## Abstract

We seek a unified and distinctive citation description of both journals and individuals. The journal impact factor has a restrictive definition that constrains its extension to individuals, whereas the *h*-index for individuals can easily be applied to journals. Going beyond any single parameter, the shape of each negative slope Hirsch curve of citations vs. rank index is distinctive. This shape can be described through five minimal parameters or ‘flags’: the *h*-index itself on the curve; the average citation of each segment on either side of *h*; and the two axis endpoints. We obtain the five flags from real data for two journals and 10 individual faculty, showing they provide unique citation fingerprints, enabling detailed comparative assessments. A computer code is provided to calculate five flags as the output, from citation data as the input. Since papers (citations) can form nodes (links) of a network, Hirsch curves and five flags could carry over to describe local degree sequences of general networks.

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## Acknowledgements

It is a pleasure to thank Mustansir Barma, Smarajit Karmakar, Prasad Perlekar and Surajit Sengupta for helpful discussions.

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## Appendix A. Properties of the IF

### Appendix A. Properties of the IF

We introduce a notation to describe citations to published papers. Citations to papers involve pairs of publication year/citation year and a suitables notation is needed to uniquely identify various citation parameters.

Consider \(n_p(y)\) papers published in the year *y*, in a publication block *y*(*A*) of duration *A* years. The total papers are \(N_p (A) = \sum _{y(A)} n_p(y)\). We define \(P_c (C; y,Y)\) as the number of citations *C* garnered in the year *Y*, in a citation block *Y*(*B*) of duration *B* years. The difference between the start of the publication block of *A* years and the start of the citation block of *B* years is the delay \(D \equiv \text {Start } (B)-\text {Start } (A) \) years, i.e., zero for \(B=A\) block coincidence.

A useful notation to describe different citation variables is the sum over all allowed *C*, *y*, *Y* that defines the total citations \(S_c(A, B; D)\):

where

We make four observations.

### 1.1 Observation 1: The *A*-year IF is not an *A*-year average

In the familiar case of a citation average, the publication/citation year blocks are equal, coincident and not sequential. Thus, \(A=B\) and \(D =0\). The total number of citations is \(N_{c,\mathrm {tot}} = \sum _C N_c (C; A,A,0) = S_c(A,A,0)\), where \(N_c (C; A,A,0) \equiv \sum _{y(A) }\sum _{Y(A)} P_c (C; y,Y))\). The average citation per paper \({\bar{C}}\) over *A* years is

Here \(N_c(C;5,5,0)\) is the 5-year citation frequency, written simply as \( N_c (C)\) and shown in figures 2 and 3. On the other hand, the *A*-year current impact factor IF(*A*) has publication/citation year blocks that are unequal and not coincident, but sequential, so \(A \ne B\) and \(D \ne 0\). The single citation year \(B=1\) commences right after *A*, and so \(D=A\). Thus

Clearly, IF\((A) \ne {{\bar{C}}}(A)\). Here, the number of papers \(N_p(A)\) (cited more than once [47]) is replaced by the number of items *N*(*A*) (with any citations).

### 1.2 Observation 2: \({I\!F}(A)\) has far fewer citations than *A*-year average

The JIF with one citation year, has restricted (and hence fewer) publication–citation year pairs. A toy-model for the 2018 citation year is illustrative. Publications in \(y =2016\) have citations \(P_c(C; 16, 16)\), \(P_c(C; 16, 17)\), \( P_c(C;16, 18)\). Publications in \(y= 2017\) have citations \(P_c (C; 17, 17), P_c (C; 17,18)\). Now suppose the number of citations are \(P_c =2000\) in the year of publication, 1500 in the second year, 500 in the third year and zero thereafter. For the JIF in the year 2018, the two allowed publication–citation pairs are \((y,Y) =(16,18),\) (17, 18) and so total citations are \( [(500) + (1500)] =2000\), for a smaller \(\mathrm {IF}(2) \equiv \mathrm {JIF} = 2000/ [100 +100] = 10\). For the average citation with the same two publication years, the pairs are \((y,Y) = (16,16), (16,17),(17,17)\) and so total citations are the larger \([(2000+ 1500) + (2000)] =5500\) cites, yielding \({{\bar{C}}} = 5500/[100 +100] =27.5\) cites >\(\mathrm {JIF} =10\) cites.

### 1.3 Observation 3: Different parameters give different rankings

In the early scientometric literature, an adjective made a difference: Current IF means one year of citations and cumulative IF means summing up several years of citations. A multiple-citation-year parameter is [11] the 5-year ‘cumulated’ impact factor with \(B=5\) years of citations, e.g. 1999–2004, from one \(A=1\) publication year of say 1999, with the same start year and so \(D=0\). It is given by \(\sim \!\!S_c (1, 5; 0) /N (1)\) and was applied to JAMA for different single years of publication.

Another parameter is the 15-year ‘cumulative’ impact factor [12, 13] for citations over \(B=15\) years, e.g. 1981–1995, to \(A= 2\) years of publications, e.g. 1981–1982, with the same start year and so \(D=0\). It is \(\sim \!\!\!S_c( 2, 15;0)/N(2)\) and is applied to generate rankings for 100 journals [12, 13] and compare: with the JIF rankings for JCR reference year 1983.

With \(\Delta R\) the difference between the two rankings for a given journal, the average ranking-change magnitude \(\langle | \Delta R | \rangle \) can be found, over subsets of the ranks. For the top 10 journals, the average \(\langle | \Delta R |\rangle = 2 \), is small, consistent with a claimed insensitivity [12, 13]. However, for all the 100 journals, the average ranking shift shows substantial shuffle, \(\langle |\Delta R|\rangle \simeq 34\). The specific JIF rankings depend on the chosen JIF parameter: other choices could give other journal rankings.

### 1.4 Observation 4: The \({I\!F}(A)\) definition makes rankings *A*-insensitive

For different durations *A*, how different are the rankings obtained from the *A*-year current impact factor of \(\mathrm {IF}(A) = S_c(A,1; A) /N (A)\)? Surprisingly, the large-*A* ranking can be close to the usual \(A=2\) ranking from the JIF. Suppose that the numerator rises as more years are included, but then flattens to a constant, for *A* larger than the half-life (‘old papers are less cited’). Suppose further, that the denominator varies as the number of years *A* in the block, or \(N (A) = A N (1)\) (‘journal size is the same, every year’). In such a case, \(\mathrm {IF}(A) \simeq (2/A) \mathrm {IF}(2)\) and the journal rankings (not values) can be *A*-insensitive, from the definition. The ranking commonality does not imply that the JIF ranking has any property of uniqueness, or of optimisation [35].

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Popli, P., Shenoy, S.R. Unified citation parameters for journals and individuals: Beyond the journal impact factor or the *h*-index alone.
*Pramana - J Phys* **96**, 189 (2022). https://doi.org/10.1007/s12043-022-02413-z

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DOI: https://doi.org/10.1007/s12043-022-02413-z