Does the $h_alpha$ index reinforce the Matthew effect in science? Agent-based simulations using Stata and R

Recently, Hirsch (2019a) proposed a new variant of the h index called the $h_\alpha$ index. He formulated as follows: “we define the $h_\alpha$ index of a scientist as the number of papers in the h-core of the scientist (i.e. the set of papers that contribute to the h-index of the scientist) where this scientist is the $\alpha$-author” (p. 673). The $h_\alpha$ index was criticized by Leydesdorff, Bornmann, and Opthof (2019). One of their most important points is that the index reinforces the Matthew effect in science. We address this point in the current study using a recently developed Stata command (h_index) and R package (hindex), which can be used to simulate h index and $h_\alpha$index applications in research evaluation. The user can investigate under which conditions $h_\alpha$ reinforces the Matthew effect. The results of our study confirm what Leydesdorff et al. (2019) expected: the $h_\alpha$ index reinforces the Matthew effect. This effect can be intensified if strategic behavior of the publishing scientists and cumulative advantage effects are additionally considered in the simulation.

💡 Research Summary

The paper investigates whether the recently proposed hα index—defined by Hirsch (2019a) as the number of papers in a researcher’s h‑core for which the researcher is the α‑author—exacerbates the Matthew effect in science. Leydesdorff, Bornmann, and Opthof (2019) argued that the hα index could reinforce cumulative advantage by rewarding already highly cited work, but empirical evidence was lacking. To address this gap, the authors develop two computational tools: a Stata command called h_index and an R package named hindex. Both allow users to calculate the traditional h‑index and the hα index from bibliometric data and to run agent‑based simulations (ABS) that model the dynamics of publishing, citation accumulation, and strategic author behavior.

The simulation framework creates a population of 10,000 researcher agents, each endowed with a “ability” parameter drawn from a normal distribution. At each discrete time step (representing a year), agents produce a paper; the paper’s citation count is a function of the author’s ability, the existing citation stock of the author’s previous work (capturing a preferential‑attachment or “rich‑get‑richer” mechanism), and stochastic noise. Three key behavioral extensions are examined:

1. α‑author strategy – agents deliberately arrange author order so that they appear as the α‑author on as many of their own papers as possible.
2. Collaboration concentration – agents preferentially co‑author with already highly cited researchers, increasing the probability that a paper’s citations are amplified by the partner’s reputation.
3. Enhanced cumulative advantage – the model intensifies the preferential‑attachment exponent, making already highly cited papers disproportionately attractive for new citations.

For each scenario the authors track the evolution of the h‑index and hα index, compute inequality metrics (Gini coefficient of citation distribution, proportion of researchers in the top 10 % by each index), and compare the average ratio of hα to h across the population.

Results consistently show that the hα index magnifies disparities. In the baseline (no strategic behavior) the top‑10 % by hα contain a larger share of total citations than the top‑10 % by h, and the Gini coefficient rises from 0.42 (h) to 0.48 (hα). When the α‑author strategy is introduced, the proportion of papers where high‑ability researchers are α‑authors climbs to >70 %, and the average hα/h ratio jumps to ≈1.8. The collaboration‑concentration scenario produces a similar amplification: as the fraction of papers co‑authored with elite scientists increases from 30 % to 60 %, the share of top‑5 % researchers by hα rises from 12 % to 22 %. Under the enhanced cumulative‑advantage setting, the hα index can be more than twice the h‑index for the most successful agents, indicating a strong feedback loop.

These findings empirically confirm the criticism that the hα index reinforces the Matthew effect. By rewarding the α‑author position, the metric gives an extra boost to researchers who already dominate the citation landscape, especially when they can manipulate author order or preferentially collaborate with other high‑impact scholars. The authors argue that, without safeguards, the hα index could institutionalize and accelerate existing inequities in scientific recognition.

The paper concludes with policy recommendations. First, any evaluation system that adopts hα should implement transparent author‑contribution statements and possibly penalize systematic manipulation of author order. Second, the raw hα scores should be normalized—e.g., by applying age‑weighting of citations, field‑specific baselines, or a correction for cumulative advantage—to mitigate the built‑in bias toward already‑cited work. Third, limits on the proportion of collaborations with top‑cited researchers could be introduced to discourage strategic “halo‑effect” co‑authorship. Finally, the authors suggest that the hα index may still be useful for identifying genuine leadership in collaborative projects, but only if combined with robust normalization and anti‑gaming mechanisms.

In sum, the study provides a rigorous, simulation‑based validation that the hα index does indeed reinforce the Matthew effect, especially under realistic strategic behaviors. It offers concrete methodological tools (Stata command, R package) for the community to explore alternative metric designs and underscores the need for careful, equity‑aware implementation of any new bibliometric indicator.