{"id":68,"date":"2018-07-09T05:10:36","date_gmt":"2018-07-09T05:10:36","guid":{"rendered":"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=68"},"modified":"2018-12-07T05:43:58","modified_gmt":"2018-12-07T05:43:58","slug":"different-models-to-explain-the-phenomena-of-growth-and-obsolescence-of-literature","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/chapter\/different-models-to-explain-the-phenomena-of-growth-and-obsolescence-of-literature\/","title":{"rendered":"Different Models to Explain the Phenomena of Growth and Obsolescence of Literature"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/8k9gj0zie3U\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>I. Objectives<\/strong>\r\n\r\n&nbsp;\r\n\r\nThe objectives of the Unit are to discuss:\r\n<ul>\r\n \t<li style=\"text-align: justify\">Different models which are useful to explain the data on growth of literature-- exponential, logistic, power, Gompertz, etc.<\/li>\r\n \t<li style=\"text-align: justify\">Obsolescence of literature<\/li>\r\n \t<li style=\"text-align: justify\">Relation between growth of sources and items, growth rates and obsolescence rates, etc.<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">II.\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Learning Outcome<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">It is an important concept and you have now learned it. At the end of this module, you have also learnt various growth models and their characteristics; also, the relations among the various models; how to identify the trend and how to compute growth rates, doubling time, etc.<\/p>\r\n&nbsp;\r\n\r\n<strong>III. Module Structure<\/strong>\r\n\r\n&nbsp;\r\n\r\n1.\u00a0 Introduction\r\n\r\n2.\u00a0\u00a0 Different Models of Growth\r\n<p style=\"padding-left: 30px\">2.1 The Exponential Model<\/p>\r\n<p style=\"padding-left: 30px\">2.2 The Logistic Model<\/p>\r\n<p style=\"padding-left: 30px\">2.3 The Power Function<\/p>\r\n<p style=\"padding-left: 30px\">2.4 The Gompertz Model<\/p>\r\n<p style=\"padding-left: 30px\">2.5 Ware\u2019s Model<\/p>\r\n3.\u00a0 Selecting a Trade Type\r\n<p style=\"padding-left: 30px\">3.1 Relationship between Growth of Sources (Journals) and Items (Articles)<\/p>\r\n4. Obsolescence of Literature\r\n<p style=\"padding-left: 30px\">4.1 Relation between Growth rate and Obsolescence rate<\/p>\r\n5.\u00a0 Summary\r\n\r\n6.\u00a0 References\r\n\r\n&nbsp;\r\n\r\n<strong>1. Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The numbers of scientific journals including the abstracting periodicals are simple indicators of scientific growth. Price in 1963 argued observed that literature doubles approximately once in fifteen years. Neelameghan (1963) analyzed the documents on the history of medicine in India for the period 1954-61, during which period Indian contribution was 65% and foreign was 30%. He studied the growth of Indian medical\u00a0<span style=\"font-size: 1em;text-align: initial\">societies and medical periodicals between 1780 and 1920. He also studied the coverage of Indian medical literature in Index Medicus and Experta Medica and it was found that they covered respectively only 38% and 13.5% of the Indian literature. Since then a number of articles were published on this topic, particularly the growth of literature in different subjects and on various growth models. The number and the growth characteristics (of articles, journals, scientists, discoveries, etc.) have been matters of some debate for considerable time. For instance, Price (1963) argued that:<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in fifty years the number of universities, labor force, population, etc. doubles;\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in twenty years GNP, discoveries, scientists, college entrants\/1000 population doubles;\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in fifteen years the number of scientific journals doubles;\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in ten years the number of articles \/ literature in a field (particularly in science) doubles; and\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in two years the number of web sites doubles\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">As observed by many, the growth of publications passes through the following four stages (Price (1963), Michael Mabe (2003) and many others):<\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: justify\">The preliminary period of growth in which the absolute increments are small although the rate of increase is large \u2013 this is the first stage;<\/li>\r\n \t<li style=\"text-align: justify\">During the period of exponential growth, the number of publications in a field double at regular intervals as a result of a high rate of growth \u2013 this is the stage 2;<\/li>\r\n \t<li style=\"text-align: justify\">The rate of growth declines but the annual increments still remain approximately constant, in stage 3; and<\/li>\r\n \t<li style=\"text-align: justify\">In the fourth stage, both the rate of increment and the absolute increase decline and eventually approach to zero.<\/li>\r\n<\/ul>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The growth curves which explain the above four steps are also referred to as S-shape curve. A typical S-shape curve is given in Figure 1. This Figure clearly shows the four different stages \u2013 in X \u2013 axis, 0 to 3 is the first stage; 3 to 5 is the second stage; 5 to 7 is the third stage and 7 to 8 is the fourth stage.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The growth generally adopts an S- shaped pattern and is symmetrical about its point of infection. Price in 1963 estimated that the number of scholarly periodical titles being published at the end of the twentieth century would exceed one million. This however has not come true.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-71 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-17.png\" alt=\"\" width=\"335\" height=\"261\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">He argued that the logistic growth of knowledge over a period of time is a result of a number of applications of intellectual innovations. A logistic curve was fitted to the cumulative number of new publications appearing every year in science. He has also studied the growth of literature covered by Physics Abstracts during 1900-1950. He observed that except for the interruptions during the two world wars the literature has been increasing at exponential rates with a doubling time of about twelve years. Meadows (1993, 1998) on the other hand observed that estimates of the number of journals varied from 10,000 in 1951 to 70,000 in 1987.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Mabe (2003) in his study also observed that journal growth rates have been remarkably consistent over time with average rates of 3.46%, since 1800. He has in fact observed:<\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: justify\">From 1900 to 1940, the number of active journal titles grew at an actual rate of 3.23%, a doubling time of twenty-two years.<\/li>\r\n \t<li style=\"text-align: justify\">From 1945 to 1976, the number of journals grew at an annual rate of 4.35%, representing a doubling time of sixteen years.<\/li>\r\n \t<li style=\"text-align: justify\">Since 1977, the number of journals grew at 3.26%. Growth rates were very high; this trend continued until mid 1970s. Mabe pointed out that the slow growth rates after the mid 1970s were due to<\/li>\r\n \t<li style=\"text-align: justify\">The oil crisis of the 1970s<\/li>\r\n \t<li style=\"text-align: justify\">The increasing public awareness of potential ecological disaster and<\/li>\r\n \t<li style=\"text-align: justify\">The turning away from nuclear technology in the 1950s.<\/li>\r\n<\/ul>\r\n<p style=\"text-align: justify\">These factors, certainly lead to a slow down of government support for research. Yamazaki (1998), with the following three assumptions<\/p>\r\n&nbsp;\r\n\r\ni. The number of journals in a given subject is growing exponentially in time,\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">ii. Concurrently each journal is also augmenting the number of papers on the subject exponentially in time, and<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">iii. The rate of growth of articles in individual journal is the same for all journals,<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div>\r\n<p style=\"text-align: justify\">Studied the number and rate of growth of scientific journals. His review critically assesses studies based on the 'ecological approach' to journal publishing growth. He concluded that the annual rate of growth of scientific journals is 1.85% from the end of the Eighteenth century. Naranan (1970) has shown that \u2018a frequency distribution (J(p)) of the number of journals with p articles is of the form<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\">J (p) \u221d p-\u03b1<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">With \u03b1 \u2248 2, this model is similar to that of Bradford\u2019s law. The mathematical concepts of growth have become popular through the largely circulated report of the Club of Rome (Donella and Others (1972)). There are several models which are very useful to apply growth of literature. Some of these models are discussed below.<\/p>\r\n&nbsp;\r\n\r\n<strong>2. Different Models of Growth<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>2.1 The Exponential Model<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">An exponential model is associated with the name of Thomas Robert Malthus (1766-1834) who first realized that any species could potentially increase in numbers according to a geometric series. Exponential growth represents an increase with a fixed proportion of total population for each unit of time. For example, if a species has non-overlapping populations (e.g., annual plants), and each organism produces \u201cb\u201d offspring, then, the initial size of the population, say \u201ca\u201d, in time t=0,1,2, is equal to (12):<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left\">Yt = a.bt<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">In this case, the growth rate is (b-1) * 100. This model is also popularly known as exponential model or log-linear model, and often expressed it as:<\/p>\r\n&nbsp;\r\n\r\nln y = ln \u03b1 + \u03b2 x,\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Here \u03b2 is the slope of curve, and measures proportional changes in y for a given absolute change in x. The model not only provides the rate of growth (the exponential parameter), but also the rate at which the size of the population doubles. The exponential growth has also been linked and compared with the size of compound interest. The exponential function assumes a convex shape in its graphical presentation. In exponential growth, the increase is proportional to population size, i.e. if the population is y at time t then<\/p>\r\n&nbsp;\r\n\r\n1\/<em>y<\/em> . <em>dy\/<\/em><em>dx<\/em> = <em>\u03b2<\/em> ,\r\n\r\n&nbsp;\r\n\r\nwhere \u03b2 is the Malthusian parameter of the population. In terms of differential equations,\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">if Y is the population, and dy\/dx its growth rate, then its relative growth rate is 1<em>y<\/em> . <em>dy<\/em><em>dx<\/em> = <em>\u03b2<\/em> If the relative growth rate is constant, it is not difficult to verify that the\u00a0<span style=\"font-size: 1em;text-align: initial\">solution to this equation is P(x) = exp(\u03b2x). When calculating or discussing relative growth rate, it is important to pay attention to the units of time being considered.