Viva the h-index

In their article ‘The inconsistency of the h-index’ Ludo Waltman and Nees Jan van Neck give three examples to demonstrate the inconsistency of the h-index. As will be explained, a little extension of their examples just illustrate the opposite, a stable feature of the h-index. For starting authors it, the h-index that is, focusses on the number of articles; for experienced authors its focus shifts towards the citation scores. This feature may be liked or not but does not make the h-index an inconsistent and inappropriate indicator, as the authors claim.

💡 Research Summary

The paper is a rebuttal to Waltman and van Eck’s claim that the h‑index is inconsistent. The author introduces a simple production‑quality framework: for any researcher, let p be the number of “relevant” publications (those that constitute the h‑index) and let q be the lowest citation count among those publications. Within this definition the h‑index is exactly the minimum of the two numbers, h = min(p, q). This relationship, termed the “h‑index law,” yields only two possible regimes. When production is lower than quality (p < q) the h‑index is limited by the number of papers; when production equals quality (p = q) the index is limited by the citation floor. The case p > q cannot occur because p is defined only over the set of papers that already satisfy the citation threshold.

Armed with this law, the author revisits the three examples used by Waltman and van Eck to illustrate alleged inconsistencies.

Example 1 compares two authors, X and Y, over successive five‑year periods. Initially X has p = 9, q = 12 while Y has p = 7, q = 15, giving h‑values of 9 and 7 respectively. After a second identical period both authors reach p = q (12 for X, 15 for Y) and their h‑indices become 12 and 15. Further identical output does not change the indices because the minimum of p and q is already fixed. The trajectory (9 → 12 → 14 → 14 for X and 7 → 14 → 15 → 15 for Y) perfectly follows the h‑index law: early on production determines h, later quality takes over.

Example 2 involves two researchers who repeatedly co‑author two papers that each receive eight citations. Initially X has p = 5, q = 5 (h = 5) and Y has p = 4, q = 6 (h = 4). After the first joint step X’s h stays at 5 while Y’s rises to 6; after the second step both reach h = 6, and after the third step both reach h = 8. Beyond that, additional identical steps have no effect. Again the early advantage of X is due to higher production, while Y’s superior citation floor later overtakes, illustrating the same law in action.

Example 3 addresses group indices. Two authors X₁ and X₂ each have seven papers with nine citations (individual h = 7). When combined into a single “author” the group has fourteen papers, each still cited nine times, yielding a group h‑index of 9. Thus the group h‑index is not the sum of the individual h‑indices; it lies between the maximum individual h and the sum, as formalised by the inequality max(hᵢ) ≤ h_G ≤ Σhᵢ. The author points out that many very different compositions can produce the same group h‑index, suggesting that the average h‑index of members may be a more informative metric for groups or journals.

The conclusion reiterates that all three examples conform to the h‑index law: quality eventually dominates production, a built‑in feature of the metric. Consequently, the h‑index rewards junior researchers for the sheer number of papers while preventing senior researchers from inflating their score by repeatedly publishing papers of the same modest impact. The alleged “inconsistency” is therefore a misinterpretation of the index’s dynamic nature rather than a flaw. The author ends with a metaphor comparing the h‑index to child’s first steps versus an adult’s tango, emphasizing that the index’s behavior is both predictable and appropriate.