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iterative weighted EM′-index

The iterative weighted EM′-index (Bihari et al. 2021) is a modification of the EM′-index (Bihari and Tripathi 2017) which uses a weighted-sum of each successive element in the vector rather than the square-root of the sum. It is an extension of the iterative weighted EM-index which includes all publications, rather than just those from the core. We begin by creating a vector (E) where the first value is E1 = h. Subsequent values of the vector, Ei+1, are determined by subtracting Ei from the citation count for all publications in the core defined by Ei, and recalculating h from these new citation counts, reranking all publications by these new citation counts as necessary (i.e., some of the publications previously in the tail of the citation distribution may advance beyond publications in the core as citations representing earlier calculations of h are “used up”). This process continues until one runs out of citations, all of the remaining publications have only a single remaining citation, or there is only a single publication left to be considered. From this vector, one calculates the index as:

$$iw_{EM^\prime}= \sum\limits_{i=1}^{n}\frac{E_i}{i},$$

where Ei and n are the ith element and length of E, respectively.

Example

Publications are ordered by number of citations, from highest to lowest. After each step, Ei is subtracted from the citations of the top Ei publications and all publications are re-ranked by this adjusted citation count for the next step.

Citations (Ci)592616111110433211100000
Rank (i)123456789101112131415161718
E1 = 6
Adjusted Citations (Ci)532010554433211100000
New Rank (i)123456789101112131415161718
E2 = 5
Adjusted Citations (Ci)48155443321110000000
New Rank (i)123456789101112131415161718
E3 = 4
Adjusted Citations (Ci)44114332111100000000
New Rank (i)123456789101112131415161718
E4 = 3
Adjusted Citations (Ci)4183321111100000000
New Rank (i)123456789101112131415161718
E5 = 3
Adjusted Citations (Ci)3853211111000000000
New Rank (i)123456789101112131415161718
E6 = 3
Adjusted Citations (Ci)3522111110000000000
New Rank (i)123456789101112131415161718
E7 = 2
Adjusted Citations (Ci)3321111100000000000
New Rank (i)123456789101112131415161718
E8 = 2
Adjusted Citations (Ci)3111111000000000000
New Rank (i)123456789101112131415161718
E9 = 1
Adjusted Citations (Ci)3011111000000000000
New Rank (i)123456789101112131415161718
E10 = 1
Adjusted Citations (Ci)2911111000000000000
New Rank (i)123456789101112131415161718
E11 = 1
Adjusted Citations (Ci)2811111000000000000
New Rank (i)123456789101112131415161718
E12 = 1
Adjusted Citations (Ci)2711111000000000000
New Rank (i)123456789101112131415161718
E13 = 1
Adjusted Citations (Ci)2611111000000000000
New Rank (i)123456789101112131415161718
E14 = 1
Adjusted Citations (Ci)2511111000000000000
New Rank (i)123456789101112131415161718
E15 = 1
Adjusted Citations (Ci)2411111000000000000
New Rank (i)123456789101112131415161718
E16 = 1
Adjusted Citations (Ci)2311111000000000000
New Rank (i)123456789101112131415161718
E17 = 1
Adjusted Citations (Ci)2211111000000000000
New Rank (i)123456789101112131415161718
E18 = 1
Adjusted Citations (Ci)2111111000000000000
New Rank (i)123456789101112131415161718
E19 = 1
Adjusted Citations (Ci)2011111000000000000
New Rank (i)123456789101112131415161718
E20 = 1
Adjusted Citations (Ci)1911111000000000000
New Rank (i)123456789101112131415161718
E21 = 1
Adjusted Citations (Ci)1811111000000000000
New Rank (i)123456789101112131415161718
E22 = 1
Adjusted Citations (Ci)1711111000000000000
New Rank (i)123456789101112131415161718
E23 = 1
Adjusted Citations (Ci)1611111000000000000
New Rank (i)123456789101112131415161718
E24 = 1
Adjusted Citations (Ci)1511111000000000000
New Rank (i)123456789101112131415161718
E25 = 1
Adjusted Citations (Ci)1411111000000000000
New Rank (i)123456789101112131415161718
E26 = 1
Adjusted Citations (Ci)1311111000000000000
New Rank (i)123456789101112131415161718
E27 = 1
Adjusted Citations (Ci)1211111000000000000
New Rank (i)123456789101112131415161718
E28 = 1
Adjusted Citations (Ci)1111111000000000000
New Rank (i)123456789101112131415161718
E29 = 1
Adjusted Citations (Ci)1011111000000000000
New Rank (i)123456789101112131415161718
E30 = 1
Adjusted Citations (Ci)911111000000000000
New Rank (i)123456789101112131415161718
E31 = 1
Adjusted Citations (Ci)811111000000000000
New Rank (i)123456789101112131415161718
E32 = 1
Adjusted Citations (Ci)711111000000000000
New Rank (i)123456789101112131415161718
E33 = 1
Adjusted Citations (Ci)611111000000000000
New Rank (i)123456789101112131415161718
E34 = 1
Adjusted Citations (Ci)511111000000000000
New Rank (i)123456789101112131415161718
E35 = 1
Adjusted Citations (Ci)411111000000000000
New Rank (i)123456789101112131415161718
E36 = 1
Adjusted Citations (Ci)311111000000000000
New Rank (i)123456789101112131415161718
E37 = 1
Adjusted Citations (Ci)211111000000000000
New Rank (i)123456789101112131415161718
E38 = 1
Adjusted Citations (Ci)111111000000000000
New Rank (i)123456789101112131415161718
E39 = 1

iwEM′ is the sum of each component of E weighted by it's order, thus iwEM′ = 6/1 + 5/2 + 4/3 + 3/4 + 3/5 + 3/6 + 2/7 + 2/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 + 1/17 + 1/18 + 1/19 + 1/20 + 1/21 + 1/22 + 1/23 + 1/24 + 1/25 + 1/26 + 1/27 + 1/28 + 1/29 + 1/30 + 1/31 + 1/32 + 1/33 + 1/34 + 1/35 + 1/36 + 1/37 + 1/38 + 1/39 = 13.7547

History

YeariwEM′
19971.0000
19984.5833
19996.8488
200010.7407
200113.7547
200218.2751
200322.7329
200428.7209
200535.3918
200640.7205
200747.3183
200852.9595
200958.4901
201063.2383
201169.9456
201276.2807
201381.0804
201484.8290
201588.4404
201691.3405
201794.6274
201897.6020
2019100.7649
2020103.9045
2021107.1883
2022110.4983
2023112.4298
2024115.1594
2025117.2593

References