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-72 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-18.png\" alt=\"\" width=\"556\" height=\"385\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">If the growth rate is relative to the size of the population, then it is generally referred to as relative growth rate. It is also called the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Exponential_growth\"><strong>exponential growth<\/strong> <\/a><strong>rate<\/strong>, or the continuous growth rate. The exponential function can be applied to both growth process as well as decay process (obsolescence studies). The examples of growth process are given below (Croxton and Cowden (1966)):<\/p>\r\n&nbsp;\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Growth of literature \u2013 growth of articles, journals, author population, universities, etc.;\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Microbiology (growth of bacteria);\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Conservation biology (restoration of disturbed populations);\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Insect rearing (prediction of yield);\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Plant or insect quarantine (population growth of introduced species); and\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Fisheries (prediction of fish dynamics). A typical exponential curve is shown below:\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The <strong>doubling time<\/strong> is the period of time required for a quantity to double in size or value. It is applied to <a href=\"http:\/\/en.wikipedia.org\/wiki\/Population_growth\">population growth, <\/a><a href=\"http:\/\/en.wikipedia.org\/wiki\/Inflation\">library <\/a>collection, number of universities or colleges or students and many other things which tend to grow over time. When the relative growth rate (not the absolute growth rate) is constant, the quantity undergoes <a href=\"http:\/\/en.wikipedia.org\/wiki\/Exponential_growth\">exponential growth <\/a>and has a constant doubling time or period which can be calculated directly from the growth rate. This time can be calculated by dividing\u00a0<span style=\"text-align: initial;font-size: 1em\">the <\/span><a style=\"text-align: initial;font-size: 1em\" href=\"http:\/\/en.wikipedia.org\/wiki\/Natural_logarithm\">natural logarithm <\/a><span style=\"text-align: initial;font-size: 1em\">of 2 by the exponent of growth, or approximated by dividing 70 by the percentage growth rate; that is:<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-73 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-19.png\" alt=\"\" width=\"592\" height=\"489\" \/>\r\n<p style=\"text-align: justify\">Given the two measurements of a growing quantity, q1 at time t1 and q2 at time t2, and assuming a constant growth rate, you can calculate the doubling time as<\/p>\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-74 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-20.png\" alt=\"\" width=\"241\" height=\"79\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The equivalent concept to doubling time for a material undergoing a constant negative relative growth rate or <a href=\"http:\/\/en.wikipedia.org\/wiki\/Exponential_decay\">exponential decay <\/a>is the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Half-life\">half-life<\/a><\/p>\r\n&nbsp;\r\n\r\nThe Calculation of Simple Percentage Growth rate\r\n\r\n&nbsp;\r\n\r\nThe percent change from one period to another is calculated from the formula:\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-75 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-21.png\" alt=\"\" width=\"193\" height=\"70\" \/>\r\n\r\nWhere:\r\nGR(%) = Percent Growth Rate\r\nyt+1 = Value at time (t+1)\r\nyt = Value at time t\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\nThe annual percentage growth rate is simply the percent growth divided by N, the number of years.\r\n\r\n&nbsp;\r\n\r\n<strong>Example<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In 1980, the number of documents in Library A was 250,000. This grew to 280,000 in 1990. What is the annual percentage growth rate of library collection in Library A?<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\nyt = 250,000,\u00a0 yt+10 = 280,000 and N = 10\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">N<\/span><span style=\"text-align: initial;font-size: 1em\">= 10<\/span>\r\n\r\n<\/div>\r\n<img class=\"size-full wp-image-76 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-22.png\" alt=\"\" width=\"267\" height=\"75\" \/>\r\n<div>\r\n\r\nThe library collection grew 12 percent between 1980 and 1990 or at a rate of 1.2 percent annually\r\n\r\n&nbsp;\r\n\r\n<strong>2.2 The Logistic Model<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The Belgian mathematician Pierre Verhulst (1838) developed the Logistic model (17). A typical logistic curve is shown in Figure 3. He suggested that the rate of population increase might depend on population density:<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\"><em>y<\/em> = <span style=\"text-decoration: underline\">a<\/span><\/p>\r\n<p style=\"text-align: center\">\u00a0 \u00a0 \u00a0 bc<sup>t<\/sup><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The algebra of the logistic family is something of a hybrid. It mixes together the behaviors of both exponentials and powers (proportions, like rational functions). The parameters b and c are simply the y-intercept and the base of the component exponential function bc t. The rate at which a logistic function falls from or rises to its limiting value is completely determined by the exponential function in the denominator, by the parameters b and c.<\/p>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone wp-image-77\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-23.png\" alt=\"\" width=\"749\" height=\"177\" \/>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div>\r\n\r\nThe curve depicts three things, as explained in section 1:\r\n\r\n&nbsp;\r\n\r\ni.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Show growth in the early stage.\r\n\r\nii.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Intermediate period of rapid growth\r\n\r\niii.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 An approach to maturity.\r\n\r\n<img class=\"size-full wp-image-78 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-24.png\" alt=\"\" width=\"339\" height=\"296\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>2.3 The Power Function<\/strong>\r\n\r\n&nbsp;\r\n\r\nIt is a log-log or double log model. It is mathematically represented as:\r\n\r\n<em>y\u00a0\u00a0 <\/em>= <em>\u03b1<\/em> <em>t<\/em> \u03b2\r\n\r\ni.e., log y = \u03b1+ \u03b2log t\r\n\r\nSometimes, the power model is also represented as yt = \u03b1+ \u03b2t \u03b3\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">where \u03b1, \u03b2 &gt; 0. For 0 &lt; \u03b3&lt; 1, the function y takes a concave shape, but without an upper limit. For \u03b3 = 1, the function y assumes a linear shape. For \u03b3 &gt; 1 the function y takes a convex shape. A typical Power model is shown in Fig. 4.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<p style=\"text-align: center\"><img class=\"size-full wp-image-79 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-25.png\" alt=\"\" width=\"567\" height=\"233\" \/><strong style=\"text-align: center;font-size: 1em\">Fig. 4: A Typical Power Model<\/strong><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>2.4 The Gompertz Model<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The Gompertz (1825) model describes a trend in which the growth increment of the logarithms is declining by a constant percentage. Thus, the natural value of trend would show a declining ratio of increment, but the ratio does not decrease by either a constant amount or a constant percentage. The equation for the Gompertz curve is<\/p>\r\n&nbsp;\r\n\r\n<em>y <\/em>=<sup><em>\u00a0<\/em><\/sup><em>ab<sup>ct<\/sup><\/em>\r\n\r\ni.e., log y = log a + (log b) c<sup>t<\/sup>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The Gompertz and logistic curves are similar in that they both can be used to describe an increasing series, which is increasing by a decreasing percentage of growth, or a decreasing series, which is decreasing, by a decreasing percentage of declines. They differ in that the Gompertz curve involves a constant ratio of successive first differences of the log y values, while the logistic curve entails a constant ratio of successive first differences of 1\/y values.<\/p>\r\n&nbsp;\r\n\r\n<strong>2.5 Ware\u2019s Model<\/strong>\r\n\r\n&nbsp;\r\n\r\nThe other equally well-known model is Ware\u2019s model. It is represented as\r\n\r\ny = \u03b4 (1-\u03c6-t) \u03b4, \u03c6 &gt; 1.\r\n<p style=\"text-align: justify\">The Figure 5 shows a typical Ware\u2019s model.<\/p>\r\n<img class=\"size-full wp-image-80 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-26.png\" alt=\"\" width=\"403\" height=\"322\" \/>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">3. Selecting a Trade Type<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">There are many trend types. It is difficult to decide which one to use. Following are the simple guidelines to select the appropriate model (Croxton and Cowden):<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">i. If the approximate trend, when plotted on semi-logarithmic paper is strait line or if the first differences of logarithms are constant, use an exponential model.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">ii.If the first differences resemble a skewed frequency curve, or if the first differences of logarithms are changing by a constant percentage, use a Gompertz model.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">iii. If the first difference resembles a normal frequency curve or if the first differences of reciprocals are changing by a constant percentage, use logistic model. Also, it the approximate trend value (or the original data), when expressed, as percentage of a selected asymptote, appears linear on arithmetic probability paper, use a logistic model.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Egghe and Rao (1992) have suggested an innovative methodology for identification and classification of growth models. They have classified the growth models based on growth functions, i.e. \u03b11 and \u03b12. They have denoted the growth function as C(t) (theoretical or concrete data). These growth rate functions may be defines as:<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u03b11(t) = C(t+1)\/C(t) and \u03b12 (t) = C (2t)\/C(t)<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">t = 1,2,3, \u2026.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">\u03b11 is called the first growth rate function and \u03b12 is called the second growth rate function. The basic idea here is that the graph of \u03b11 and \u03b12 are much more different than the corresponding graphs of other growth models. The theoretical relationship between \u03b11 and \u03b12 has been worked out to be:<\/span><\/p>\r\n\r\n<div>\r\n\r\n\u03b12(t) = \u03b11(2t-1) \u03b11(2t-2) \u2026. \u03b11(t)\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">According to them, the method of determining the growth goes back to the intrinsic growth rate properties of the data for a good understanding of what is really going on. The authors have also suggested that in order to get a simple clue for the selection of the best model, the plot of two growth rate functions for different mathematical models (namely; exponential, power, linear, logistic and Gompertz) may be drawn and visualized. These graphs can be classified as: Type 1 \u2013 increasing, Type 2 \u2013 Constant, type 3 \u2013 decreasing, Type 4 \u2013 increasing and then decreasing, as shown in Table 2.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-81 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-27.png\" alt=\"\" width=\"472\" height=\"244\" \/>\r\n\r\n<\/div>\r\n<div>\r\n<p style=\"text-align: center\"><strong>Table 2: Classification of growth models using growth rate function<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Wolfram et.al (1990) explored the Linear, Exponential and Power model to the growth of publications in a period of 20 years, as reflected in the databases belonging to science, technology, social science and humanities. They found that, in most cases, the mathematical model that provided the best fit to the observed data was a power model, rather than an exponential, logistic or a linear model; and they concluded, \u201cThe breakdown in exponential growth is well underway. The power model was in particular best, because it has the advantages of modeling the growth behavior of both the linear and exponential models.