Applicability of Lotka s Law and Authorship pattern in the field of Mathematical Science Research: A Scientometric Study Rajani, S. Research Scholar Rani Channamma University, Belagavi, Karnataka rajani@bub.ernet.in Ravi, B. Deputy Librarian University of Hyderabad, Hyderabad, Andhra Pradesh bravicta@gmail.com Abstracts This paper depicts an authorship pattern of Indian research productivity in the web of science database during 1999-2014. The data downloaded from indexed SCI expanded in the web of Science database by the publisher of Thomson Reuters, and the software application is used for the analysis called Histcite software. It is also studied by applying the application of Lotka s Law method and degree of collaboration of authorship pattern. The data reveals that the totally 16,316 records with 38,104 authors are available across the globe in this studied. The study found that the multiple authors are high percentage i.e 82.7% and predominant to single authorship. The degree of the collaboration average number of the mean value is 0.93%. The study reveals that the highest number of contribution and also the Institutional Affiliation from Indian research productivity with co-authorship, TGLCS and H- index value. Keywords: Lotka s Law, Authorship Pattern, Mathematics literature, Web of Science Database, Scientometrics Introduction: There are numerous authors published the scholarly publication of different fields by the various publishers of the database. The three top laws are called Lotka s law, Bradford laws and Zipfs law in the Scientometric Studies. Lotka s law is one of the three major laws of bibliometrics that mainly explain the literature distribution of various authors productivity in a given field (Lotka, 1926). Cronin (2001) states that the authorship as undisputed coin of the real in academia and absolutely central to the operation of the academic reward system. 464
In this study used the Lotka s Law method, they are a multiple number of authors productivity have been analyzed by using the application of Lotka s Law method of the different literature of various databases. Review of Literature: Review of previous work has done by many authors, and some of them are listed below. According to Roman, E. (Jan 2000) has carried on the new measure of connectedness between research areas, namely, the bidirectional migration of authors between 39 subfields as seen from Mathematical Reviews for 17 years (1959-75). The inter-related structure of Mathematics is then mapped using migration data in the appropriate range. The data found that the resulting map to be a good reflection of the disciplinary variation in the field of Mathematics. Ueno, Sen et al. (Mar 2006) study the growth of scientific papers in science and technology during the 90s. They analyzed the publication of scientific papers and the factor of increasing articles between Japan-China relationships in co-authored papers. The findings revealed that the rating of growth documents production became the sixth place in the world by sharing articles with increased more than three times, and the papers production has concentrated on priority universities since the implementation of the top priority policy (211 projects). Further this study Japan-China relationship rapidly deepens in the high shared fields such as chemistries, materials, physics and engineering, while in mathematics in which China exceeds Japan, Japan-China relationship is not healthy. Sangam, S. L. and Savanur, Kiran. (Jun 2006) has devoted to the mathematics research of Prof. N. Rudraiah of Bangalore University, Bangalore his work on 43 years of productive life in various fields of applied mathematics like fluid mechanics, magnetohydrodynamics, electrodynamics and smart materials of nanostructures. In his work found that the collaborated with 102 colleagues and students has published 271 papers during 1962-2004. The collaboration co-efficient was 0.54 and Highest collaborations were with M. Venkatachalappa (31), and B.C. Chandrasekhara (21). However, the core journals publishing his papers were Indian Journal of Pure and Applied Mathematics, Current Science, 465
