Bibliometric Laws:
Utility and Application
Learning Objectives
Concept of Bibliometric Laws
Understanding and Mathematical
Description of three laws of bibliometrics.
Application and Use of Bibliometric Laws.
There are three fundamental laws which are widely used and discussed in Bibliometrics:
1.
Lotka’s inverse Square Law of productivity.
2.
Bradford’s law of Scattering of Scientific papers.
3.
Zipf’s Law of Word Occurrence.
Lotka’s Law
Alfred J. Lotka was a mathematician, supervisor of mathematical research in the Statistical Bureau of the Metropolitan Life Insurance Company.
Lotka (1926) proposed inverse square law which correlates contributors of scientific papers with their contributions.
Lotka checked the decennial index of ‘Chemical Abstracts’
1907-1916 and counted number of names against which appeared 1,2,3 etc. entries. He tabulated the data for 6,891 names, beginning with letter ‘A’ and ‘B’. A graph was plotted on a logarithmic scale, the number of authors against the number of contributions made by each other and it was found that in each case the points were closely scattered about a straight line, having a slope approximately two to one.
On the basis of the data, Lotka deduced an equation, for the relation between the frequency
‘y’ of persons making ‘x’ contributions as x
ny = constant.
And for the special case n=2, the constant is
0.6079, it was summarized as for every 100
authors contributing one article, 25 will
contribute two articles, about 11 will contribute 3
articles and 6 will contribute 4 articles and so on.
So, the Law is easily interpreted as:
For every 100 authors contributing one article, 25 will contribute 2, 11 will contribute 3, and 6 will contribute 4 each. We can see a general decrease in performance among a body of authors following 1:n
2pattern
Ranking Of Authors
No. of Authors No. of Articles
100 1
25 2
11 3
6 4
4 5
Alternatively it can also be put as:
The number of authors making n contributions to the literature is about 1/n2 of those making one
– 60% of authors make one contribution – 15% of authors make two contributions – <7% of authors make three contributions – <4% of authors make four contributions – <2.5% of authors make five contributions – 1.25% of authors make six contributions – <1% of authors make seven contributions
Mathematically the law can be interpreted as:
x
n1/y x
n= c1/y x
n*y = c
The total number of authors y in a given subject, each producing x publications, is inversely proportional to some exponential function n of x.
Where:
x = number of publications
y = no. of authors credited with x publications n = constant (equals 2 for scientific subjects)
c = constant inverse square law of scientific productivity
Bradford’s Law of Scattering
Samuel Clement Bradford (1934) examined two bibliographies prepared in the Science Library on Applied Geophysics (1928-31) and Lubrication (1931-32) and he prepared lists of journals arranged by decreasing order of source items contributed by the journals to the bibliographies.
He noticed that in each subject, there were a few productive sources, large number of sources which were moderately productive and still a large number of sources of constantly diminishing productivity. In the list of periodicals ranked by diminishing productivity, -
-Bradford identified three groups of periodicals, produced approximately the same number of articles on the subject, but the number of periodicals in these three equi- productive zones increased by a constant factor.
Based on this, the law was stated as:
“If scientific periodicals are arranged in the order of decreasing productivity of articles on a given subject that 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,,,,,, 39.”
One of Bradford's hypotheses was that “references are scattered throughout all periodicals with a frequency approximately related inversely to the scope”.
On this hypothesis, the aggregate of periodicals can be divided into classes according to relevance of scope to the subject concerned, but the more remote classes will, in the aggregate, produce as many references as the more related classes.
Observations revealed three ‘rough’ zones or groupings which Bradford's graded as:
1) Those producing more than 4 references a year.
2) Those producing more than 1 and not more than 4 a year.
3) Those producing 1 or less a year.
Bradford found that “the groups thus produce about the same proportion of references in each case, and the number of constituents increases from group to group, by a multiplier which though by no means constant, approximates fairly closely to the number 3, especially for the two larger groups.
The law of distribution of papers on a given subject in scientific periodicals may thus be stated as:
If 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 numbers of periodicals in the nucleus and succeeding zones will be as:
1:n:n2
Bradford also plotted graph of the cumulative number of source items R(n) verses the logarithm values of the cumulative number of journals (log n) such a graph, is sometimes called as Bradford’s Bibliograph.
The graph being as a rising curves API, and then continues as a straight line. The rising part of the graph represents the nucleus of highly productive journals. The points P1, P2 and P3 on the bibliographs are the boundaries of three equi-productive zones in which the same number of years.