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Egghe and Rao (1992) clarified the formal distinctions between the four models that Wolfram et al. examined, pointing out that any linear model should more properly be recognized as a power model of a special kind, and introducing two other comparable models, the Gompertz and Ware functions, that Wolfram et al. did not consider. Revisiting the data collected in the earlier study, Egghe and Rao observed that an exponential model was never the best fit. Indeed, they have shown that such a model could never have been expected to provide the best fit, given that the rate of growth in every database declined steadily over the years studied. They also found that a power model fitted best in cases of convex growth and that a Gompertz model generally fitted best in cases of S- shaped growth. Egghe and Rao\u2019s findings suggested that, in modeling the growth of literature, the choice between an exponential and a logistic function may have always been a false one, and that we should instead be asking whether growth is best described by a power law or a Gompertz function.<\/p>\r\n&nbsp;\r\n\r\n<strong>3.1 Relationship between Growth of Sources (Journals) and Items (Articles)<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Egghe (2005) observed that growth rates of sources (journals) usually are different from growth rates of items (articles); further he argued, \u201cThe references in publications grow with a rate that is different (usually higher) from the growth rate of the publications themselves.\u201d His study showed that Naranan\u2019s model (exponential model: y=act) hardly fits the empirical data. He showed that the \"simple\" 2-dimensional informetrics models of source-item relations are not able to explain this. He has further shown that a linear 3-dimensional informetrics (i.e. adding a new source set) is capable to model disproportionate growth. The explanation consists of \u201cdefining\u201d a set of \u201csuper sources\u201d which produce the original sources but which also attach the items into the original sources. In this way, disproportionate growth of references versus articles can be explained by looking at authors. In the same way, disproportionate growth of articles versus journals (a new dataset is compiled from the database Econlit) can be explained by considering journal publishers. Formulae of such different growth rates are presented using Lotkaian informetrics and new and existing data sets are presented and interpreted in terms of the used linear 3-dimensional mode.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Sahoo (2006) in his thesis compared the growth of the journals with that of the articles, in the area of software studies. He observed that the correlation coefficient (r) between the number of journals and the articles is 0.9811. That is, 96% of the variation in y (articles) is due to the variation in x (journals) \u2013 <\/span><strong style=\"font-size: 1em\">higher the number of<\/strong> <strong style=\"font-size: 1em\">journals, the higher the number of articles<\/strong><span style=\"font-size: 1em\">. However, for the case of World literature he has observed that the correlation coefficient between number of journals and the number of articles published is only 0.8517. Unlike India, the correlation is not so high for the world literature. Figures 6 and 7 shows the growth rate curves for journals and articles published by the journals for India and World literature respectively from 1990 to 2003. It has been observed that the growth rates of journals have been decreased with the decrease of growth rate of articles. However, for the \u2018India data for the years 1995 and 2001, it has been observed that when there is a negative growth in journals, it does not show any negative growth for the article. From these observations, Sahoo concluded that only in some cases the growth of journals affect the growth of the number of articles. The growth rate of journals and articles are not same; for the Indian literature average growth rate is 9% for both journals and the articles and for the world literature the average growth rate of journals is 3% and average growth rate of articles is 9%.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">In another study, Ravichandra Rao and Divya (2010) studied the growth of literature in Malaria Research. They also studied the relation among journals, articles and authors. Their study suggests that the journals, articles, and authors increase approximately exponentially. The number of articles has increased from 3,996 to 57, 627 from 55-65 to 96-05. Also the number of journals has been increased 503 to 3,072 from 55-65 to 96-05. The R2 value for the trend for journals, articles, and authors are 0.9502, 0.9475 and 0.9651 respectively; the low R2 value are perhaps due to the less number of data sets; the Figures 8-10 are much more convincing that the data on journals, articles, and authors increase exponentially. Under the assumption that the data confirm to exponential model, the growth rates have been computed; the growth rates of the journals, articles and authors are 5.31%, 7.38%, and 10.06% respectively. The most important observation is that the number of least productive journal has been increased to 2,951 from 463. This perhaps due to<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Interdisciplinary nature of research in Malaria and related topics\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 May be an incomplete bibliography\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 High growth rates (exponential in nature!) of journals and articles\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-82 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-28.png\" alt=\"\" width=\"379\" height=\"220\" \/>\r\n\r\n<\/div>\r\n<div>\r\n<p style=\"text-align: center\"><strong>Fig. 8: Growth curve of journals<\/strong><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-83 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-29.png\" alt=\"\" width=\"353\" height=\"461\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">An attempt was also made to analyze the chemical literature based on the data from Chemical Abstracts. The Fig. 11 shows the growth trend with its R2. The growth rate is 5.98.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-84 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-30.png\" alt=\"\" width=\"367\" height=\"267\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">4. Obsolescence of Literature<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">\u2018Obsolete\u2019 generally means out of date or no longer in use. The process of being obsolete is known as obsolescence. It is also often referred to as \u2018phenomenon of replacement.\u2019 The term obsolescence is used for the first time by Gross and Gross in 1927. The authors analyzed the references in the 1926 volume of the journal of chemical literature and observed that the number of references falls to one-half in\u00a0<\/span><span style=\"font-size: 1em\">fifteen years. Obsolescence is thus a characteristic of scientific and technical literature. Burton and Kebler (1960) are the first to use the term \u2018half-life\u2019 in 1960. It is defined as \u2018the time during which one-half of all the currently active literature published.\u2019 It is the period of time needed to account for one-half all the citations received by a group of publications. The concept of half-life is always discussed in the context of diachronous studies. More precisely, Line and Sandison (1974) refer to diachronous studies in those that follow the use of particular items through successive observations at different points in time, whereas synchronous studies are concerned with the plotting the age distribution of material used at one point in time.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">However, there is no reason suppose that the half-life for some subject is the same as the median citation age in that subject. Half-life in the context of synchronous data is referred to as median age of the citations \/ references. The use of literature may decline much faster with data of ephemeral relevance, if it is in the form of reports, thesis, advance communication or pre-print and in the context of advancing technology. However, the use of literature may decline slowly when it is descriptive (e.g., taxonomic botany) and critical (e.g., literary criticism); it may also decline if it deals with concepts (e.g., philosophy).<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Brookes in one of his articles (1970) argued that if growth rates of literature and contributors are equal then the obsolescence rate remain constant. In this sense growth and obsolescence are related. Ravichandra Rao and Meera (1991) have studied the relation between growth and obsolescence of literature, particularly in mathematics. Gupta (1998) studied the relationship between growth rates and obsolescence rates and half-life of theoretical population genetics literature. He explored the application of lognormal distribution to the age distribution of citations over a period of time.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">In the analysis of obsolescence, Brookes argued that the geometric distribution expresses the idea that when a reference is made to particular periodical of age t years. The geometric distribution is given by<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n(1-a) a<sup>t-1<\/sup> ,\r\n\r\n\u2018a (&lt; 1)\u2019 is a parameter \u2013 the annual aging factor; it is assumed to be constant over all values of t.\r\n\r\nLet U = 1 + a2 + a3 + a4 + \u2026. + at + \u2026.\r\n\r\ni.e., U = 1\/(1-a).\r\n\r\nSimilarly if U(t) = at + at+1 + at+1 + at+2 + \u2026\u2026\u00a0 = at (U(0),\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">then\u00a0 U(t)\/U(0)\u00a0 = at.<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Using this relation, by graphical method, we can compute half-life as well as \u2018a\u2019.<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">4.1 Relation between Growth rate and Obsolescence rate<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">If we assume the literature is growing exponentially at an annual rate of g, we then have<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">R(T) = R(0)egT,<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">where R(T) is the number of references made to the literature during the year T.<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">We also have<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">U(0) = R(0)\/(1-a0) and U(T)\u00a0 =\u00a0 R(T) )\/(1-aT)<\/span>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">where a0 and aT are the annual aging factors corresponding to the years 0 and T respectively. Under the assumption that utility remains constant (U(0) = U(T)) , we then have<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">R(0)\/(1-a0) = R(T) )\/(1-aT)<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">By substituting the value of R(T), we thus have a relation between the growth and the obsolescence:<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">e<\/strong><sup style=\"text-align: initial\"><strong>gT<\/strong><\/sup><strong style=\"text-align: initial;font-size: 1em\">\u00a0 = (1-a<\/strong><strong style=\"text-align: initial;font-size: 1em\">T<\/strong><strong style=\"text-align: initial;font-size: 1em\">)\/(1- a<\/strong><strong style=\"text-align: initial;font-size: 1em\">0<\/strong><strong style=\"text-align: initial;font-size: 1em\">)<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">However, Egghe and Ravichandra Rao (1992) showed that the obsolescence factors (aging factors) \u2018a\u2019 is not a constant, but merely a function of time. The authors have also shown that the function \u2018a\u2019 has a minimum which is obtained at a time t later than the time at which the maximum of the number of citations is reached.