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Bounds, Decompositions and Null Behaviour of KRATOS: A Mathematical Specification of a Recognition-Comparability Diagnostic
Maria Dolores Gonzalez, Alberto Barbado
TL;DR
Composite indicators are often reported without specifying their attainable joint values, input responses, or behavior under calibration resampling. This note gives a mathematical specification of KRATOS and derives properties of its components, composite, and permutation nulls. It shows how recognition alignment, the composition ceiling, and null dispersion depend on group means, participation imbalance, group size, and citation-weight variation.
Problem
Composite indicators are often reported without specifying their attainable joint values, responses to input changes, or behavior under calibration resampling.
Source: §1 · p. 2Method
Results
Recognition alignment is sensitive to group mean citation rates; ¯A ≥ 1−2dTV(p,e), and null dispersion grows with heavy-tailed weights and small groups.
Source: §5 · pp. 4–5, §7 · p. 8Takeaways & Limitations
KJI’s “Justice” label carries no normative meaning, and substantive interpretations of differences between corpora belong to empirical work.
Source: §6 · p. 6, §9 · p. 9The specification does not justify the group choices, uniform reference, or use of citation weight as a proxy for recognition.
Source: §9 · p. 9
Abstract
from arXiv · showhide
KRATOS is a group-structured bibliometric diagnostic that compares the distribution of documents (participation) with the distribution of citation weight (recognition) over a fixed, finite universe of analytical groups. This note gives its complete mathematical specification and derives the properties that govern its interpretation. Beyond bounds and equality conditions for each component, we show that the recognition-alignment score depends on citation data only through ratios of group mean citation rates to the corpus mean; that the composition ceiling of the participation--recognition factor is an affine function of the total variation distance to the uniform reference; and that the factor itself satisfies Fr\'echet-type bounds in terms of this ceiling and a participation-weighted recognition score. We establish an exact logarithmic decomposition of the composite index, an ordering between the primary and a reciprocal-symmetric recognition score, invariance properties, and closed-form first and second moments of the recognition ratios under global and stratified permutation nulls. These results explain why composite orderings can be sensitive to small groups, heavy-tailed citation distributions and metadata reassignment. All results are verified numerically with an accompanying script. The specification concerns measurement structure only; it does not define a measure of epistemic change or justice.
1 Introduction
This note formalizes what KRATOS’s composite components can jointly attain, how they respond to input changes, and how they behave under calibration resampling. It derives properties of the composite and its null behavior, while noting that the contribution is their joint specification rather than new individual building blocks.
1 Introduction
- KRATOS’s empirical application to four bibliographic corpora and its conditional permutation calibration are reported in a companion manuscript.The frozen implementation is archived at González Barbado and Barbado González.Source: §1 · pp. 1–2
- The note addresses the lack of explicit accounts of composite indicators’ attainable values, input responses, and behavior under calibration resampling.For KRATOS, this structure helps explain why cross-corpus KJI orderings change under metadata reassignment and resampling.Source: §1 · p. 2
- It characterizes the participation–recognition factor, decomposes KJI logarithmically, orders alternative recognition scores, and derives closed-form null moments for global and stratified permutations.The factor results include a recognition score based on relative group mean citation rates, a composition ceiling linked to total variation, and bounds involving that ceiling and a recognition term.Source: §1 · p. 2
- The contribution is the joint specification and the properties of the resulting composite, not new individual building blocks.The note identifies normalized Shannon entropy and classical sampling theory as established foundations.Source: §1 · p. 2
2 Domain and notation
KRATOS is defined over a fixed finite universe of analytical groups, with unresolved records kept outside that universe and calculations restricted to resolved documents. It compares document participation with citation recognition using nonnegative citation weights and a uniform participation reference.
2 Domain and notation
- The analytical universe has G≥2 groups; unresolved assignments remain in an audit state, while empty groups remain in the universe and calculations use resolved documents only.Coverage is κ = N_R/N when N > 0.Source: §2 · p. 2
- Each resolved document is assigned to one group and carries a nonnegative citation weight; n_u counts the documents assigned to group u.Source: §2 · p. 2
- Participation lies on the {n/N_R} lattice in applications, while any recognition vector in the simplex is realizable by suitable nonnegative weights; the participation reference is uniform at p*=1/G.For groups with n_u>0, the group mean citation weight is ¯x_u=c_u/n_u, and the corpus mean is ¯x=C_R/N_R.Source: §2 · p. 2
3 Distributional evenness
The section defines distributional evenness as normalized Shannon entropy over the group universe and establishes its range, extremal configurations, and dependence on universe size.
