REVIEW 1 major objections 6 minor 13 references
Mathematical reflections on modified fractional counting
T0 review · 1 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Every MFC$_k$ value for an author is the weighted geometric average of complete-normalized fractional counting and full counting, with the weight determined by $k$ and no dependence on the number of authors $N$.
desk verdict A clean, clearly scoped math note: Theorem 1 is essentially a restatement of the definition, but the institutional counting sequences and majorization analysis give it real value for bibliometrics. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the parametric credit formula $MFC_k = (1/N)^{1/k}$; Theorem 1 follows from rewriting it as $(1/N)^{1/k} = (1/N)^\lambda$ with $\lambda = 1/k$, which is exactly the weighted geometric average of $1/N$ and $1$. The institutional part is built on a binary incidence matrix $a_{ij} \in \{0,1\}$ recording which authors of institute $S$ appear in publication $p_j$, from which the counts $Y_j$ and $N_j$ are read; the three score formulas (14), (15), and (16) are successive choices of how to take $k$-th roots of relative contributions. The majorization order, defined by Lorenz-curve dominance, is what lets the paper compare total publication scores before and after reallocating author counts among institutes.
What would settle it
The author-level identity is algebraic and cannot fail, so the testable claim is the byline occurrence model: record a publication in which several co-authors each list two affiliations, so that summing institute-by-institute participations exceeds the paper's $N_j$, and check whether formula (15) assigns credits consistent with the intended meaning. A direct dataset test would compare MFC scores computed from bylines against scores based on author-reported contributions, looking for cases where the two orderings of institutes disagree.
Extended reading notes
Core claim
The paper's Theorem 1 states that with $MFC_1 = 1/N$ and $MFC_\infty = 1$, for every $\lambda \in [0,1]$ the weighted geometric average $G_\lambda = (MFC_1)^\lambda (MFC_\infty)^{1-\lambda}$ equals $MFC_{1/\lambda}$. In words, the whole MFC family is exactly the geometric interpolation between complete-normalized fractional counting and full counting, with interpolation parameter $1/k$; arithmetic and harmonic averages cannot play this role independently of $N$. For institutions, formula (15), the MFC actually used by Sivertsen, Rousseau and Zhang, is shown to connect complete-fractionalized counting at $k=1$ to whole counting at $k=\infty$, while the alternative CMFC connects complete-fractionalized to complete counting, and PMFC connects whole-fractionalized to whole counting. The paper also proves replication invariance for MFC (scaling all authors by a constant leaves an institute's MFC score unchanged) and uses the majorization order to show that an institute's percentage share is larger when the distribution of the other institutes' contributions is more unequal.
Load-bearing premise
The paper assumes a publication's credit is fully determined by how many times an author or an institute appears in the byline, with every author carrying equal weight and authors with multiple affiliations ignored; if credit should reflect actual intellectual contribution, or if multi-affiliated authors are treated differently, the formulas describe a different quantity.
Editorial extensions
If this is right
- For an individual author, MFC$_k$ is interpretable exactly as a geometric average with weight $1/k$ on fractional counting, independent of $N$, so choosing $k$ is equivalent to choosing a desired balance between fractional and full credit.
- The sum of author credits in a single publication increases with $k$ from $1$ to $N$, according to $N^{(k-1)/k}$.
- MFC for an institute is invariant under proportional scaling of author counts: participating with 1 of 4 authors scores the same as participating with 3 of 12 authors.
- If all entities add the same number of co-authors, entities that were below the average participation gain score and entities above the average lose score.
- Under MFC, a given institute's percentage of a publication's total score is higher when the roles of the other institutes are more unequal.
Reading between the lines
- Because Theorem 1 identifies MFC exactly with geometric interpolation, practical debates about the right value of $k$ can be reframed as debates about how much weight to put on fractional versus full credit; if an evaluator can state that weight, $k = 1/\lambda$ follows directly.