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Egghe (1993) also developed a model to study influence of growth on obsolescence. He found different results for the synchronous and diachronus study. He argued that for an increase of growth implies an increase of the obsolescence for the synchronous case and for the diachronous case, it is quite opposite. In order to derive the relation, he also assumed the exponential models for growth as well as for obsolescence. In another paper, for the diachronous aging distribution and based on a decreasing exponential model, Egghe derived first citation distribution. In his study he assumed the distribution of the total number of citations received confirms to a classical Lotka\u2019s function. The first citation distribution is given by<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n\u03c6 (t1) = \u03b3 (1- a t1)\u03b1-1\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">where \u03b3the fraction of papers is that eventually get cited; t1 is the time of the citation, \u2018a\u2019 is the aging rate and \u03b1 is Lotka\u2019s exponent. Egghe and Ravichandra Rao in their study in 2002 observed that the cumulative distribution of the age of the most recent reference distribution is the dual variant of the first citation distribution. This model is different from the first citation distribution. In another study, Egghe and Rao have shown the general relation between the first citation distribution and the general citation age distribution. They have shown that if Lotka\u2019s exponent \u03b1 = 2, both distributions are the same. In the same study, they have argued that the distribution of\u00a0<span style=\"font-size: 1em;text-align: initial\">n<sup>th<\/sup> citation is similar to that of the first citation distribution. Egghe, Rao and Rousseau (1995) studied the influence of production on utilization function. Assuming an increasing exponential function for production and a decreasing one for aging, the authors have shown that in the synchronous case, the larger the increase in production, the larger the obsolescence; however, for the diachronous case it is quite opposite. This proof is different from the earlier one derived by Egghe (1993).<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">5. Summary<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Most of the studies on growth of literature or obsolescence of literature are empirical in nature. Once we identify a suitable model to explain the empirical data on growth of literature or obsolescence of literature, we can apply regression analysis (to fit either the linear or non-linear model to the observed data). Then, by applying the appropriate statistical tests (usually, by considering the R2 value), we may choose a model to explain the data.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Any study in scientometrics must be based on certain guidelines\/ methodologies; in order to accept its validity as well as its generalization of the result. The general guidelines are:<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Identify the general problem(s)\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Conduct literature search\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Decide the design methodology\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Collect the data either for the population or for a sample\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Analyze the data\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Report the result\r\n\r\n\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Refine the hypotheses\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In Step 5, to analyze the data we may adopt one of the models which are discussed in this Unit. Scientometric techniques have evolved over time and are continuing to do so. The counting of papers with attribution (by country, by institution, by author, etc.) is a part of the scientometrics.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">It has been argued that the growth of entities (such as journals, articles, authors, etc.) follows exponential model; however, it has been observed that it is not always true; so there is need for to identify a suitable model to explain the data related to growth of scientometric entities in general and also to study the data on obsolescence of literature.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Different Models to Explain the Phenomena of Growth and Obsolescence of Literature<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/8k9gj0zie3U\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n<strong>6.\u00a0 <\/strong><strong>References<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<ul>\r\n \t<li style=\"text-align: justify\">Brookes, B.C. (1970) \u201cObsolescence of special library periodicals: Sampling Errors and Utility contours\u201d. Journal of the American Society for Information Science, 21: 320-9.<\/li>\r\n \t<li style=\"text-align: justify\">Burton, R.E. and Kebler, R.W. (1960) The half-life of some scientific and technical literature. American Documentation, 11: 18-22.<\/li>\r\n \t<li style=\"text-align: justify\">Croxton, F.E. and Cowden, D.J. (1966) \u201cApplied general statistics\u201d. Prentice-Hall of India (Private) Ltd. New Delhi.<\/li>\r\n \t<li style=\"text-align: justify\">\u201cDefinition of Gross domestic product\u201d. http:\/\/www.wordiq.com\/definition\/Gross_ domestic_product.<\/li>\r\n \t<li style=\"text-align: justify\">Donella, H. Meadows; Dennis L. Meadows; Jorgen Randers and William W. Behrens III. (1972) The Limits to Growth. Club of Rome.<\/li>\r\n \t<li style=\"text-align: justify\">Egghe, L. (1993) \u201cOn the Influence of Growth on Obsolescence\u201d. Scientometrics, 27(2); 195-214.<\/li>\r\n \t<li style=\"text-align: justify\">Egghe, L. (2005) \u201cAn explanation of disproportionate growth using linear 3-dimensional informetrics and its relation with the fractal dimension\u201d. Scientometrics. 63(2); 277-296.<\/li>\r\n \t<li style=\"text-align: justify\">Egghe, L. and Rao, I.K.R. (1992) \u201cClassification of growth models based on growth rates and its applications\u201d. Scientometrics, 25: 5-46.<\/li>\r\n \t<li style=\"text-align: justify\">Egghe, L. and Ravichandra Rao, I.K. (1992) \u201cCitation age data and the obsolescence function: fits and explanations\u201d. Information Processing and Management, 28(2): 201-17.<\/li>\r\n \t<li style=\"text-align: justify\">Egghe, L. and Ravichandra Rao, I.K. (1995) \u201cOn the influence of production on utilization functions: Obsolescence or increased use?\u201d Scientometrics, 34(2): 285-315)<\/li>\r\n \t<li style=\"text-align: justify\">Egghe, L. and Ravichandra Rao, I.K. (2002) \u201cTheory and Experimentation on the most recent reference distribution\u201d. Scientometrics. 53(3): 371-387.<\/li>\r\n \t<li style=\"text-align: justify\">Exponential Model. http:\/\/www.ento.vt.edu\/~sharov\/ PopEcol\/lec5\/exp.html.<\/li>\r\n \t<li style=\"text-align: justify\">Gompertz, B. (1825) \u201cOn the Nature of the Function Expressive of the Law of Human Mortality and on a New Mode of Determining the Value of Life Contingencies\u201d. Philosophical Transactions of the Royal Society of London, 115: 513-585.<\/li>\r\n \t<li style=\"text-align: justify\">Gross, P.L.K. and Gross, E.M. (1927) \u201cCollege Libraries and chemical education\u201d. Science, 66: 1229-34.<\/li>\r\n \t<li style=\"text-align: justify\">Gupta, B.M. (1998) \u201cGrowth and Obsolescence off Literature in Theoretical Population Geneticsss\u2026\u201d Scientometrics, 42(3):335-347.<\/li>\r\n \t<li style=\"text-align: justify\">Line (Maurice B.) and Sandison. A. (1974) Obsolescence and change in the use of literature with time. Journal of Documentation. 30(3): 283-350.<\/li>\r\n \t<li style=\"text-align: justify\">Logistic\u00a0 \u00a0 Functions. families\/1_81.html.http:\/\/www.wmueller.com\/precalculus\/<\/li>\r\n \t<li style=\"text-align: justify\">Mabe, Michael. (2003) \u201cThe growth and number of journals\u201d. Serials. 16(2): 191-7.<\/li>\r\n \t<li style=\"text-align: justify\">Malta, Kaushik; et al. (2005) \u201cScaling phenomena in the growth dynamics of scientific output\u201d. Journal of the American Society for Information Science and Technology, 56(9): 893-902.<\/li>\r\n \t<li style=\"text-align: justify\">Meadows, A.J. (1993) In Woodward and Pilling. The International Serials Industry. Aldershot, Gower, pp 24-7.<\/li>\r\n \t<li style=\"text-align: justify\">Meadows, A.J. (1998) \u201cCommunicating Research. Academic Press. London and San Diego Press. p. 15-6.<\/li>\r\n \t<li style=\"text-align: justify\">Menard, H.W. (1974) Science: growth and change. Harvard University Press, Cambridge, Mass.<\/li>\r\n \t<li style=\"text-align: justify\">Naranan,\u00a0 S.\u00a0 (1970.)\u00a0 \u201cBradford\u2019s\u00a0 law\u00a0 of\u00a0 bibliography\u00a0 of\u00a0 science:\u00a0 an interpretation\u201d. Nature. 227(5258): 631-632.<\/li>\r\n \t<li style=\"text-align: justify\">Neelameghan, A. (1963) Documentation of the History of Medicine in India. Ann Lib Sc. Doc., 10(3\/4): 116-42.<\/li>\r\n \t<li style=\"text-align: justify\">Price Derek De Solla. (1963) \u201cLittle Science, Big Science\u201d. Columbia University Press. New York (GS201.)<\/li>\r\n \t<li style=\"text-align: justify\">Ravichandra Rao, I.K. and Meera, B.M. (1991) \u201cGrowth and obsolescence of literature: an empirical study\u201d. <strong style=\"text-align: initial;font-size: 1em\">In<\/strong><span style=\"text-align: initial;font-size: 1em\"> I.K.R. Rao ed. Informetrics \u2013 91. Sarada Ranganatahan Endowment for library. Bangalore. p. 377-394.<\/span><\/li>\r\n \t<li style=\"text-align: justify\">Ravichandra Rao I.K. and Divya Srivastava (2010). \u2018Growth of Journals, Articla and authors\u201d. Journal of Informetrics, 4(3): 249-256.<\/li>\r\n \t<li style=\"text-align: justify\">Robert Malthus, Thomas. (1798) \u201cAn essay on the Principle of Population\u201d.<\/li>\r\n \t<li style=\"text-align: justify\">Sahoo, Bibhuti Bhusan. (2006) \u201cScientometric Study of Literature in Software Studies in India with a comparison to the World Literature\u201d. Thesis Submitted to the Dept. of Library and Information Science of the Pune University for the Degree of Doctor of Philosophy in Library and Information Science. Guide: I. K. Ravichandra Rao, Documentation Research and Training Center, Indian Statistical Institute, Bangalore.<\/li>\r\n \t<li style=\"text-align: justify\">Wolfram D; Chu, C.M. and Liu, Xin (1990). \u201cGrowth of Knowledge: Bibliometric analysis using online databases data in L Egghe and R Rouseau, Eds. Informetrics 89\/90. 355-72.<\/li>\r\n \t<li style=\"text-align: justify\">Yamazaki, Shigeaki. (1987) \u201cA critical review of some ecological studies on scientific journals [in Japanese]\u201d. Journal of Information Processing and Management, 29(10): 863-870.<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/8k9gj0zie3U\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>I. Objectives<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>The objectives of the Unit are to discuss:<\/p>\n<ul>\n<li style=\"text-align: justify\">Different models which are useful to explain the data on growth of literature&#8211; exponential, logistic, power, Gompertz, etc.<\/li>\n<li style=\"text-align: justify\">Obsolescence of literature<\/li>\n<li style=\"text-align: justify\">Relation between growth of sources and items, growth rates and obsolescence rates, etc.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">II.\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Learning Outcome<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">It is an important concept and you have now learned it. At the end of this module, you have also learnt various growth models and their characteristics; also, the relations among the various models; how to identify the trend and how to compute growth rates, doubling time, etc.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>III. Module Structure<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>1.\u00a0 Introduction<\/p>\n<p>2.\u00a0\u00a0 Different Models of Growth<\/p>\n<p style=\"padding-left: 30px\">2.1 The Exponential Model<\/p>\n<p style=\"padding-left: 30px\">2.2 The Logistic Model<\/p>\n<p style=\"padding-left: 30px\">2.3 The Power Function<\/p>\n<p style=\"padding-left: 30px\">2.4 The Gompertz Model<\/p>\n<p style=\"padding-left: 30px\">2.5 Ware\u2019s Model<\/p>\n<p>3.\u00a0 Selecting a Trade Type<\/p>\n<p style=\"padding-left: 30px\">3.1 Relationship between Growth of Sources (Journals) and Items (Articles)<\/p>\n<p>4. Obsolescence of Literature<\/p>\n<p style=\"padding-left: 30px\">4.1 Relation between Growth rate and Obsolescence rate<\/p>\n<p>5.\u00a0 Summary<\/p>\n<p>6.\u00a0 References<\/p>\n<p>&nbsp;<\/p>\n<p><strong>1. Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The numbers of scientific journals including the abstracting periodicals are simple indicators of scientific growth. Price in 1963 argued observed that literature doubles approximately once in fifteen years. Neelameghan (1963) analyzed the documents on the history of medicine in India for the period 1954-61, during which period Indian contribution was 65% and foreign was 30%. He studied the growth of Indian medical\u00a0<span style=\"font-size: 1em;text-align: initial\">societies and medical periodicals between 1780 and 1920. He also studied the coverage of Indian medical literature in Index Medicus and Experta Medica and it was found that they covered respectively only 38% and 13.5% of the Indian literature. Since then a number of articles were published on this topic, particularly the growth of literature in different subjects and on various growth models. The number and the growth characteristics (of articles, journals, scientists, discoveries, etc.) have been matters of some debate for considerable time. For instance, Price (1963) argued that:<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in fifty years the number of universities, labor force, population, etc. doubles;<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in twenty years GNP, discoveries, scientists, college entrants\/1000 population doubles;<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in fifteen years the number of scientific journals doubles;<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in ten years the number of articles \/ literature in a field (particularly in science) doubles; and<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Once in two years the number of web sites doubles<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As observed by many, the growth of publications passes through the following four stages (Price (1963), Michael Mabe (2003) and many others):<\/p>\n<ul>\n<li style=\"text-align: justify\">The preliminary period of growth in which the absolute increments are small although the rate of increase is large \u2013 this is the first stage;<\/li>\n<li style=\"text-align: justify\">During the period of exponential growth, the number of publications in a field double at regular intervals as a result of a high rate of growth \u2013 this is the stage 2;<\/li>\n<li style=\"text-align: justify\">The rate of growth declines but the annual increments still remain approximately constant, in stage 3; and<\/li>\n<li style=\"text-align: justify\">In the fourth stage, both the rate of increment and the absolute increase decline and eventually approach to zero.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The growth curves which explain the above four steps are also referred to as S-shape curve. A typical S-shape curve is given in Figure 1. This Figure clearly shows the four different stages \u2013 in X \u2013 axis, 0 to 3 is the first stage; 3 to 5 is the second stage; 5 to 7 is the third stage and 7 to 8 is the fourth stage.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The growth generally adopts an S- shaped pattern and is symmetrical about its point of infection. Price in 1963 estimated that the number of scholarly periodical titles being published at the end of the twentieth century would exceed one million. This however has not come true.<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-71 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-17.png\" alt=\"\" width=\"335\" height=\"261\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-17.png 335w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-17-300x234.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-17-65x51.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-17-225x175.png 225w\" sizes=\"auto, (max-width: 335px) 100vw, 335px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">He argued that the logistic growth of knowledge over a period of time is a result of a number of applications of intellectual innovations. A logistic curve was fitted to the cumulative number of new publications appearing every year in science. He has also studied the growth of literature covered by Physics Abstracts during 1900-1950. He observed that except for the interruptions during the two world wars the literature has been increasing at exponential rates with a doubling time of about twelve years. Meadows (1993, 1998) on the other hand observed that estimates of the number of journals varied from 10,000 in 1951 to 70,000 in 1987.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Mabe (2003) in his study also observed that journal growth rates have been remarkably consistent over time with average rates of 3.46%, since 1800. He has in fact observed:<\/p>\n<ul>\n<li style=\"text-align: justify\">From 1900 to 1940, the number of active journal titles grew at an actual rate of 3.23%, a doubling time of twenty-two years.<\/li>\n<li style=\"text-align: justify\">From 1945 to 1976, the number of journals grew at an annual rate of 4.35%, representing a doubling time of sixteen years.<\/li>\n<li style=\"text-align: justify\">Since 1977, the number of journals grew at 3.26%. Growth rates were very high; this trend continued until mid 1970s. Mabe pointed out that the slow growth rates after the mid 1970s were due to<\/li>\n<li style=\"text-align: justify\">The oil crisis of the 1970s<\/li>\n<li style=\"text-align: justify\">The increasing public awareness of potential ecological disaster and<\/li>\n<li style=\"text-align: justify\">The turning away from nuclear technology in the 1950s.<\/li>\n<\/ul>\n<p style=\"text-align: justify\">These factors, certainly lead to a slow down of government support for research. Yamazaki (1998), with the following three assumptions<\/p>\n<p>&nbsp;<\/p>\n<p>i. The number of journals in a given subject is growing exponentially in time,<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">ii. Concurrently each journal is also augmenting the number of papers on the subject exponentially in time, and<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">iii. The rate of growth of articles in individual journal is the same for all journals,<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<p style=\"text-align: justify\">Studied the number and rate of growth of scientific journals. His review critically assesses studies based on the &#8216;ecological approach&#8217; to journal publishing growth. He concluded that the annual rate of growth of scientific journals is 1.85% from the end of the Eighteenth century. Naranan (1970) has shown that \u2018a frequency distribution (J(p)) of the number of journals with p articles is of the form<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">J (p) \u221d p-\u03b1<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">With \u03b1 \u2248 2, this model is similar to that of Bradford\u2019s law. The mathematical concepts of growth have become popular through the largely circulated report of the Club of Rome (Donella and Others (1972)). There are several models which are very useful to apply growth of literature. Some of these models are discussed below.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>2. Different Models of Growth<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>2.1 The Exponential Model<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">An exponential model is associated with the name of Thomas Robert Malthus (1766-1834) who first realized that any species could potentially increase in numbers according to a geometric series. Exponential growth represents an increase with a fixed proportion of total population for each unit of time. For example, if a species has non-overlapping populations (e.g., annual plants), and each organism produces \u201cb\u201d offspring, then, the initial size of the population, say \u201ca\u201d, in time t=0,1,2, is equal to (12):<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left\">Yt = a.bt<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In this case, the growth rate is (b-1) * 100. This model is also popularly known as exponential model or log-linear model, and often expressed it as:<\/p>\n<p>&nbsp;<\/p>\n<p>ln y = ln \u03b1 + \u03b2 x,<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Here \u03b2 is the slope of curve, and measures proportional changes in y for a given absolute change in x. The model not only provides the rate of growth (the exponential parameter), but also the rate at which the size of the population doubles. The exponential growth has also been linked and compared with the size of compound interest. The exponential function assumes a convex shape in its graphical presentation. In exponential growth, the increase is proportional to population size, i.e. if the population is y at time t then<\/p>\n<p>&nbsp;<\/p>\n<p>1\/<em>y<\/em> . <em>dy\/<\/em><em>dx<\/em> = <em>\u03b2<\/em> ,<\/p>\n<p>&nbsp;<\/p>\n<p>where \u03b2 is the Malthusian parameter of the population. In terms of differential equations,<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">if Y is the population, and dy\/dx its growth rate, then its relative growth rate is 1<em>y<\/em> . <em>dy<\/em><em>dx<\/em> = <em>\u03b2<\/em> If the relative growth rate is constant, it is not difficult to verify that the\u00a0<span style=\"font-size: 1em;text-align: initial\">solution to this equation is P(x) = exp(\u03b2x). When calculating or discussing relative growth rate, it is important to pay attention to the units of time being considered.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-72 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-18.png\" alt=\"\" width=\"556\" height=\"385\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-18.png 556w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-18-300x208.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-18-65x45.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-18-225x156.png 225w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-18-350x242.png 350w\" sizes=\"auto, (max-width: 556px) 100vw, 556px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If the growth rate is relative to the size of the population, then it is generally referred to as relative growth rate. It is also called the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Exponential_growth\"><strong>exponential growth<\/strong> <\/a><strong>rate<\/strong>, or the continuous growth rate. The exponential function can be applied to both growth process as well as decay process (obsolescence studies). The examples of growth process are given below (Croxton and Cowden (1966)):<\/p>\n<p>&nbsp;<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Growth of literature \u2013 growth of articles, journals, author population, universities, etc.;<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Microbiology (growth of bacteria);<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Conservation biology (restoration of disturbed populations);<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Insect rearing (prediction of yield);<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Plant or insect quarantine (population growth of introduced species); and<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Fisheries (prediction of fish dynamics). A typical exponential curve is shown below:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The <strong>doubling time<\/strong> is the period of time required for a quantity to double in size or value. It is applied to <a href=\"http:\/\/en.wikipedia.org\/wiki\/Population_growth\">population growth, <\/a><a href=\"http:\/\/en.wikipedia.org\/wiki\/Inflation\">library <\/a>collection, number of universities or colleges or students and many other things which tend to grow over time. When the relative growth rate (not the absolute growth rate) is constant, the quantity undergoes <a href=\"http:\/\/en.wikipedia.org\/wiki\/Exponential_growth\">exponential growth <\/a>and has a constant doubling time or period which can be calculated directly from the growth rate. This time can be calculated by dividing\u00a0<span style=\"text-align: initial;font-size: 1em\">the <\/span><a style=\"text-align: initial;font-size: 1em\" href=\"http:\/\/en.wikipedia.org\/wiki\/Natural_logarithm\">natural logarithm <\/a><span style=\"text-align: initial;font-size: 1em\">of 2 by the exponent of growth, or approximated by dividing 70 by the percentage growth rate; that is:<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-73 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-19.png\" alt=\"\" width=\"592\" height=\"489\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-19.png 592w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-19-300x248.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-19-65x54.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-19-225x186.png 225w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-19-350x289.png 350w\" sizes=\"auto, (max-width: 592px) 100vw, 592px\" \/><\/p>\n<p style=\"text-align: justify\">Given the two measurements of a growing quantity, q1 at time t1 and q2 at time t2, and assuming a constant growth rate, you can calculate the doubling time as<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-74 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-20.png\" alt=\"\" width=\"241\" height=\"79\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-20.png 241w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-20-65x21.