Weighted Hir sch index

Institute of Information Technology of ANAS

Rahila Hasanova

23.05.2013

"New Challenges in the European Area: International Baku Forum of Young Scientists

A scientist has index h if h of his or her Np papers have at least h citations each and the other (Np – h) papers have ≤h citations each.

2

first h papers

More than h citations

Papers

citations= papers= h

J.E.Hirsch, An index to quantify an individual’s scientific research output // Proceedings of the National Academy of Sciences of the USA, 2005, vol.102, no.46, pp.16569-16572.

Quantity (publications) and impact (citations). Awarding prizes Single-number criteria Easy to understand

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Differences among fields. Scientist's career.. Citation to journal articles… Decrease…. Highly cited papers….. Self-citations…..

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Solomon H. Snyder: h = 191

 Hector Garcia-Molina: h = 88 

Suppose that, the scientific researcher has 9 papers and these papers received 20, 15, 7, 5, 5, 5, 4, 2, 2 citations accordingly.

5

Articles’ rank of the researcher

1 2 3 4 5 6 7 8 9

Number of citations

20 15 7 5 5 5 4 2 2

Hirsch core

h = 5

The highest rank such that the top g papers have, together, at least g2 citations. This also means that the top g+1 have less than (g+1)2 papers.

Example. Suppose that, the scientific researcher has 9 papers and these papers received 20, 15, 7, 5, 5, 5, 4, 2, 2 citations accordingly.

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Let's compare his h-and g-indices.

Articles’ rank of a researcher

1 2 3 4 5 6 7 8 9

Number of citations

20 15 7 5 5 5 4 2 2

L.Egghe, Theory and practice of the g-index // Scientometrics, 2006, vol.69, no.1, pp.131-152.

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Citations for each paper

The rank of the paper

Sum of citations

The square rank of the

paper

20 1 20 1

15 2 35 4

7 3 42 9

5 4 47 16

5 5 52 25

5 6 57 36

4 7 61 49

2 8 63 64

2 9 65 81

h-core

g-core

h = 5, g = 7

m-index the median number of citations received by papers in the Hirsch core.

Example. Suppose that, the scientific researcher has 8 papers and these papers received 23, 20, 19, 12, 7, 4, 3 and 1 citations accordingly.

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Articles’ rank of the researcher 1 2 3 4 5 6 7 8

Number of citations 23 20 19 12 7 4 3 1

h = 5

m = 19

Papers including to Hirsch core

Q.L.Bornmann, R.Mutz, and H.D.Daniel, Are there better indices for evaluation purposes than the h-index using data from biomedicine // Journal of the American Society for Information Science and Technology, 2008, vol.59, no.5, pp.830-837.

Arithmetic mean

9

2

22 mhRhm

Root mean square

Geometric mean

mh

hmHhm

2

Harmonic mean

hmGhm

2

mhAhm

*Aliguliyev, R., Hasanova, R., The evaluation of the scientific output of researchers / The 3rd International Conference “Problems of Cybernetics and Informatics”, 6-8 Sept. 2010, Baku: http://www.pci2010.science.az/1/35.pdf **Jipa S., Gorghiu L.M., Dumitrescu C., Oros C. Research output new evaluation of chemistry group in Valahia university with the use of various bibliometric indicators // Journal of Science and Arts, 2012, no. 3, vol. 20, pp. 335-342. http://www.icstm.ro/DOCS/josa/josa_2012_3/b_06_Silviu_Jipa.pdf

For compare researchers with the same h-index, to the number of citations coming to each paper in Hirsch core, added arithmetic mean of authors’ h-indexes (who cited these researchers’ papers) as weighted coefficient.

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Assume that,

I scientific researcher has 6 papers and these papers received citations accordingly.

II scientific researcher has 6 papers and these papers received 6, 3, 2, 0, 0, 0 citations accordingly.

