# Michael S. Rosenberg’s Laboratory

Computational Evolutionary Biology & Bioinformatics

E-mail: msr@asu.edu
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## w-index (Wohlin)

Wohlin's w-index (Wohlin 2009) is similar to others that try to address the issue where not all citations are included in the h-index and that many different distributions of citations can have identical h-indices. Unlike the other indices, however, it does not start with h, and instead uses a somewhat complicated procedure of dividing papers into classes based on the number of citations. Rather than give publications more weight for every citation, this index gives more weight to citations as the publication moves from one class to the next. Publications with fewer than five citations are ignored (given a weight of 0). The first class represents publications with 5-9 citations; each subsequent class has a width double that of the previous class, thus the second class represents 10-19 citations, the third class 20-39 citations, etc. This structure was chosen (other classification schemes could be substituted) because citations curves are usually skewed with many publications with relatively smaller numbers of citations, and few publications with relative large numbers of citations.

Class
(c)
Citation Rangeln (lower limit)
(Tc)
Cumulative
Sum of Tc
(Vc)
00–40.00000.0000
15–91.60941.6094
210–192.30263.9120
320–392.99576.9078
440–793.688910.5966
580–1594.382014.9787
6160–3195.075220.0538
7320–6395.768325.8222
8640–12796.461532.2836
91280–25597.154639.4382
102560–51197.847847.2860
etc.

To calculate the metric, for each of the c' classes, one can count the number of publications within the cth class, Xc. Skewed distributions are often normalized using a logarithmic transform. Therefore, one calculates the natural logarithm of the lower limit of each class as Tc = ln(Bc) where Bc is the lower limit of the cth class. One can also calculate Vc as the cumulative sum of Tc for all classes from 1 to c. The w-index is then calculated as

$$w=\sum\limits_{c=1}^{c^\prime}{X_c V_c}$$

The w-index increases as a publication moves from one class to the next. Moving between larger classes gives more weight than moving between smaller classes. Because it considers citations more broadly, the w-index is more fine-grained than the h-index.

### Example

Publications are ordered by number of citations, from highest to lowest.

 Citations (Ci) 42 36 14 11 9 9 3 2 2 2 1 1 1 0 0 0

Class
(c)
Citation
Range
Count of Pubs
in Range
(Xc)
ln (lower limit)
(Tc)
Cumulative
Sum of Tc
(Vc)
XcVc
00–4100.00000.00000.0000
15–921.60941.60943.2189
210–1922.30263.91207.8240
320–3912.99576.90786.9078
440–7913.688910.596610.5966

The index is the sum of XcVc, thus w = 28.5473.

Yearw
19970.0000
19981.6094
19995.5215
200017.2575
200128.5473
200249.0474
200364.6955
2004105.0026
2005140.4814
2006180.5654
2007224.8082
2008251.2998
2009297.8689
2010340.7255
2011376.6743
2012411.0136
2013440.7478
2014469.3425
2015501.1561
2016525.1220
2017549.3347

## References

• Wohlin, C. (2009) A new index for the citation curve of researchers. Scientometrics 81(2):521–533.