← Back to introduction

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 Ci ≤ 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 ci is the number of self-citations to the ith 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)592616111110433211100000
Self Citations (ci)210203321110000000
Rank (i)123456789101112131415161718
h = 6
ai000000001112334567
qi0000001/21/31/31/41/51/51/51/61/61/61/61/6
ciqi0.0000.0000.0000.0000.0000.0001.5000.6670.3330.2500.2000.0000.0000.0000.0000.0000.0000.000

The sum of ciqi is 2.9500, thus Q is 0.1639.

History

YearQ
19970.3333
19980.0625
19990.1111
20000.0952
20010.1639
20020.3553
20030.2096
20040.1876
20050.1604
20060.1784
20070.2736
20080.1356
20090.1536
20100.0944
20110.0782
20120.0438
20130.0790
20140.0610
20150.0572
20160.0536
20170.0841
20180.0803
20190.0737
20200.0442
20210.0361
20220.0615
20230.0275
20240.0442
20250.0339

References