Source-linked AI summary
A modification of the h-index: the hm-index accounts for multi-authored manuscripts
Michael Schreiber
TL;DR
Because the h-index does not account for multiple authorship, the paper evaluates a fractional publication-counting modification called the hm-index. Across fictitious and empirical examples, it finds this approach preferable to fractional citation counting and h-index normalization by average h-core authorship.
Problem
The h-index does not account for multiple authorship, motivating a measure that incorporates co-authorship appropriately.
Method
The paper evaluates the hm-index by fractionalising publication counts across three fictitious model cases and one empirical dataset.
Results
The hm-index is found preferable to fractionalised citation counting and normalizing the h-index by mean h-core authorship, which excessively reduces the index.
Takeaways & Limitations
The hm-index appears to be a fair way to account for multiple authorship when assessing researchers.
Takeaways & Limitations
Fractional citation counting is impractical because it requires rearranging the papers.
Abstract
from arXiv · showhide
In order to take multiple co-authorship appropriately into account, a straightforward modification of the Hirsch index was recently proposed. Fractionalised counting of the papers yields an appropriate measure which is called the hm-index. The effect of this procedure is compared in the present work with other variants of the h-index and found to be superior to the fractionalised counting of citations and to the normalization of the h-index with the average number of authors in the h-core. Three fictitious examples for model cases and one empirical case are analysed.
1. Introduction
The introduction proposes the hm-index, which accounts for multiple authorship through fractionalized paper counting. It argues that this modification avoids problems of citation fractionalization and average-author normalization, making it more appropriate for the Hirsch index.
- 1. Introduction: The h-index measures a researcher’s impact as the highest number of papers cited at least h times but does not account for multiple co-authorship.
- 1. Introduction: Normalizing h by the average number of authors can disadvantage researchers with highly co-authored papers and can even decrease when such a paper gains citations.
- 1. Introduction: Fractionalized citation counting can exclude highly cited multi-authored papers from the core because publications must be reordered by citations per author.
- 1. Introduction: The hm-index uses fractionalized paper counts to define a reduced or effective rank of papers cited at least hm times, while preserving the citation-based core.The reduced number of papers is interpreted as an effective rank, with other papers cited no more than hm times.
- 1. Introduction: The manuscript compares the hm-index with other Hirsch-index variants using three fictitious model cases and an empirical citation record, arguing that it is more appropriate for multiple authorship.
2. Three fictitious examples for model cases
Three constructed examples show how fractional counting of publications produces hm-index values that differ from hI- and hf-index values. Across extreme cases, hm incorporates additional papers and remains stable under ranking ambiguities that can substantially alter hI.
- 2. Three fictitious examples for model cases: In the first example, fractional publication counting gives hm = 3, adding two papers to the hm-core.The corresponding indices are h = 5, hI = 2.08, and hf = 4.
- 2. Three fictitious examples for model cases: In the second extreme example, hm = 7 includes all 11 papers, whereas hf = 4 derives only from the last four papers.Papers beyond rank 11 do not contribute to hm.
- 2. Three fictitious examples for model cases: In the third example, the indices are h = 8, hf = 8, hm = 6, and hI = 5.33.The first eight papers contribute to h, hf, and hm, while the ninth and later papers do not initially contribute.
- 2. Three fictitious examples for model cases: Increasing the ninth paper’s citations to c = 7 makes it contribute slightly to hm, while c = 8 leaves all indices unchanged.Its contribution remains small because the paper has many authors.
3. An empirical example
In the empirical citation record, fractional paper counting reduced the Hirsch index from h = 28 to hm = 18.48, while author-number normalization produced a much smaller hI = 9.69. The hf-index was 18 but required substantially more homograph checking than hm.
- An empirical example: Fractional paper counting yielded hm = 18.48 from h = 28, with the hm-core expanding to 43 publications.The reduced numbers for the 28 papers in the h-core gave reff = 11.53 before the hm-core was determined.
- An empirical example: The hI-index’s extreme reduction was mitigated by square-root normalization in the hP-index, although the same problems remained in weaker form.The hP-index evaluates an author’s “pure” contribution when papers are counted fractionally.
- An empirical example: Dividing each paper’s citations by its author count yielded hf = 18, but the rearrangement placed a paper originally ranked r = 44 in the hf-core.This ranking change made the precision problem more severe than for the h-index and hm-index.
- An empirical example: For hf, all 49 papers with at least 18 citations potentially required checking, and some checks proved unnecessary only after rearrangement.This increased the precision problem substantially compared with hm.
4. Further discussion and summary
The hm-index fairly accounts for multiple authorship by fractionalizing publications without rearranging citation records, while enabling straightforward aggregation across researchers. It is preferred to alternative variants, although broader empirical validation and caution against single-number evaluation remain necessary.
- Summary: Fractionalizing publications preserves paper order and permits direct hm-index determination, avoiding unnecessary homograph checks despite additional papers entering the hm-core.The method counts each paper according to the inverse of its number of authors.
- Summary: The hm-index is presented as the fairest way to account for multiple authorship when individual co-author contributions are unknown.Its calculation requires somewhat more effort than the h-index, but the authors consider that effort worthwhile.
- Limitations: Further empirical testing across research fields is needed before using the hm-index for evaluation, and scientific achievement should not be measured by a single number.Excluding self-citations is possible but would require a substantially larger database, so that analysis is left for future work.
- Summary: The hm-index avoids citation-record rearrangement, sensitivity to extreme co-author counts, decreases with increasing citations, and displacement of highly cited papers from the core.These properties distinguish it from the hf-index and hI-index.
- Summary: The hm-index enables straightforward aggregation of data sets across several people, counting a jointly authored paper fractionally for each researcher rather than fully for each h-index.A jointly authored paper contributes two times one half to the researchers’ hm-indices when sufficiently cited.
number of citations
The section compares citation-count-based indices using publication-level and effective-rank representations. Figures and fictitious datasets illustrate how fractionalized citations and author-adjusted ranks determine index contributions.
- number of citations: The citation profile of M. Schreiber’s 62 most-cited publications reaches h = 28, while effective-rank histograms compress the papers toward the left.Figure 1 contrasts original citation counts by rank with effective-rank representations for the h-, hm-, and hI-indices.
- number of citations: Fractionalized citation counts divide each publication’s citations by its number of authors, producing an hf-index of hf = 18 in the illustrated dataset.The rearranged quotients are shown in the lower histogram after sorting them in decreasing order.
- number of citations: The fictitious model datasets calculate hf from c(r)/a(r) and hm from the effective rank reff(r) = reff(r-1)+1/a(r).Boldface identifies the papers contributing to the h-, hf-, and hm-indices in the first model dataset.
- number of citations: Tables 2 and 3 repeat the model-data analysis for two additional fictitious publication datasets.Both tables use the same format as Table 1.
- number of citations: Table 4 varies the citation count of the last paper in Table 3 and reports the resulting effective rank and index values.Boldface marks when the changed paper contributes to each respective index.