Normalizing the g-index

Abstract

Recently, a normalized version of the g-index has been proposed as an impact and concentration measure: the so-called s measure. The main problem with this measure is that—somewhat paradoxically—it may be that s = 1 even in the case in which citations are perfectly evenly spread across articles. We prove that the measure s can and should be improved by a different choice of a normalizing term. This is done by defining the latter as a function of the citation count of the single most cited paper. The new index presented here does not suffer from insensitivity to citation transfers within the g-core.

Appendix: If we add a “fictitious” publications with 0 citations, the concentration increases

Let X ^* = (x ^*₁ , x ^*₂ , …, x ^* _T+1 ) be the (T + 1)-dimensional vector (x ₁, x ₂, …, x _T , 0) (i.e. the (vector resulting from the T-dimensional vector X by adding an article with 0 citations). Since, by construction, x ^*₍₁₎  = 0, x ^*_(i)  = x _(i−1), i = 2, …, T + 1, and Q ^* _i  = Q _i−1, i = 1, …, T + 1. Then

$$ L_{{X^{*} }} \left( t \right) = Q_{i - 1} - q_{i - 1} i + q_{i - 1} \left( {T + 1} \right)t, $$

for every $t \in \left[ {\frac{i - 1}{T + 1},\frac{i}{T + 1}} \right]$, i = 1, …, T + 1. We can prove that $L_{{X^{*} }} \left( t \right) \le L_{X} \left( t \right)$, for the generic point t, t ∊ [F _i−1, F _i ], i = 1, …, T, with a strict inequality at least for some value of t. Since $F_{i - 1} < \frac{i}{T + 1} < F_{i}$, we can distinguish two cases: (a) $F_{i - 1} \le t \le \frac{i}{T + 1}$ and (b) $\frac{i}{T + 1} \le t \le F_{i}$.

• (a) In this case, we have to prove the inequality

  $$ Q_{i - 1} - q_{i - 1} i + q_{i - 1} \left( {T + 1} \right)t \le Q_{i} - q_{i} i + q_{i} Tt, $$

  that is equivalent to

  $$ x_{{\left( {i - 1} \right)}} \left( {t\left( {T + 1} \right) - i} \right) \le x_{(i)} \left( {1 - i + Tt} \right). $$

  Now, since $t \le \frac{i}{T + 1}$, we have x _(i−1)(t(T + 1) − i) ≤ 0. Besides, since $\frac{i - 1}{T} \le t$, we have x _(i)(1 − i + Tt) ≥ 0. Hence the inequality is proved for case (a).

• (b) In this case, we have to prove the inequality

  $$ Q_{i} - q_{i} i + q_{i} \left( {T + 1} \right)t \le Q_{i} - q_{i} i + q_{i} Tt, $$

  that is equivalent to

  $$ - i - 1 + Tt + t \le - i + Tt, $$

  which is clearly fulfilled, with strict inequality for every t in the interval $\left( {\frac{T}{T + 1},1} \right)$. Hence our thesis is proved.

  More generally, on adding a set of k ≥ 1 fictitious publications with 0 citations to a non-null T-dimensional vector, we always increase the concentration and, as k tends to infinity, each well-defined normalized measure of concentration tends to 1.