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-20-225x74.png 225w\" sizes=\"auto, (max-width: 241px) 100vw, 241px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The equivalent concept to doubling time for a material undergoing a constant negative relative growth rate or <a href=\"http:\/\/en.wikipedia.org\/wiki\/Exponential_decay\">exponential decay <\/a>is the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Half-life\">half-life<\/a><\/p>\n<p>&nbsp;<\/p>\n<p>The Calculation of Simple Percentage Growth rate<\/p>\n<p>&nbsp;<\/p>\n<p>The percent change from one period to another is calculated from the formula:<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-75 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-21.png\" alt=\"\" width=\"193\" height=\"70\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-21.png 193w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-21-65x24.png 65w\" sizes=\"auto, (max-width: 193px) 100vw, 193px\" \/><\/p>\n<p>Where:<br \/>\nGR(%) = Percent Growth Rate<br \/>\nyt+1 = Value at time (t+1)<br \/>\nyt = Value at time t<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>The annual percentage growth rate is simply the percent growth divided by N, the number of years.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In 1980, the number of documents in Library A was 250,000. This grew to 280,000 in 1990. What is the annual percentage growth rate of library collection in Library A?<\/p>\n<\/div>\n<div>\n<p>yt = 250,000,\u00a0 yt+10 = 280,000 and N = 10<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">N<\/span><span style=\"text-align: initial;font-size: 1em\">= 10<\/span><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-76 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-22.png\" alt=\"\" width=\"267\" height=\"75\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-22.png 267w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-22-65x18.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-22-225x63.png 225w\" sizes=\"auto, (max-width: 267px) 100vw, 267px\" \/><\/p>\n<div>\n<p>The library collection grew 12 percent between 1980 and 1990 or at a rate of 1.2 percent annually<\/p>\n<p>&nbsp;<\/p>\n<p><strong>2.2 The Logistic Model<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The Belgian mathematician Pierre Verhulst (1838) developed the Logistic model (17). A typical logistic curve is shown in Figure 3. He suggested that the rate of population increase might depend on population density:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><em>y<\/em> = <span style=\"text-decoration: underline\">a<\/span><\/p>\n<p style=\"text-align: center\">\u00a0 \u00a0 \u00a0 bc<sup>t<\/sup><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The algebra of the logistic family is something of a hybrid. It mixes together the behaviors of both exponentials and powers (proportions, like rational functions). The parameters b and c are simply the y-intercept and the base of the component exponential function bc t. The rate at which a logistic function falls from or rises to its limiting value is completely determined by the exponential function in the denominator, by the parameters b and c.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-77\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-23.png\" alt=\"\" width=\"749\" height=\"177\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-23.png 571w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-23-300x71.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-23-65x15.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-23-225x53.png 225w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-23-350x83.png 350w\" sizes=\"auto, (max-width: 749px) 100vw, 749px\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<p>The curve depicts three things, as explained in section 1:<\/p>\n<p>&nbsp;<\/p>\n<p>i.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Show growth in the early stage.<\/p>\n<p>ii.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Intermediate period of rapid growth<\/p>\n<p>iii.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 An approach to maturity.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-78 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-24.png\" alt=\"\" width=\"339\" height=\"296\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-24.png 339w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-24-300x262.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-24-65x57.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-24-225x196.png 225w\" sizes=\"auto, (max-width: 339px) 100vw, 339px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>2.3 The Power Function<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>It is a log-log or double log model. It is mathematically represented as:<\/p>\n<p><em>y\u00a0\u00a0 <\/em>= <em>\u03b1<\/em> <em>t<\/em> \u03b2<\/p>\n<p>i.e., log y = \u03b1+ \u03b2log t<\/p>\n<p>Sometimes, the power model is also represented as yt = \u03b1+ \u03b2t \u03b3<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">where \u03b1, \u03b2 &gt; 0. For 0 &lt; \u03b3&lt; 1, the function y takes a concave shape, but without an upper limit. For \u03b3 = 1, the function y assumes a linear shape. For \u03b3 &gt; 1 the function y takes a convex shape. A typical Power model is shown in Fig. 4.<\/p>\n<\/div>\n<div>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-79 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-25.png\" alt=\"\" width=\"567\" height=\"233\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-25.png 567w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-25-300x123.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-25-65x27.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-25-225x92.png 225w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-25-350x144.png 350w\" sizes=\"auto, (max-width: 567px) 100vw, 567px\" \/><strong style=\"text-align: center;font-size: 1em\">Fig. 4: A Typical Power Model<\/strong><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>2.4 The Gompertz Model<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The Gompertz (1825) model describes a trend in which the growth increment of the logarithms is declining by a constant percentage. Thus, the natural value of trend would show a declining ratio of increment, but the ratio does not decrease by either a constant amount or a constant percentage. The equation for the Gompertz curve is<\/p>\n<p>&nbsp;<\/p>\n<p><em>y <\/em>=<sup><em>\u00a0<\/em><\/sup><em>ab<sup>ct<\/sup><\/em><\/p>\n<p>i.e., log y = log a + (log b) c<sup>t<\/sup><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The Gompertz and logistic curves are similar in that they both can be used to describe an increasing series, which is increasing by a decreasing percentage of growth, or a decreasing series, which is decreasing, by a decreasing percentage of declines. They differ in that the Gompertz curve involves a constant ratio of successive first differences of the log y values, while the logistic curve entails a constant ratio of successive first differences of 1\/y values.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>2.5 Ware\u2019s Model<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>The other equally well-known model is Ware\u2019s model. It is represented as<\/p>\n<p>y = \u03b4 (1-\u03c6-t) \u03b4, \u03c6 &gt; 1.<\/p>\n<p style=\"text-align: justify\">The Figure 5 shows a typical Ware\u2019s model.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-80 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-26.png\" alt=\"\" width=\"403\" height=\"322\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-26.png 403w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-26-300x240.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-26-65x52.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-26-225x180.png 225w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-26-350x280.png 350w\" sizes=\"auto, (max-width: 403px) 100vw, 403px\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">3. Selecting a Trade Type<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">There are many trend types. It is difficult to decide which one to use. Following are the simple guidelines to select the appropriate model (Croxton and Cowden):<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">i. If the approximate trend, when plotted on semi-logarithmic paper is strait line or if the first differences of logarithms are constant, use an exponential model.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">ii.If the first differences resemble a skewed frequency curve, or if the first differences of logarithms are changing by a constant percentage, use a Gompertz model.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">iii. If the first difference resembles a normal frequency curve or if the first differences of reciprocals are changing by a constant percentage, use logistic model. Also, it the approximate trend value (or the original data), when expressed, as percentage of a selected asymptote, appears linear on arithmetic probability paper, use a logistic model.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Egghe and Rao (1992) have suggested an innovative methodology for identification and classification of growth models. They have classified the growth models based on growth functions, i.e. \u03b11 and \u03b12. They have denoted the growth function as C(t) (theoretical or concrete data). These growth rate functions may be defines as:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u03b11(t) = C(t+1)\/C(t) and \u03b12 (t) = C (2t)\/C(t)<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">t = 1,2,3, \u2026.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">\u03b11 is called the first growth rate function and \u03b12 is called the second growth rate function. The basic idea here is that the graph of \u03b11 and \u03b12 are much more different than the corresponding graphs of other growth models. The theoretical relationship between \u03b11 and \u03b12 has been worked out to be:<\/span><\/p>\n<div>\n<p>\u03b12(t) = \u03b11(2t-1) \u03b11(2t-2) \u2026. \u03b11(t)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">According to them, the method of determining the growth goes back to the intrinsic growth rate properties of the data for a good understanding of what is really going on. The authors have also suggested that in order to get a simple clue for the selection of the best model, the plot of two growth rate functions for different mathematical models (namely; exponential, power, linear, logistic and Gompertz) may be drawn and visualized. These graphs can be classified as: Type 1 \u2013 increasing, Type 2 \u2013 Constant, type 3 \u2013 decreasing, Type 4 \u2013 increasing and then decreasing, as shown in Table 2.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-81 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-27.png\" alt=\"\" width=\"472\" height=\"244\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-27.png 472w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-27-300x155.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-27-65x34.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-27-225x116.png 225w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-27-350x181.png 350w\" sizes=\"auto, (max-width: 472px) 100vw, 472px\" \/><\/p>\n<\/div>\n<div>\n<p style=\"text-align: center\"><strong>Table 2: Classification of growth models using growth rate function<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Wolfram et.al (1990) explored the Linear, Exponential and Power model to the growth of publications in a period of 20 years, as reflected in the databases belonging to science, technology, social science and humanities. They found that, in most cases, the mathematical model that provided the best fit to the observed data was a power model, rather than an exponential, logistic or a linear model; and they concluded, \u201cThe breakdown in exponential growth is well underway. The power model was in particular best, because it has the advantages of modeling the growth behavior of both the linear and exponential models.