International Journal of Heat and Mass Transfer, Acta Mechanica, Journal of Fluid Machanics, Proc. Royal Cambridge Society of London, Physics of Fluid. Objectives of the Study: Some of the Specific aims of the study are: To observe and analysis of Authorship collaboration in the field of Mathematical Science Research; To know the prolific authorship pattern; To prepare the ranking list of institutional affiliation of authors in the field of Mathematical Science Research; To determine the application of Lotka s Law of Research productivity in India. Methodology The data was collected from Indian Mathematical Science Research productivity around 16,317 indexed in SCI expanded a database of a web of science database published by the Thomson Reuters during 1999-2014. In this study analyzed the data by using the Histcite software application from the web of science publisher. Some of the analytical indicators tools are a year wise distribution of Indian productivity, applied the method of Lotka s Law, the prolific author to know the highest publication of an article, ranking list of institution affiliation of authors, and calculated the degree of authors collaboration and so on. The graphical represents the bibliographic coupling of authors and bibliographic coupling of source utilized from the software called VOS Viewer of Bibliometric mapping software. Data Analysis: Year wise distribution of publication The data was collected from the Web of Science Database indexed in SCI expanded database during 1999 to 2014. Table-1 and Graph-1 represent the years of allocation of Indian Publications. 466
It s identified that around 16,317 of records available in these research publications. In Table 1 shows that the highest publications are 1614 articles in 2013 with 3488 Global Citation Scores and lowest publications are 576 articles in the year 1999 with 5344 Global Citation Scores. It also found that observing by TLCS (Total Local Citation Scores) overall period 93,696 citation scores measured, in the year 2009 with 8084 GLC having highest citation score and its h-index value is 34 and the lowest citation score with 1765 in the year 2014 having h-index value are 11. Sl. No. Table-1 Yearwise distribution of Indian publications Year Indian % Cumulative Cum TLCS TGCS Citing Output of % articles Publications Average Citation Per Article 1. 1999 576 3.53 576 3.53 777 5344 5149 9.28 31 2. 2000 623 3.81 1199 7.34 821 7198 7047 11.55 35 3. 2001 587 3.59 1786 10.94 907 5679 5370 9.67 35 4. 2002 653 4.00 2439 14.94 562 5242 4986 8.02 29 5. 2003 660 4.04 3099 18.99 793 5967 5690 9.04 36 6. 2004 629 3.85 3728 22.84 811 5503 5214 8.75 31 7. 2005 741 4.54 4469 27.39 882 6463 6054 8.72 33 8. 2006 769 4.71 5238 32.10 978 5705 5395 7.42 32 9. 2007 1051 6.44 6289 38.54 1053 6567 6124 6.25 34 10. 2008 1166 7.14 7455 45.69 1169 6599 6112 5.66 31 11. 2009 1367 8.37 8822 54.06 1190 8084 7191 5.91 34 12. 2010 1276 7.82 10098 61.89 947 7065 6314 5.50 31 13. 2011 1494 9.15 11592 71.04 1070 7536 6279 5.04 32 14. 2012 1530 9.37 13122 80.42 675 5491 4774 3.95 27 15. 2013 1614 9.89 14736 90.31 330 3488 3082 2.16 18 16. 2014 1580 9.68 16316 100 100 1765 1559 1.12 11 Total 16,316 13065 93696 h- index 467
Authorship Pattern Fig-1: Year-wise distribution of Indian Research productivity Table 2 studied the authorship pattern of mathematics literature during 1999 to 2014. It s found that the data totally 38,104 authors have contributed 16,317 published research output during the period. Out of 16,317 articles, 7640 (47%) of articles are written by double authors have got the first position and followed by the 4,282 (26.2%) of articles are written by three authors. 2794 (17.12%) of articles have contributed by the single author, whereas in four authors have contributed 1156 (7.08 %) and remaining of articles are very less contributed in this study. 468
Year Single Author Double Authors Table-2 Shows Authorship Patterns in the field of Mathematics Research Output Third Four Five Six Seven Eight Nine Ten and Authors Authors Authors Authors Authors Authors Authors Above Authors 1999 148 297 100 23 4 1 2 - - 1 1,187(3.11) 2000 166 290 136 22 8 - - - - 1 1,305(3.42) 2001 134 304 124 15 4 2 2 1-1 1,262(3.31) 2002 152 330 132 30 4 2 1 1-1 1,385(3.63) 2003 162 316 134 31 14 1 1-1 - 1,412(3.70) 2004 155 286 140 29 14 4 - - - 1 1,370(3.59) 2005 121 377 177 46 11 5 3 1 - - 1,704(4.47) 2006 131 374 186 53 14 5 3 1-2 1,801(4.72) 2007 180 506 285 59 20 1 - - - - 2,389(6.26) 2008 186 580 309 59 11 10-8 - 3 2,734(7.17) 2009 207 639 384 100 28 6 2 1 - - 3,235(8.48) 2010 211 613 342 85 18 5 1 - - 1 2,941(7.71) 2011 233 672 421 124 28 11 5 - - - 3,577(9.38) 2012 203 671 442 158 39 15 1 1 - - 3,803(9.98) 2013 230 713 466 146 43 10 4 - - 2 3,967(10.41) 2014 175 672 504 176 35 14 1 1-2 4,032(10.58) Total 2,794 (7.33) 15,280 (40.10) 12,846 (33.71) 4,624 (12.13) 1,475 (3.87) 552 (1.44) 182 (0.47) 120 (0.31) 9 (0.02) 222 (0.58) Total 38,104 469