3 Distributional evenness
- H′(z) is normalized Shannon entropy over G categories, using 0 ln 0 = 0; it is Pielou’s evenness.Source: §3 · p. 3
- For every simplex distribution, 0≤H′(z)≤1; it is zero exactly at simplex vertices, one exactly at the uniform vector, and invariant to coordinate permutations.Source: §3 · p. 3
- Evenness values over different universe sizes are not directly comparable merely because they share the [0,1] range; empty groups affect the value through G.Changing the group universe defines a different measurement specification.Source: §3 · p. 3
4 Knowledge Contribution Diversity Index
The Knowledge Contribution Diversity Index combines document- and citation-distribution evenness using a fixed analytic weight. Its bounds and continuity depend on the component values and the weight’s position, and geometric aggregation limits compensation between components.
4 Knowledge Contribution Diversity Index
- The primary specification fixes λ=1/2 analytically rather than estimating it.Source: §4 · p. 3
- For 0<λ<1, KCDIλ is non-decreasing in each evenness component and equals zero whenever either component is zero.It does not exceed the corresponding weighted arithmetic mean, and the gap widens as the components diverge.Source: §4 · p. 3
5 Participation–recognition alignment
KRATOS’s participation–recognition factor combines a participation-based composition ceiling with group-level recognition alignment. The section characterizes their bounds, distance interpretations, and sensitivities.
5 Participation–recognition alignment
- A_u and S_u lie in [0,1]: A_u is maximal at uniform participation, while S_u equals 1 when recognition matches participation and falls to 0 when the relative rate reaches 2.For positive participation, S_u is piecewise determined by the group-to-corpus mean citation-rate ratio r_u.Source: §5 · p. 4
- S_u depends on citation weights only through the group-to-corpus mean citation-rate ratio; in a small group, one highly cited document can push that ratio above 2 and set S_u to 0.The score therefore captures parity in group means, regardless of how citations are distributed among the remaining documents.Source: §5 · p. 4
- The composition ceiling Ā is at least 1 − 2 dTV(p,e), with equality iff max_u p_u ≤ 2/G; in particular, Ā = 1 iff p = e.Source: §5 · p. 5
- P lies between 0 and Ā, with P = Ā iff recognition aligns wherever A_u > 0; its lower bound also depends on participation-weighted recognition and total variation distances.For fixed participation, P = Ā at s = p and P = 0 at some simplex vertex; the two-distance bound is P ≥ 1 − 2dTV(p,e) − 2dTV(p,s).Source: §5 · p. 5
- The entropy-based evenness measure and Ā are not interchangeable: the former summarizes the whole distribution, while Ā ignores groups at least twice the uniform share.The paper recommends reporting both quantities.Source: §5 · pp. 5–6
- The reciprocal-symmetric recognition score is symmetric under r_u ↔ 1/r_u, and switching to it can only lower P and KJI.Any resulting reordering across corpora reflects how far their ratios lie from 1.Source: §5 · p. 6
6 The composite index
KJI combines KCDI with the participation–recognition factor P, with bounds and a uniformity condition governing its maximum. Its logarithmic decomposition allocates index changes among components, while invariance results clarify which input changes matter.
6 The composite index
- KJIλ equals KCDIλ × P and lies between 0 and KCDIλ min(Ā, PS), which is itself at most KCDIλ ≤ 1.The gap KCDIλ − KJIλ equals KCDIλ(1 − P).Source: §6 · p. 6
- KJIλ reaches 1 exactly when both participation and citation-weight distributions are uniform, provided NR > 0 and CR > 0.The proof links the maximum to P = 1 and then to p = s = e.Source: §6 · p. 6
- For two configurations with positive H′C and P, the exact logarithmic decomposition allocates the change in KJI among evenness, composition, and conditional recognition.The decomposition is an accounting identity, not a causal attribution, and its terms are not statistically independent.Source: §6 · pp. 6–7
- Permuting group labels only permutes group-level quantities, while adding or removing audit-state documents changes coverage κ but no other quantity.Coverage must therefore be reported alongside the indices, which are blind to how much of the corpus is resolved.Source: §6 · p. 7
- Rescaling all citation weights by one positive constant leaves the recognition and composite quantities unchanged, but group-, year-, or document-specific rescaling need not.The companion study’s year-normalised weights change s because publication cohorts receive different rescaling constants.Source: §6 · p. 7
7 Behaviour under permutation nulls
The section derives recognition-ratio moments under global and stratified permutation nulls. Global shuffling leaves participation fixed; stratification changes the null centre and removes between-stratum variation.
