Evolutionary Biological Data Science Laboratory

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 Range	ln (lower limit) (Tc)	Cumulative Sum of Tc (Vc)
0	0–4	0.0000	0.0000
1	5–9	1.6094	1.6094
2	10–19	2.3026	3.9120
3	20–39	2.9957	6.9078
4	40–79	3.6889	10.5966
5	80–159	4.3820	14.9787
6	160–319	5.0752	20.0538
7	320–639	5.7683	25.8222
8	640–1279	6.4615	32.2836
9	1280–2559	7.1546	39.4382
10	2560–5119	7.8478	47.2860
etc.

To calculate the metric, for each of the c' classes, one can count the number of publications within the c^th class, X_c. Skewed distributions are often normalized using a logarithmic transform. Therefore, one calculates the natural logarithm of the lower limit of each class as T_c = ln(B_c) where B_c is the lower limit of the c^th class. One can also calculate V_c as the cumulative sum of T_c 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)	57	26	16	12	11	10	4	3	2	1	1	1	1	0	0	0	0	0

Class (c)	Citation Range	Count of Pubs in Range (Xc)	ln (lower limit) (Tc)	Cumulative Sum of Tc (Vc)	XcVc
0	0–4	12	0.0000	0.0000	0.0000
1	5–9	0	1.6094	1.6094	0.0000
2	10–19	4	2.3026	3.9120	15.6481
3	20–39	1	2.9957	6.9078	6.9078
4	40–79	1	3.6889	10.5966	10.5966

The index is the sum of X_cV_c, thus w = 33.1525.

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