Ranking of Prolific Authors In Table -3 enumerates the ranking of the authorship pattern in the field of mathematics literature based on their highest value of overall the past 10 years during the study. It also studied each authors with Total Local Citation Score, Total Global Citation Score, Citing articles, Co-authorship, Citation Per paper and H-index of the value and it is listed out only top 10 of most of the research productivity of authors. The results show that the Biswas I from Maharashtra has 282 records with 632 global citation score and he collaborated with coauthorship is 89; its h-index value is 11 has got first position; followed by the Kumar, S from Surat has 132 records with 638 TGCS and h-index value is 13 and so on.table-3 Ranking of Prolific Authors Sl. No. Author State Recs Ran k 1. Biswas I Mahara stra (India) TL CS TGC S Coautho rship CA CPP h- index 282 1 352 632 89 382 2.24 11 2 Kumar S Surat 132 2 76 638 93 586 4.25 13 3 Kumar A Punjab 127 3 53 472 65 556 4.27 13 4 Balachandran Coimba 125 4 236 1129 49 629 9.06 17 K tore 5 Ponnusamy S Chenna 114 5 207 498 85 265 4.39 10 i 6 Kundu D Uttar 110 6 270 1743 85 867 15.96 23 Pradesh 7 Gupta V New 107 7 161 658 22 348 5.33 12 Delhi 8 Ansari QH Uttar 103 8 208 1874 114 979 18.22 27 Pradesh 9 Das S Uttar 102 9 51 548 81 542 5.33 11 Pradesh 10 Kumar R Austria 101 10 62 380 35 348 3.83 9 Author productivity and Lotka s Law Lotka's law determine the publication of authorship pattern of the frequency in a given field. It states that "the number (of authors) making n contributions is about 1/n² of those making one; and the proportion of all contributors, that make a single contribution, is about 60%" 470
(Potter, 1988). According to the Lotka s empirical law of scientific productivity it states that y is the number of authors and X is the number of papers whereas n and c are constant. This formula is also called inverse square Law, as per the mathematical expression summarized as x n y=c where (n and c are constants) As per presented by the Bookstein state that the lotak s law of the author productivity is summarised as below C a n = where n=1,2,3 and C> o n α By applying the mathematical method as n is the value of 1,2,3,4.and so on, α is the value of productivity constants, a represents the probability of authors producing n contributions of each and C and α are the two constants of this particular data. Applicability of Lotka s Law of author productivity in the Mathematical Science Research set of data. Table no -4 depicts the authors productivity using histcite software application during the years. Out of 16,316 articles contributed totally 38,027 authorship in this studied. Where 2799 contributors have one article each, 7656 authors contributed two articles, and 4268 authors have four articles each and the credit it shows on the every level so on. It is observed from the data 2799 authors have contributed one paper each, the value of c is the constants considered as c=1,2,3,4. In the above equation is considered as the values of an obtained from the results C C C C log ------- a n = or n α = ---------- α log n = log ------ α = a n n α a n a n ----------------------- -0.44 Where ( C=2799 a n =7657 n=2) α = ------------ α = -1.46 0.30103 So, C= 2799 α= -1.46 a n =? where n=1,2,3,.and so on C 2799 (a n ) 1 = ------ = 2799 (a n ) 2 = ----------- = 7,717 and so on n -1.46 2-1.46 log n 471
Table -4 depicts that the author productivity distribution of Lotka s Law with the calculated the value of exponent (α= -1.46). The result of the present study its shows that the increasing in the infinite value for the research productivity in the field of mathematical science research. No. of Papers (A) No. of Authors (B) Table-4 Author Productivity % Authorship (AxB) % Expected Authors (α= -1.46) 1 2799 17.15 2799 7.36 2,799 2 7657 46.94 15314 40.27 7,717 3 4268 26.16 12804 33.67 13,918 4 1154 7.07 4616 12.13 21,188 5 295 1.80 1475 3.87 29,340 6 95 0.56 570 1.49 38,290 7 26 0.16 182 0.47 47,961 8 8 0.05 64 0.17 58,313 9 1 0.01 9 0.02 69,282 10 3 0.02 30 0.07 80,663 11 2 0.012 22 0.05 92,682 12 1 0.01 12 0.03 1,05,344 13 4 0.02 52 0.14 1,18,602 21 1 0.01 21 0.06 2,39,231 23 1 0.01 23 0.06 2,72,541 34 1 0.01 34 0.09 4,81,756 16,316 100 38027 100 16,79,627 Author Collaboration According to the K. Subramanyam suggested that the mathematical formula for the Degree of collaboration can be expressed as: C = Nm Nm+Ns Where C is the Degree of Collaboration, Nm is the Number of Multiple Authored papers in the discipline and Ns is the number of Single authored paper in the discipline. 472