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For the first researcher h=2

For the second researcher h=2

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Rank I researcher Rank II researcher

1 <5 1 <6

2 <4 * 2 <3 *

3 >2 3 >2

4 >1 4 >1

5 >1 5 >1

6 >0 6 >1

Let’s sort number of citations coming to the papers of both researchers in decreasing order:

*Here the power of citations coming to the papers is not taken account (for ex., h-index of citation’s author, impact factor of the journal from where citation coming and etc.).

For I researcher h=2, g=3, m=4.5

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Number of citations

RankSum of

citationsSquare

rank

5 1 5 1

4 2 9 4

2 3 11 9

1 4 12 16

1 5 13 25

0 6 13 36

Let’s sort number of citations coming to the papers of both researchers in decreasing order :

*Here the power of citations coming to the papers is not taken account (for ex., h-index of citation’s author, impact factor of the journal from where citation coming and etc.).

For II researcher h=2, g=3, m=4.5

Number of citations

RankSum of

citationsSquare

rank

6 1 6 1

3 2 9 4

2 3 11 9

1 4 12 16

1 5 13 25

1 6 14 36

Here,

For the first researcher’s h-index his the first and the second paper gives 1 point each. So,

h(I) = 1 (for number of citations coming to the first paper) + 1 (for number of citations coming to the second paper) = 2

For the second researcher’s h-index his the first and the second paper gives 1 point each. So,

h(II) = 1 (for number of citations coming to the first paper) + 1 (for number of citations coming to the second paper) = 2

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Number of citation of the first paper for I researcher is 5,

number of citations of the second paper is 4.

I paper

I citation

II citation

III citation

IV citation

V citation

II paper

I citation

II citation

III citation

IV citation

*Here number of citations >= 2, that is way they are taken into account.

h=2

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H-index of authors of citations coming to the first paper of I researcher is equal to 4, 3, 2, 0, 0 accordingly.

H-index of authors of citations coming to the second paper of I researcher is equal to 7, 6, 1, 1 accordingly.

I paper

I citation

II citation

III citation

IV citation

V citation

II paper

I citation

II citation

III citation

IV citation

h=4 h=3 h=2

h=0 h=0

h=7 h=6

h=1 h=1

*Here h-index of each citation’s author is taken account. Thus, if any of citations has 2 or more co-authors, then h-index is calculated as arithmetic mean of these authors’ h-index. Self-citation is calculated as h=0.

Normally, the first and the second papers give 1 point each to h-index of the I researcher. Here we include weighted coefficients to these points:

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55.54

11671

5

002341

Ih

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Number of citation of the first paper for II researcher is 6,

number of citations of the second paper is 3.

I paper

I citation

II citation

III citation

IV citation

VI citation

II paper

II citation

III citation

V citation

I citation

*Here number of citations >= 2, that is way they are taken into account.

h=2

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H-index of authors of citations coming to the first paper of II researcher is equal to 6, 6, 1, 1, 0, 0 accordingly.

H-index of authors of citations coming to the second paper of II researcher is equal to 1, 1, 0 accordingly.

I paper

I citation

II citation

III citation

IV citation

VI citation

II paper

II citation

III citation

V citation

I citation

h=6 h=6 h=1

h=1

h=0h=0

h=1

h=1

h=0

*Here h-index of the citation’s author is taken account. Thus, if any of citations has 2 or more co-authors, then h-index is calculated as arithmetic mean of these authors’ h-index. Self-citation is calculated as h=0.

Normally, the first and the second papers give 1 point each to h-index of the II researcher. Here we include weighted coefficients to these points:

20

33.33

0111

6

0011661

IIh

*As seems from example, h(I)>h(II). This means that, the power of citations of the I researcher’s papers is higher than the power of citations of the II researcher’ papers.

h(I) = 5.55

h(II) = 3.33

Institute of Information Technology of ANAS

“Weighted Hirsch index”, Rahila Hasanova

23.05.2013