- The paper's three formulas define a two-dimensional square of counting methods indexed by participation versus contribution and fractional versus full; future indicators could interpolate along other paths in that square, or use different $k$ values for different institutes.
- The author-weighted extension sketched in Section 6, with first or corresponding authors weighted more heavily, could be tested against author-reported contribution scores; if such weights are available, the paper's inequalities and majorization properties would carry over unchanged.
- The majorization result implies an evaluation-policy hazard the paper leaves implicit: comparing institutes by MFC percentages across publications with different co-author distributions may reward institutes whose collaborators are concentrated in one or two other institutes.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper formalizes the position of modified fractional counting (MFC) between full counting and complete-normalized fractional counting. It defines MFC_k = (1/N)^{1/k} for an N-author publication and proves (Theorem 1) that this value is the weighted geometric average of MFC_1 and MFC_∞ with weights (λ, 1−λ), where k = 1/λ; this makes the interpolation claim exact and independent of N. Propositions 1 and 2 show that arithmetic and harmonic averages cannot play this role. The second half of the paper works under an explicit byline-incidence model and introduces three institutional score sequences: CMFC (Eq. 14), MFC (Eq. 15), and PMFC (Eq. 16), which connect different pairs of counting extremes. Proposition 3 compares MFC and CMFC; Propositions 4–8 collect replication and monotonicity properties; Section 5 uses majorization to explain how the target institute's percentage under MFC depends on the evenness of the other institutes' contributions.
Significance. If the claims are taken with the stated scope, the paper is a useful mathematical clarification. Its main merit is showing that MFC is exactly a geometric interpolation family, so that the sense in which intermediate values of k lie between the extremes is made precise. The institutional part is honest about its assumptions: multiple affiliations and intellectual contribution are explicitly set aside, and the three counting sequences are clearly delineated. The majorization explanation in Section 5 is a genuinely nice result that adds explanatory value beyond the definitional identities. The paper is also easy to verify: the proofs are elementary and mostly fully presented, and the limitations are stated where they apply. The formal errors I found are local, but one of them affects a stated proposition and should be corrected before publication.
major comments (1)
- [Section 3.2, Proposition 3 (Eq. (17) and (18))] The claimed equivalence "MFC_k(S) ≤ CMFC_k(S) ⇔ k ≥ 1" is false without a non-degeneracy condition. If every publication in P has at most one author from S, then Y_j ∈ {0,1} and formula (14) equals formula (15) for every k, so the inequality also holds for k < 1; conversely, if some publication has Y_j ≥ 2 and k < 1, then inequality (18) fails and MFC_k(S) > CMFC_k(S). The proposition should be restated with an explicit caveat, such as "when some publication has Y_j ≥ 2, the inequality holds exactly for k ≥ 1," or the statement should be limited to the paper's standing assumption k ≥ 1.
minor comments (6)
- [Section 2, Theorem 1] At λ = 0 the expression MFC_{1/λ} is undefined; the theorem should either restrict λ to (0,1] and treat λ = 0 by the stated limit MFC_∞ = 1, or add the convention MFC_{1/0} = MFC_∞.
- [Section 3.2, Eq. (14)] The simplification ∑_i (a_ij/N_j)^{1/k} = (∑_i a_ij)/N_j^{1/k} should note explicitly that it uses a_ij ∈ {0,1}; without this binary property the step would not be valid.
- [Section 3.1, Remark 2] In the k = 2 comparison the text writes CMFC1(S) = 2/3 where CMFC2(S) = 2/√9 is meant, and in the k = 3 comparison it writes PMFC2 and CMFC1 where PMFC3 and CMFC3 are meant.
- [Table 2] The entry "1/3/" in the Fractionalized-whole row has a stray slash and should read "1/3".
- [Section 6, Example 1] The cross-reference "see (22)" for the author-weight example does not point to an equation in the paper; it should refer to the definition b_j = Y_j/N_j, for instance Eq. (13) or the display preceding it.
- [Section 4, Corollary 3] The proof invokes Markov's inequality to conclude Md/2 ≤ μ; one sentence explaining that this is the standard bound median ≤ 2 mean for nonnegative data would make the step clearer.
