulation.<\/p>\r\n \r\n\r\n9. Bradford-Zipf Distribution<\/strong>\r\n\r\n \r\n

Kendall [12], a statistician by profession, also studied Bradford distribution using 1,763 references on operational research pertaining to 370 journals. For the sake of comparison \u20181465 references to statistical methodology (covering the period 1925-39)\u2019 were used. The graph plotted following Bradford\u2019s method produced a curve which was remarkable for its linearity. He also noticed that the Law is similar to, but not identical with the Zipf\u2019s law. Let us consider the data given in Table 4. The data set provides the typical Bradford distribution.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Rank\u00a0Periodical\/s<\/strong><\/td>\r\nNo. of\u00a0article\/s<\/strong><\/td>\r\nNo. of\u00a0total<\/strong><\/td>\r\nCumulative<\/strong><\/td>\r\n<\/tr>\r\n
1<\/td>\r\n1<\/td>\r\n20<\/td>\r\n20<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n1<\/td>\r\n14<\/td>\r\n34<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n1<\/td>\r\n12<\/td>\r\n46<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n1<\/td>\r\n11<\/td>\r\n57<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n1<\/td>\r\n10<\/td>\r\n67<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n1<\/td>\r\n9<\/td>\r\n76<\/td>\r\n<\/tr>\r\n
9<\/td>\r\n3<\/td>\r\n8<\/td>\r\n100<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n1<\/td>\r\n7<\/td>\r\n107<\/td>\r\n<\/tr>\r\n
12<\/td>\r\n2<\/td>\r\n6<\/td>\r\n119<\/td>\r\n<\/tr>\r\n
14<\/td>\r\n2<\/td>\r\n5<\/td>\r\n129<\/td>\r\n<\/tr>\r\n
15<\/td>\r\n1<\/td>\r\n4<\/td>\r\n133<\/td>\r\n<\/tr>\r\n
25<\/td>\r\n10<\/td>\r\n3<\/td>\r\n163<\/td>\r\n<\/tr>\r\n
40<\/td>\r\n15<\/td>\r\n2<\/td>\r\n193<\/td>\r\n<\/tr>\r\n
84<\/td>\r\n44<\/td>\r\n1<\/td>\r\n237<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTable 4\u2013 A data set following Bradford distribution.<\/strong>\r\n\r\n<\/div>\r\n
<\/div>\r\n
Inverting the columns 1 and 3 of Table 4 and multiplying the numbers of each row we get the following result (Table 5).The number in the second column may be considered as frequency.<\/span><\/div>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u00a0Rank i.e.<\/strong><\/td>\r\nNo. of article\/s, Frequency<\/strong><\/td>\r\nRank x Frequency<\/strong><\/td>\r\n<\/tr>\r\n
84<\/td>\r\n1<\/td>\r\n84<\/td>\r\n<\/tr>\r\n
40<\/td>\r\n2<\/strong><\/td>\r\n80<\/td>\r\n<\/tr>\r\n
25<\/td>\r\n3<\/strong><\/td>\r\n75<\/td>\r\n<\/tr>\r\n
15<\/td>\r\n4<\/td>\r\n60<\/td>\r\n<\/tr>\r\n
14<\/td>\r\n5<\/td>\r\n70<\/td>\r\n<\/tr>\r\n
12<\/td>\r\n6<\/td>\r\n72<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n7<\/td>\r\n70<\/td>\r\n<\/tr>\r\n
9<\/td>\r\n8<\/td>\r\n72<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n9<\/td>\r\n54<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n10<\/td>\r\n50<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n11<\/td>\r\n44<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n12<\/td>\r\n36<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n14<\/td>\r\n28<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n20<\/td>\r\n20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTable 5 \u2013 Partly inverted form of Table 4<\/strong>\r\n\r\n<\/div>\r\n
<\/div>\r\n
The figure in the third column clearly indicates that they by and large follow Zipf\u2019s law<\/span>.<\/strong> The two distributions are in fact very close, hence they are often referred to as Bradford-Zipf distribution. The linearity of the Bradford Bibliograph indicates a true Zipf situation.<\/span><\/div>\r\n
<\/div>\r\n
10.\u00a0 <\/strong>Characteristics of Bibliometric Distribution<\/strong><\/div>\r\n