Measuring Citation Diffusion of Selective Indian Physics and Astronomy Journals by Citation Swing Factor (CSF)

The h-index, introduced by Hirsch, is based on the mutual variation between the number of cited and source items. The continuous citation accumulation process over time results in diffusion of cited items from the h-core zone to the adjacent citation-asymmetric h-excess and/or h-tail zones. The indicator Citation Swing Factor (CSF) has recently been developed to measure this diffusion process quantitatively on the basis of h-core citations, excess citations and total citations. CSF is defined as the ratio of change in FHE to change in FET, where FHE (Fractional H-core to Excess citation) indicates the ratio of h-core citations to excess citations and FET (Fractional Excess to Total citation) indicates the ratio of excess citations to total citations. The observed or experimental value of CSF as followed from the basic definition, i.e. the ratio of change in FHE to change in FET over consecutive years, results (-R3/he2) that was obtained on the basis of a theoretical calculation, where R2, h2 and e2 indicate total citations, h-core citations and excess citations respectively. The later expression indicates the expected or theoretical value of CSF. This paper found observed values of CSF for fifteen esteemed Indian physics journals over the last decade (2010-2019) and compared it with the respective theoretical values. The average error over all journals for ten years is found 2.94% indicating close proximity between theoretically expected and practically observed values. Only one journal, viz. Bulletin of the Astronomical Society of India shows large discrepancy between expected and observed values with an average error of 14.3%.


INTRODUCTION
The citation analysis is a tool for quantitative studies of science research output. Pinski and Narin [1,2] were first who applied citation analysis in a systematic way to assess institutions using a standard methodology. Eugene Garfield [3][4][5][6][7] illustrated in several articles the potentialities of citation analysis in the evaluation of research faculty. According to Price, [8] citation patterns in research articles indicate the research front in a particular subject domain. The citation is a recognition of intellectual works that is reckoned as principal rewards of science. [9] Other viewpoints recommend that publishing papers is a means of protecting an individual's intellectual property rights, or it is a way to convince others to accept certain ideas. [10] The usual citation-based metrics like h-index or Eigenfactor have recently been complemented by alternative indicators, [11] mostly due to the rise of the social web and its fast uptake by scholars. [12] The accretion of citation by papers, though varies widely across disciplines, yet the diffusion of citation shows the S-curve for cumulative citations in all major science disciplines that is reckoned as a general citation diffusion model introduced by Price. [8] Citation Swing Factor: An Indicator to Measure Citation Diffusion The h-index of Hirsch is very well-known nowadays. A scientist has h-index equal to H if the top H of his/her N publications from a ranked list have at least H citations each. [13] Besides, there are numbers of indices developed so far known as h-type indices. [14] One of the major objectives of h-type indices were to normalise h-index by dividing number of publications or the age of citation (time normalization). An author or journal once receives one citation enters in the domain of the cited vs. citing graph (Figure 1) through the tail zone that is the entry point. The number of citations received may be increased in due course of time causing the said cited item gradually shifting from the tail zone towards h-core zone and h-excess zone eventually. Such a movement of a cited item in the cited vs. citing graph (Figure 1) may be termed as same consequently with the respective calculated theoretical values. The main objective of this study is to practically testify the formulation of the new indicator Citation Swing Factor.

Formulation of the Problem
The excess citations received by all articles in the h-core zone, which is denoted by e 2 (Figure 1), may be represented as: Where c j are the citations received by the j th paper and e 2 denotes the excess citations within the h-core zone. Assuming, It is obtained, d 2 = e 2 + h 2 ; Here e ≥ 0 and e is a real number.
The relationship between h and e, as expressed in equation (3) and equation (4), instantly depicts a plane spanned by two axes, h and e, or h-e plane. Now, an arbitrary point in the h-e plane represents the overall information of citations received by all papers in the h-core. It is interesting to point out that the Euclidean distance between the origin and the point P(h,e) is equal to The X-axis and Y-axis represent a number of publications and citations, respectively. The area under the rectangular hyperbolic curve (Figure 1) represents the total number of citations received, which is segmentized into three components. The h-core citation is represented by the shaded square (h 2 ) area, while the total excess citation is scattered outside the shaded square area, under the curve ( Figure 1) separated into two segments by the h 2 -zone, viz. upper h-core zone and lower h-core zone. The upper and lower h-core zones residing adjacent to Y-axis and X-axis, together represent the total number of excess citations or net excess citations over h-core citations. The number of citations in the lower h-core zone is also known as Tail Citation. [16,17] Broadly speaking, tail citation also belongs to the category of excess citation, but as it consists of a large number of publications received a low number of citations (1, 2, 3 ....), therefore the name 'Tail' resembling trough of the graph. The citations in the upper h-core zone are also known as an h-core excess citation [16,17] that distinguishes it from its 'Tail' counterpart. The h-core citation indicates the cluster of h-h citations vs. papers, which is the result of the accumulation of 'h' citations over at least 'h' papers. Larger the area of h 2 (Figure 1), the value of h-index will be proportionately greater. For any fixed total number of citations, the steady increase in the value diffusion of cited item. The indicator Citation Swing Factor (CSF) has recently been developed to measure this diffusion quantitatively, [15] which is the ratio of change in FHE (dθ) to change in FET (dε). The parameters FHE and FET indicate the fraction of h-core to excess citations and fraction of excess to total citations respectively and equivalent to the fractional h-core citations over fractional excess citations. The observed or experimental value of CSF that is followed from the basic definition is represented as (dθ/dε). Here both θ and ε are continuous variables and consequently, both FHE and FET are also continuous variables. The differentiation of θ with respect to ε yielded the value (-R 3 /he 2 ), where R 2 , h 2 and e 2 indicate total citations, h-core citations and excess citations respectively. [16,17] The indicator CSF thus points out the shift of h-core citations with respect to fold of excess citations to total citations, which in turn, Figures the citation shift from h-core to h-excess zone.