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Egghe and Rao (1992) clarified the formal distinctions between the four models that Wolfram et al. examined, pointing out that any linear model should more properly be recognized as a power model of a special kind, and introducing two other comparable models, the Gompertz and Ware functions, that Wolfram et al. did not consider. Revisiting the data collected in the earlier study, Egghe and Rao observed that an exponential model was never the best fit. Indeed, they have shown that such a model could never have been expected to provide the best fit, given that the rate of growth in every database declined steadily over the years studied. They also found that a power model fitted best in cases of convex growth and that a Gompertz model generally fitted best in cases of S- shaped growth. Egghe and Rao\u2019s findings suggested that, in modeling the growth of literature, the choice between an exponential and a logistic function may have always been a false one, and that we should instead be asking whether growth is best described by a power law or a Gompertz function.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3.1 Relationship between Growth of Sources (Journals) and Items (Articles)<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Egghe (2005) observed that growth rates of sources (journals) usually are different from growth rates of items (articles); further he argued, \u201cThe references in publications grow with a rate that is different (usually higher) from the growth rate of the publications themselves.\u201d His study showed that Naranan\u2019s model (exponential model: y=act) hardly fits the empirical data. He showed that the &#8220;simple&#8221; 2-dimensional informetrics models of source-item relations are not able to explain this. He has further shown that a linear 3-dimensional informetrics (i.e. adding a new source set) is capable to model disproportionate growth. The explanation consists of \u201cdefining\u201d a set of \u201csuper sources\u201d which produce the original sources but which also attach the items into the original sources. In this way, disproportionate growth of references versus articles can be explained by looking at authors. In the same way, disproportionate growth of articles versus journals (a new dataset is compiled from the database Econlit) can be explained by considering journal publishers. Formulae of such different growth rates are presented using Lotkaian informetrics and new and existing data sets are presented and interpreted in terms of the used linear 3-dimensional mode.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Sahoo (2006) in his thesis compared the growth of the journals with that of the articles, in the area of software studies. He observed that the correlation coefficient (r) between the number of journals and the articles is 0.9811. That is, 96% of the variation in y (articles) is due to the variation in x (journals) \u2013 <\/span><strong style=\"font-size: 1em\">higher the number of<\/strong> <strong style=\"font-size: 1em\">journals, the higher the number of articles<\/strong><span style=\"font-size: 1em\">. However, for the case of World literature he has observed that the correlation coefficient between number of journals and the number of articles published is only 0.8517. Unlike India, the correlation is not so high for the world literature. Figures 6 and 7 shows the growth rate curves for journals and articles published by the journals for India and World literature respectively from 1990 to 2003. It has been observed that the growth rates of journals have been decreased with the decrease of growth rate of articles. However, for the \u2018India data for the years 1995 and 2001, it has been observed that when there is a negative growth in journals, it does not show any negative growth for the article. From these observations, Sahoo concluded that only in some cases the growth of journals affect the growth of the number of articles. The growth rate of journals and articles are not same; for the Indian literature average growth rate is 9% for both journals and the articles and for the world literature the average growth rate of journals is 3% and average growth rate of articles is 9%.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">In another study, Ravichandra Rao and Divya (2010) studied the growth of literature in Malaria Research. They also studied the relation among journals, articles and authors. Their study suggests that the journals, articles, and authors increase approximately exponentially. The number of articles has increased from 3,996 to 57, 627 from 55-65 to 96-05. Also the number of journals has been increased 503 to 3,072 from 55-65 to 96-05. The R2 value for the trend for journals, articles, and authors are 0.9502, 0.9475 and 0.9651 respectively; the low R2 value are perhaps due to the less number of data sets; the Figures 8-10 are much more convincing that the data on journals, articles, and authors increase exponentially. Under the assumption that the data confirm to exponential model, the growth rates have been computed; the growth rates of the journals, articles and authors are 5.31%, 7.38%, and 10.06% respectively. The most important observation is that the number of least productive journal has been increased to 2,951 from 463. This perhaps due to<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Interdisciplinary nature of research in Malaria and related topics<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 May be an incomplete bibliography<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 High growth rates (exponential in nature!) of journals and articles<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-82 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-28.png\" alt=\"\" width=\"379\" height=\"220\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-28.png 379w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-28-300x174.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-28-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-28-225x131.png 225w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-28-350x203.png 350w\" sizes=\"auto, (max-width: 379px) 100vw, 379px\" \/><\/p>\n<\/div>\n<div>\n<p style=\"text-align: center\"><strong>Fig. 8: Growth curve of journals<\/strong><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-83 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-29.png\" alt=\"\" width=\"353\" height=\"461\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-29.png 353w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-29-230x300.png 230w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-29-65x85.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-29-225x294.png 225w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-29-350x457.png 350w\" sizes=\"auto, (max-width: 353px) 100vw, 353px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">An attempt was also made to analyze the chemical literature based on the data from Chemical Abstracts. The Fig. 11 shows the growth trend with its R2. The growth rate is 5.98.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-84 aligncenter\" src=\"http:\/\/liscp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/19\/2018\/07\/1-30.png\" alt=\"\" width=\"367\" height=\"267\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-30.png 367w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-30-300x218.png 300w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-30-65x47.png 65w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-30-225x164.png 225w, https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-content\/uploads\/sites\/19\/2018\/07\/1-30-350x255.png 350w\" sizes=\"auto, (max-width: 367px) 100vw, 367px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">4. Obsolescence of Literature<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">\u2018Obsolete\u2019 generally means out of date or no longer in use. The process of being obsolete is known as obsolescence. It is also often referred to as \u2018phenomenon of replacement.\u2019 The term obsolescence is used for the first time by Gross and Gross in 1927. The authors analyzed the references in the 1926 volume of the journal of chemical literature and observed that the number of references falls to one-half in\u00a0<\/span><span style=\"font-size: 1em\">fifteen years. Obsolescence is thus a characteristic of scientific and technical literature. Burton and Kebler (1960) are the first to use the term \u2018half-life\u2019 in 1960. It is defined as \u2018the time during which one-half of all the currently active literature published.\u2019 It is the period of time needed to account for one-half all the citations received by a group of publications. The concept of half-life is always discussed in the context of diachronous studies. More precisely, Line and Sandison (1974) refer to diachronous studies in those that follow the use of particular items through successive observations at different points in time, whereas synchronous studies are concerned with the plotting the age distribution of material used at one point in time.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">However, there is no reason suppose that the half-life for some subject is the same as the median citation age in that subject. Half-life in the context of synchronous data is referred to as median age of the citations \/ references. The use of literature may decline much faster with data of ephemeral relevance, if it is in the form of reports, thesis, advance communication or pre-print and in the context of advancing technology. However, the use of literature may decline slowly when it is descriptive (e.g., taxonomic botany) and critical (e.g., literary criticism); it may also decline if it deals with concepts (e.g., philosophy).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Brookes in one of his articles (1970) argued that if growth rates of literature and contributors are equal then the obsolescence rate remain constant. In this sense growth and obsolescence are related. Ravichandra Rao and Meera (1991) have studied the relation between growth and obsolescence of literature, particularly in mathematics. Gupta (1998) studied the relationship between growth rates and obsolescence rates and half-life of theoretical population genetics literature. He explored the application of lognormal distribution to the age distribution of citations over a period of time.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">In the analysis of obsolescence, Brookes argued that the geometric distribution expresses the idea that when a reference is made to particular periodical of age t years. The geometric distribution is given by<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>(1-a) a<sup>t-1<\/sup> ,<\/p>\n<p>\u2018a (&lt; 1)\u2019 is a parameter \u2013 the annual aging factor; it is assumed to be constant over all values of t.<\/p>\n<p>Let U = 1 + a2 + a3 + a4 + \u2026. + at + \u2026.<\/p>\n<p>i.e., U = 1\/(1-a).<\/p>\n<p>Similarly if U(t) = at + at+1 + at+1 + at+2 + \u2026\u2026\u00a0 = at (U(0),<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">then\u00a0 U(t)\/U(0)\u00a0 = at.<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Using this relation, by graphical method, we can compute half-life as well as \u2018a\u2019.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">4.1 Relation between Growth rate and Obsolescence rate<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">If we assume the literature is growing exponentially at an annual rate of g, we then have<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">R(T) = R(0)egT,<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">where R(T) is the number of references made to the literature during the year T.<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">We also have<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">U(0) = R(0)\/(1-a0) and U(T)\u00a0 =\u00a0 R(T) )\/(1-aT)<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: justify;font-size: 1em\">where a0 and aT are the annual aging factors corresponding to the years 0 and T respectively. Under the assumption that utility remains constant (U(0) = U(T)) , we then have<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">R(0)\/(1-a0) = R(T) )\/(1-aT)<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">By substituting the value of R(T), we thus have a relation between the growth and the obsolescence:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">e<\/strong><sup style=\"text-align: initial\"><strong>gT<\/strong><\/sup><strong style=\"text-align: initial;font-size: 1em\">\u00a0 = (1-a<\/strong><strong style=\"text-align: initial;font-size: 1em\">T<\/strong><strong style=\"text-align: initial;font-size: 1em\">)\/(1- a<\/strong><strong style=\"text-align: initial;font-size: 1em\">0<\/strong><strong style=\"text-align: initial;font-size: 1em\">)<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">However, Egghe and Ravichandra Rao (1992) showed that the obsolescence factors (aging factors) \u2018a\u2019 is not a constant, but merely a function of time. The authors have also shown that the function \u2018a\u2019 has a minimum which is obtained at a time t later than the time at which the maximum of the number of citations is reached.