There are three different kinds of scattering:
Lexical scattering:
It is the scattering of words in texts and in collections of texts.
Semantic scattering:
It is the scattering of concepts in texts and in collections of texts.
Subject scattering:
It is the scattering of items useful to a given task or problem.
Applications
Nisonger argues in his textbook ‘Management of Serials in Libraries’
that the following points are some of the “most obvious potentials” of Bradford law:
Selection/deselection.
Defining the core.
Collection evaluation and Collection Development.
The law of diminishing returns.
Calculation of cost at various coverage.
Setting priorities among journals.
Zipf’s law of Word occurrence
George Kingsley Zipf in1949 stated that a relation between the rank of a word and the frequency of its appearance in a long text. If ‘r’ is the rank of a word and ‘f’ is its frequency, then mathematically Zipf’s law can be stated as:
rf = c, where ‘c’ is a constant
The law described that in a long textual matter, if words are arranged in their decreasing order of frequency, then the rank of any given word of the text will be inversely proportional to the frequency of occurrence of the word.
George Kingsley Zipf (1902-1950) was a Professor of Linguistic at Harvard. This law is of the observation that frequency of words within a text or occurrence of some event (P), as a function of the rank (i) when the rank is determined by the above frequency of occurrence, is a power-law function Pi~1/ia with the exponent a close to unity.
The most famous example of Zipfs law is the frequency of English words. The second example Zipf showed in his book was the population of cities (or population of communities). The population of the city as plotted as a function of the rank (the most popular city is ranked number one, etc) is a power-law function with exponent close to '1'.
The income or revenue of a company as a function of the rank is also an example of the Zipfs law. This should is also called as the Pareto's law because Pareto observed the same.
More precisely the Law states that in a relatively lengthy text, if you "list the words occurring within that text in order of decreasing frequency, the rank of a word on that list multiplied by its frequency will equal a constant".
The equation for this relationship is:
r x f = k
(where r is the rank of the word, f is the frequency, and k is the constant)
Zipf illustrated his law with an analysis of James Joyce's Ulysses.
“He showed that the tenth most frequent word occurred 2,653 times, the hundredth most frequent word occurred 265 times, the two hundredth word occurred 133 times, and so on. Zipf found, then that the rank of the word multiplied by the frequency of the word equals a constant that is approximately 26,500”.
The following table shows distribution of words inversely proportional to the frequency of occurrence of the word
Ranking of Word Occurrence
Rank (r) Frequency (f) Product (c)
1 400 400
2 200 400
3 133 399
4 100 400
5 80 400
Mathematically, the law can be interpreted as:
R 1/f R= c1/f r * f = c
Where:
r = rank (in terms of frequency)
f = frequency (no. of times the given word is used in the text) c = constant for the given text
For a given text, the rank of a word multiplied by the frequency is a constant.
Uses for Zipfian Distribution
Used in Automatic Indexing in many ways:
– Likelihood of document relevance can be determined to some degree by frequency of terms
– The most/least frequent terms in a document are not likely to produce relevant retrieval
– Terms in the middle range of frequency are most likely to produce relevant retrieval.
– This can be applied by counting all of the words in a document (minus some words in a stop list - common words (the, therefore...)) with the most frequent occurrences representing the subject matter of the document.
Could also use relative frequency (more often than expected) instead of absolute frequency.
Application of Bibliometric Laws
To quantify research and growth of different area of knowledge.
To estimate comprehensiveness of secondary periodicals.
To identify users and authorship of documents of various subjects.
To measure usefulness of ad hoc and retrospective SDI services.
Experimental models correlating or bypassing the existing models.
Identification of core journals in different disciplines to formulate a need based acquisition policy within the limited budgetary provision.
To initiate effective multilevel network system.
To regulate inflow of information and the communication.
To develop norms of standardization.
References
Bradford, S.C. (1934) "Sources information on specific subjects", Engineering (26), pp.85-86.
Potter, W. G. (1988): "Of Making Many Books There Is No End:
Bibliometrics and Libraries", Journal of Academic Librarianship, Vol.14 No. (4) pp: 238a-238c.
Nisonger T.E. (1998). Management of Serials in Libraries. Englewood, CO: Libraries Unlimited.
Birger, H. and J. Nicolaisen . Bradford’s Law of Scattering: Ambiguities in the Concept of "Subject“ .
Acknowledgments
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