Aims of the Study
A new indicator Citation Swing Factor (CSF) has recently been developed [15] to measure the diffusion of cited items (authors, journals, institutions etc.) from h-tail zone to h-core zone and subsequently from h-core zone to h-excess zone by continuously receiving citations. It is axiomatic that the citation accretion process is an incessant time-dependent phenomenon, which results a shift of cited items from the h-tail to h-excess zones via the central h-core zone. The h-tail and h-excess zones are asymmetric while the h-core zone is a symmetric zone, as h-tail zone represents large number of low-cited papers whereas the h-excess zone represents small number of high-cited papers. But the h-core zone, a squareshaped symmetric box, represents 'h' number of papers received 'h' citations. This paper aims to find out observed values of CSF for fifteen esteemed Indian physics and astronomy journals over the last decade (2010-2019) and to compare the of h-core citations (h 2 ) would gradually reduce the excess citation (e 2 ). The h-core citations echo the concentration of citations through clustering over 'h' number of core values. On the contrary, the excess citations portray the scattering of citations outside the h-core or h 2 domain. The relative share of h-core citation and excess citation in the corpus of total citation depicts the relative centralization and scattering phenomenon of citation over time.

Hypothesis Formulated
The following ten null hypotheses (H 0 ) grouped into four categories have been formulated for this study. The first and second categories including two null hypotheses each state the constancy of FET and FHE respectively both for eleven consecutive years (2009-2019) and twelve journals respectively. The third category including four null hypotheses state the constancies of CSF(O) and CSF(E) both for eleven consecutive years (2009-2019) and twelve journals. The fourth category including two null hypotheses has stated the equalities between CSF(O) and CSF(E) both for eleven consecutive years (2009-2019) and twelve journals.
It is to be noted that, although the numerical values of FET, FHE, CSF(O) and CSF(E) presented in Table 1 to Table 4 include fifteen journals, but the following ten null hypotheses are tested for twelve journals only. As the Scopus has not indexed the 2018-19 data for Indian Journal of Radio and Space Physics and the 2010-14 data for Journal of Vibrational Engineering and Technologies, and also the journal entitled Bulletin of the Astronomical Society of India, was discontinued on and from 2015, therefore the data for these three journals are not fully comprehensive over the stipulated time span. The testing of hypotheses has been executed by ANOVA (F-Test) method and T-test method, which are presented in Table 5 and Table 6. The Hypothesis No. 4 is tested by T-test for the sample mean. The results of the testing of hypotheses for the eleven consecutive years from 2009 to 2019 are presented in Table 5, while the same for the twelve said journals are presented in Table 6.

SCOPE AND METHODOLOGY
This paper has found out the observed numerical values of Citation Swing Factor or (dθ/dε) (as deduced in equation (10)) for fifteen esteemed Indian physics journals over the last decade (2010-2019) and compared it with the respective theoretical numerical values or Of the fifteen journals, eight journals belong to core domain of physics and astronomy (S. No. 1, 3, 5, 6, 7, 8, 10 and 13), while five journals belong to allied interdisciplinary areas of physics but publish articles on physics regularly (S. No. 2, 4, 9, 11 and 12). The last two journals belong to entire natural science discipline but publish physics articles on regular basis. These two journals are very old and esteemed Indian science journals.
The number of papers published in each journal from 2009 to 2019 along with total citations, h-core citations and excess citations are noted down at first. The number of published papers along with the corresponding number of citations received for each of the fifteen said journals for all the consecutive years from 2009 to 2019 have been collected from Scopus database. On the basis of these data, the h-core and h-excess citations are calculated to find out FET and FHE. The annual changes in the values of FHE and FET yielded dθ and dε respectively. The ratio of dθ to dε or dθ/dε gives the observed value of CSF, which is compared with the theoretical value, i.e. -R 3 /he 2 , where R 2 , h 2 and e 2 indicate total citations, h-core citations and h-core excess citations respectively. [16,17] The titles of the physics and astronomy journals published from India selected for this study is furnished below with the respective abbreviations given in the adjacent parenthesis.

RESULTS AND ANALYSIS
The numerical data representing both temporal variations and journal-wise variations of FET, FHE, CSF(O), CSF(E) and Percentage Errors of fifteen journals are presented in Table 1 Table 8 to Table 22 (Appendix).        The value of r xy is little dispersed from one here, but still it indicates 'Strong correlation' and close proximity between CSF(O) and CSF(E)       In this paper, the values of a new indicator viz. Citation Swing Factor (CSF) are calculated for fifteen selected Indian physics and astronomy journals. The close proximity between practically observed and theoretically expected values justifies the theoretical background of the concept of CSF. It is found that CSF remains nearly constant for all journals with very little variation. Now, whether it may be considered as a parameter for the journals will depend on the results of further studies with other core journals from other subject domains. This study may also be extended for authors. Further studies are also required to testify whether CSF remains constant for a subject over a stipulated time period, or varies for different subjects.