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Egghe (1993) also developed a model to study influence of growth on obsolescence. He found different results for the synchronous and diachronus study. He argued that for an increase of growth implies an increase of the obsolescence for the synchronous case and for the diachronous case, it is quite opposite. In order to derive the relation, he also assumed the exponential models for growth as well as for obsolescence. In another paper, for the diachronous aging distribution and based on a decreasing exponential model, Egghe derived first citation distribution. In his study he assumed the distribution of the total number of citations received confirms to a classical Lotka\u2019s function. The first citation distribution is given by<\/span><\/p>\n<\/div>\n<div>\n<p>\u03c6 (t1) = \u03b3 (1- a t1)\u03b1-1<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">where \u03b3the fraction of papers is that eventually get cited; t1 is the time of the citation, \u2018a\u2019 is the aging rate and \u03b1 is Lotka\u2019s exponent. Egghe and Ravichandra Rao in their study in 2002 observed that the cumulative distribution of the age of the most recent reference distribution is the dual variant of the first citation distribution. This model is different from the first citation distribution. In another study, Egghe and Rao have shown the general relation between the first citation distribution and the general citation age distribution. They have shown that if Lotka\u2019s exponent \u03b1 = 2, both distributions are the same. In the same study, they have argued that the distribution of\u00a0<span style=\"font-size: 1em;text-align: initial\">n<sup>th<\/sup> citation is similar to that of the first citation distribution. Egghe, Rao and Rousseau (1995) studied the influence of production on utilization function. Assuming an increasing exponential function for production and a decreasing one for aging, the authors have shown that in the synchronous case, the larger the increase in production, the larger the obsolescence; however, for the diachronous case it is quite opposite. This proof is different from the earlier one derived by Egghe (1993).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">5. Summary<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Most of the studies on growth of literature or obsolescence of literature are empirical in nature. Once we identify a suitable model to explain the empirical data on growth of literature or obsolescence of literature, we can apply regression analysis (to fit either the linear or non-linear model to the observed data). Then, by applying the appropriate statistical tests (usually, by considering the R2 value), we may choose a model to explain the data.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Any study in scientometrics must be based on certain guidelines\/ methodologies; in order to accept its validity as well as its generalization of the result. The general guidelines are:<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Identify the general problem(s)<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Conduct literature search<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Decide the design methodology<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Collect the data either for the population or for a sample<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Analyze the data<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Report the result<\/p>\n<p>\u2022\u00a0\u00a0\u00a0\u00a0\u00a0 Refine the hypotheses<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In Step 5, to analyze the data we may adopt one of the models which are discussed in this Unit. Scientometric techniques have evolved over time and are continuing to do so. The counting of papers with attribution (by country, by institution, by author, etc.) is a part of the scientometrics.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">It has been argued that the growth of entities (such as journals, articles, authors, etc.) follows exponential model; however, it has been observed that it is not always true; so there is need for to identify a suitable model to explain the data related to growth of scientometric entities in general and also to study the data on obsolescence of literature.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Different Models to Explain the Phenomena of Growth and Obsolescence of Literature<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/8k9gj0zie3U\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>6.\u00a0 <\/strong><strong>References<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify\">Brookes, B.C. (1970) \u201cObsolescence of special library periodicals: Sampling Errors and Utility contours\u201d. Journal of the American Society for Information Science, 21: 320-9.<\/li>\n<li style=\"text-align: justify\">Burton, R.E. and Kebler, R.W. (1960) The half-life of some scientific and technical literature. American Documentation, 11: 18-22.<\/li>\n<li style=\"text-align: justify\">Croxton, F.E. and Cowden, D.J. (1966) \u201cApplied general statistics\u201d. Prentice-Hall of India (Private) Ltd. New Delhi.<\/li>\n<li style=\"text-align: justify\">\u201cDefinition of Gross domestic product\u201d. http:\/\/www.wordiq.com\/definition\/Gross_ domestic_product.<\/li>\n<li style=\"text-align: justify\">Donella, H. Meadows; Dennis L. Meadows; Jorgen Randers and William W. Behrens III. (1972) The Limits to Growth. Club of Rome.<\/li>\n<li style=\"text-align: justify\">Egghe, L. (1993) \u201cOn the Influence of Growth on Obsolescence\u201d. Scientometrics, 27(2); 195-214.<\/li>\n<li style=\"text-align: justify\">Egghe, L. (2005) \u201cAn explanation of disproportionate growth using linear 3-dimensional informetrics and its relation with the fractal dimension\u201d. Scientometrics. 63(2); 277-296.<\/li>\n<li style=\"text-align: justify\">Egghe, L. and Rao, I.K.R. (1992) \u201cClassification of growth models based on growth rates and its applications\u201d. Scientometrics, 25: 5-46.<\/li>\n<li style=\"text-align: justify\">Egghe, L. and Ravichandra Rao, I.K. (1992) \u201cCitation age data and the obsolescence function: fits and explanations\u201d. Information Processing and Management, 28(2): 201-17.<\/li>\n<li style=\"text-align: justify\">Egghe, L. and Ravichandra Rao, I.K. (1995) \u201cOn the influence of production on utilization functions: Obsolescence or increased use?\u201d Scientometrics, 34(2): 285-315)<\/li>\n<li style=\"text-align: justify\">Egghe, L. and Ravichandra Rao, I.K. (2002) \u201cTheory and Experimentation on the most recent reference distribution\u201d. Scientometrics. 53(3): 371-387.<\/li>\n<li style=\"text-align: justify\">Exponential Model. http:\/\/www.ento.vt.edu\/~sharov\/ PopEcol\/lec5\/exp.html.<\/li>\n<li style=\"text-align: justify\">Gompertz, B. (1825) \u201cOn the Nature of the Function Expressive of the Law of Human Mortality and on a New Mode of Determining the Value of Life Contingencies\u201d. Philosophical Transactions of the Royal Society of London, 115: 513-585.<\/li>\n<li style=\"text-align: justify\">Gross, P.L.K. and Gross, E.M. (1927) \u201cCollege Libraries and chemical education\u201d. Science, 66: 1229-34.<\/li>\n<li style=\"text-align: justify\">Gupta, B.M. (1998) \u201cGrowth and Obsolescence off Literature in Theoretical Population Geneticsss\u2026\u201d Scientometrics, 42(3):335-347.<\/li>\n<li style=\"text-align: justify\">Line (Maurice B.) and Sandison. A. (1974) Obsolescence and change in the use of literature with time. Journal of Documentation. 30(3): 283-350.<\/li>\n<li style=\"text-align: justify\">Logistic\u00a0 \u00a0 Functions. families\/1_81.html.http:\/\/www.wmueller.com\/precalculus\/<\/li>\n<li style=\"text-align: justify\">Mabe, Michael. (2003) \u201cThe growth and number of journals\u201d. Serials. 16(2): 191-7.<\/li>\n<li style=\"text-align: justify\">Malta, Kaushik; et al. (2005) \u201cScaling phenomena in the growth dynamics of scientific output\u201d. Journal of the American Society for Information Science and Technology, 56(9): 893-902.<\/li>\n<li style=\"text-align: justify\">Meadows, A.J. (1993) In Woodward and Pilling. The International Serials Industry. Aldershot, Gower, pp 24-7.<\/li>\n<li style=\"text-align: justify\">Meadows, A.J. (1998) \u201cCommunicating Research. Academic Press. London and San Diego Press. p. 15-6.<\/li>\n<li style=\"text-align: justify\">Menard, H.W. (1974) Science: growth and change. Harvard University Press, Cambridge, Mass.<\/li>\n<li style=\"text-align: justify\">Naranan,\u00a0 S.\u00a0 (1970.)\u00a0 \u201cBradford\u2019s\u00a0 law\u00a0 of\u00a0 bibliography\u00a0 of\u00a0 science:\u00a0 an interpretation\u201d. Nature. 227(5258): 631-632.<\/li>\n<li style=\"text-align: justify\">Neelameghan, A. (1963) Documentation of the History of Medicine in India. Ann Lib Sc. Doc., 10(3\/4): 116-42.<\/li>\n<li style=\"text-align: justify\">Price Derek De Solla. (1963) \u201cLittle Science, Big Science\u201d. Columbia University Press. New York (GS201.)<\/li>\n<li style=\"text-align: justify\">Ravichandra Rao, I.K. and Meera, B.M. (1991) \u201cGrowth and obsolescence of literature: an empirical study\u201d. <strong style=\"text-align: initial;font-size: 1em\">In<\/strong><span style=\"text-align: initial;font-size: 1em\"> I.K.R. Rao ed. Informetrics \u2013 91. Sarada Ranganatahan Endowment for library. Bangalore. p. 377-394.<\/span><\/li>\n<li style=\"text-align: justify\">Ravichandra Rao I.K. and Divya Srivastava (2010). \u2018Growth of Journals, Articla and authors\u201d. Journal of Informetrics, 4(3): 249-256.<\/li>\n<li style=\"text-align: justify\">Robert Malthus, Thomas. (1798) \u201cAn essay on the Principle of Population\u201d.<\/li>\n<li style=\"text-align: justify\">Sahoo, Bibhuti Bhusan. (2006) \u201cScientometric Study of Literature in Software Studies in India with a comparison to the World Literature\u201d. Thesis Submitted to the Dept. of Library and Information Science of the Pune University for the Degree of Doctor of Philosophy in Library and Information Science. Guide: I. K. Ravichandra Rao, Documentation Research and Training Center, Indian Statistical Institute, Bangalore.<\/li>\n<li style=\"text-align: justify\">Wolfram D; Chu, C.M. and Liu, Xin (1990). \u201cGrowth of Knowledge: Bibliometric analysis using online databases data in L Egghe and R Rouseau, Eds. Informetrics 89\/90. 355-72.<\/li>\n<li style=\"text-align: justify\">Yamazaki, Shigeaki. (1987) \u201cA critical review of some ecological studies on scientific journals [in Japanese]\u201d. Journal of Information Processing and Management, 29(10): 863-870.<\/li>\n<\/ul>\n<\/div>\n","protected":false},"author":3,"menu_order":8,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["dr-i-k-ravichandra-rao"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-68","chapter","type-chapter","status-publish","hentry","contributor-dr-i-k-ravichandra-rao"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/pressbooks\/v2\/chapters\/68","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":10,"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/pressbooks\/v2\/chapters\/68\/revisions"}],"predecessor-version":[{"id":332,"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/pressbooks\/v2\/chapters\/68\/revisions\/332"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/pressbooks\/v2\/chapters\/68\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/wp\/v2\/media?parent=68"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/pressbooks\/v2\/chapter-type?post=68"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/wp\/v2\/contributor?post=68"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/liscp10\/wp-json\/wp\/v2\/license?post=68"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}