Circularity Check
No significant circularity: Theorem 1 is a transparent algebraic identity from the stated definitions, and the institutional formulas are derived from an explicitly scoped byline model.
full rationale
The paper's central Theorem 1 is an immediate algebraic consequence of its own definitions, not a circular use of a conclusion as a premise. Equation (1) defines MFC_k as (1/N)^(1/k); equation (7) defines G_lambda as (MFC_1)^lambda * (MFC_infinity)^(1-lambda). Substituting MFC_1 = 1/N and MFC_infinity = 1 gives G_lambda = (1/N)^lambda, which is exactly MFC_k with k = 1/lambda. This is a definitional identity, stated as a mathematical observation rather than an empirical prediction; it is not used as an input to prove itself. The institutional results in formulas (14)-(16) are derived from the explicitly stated byline incidence model in equation (12), with the single-affiliation simplification openly acknowledged in Section 3.1 and revisited in Section 6. The recalled Propositions 5-7 from Sivertsen et al. (2025) are background material and are not load-bearing for the new derivations. No fitted parameter is renamed as a prediction, and no claimed result depends on a self-citation chain for its validity. The only notable defect is a notational edge case: MFC_(1/lambda) is undefined at lambda = 0 unless one adopts the limit convention MFC_infinity, but this is an editorial clarification, not circularity.
Assumptions & free parameters
assumptions (5)
- standard math Weighted arithmetic, geometric, and harmonic averages as defined in Hardy et al. (1934).
- domain assumption The MFC family is defined by MFC_k=(1/N)^(1/k) for individuals and MFC_k(S)=sum_j (Y_j/N_j)^(1/k) for institutes.
- domain assumption A publication can be represented as a binary incidence matrix; multiple affiliations and intellectual contribution weights are ignored.
- standard math Lorenz curves and majorization order, and the Schur-concavity of sums of concave functions, from Hardy et al. (1934), Patil and Taillie (1982), and Marshall et al. (2011).
- standard math For nonnegative arrays, median is at most twice the mean.
Cite this review
Pith. "Pith review of Mathematical reflections on modified fractional counting." pith.science (2026). https://pith.science/paper/KCFPEFIE
@misc{pith2026250612057,
author = {Pith},
title = {Pith review of: Mathematical reflections on modified fractional counting},
year = {2026},
howpublished = {\url{https://pith.science/paper/KCFPEFIE}},
note = {Machine review of arXiv:2506.12057}
}
read the original abstract
We make precise what is meant by stating that modified fractional counting (MFC) lies between full counting and complete-normalized fractional counting by proving that for individuals, the MFC-values are weighted geometric averages of these two extremes. There are two essentially different ways to consider the production of institutes in multi-institutional articles, namely participation and actual number of contributions. Starting from an idea published by Sivertsen, Rousseau and Zhang in 2019 we present three formulae for measuring the production of institutes in multi-institutional articles. It is shown that the one proposed by Sivertsen, Rousseau and Zhang is situated between the two other ways. Less obvious properties of MFC are proven using the majorization order.
Reference graph
Works this paper leans on
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Marshall, A.W., Olkin, I., & Arnold, B.C. (2011). Inequalities: Theory of Majorization and its Applications. New York: Springer
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Patil, G.P ., & Taillie, C. (1982). Diversity as a concept and its measurement. Journal of the American Statistical Association, 77, 548-561
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Sivertsen, G. (2016). A bibliometric indicator with a balanced representation of all fields. In: International Conference on Science and Technology Indicators, STI Conference 2016- Valéncia (Spain)
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Sivertsen, G., Rousseau, R., & Zhang , L. (2019) . Measuring scientific contributions with modified fractional counting. Journal of Informetrics , 13(2), 679-694. 23
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(2025).The motivations for and effects of modified fractional counting
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2022 doi
Reviewed August 7, 2026 · model on record in the stance chip above.
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