Table -5 yearwise degree of collaboration Year Single Authors (Ns) Multiple Authored (Nm) Total Degrees of No. of % No. of % Collaboration Output Output 1999 148 5.30 1039 2.94 1187 0.87 2000 166 5.94 1139 3.22 1305 0.87 2001 134 4.80 1128 3.19 1262 0.89 2002 152 5.44 1233 3.49 1385 0.89 2003 162 5.80 1250 3.54 1412 0.88 2004 155 5.55 1215 3.44 1370 0.88 2005 121 4.33 1583 4.48 1704 0.92 2006 131 4.55 1670 4.72 1801 0.92 2007 180 6.45 2209 6.25 2389 0.92 2008 186 6.66 2548 7.21 2734 0.93 2009 207 7.41 3028 8.57 3235 0.93 2010 211 7.56 2730 7.73 2941 0.92 2011 233 8.35 3344 9.47 3577 0.93 2012 203 7.27 3600 10.19 3803 0.94 2013 230 8.24 3737 10.58 3967 0.94 2014 175 6.27 3857 10.92 4032 0.95 2,794(7.33) 100 35,310(92.66) 100 38,104 14.82 (0.92) In this study, where 35,310 C= = 0.93% 35,310+2794 Hence the degree of Collaboration is 0.93% In table 5 depicts the degree of collaboration is measured between single authored and multiple authored of research productivity during the period (1999-2014). When we calculate the each year degree of collaboration overall past 10 years and we get the average value of the degree of collaboration is 0.93%. Out of 16316 research productivity were 2,794 (7.33%) collaborated by the single authored whereas in multiple authored were collaborated among 35,310(92.6%). 473
Figure-2 Shows Bibliographic Coupling of indian Authors Fig: 3 shows co-occurrence of authors in indian research productivity 474
Institutional Affiliation of contributors Table -6 represents the distribution of Institutional affiliation in the field of mathematics literature indexed in a data source. It shows that 16316 contributors are affiliated with 4561 institutions across the globe. The list was prepared for top most of the institutions based on the highest number of contributors were affiliated to institutions with TLCS, TGCS, citing articles, Citation per paper and h-index of the respective institution. It is observed from the table that Indian Institute Technology have 25382 (15.6%) articles with 20 487 Total Global Citation Score, its h-index value is 50 and the citation per paper is 8.15% has got topper and followed by the Indian Statistics Institute has 1371(8.4%) with 8,264 TGCS and so on. Table-6 Distribution of Institutional Affiliation contributors in mathematics literature Sl. No. Institution Public ations % TLCS TGC S Citing Articles CP P h- index 1. Indian Inst Technol 2538 15.6 2337 20487 16607 8.15 50 2. Indian Stat Inst 1371 8.4 1157 8264 6703 6.09 36 3. Tata Inst 990 6.1 1096 4123 3019 4.19 22 Fundamental Res 4. Indian Inst Sci 789 4.8 635 4450 3801 5.68 28 5. Aligarh Muslim 528 3.2 956 4635 2399 8.81 32 Univ 6. Univ Delhi 480 2.9 485 2032 1367 4.28 19 7. Inst Math Sci 387 2.4 298 1653 1431 4.27 17 8. Univ Calcutta 282 1.7 189 1229 1000 4.41 16 9. Banaras Hindu Univ 273 1.7 164 1387 1211 5.13 16 10. Jadavpur Univ 261 1.6 174 1916 1664 7.43 20 475
Figure -5 Shows Bibliographic Coupling of Institution Conclusion: This study is discussed with the authorship pattern in the field of mathematics Indian literature during the limited periods. The data applied by the method of Loutkas law of Indian research productivity, and it is observed from the data that a maximum number of multiple authored of research productivity were contributed in this study. It is found that the degree of collaboration of authorship ranges from 0.82 to 0.95, and the average number of the mean value is 0.93%. We have seen all the authors categories of first authors,' second authors Lotka s inverse law the parameters of α = -1.46 respectively. Reference: 1. Nwagwu, Williams (2006). A bibliometric analysis of productivity patterns of biomedical authors of Nigeria during 1967 2002. Scientometrics, 69(2), 259-269. 2. Tsai, Hsu-Hao and Chi, Yen-Ping (2012). An empirical study of research trends and forecasts: Customer relationship management. African Journal of Business Management, Vol. 6 no.4, pp. 1418-1427. 3. Gupta, B. M and Karisiddippa, C.R. (1999). Collaboration and Author Productivity: A Study with a New Variable in Lotka s law. Scientometrics, Vol. 44 no. 1, pp. 129-134. 4. Tsai, Hsu-Hao (2015). The research trends forecasted by bibliometric methodology: a case study in e-commerce from 1996 to July 2015. Scientometrics, Vol. 105, pp.1079 1089. 5. Gupta, D.K (1987). Lotka s law and Productivity Patterns of Entomological Research in Nigeria for the period, 1900-1973. Scientometrics, Vol. 12 no. 1-2, pp. 33-46. 6. Ranganathan, G (2015). Scientometric Analysis of Authorship Trends and Collaborative Research in Horticulture. International Journal of Next Generation Library and Technologies, Vol. 1 no.3, pp. 1-11. Follow us on: IRJLIS, Facebook, Twitter 476