Evolutionary Biological Data Science Laboratory

q-index

The q-index (Bartneck and Kokkelmans 2011) is an attempt to measure whether an author is manipulating their own h-index through self-citation. It calculates a potential cost of a self-citation for each publication whose citation count is equal or less than the h, sums these costs for all self-citations, then normalizes by the total number of publications. Essentially, publications with a citation count equal to h generate a cost of 1 per self-citation, with each subsequent publication in rank order generating costs of 1/2, 1/3, 1/4, etc., with ties generating equal values. The potential costs are calculated as:

$$q_i=\left|\begin{matrix} 0 & \text{if }C_i > h \text{ or } i < h \\ \frac{1}{i+1-a_i-h} & \text{if }C_i \leq h \text{ and } i \geq h \end{matrix}\right.,$$

where

$$a_i=\left|\begin{matrix} 0 & \text{if }i \leq h \\ a_{i-1} & \text{if }i>h \text{ and } C_i \neq C_{i-1}\\ 1+a_{i-1} & \text{if }i>h \text{ and } C_i = C_{i-1} \end{matrix}\right. .$$

The total self-citations for any publication where C_i ≤ h are multiplied by the associated cost and these values are summed and averaged for all publications.

$$Q=\frac{1}{P}\sum\limits_{i=1}^{P}{c_i q_i}.$$

where c_i is the number of self-citations to the i^th publication.

Simulations indicate that selective citing to boost h may lead to a q-index, greater than one, while lower values are found when self-citation is random or fair.

Example

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

Citations (Ci)	47	26	19	15	11	10	4	3	2	1	1	1	1	0	0	0	0	0
Self Citations (ci)	2	1	0	0	2	3	3	2	1	1	1	0	0	0	0	0	0	0
Rank (i)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
						h = 6
ai	0	0	0	0	0	0	0	0	0	0	1	2	3	3	4	5	6	7
qi	0	0	0	0	0	0	1/2	1/3	1/4	1/5	1/5	1/5	1/5	1/6	1/6	1/6	1/6	1/6
ciqi	0.000	0.000	0.000	0.000	0.000	0.000	1.500	0.667	0.250	0.200	0.200	0.000	0.000	0.000	0.000	0.000	0.000	0.000

The sum of c_iq_i is 2.8167, thus Q is 0.1565.

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