7 Behaviour under permutation nulls
- Under global permutation, E[su] = pu and E[ru] = 1, while Var(ru) = (NR − nu)/(nu(NR − 1)) for groups with nu > 0.CR, p, H′D, (Au), and Ā remain invariant under the shuffle.Source: §7 · p. 7
- Under stratified permutation, E[su] generally differs from pu and equals the share group u would receive if its documents earned their stratum means.CR and (Au) are invariant under this null.Source: §7 · p. 7
- Heavy-tailed citation weights and small groups widen the null for ru and Su, making fixed-size departures harder to resolve.Null dispersion grows with the squared coefficient of variation and approximately with 1/nu.Source: §7 · p. 8
- Stratification removes between-stratum citation variation, and few documents per stratum limit attainable variation; strata with nh = 1 contribute none.Calibration of P is conditional on Ā because only the recognition branch varies under the null.Source: §7 · p. 8
- The assignment count overstates the number of distinct statistic values when citation weights repeat.The count is of label-distinct assignments.Source: §7 · p. 8
8 Boundary conventions and computation
The computation restricts analysis to resolved documents, applies boundary conventions, and computes the shares and indices in sequence. Definition-based assertions serve as implementation regression tests.
8 Boundary conventions and computation
- With no resolved documents, κ is zero and the resolved-set evenness quantities are assigned boundary values; with resolved documents but no citations, citation-dependent scores are zero.The latter zeros encode absence of citation information, not maximal concentration.Source: §8 · p. 8
- The conventions use 0 ln 0 = 0, set Su = 0 when pu = 0 without evaluating su/pu, and evaluate λ endpoints directly..Source: §8 · p. 8
- The algorithm takes group or audit labels, nonnegative citation weights, and λ, then restricts the corpus to resolved documents and computes counts, coverage, and citation totals.It then computes p and s unless a boundary convention applies, followed by the evenness quantities and indices.Source: §8 · p. 8
- The implementation checks P ≤ min(Ā, PS), P ≥ Ā + PS − 1, and KJIλ ≤ KCDIλ as regression invariants, not empirical findings.These assertions follow from the definitions and the cited theorems.Source: §8 · p. 8
9 Scope
KRATOS characterizes a finite-group functional of participation and recognition distributions, but its specification does not validate the measurement choices or support a substantive interpretation of corpus differences.
9 Scope
- The specification does not justify the chosen groups, uniform reference, or citation weight as a proxy for recognition; changing any defines a different specification.The results concern measurement structure, not epistemic change or justice.Source: §9 · p. 9
- The components discard document-level citation structure, and recognition alignment depends on group means, inheriting their sensitivity to extreme values.Source: §9 · p. 9
- KJI combines dependent components conjunctively, so interpreting corpus differences requires empirical work rather than treating the index as independent sources.Source: §9 · p. 9
A Numerical verification
An accompanying Python/NumPy script checks the stated identities, inequalities, counterexample, and permutation moments numerically. The checks found no deterministic violations and agreement between simulated and closed-form null moments up to Monte Carlo error.
A Numerical verification
- 20,000 random configurations for each G in {2, 3, 4, 6, 10} checked every Sections 4–6 inequality and identity, alongside Proposition 5.11’s counterexample.Configurations used Dirichlet draws of varying concentration and randomly emptied groups.Source: A Numerical verification · p. 9
- 40,000 Monte Carlo permutations of heavy-tailed Pareto weights checked the moments in Propositions 7.1 and 7.2.Source: A Numerical verification · p. 9
- No deterministic result was violated, and simulated null moments agreed with closed forms up to Monte Carlo error; Table 1 reports the global case.Source: Table 1 · p. 9, A Numerical verification · p. 9