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Transitivity in Inhomogeneous Random Tournaments

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Pith's one-line read The number of circular triads in a W-random tournament follows one of three fluctuation regimes set by the regularity and uniformity of the tournamenton W, and a multiplier bootstrap produces valid confidence intervals for the consistency c

desk verdict The paper splits the circular-triad count into three regimes under a general tournamenton and supplies a multiplier bootstrap plus regime test that together give asymptotically valid CIs for the consistency coefficient. read the letter →

arxiv 2606.02340 v1 pith:TPMQMZ4O submitted 2026-06-01 math.PR math.COmath.STstat.TH

Sayak Chatterjee , Bhaswar B. Bhattacharya This is my paper
classification math.PRmath.COmath.STstat.TH
keywords W-randomtournamentscirculartriadsconsistencycoefficienttournamentonsmultiplierbootstraptransitivityquasirandomnessinhomogeneousmodels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes how the number of circular triads, which quantifies departures from transitivity in pairwise comparison data, fluctuates when a tournament is generated from an arbitrary tournamenton W. Three regimes of fluctuation appear as the number of vertices grows, each tied to whether W satisfies regularity or uniformity conditions. A tournamenton multiplier bootstrap is proved to track the limiting distribution of the triad count in the appropriate regime. When this bootstrap is combined with separate tests that detect regularity and uniformity, the resulting procedure yields confidence intervals for the Kendall-Smith consistency coefficient that remain asymptotically valid for any tournamenton. The work also supplies structural descriptions of those tournamentons that force the limiting distribution into a degenerate form.

What carries the argument

The W-random tournament model, in which each directed edge is chosen independently with probability given by a fixed measurable tournamenton W from the unit square to the unit interval, together with the associated tournamenton multiplier bootstrap.

What would settle it

Generate many independent W-random tournaments of growing size n for a fixed, known W that is neither regular nor uniform; compute the empirical distribution of the suitably centered and scaled circular-triad count and check whether it converges to the non-degenerate limit predicted by the theory or is well-approximated by the bootstrap.

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Extended reading notes

Core claim

For a W-random tournament on n vertices, the number of circular triads exhibits three different fluctuation regimes, determined by suitable notions of regularity and uniformity of W; a tournamenton multiplier bootstrap consistently approximates the limiting distribution in the relevant regime, and combining it with tests for regularity and uniformity yields an algorithm for asymptotically valid confidence intervals for the consistency coefficient for all tournamentons. Structural characterizations are obtained for tournamentons for which the limiting distribution exhibits specific degeneracies.

Load-bearing premise

The observed tournament arises from independent edge directions whose probabilities are supplied by a single fixed measurable function W on the unit square.

Editorial extensions

If this is right

  • The circular-triad count admits a limiting distribution that depends on the regularity and uniformity properties of W and can be consistently approximated by the multiplier bootstrap.
  • Asymptotically valid confidence intervals for the consistency coefficient can be constructed for every possible tournamenton by first testing regularity and uniformity and then applying the bootstrap in the identified regime.
  • Structural characterizations identify the precise classes of tournamentons that produce degenerate limiting distributions for the circular-triad count.
  • The fluctuation results supply new characterizations in the theory of tournament quasirandomness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The inferential procedure could be applied directly to large empirical paired-comparison datasets from ranking or voting contexts once the W-random model is accepted.
  • The multiplier bootstrap technique may extend to other subgraph-count statistics in directed inhomogeneous random models beyond circular triads.
  • The structural conditions for degeneracy could serve as diagnostic tools for detecting when an observed tournament is close to quasirandom.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper studies fluctuations of the number of circular triads (directed 3-cycles) in W-random tournaments on n vertices, where edge directions are chosen independently according to a measurable tournamenton W:[0,1]^2→[0,1]. It identifies three distinct asymptotic regimes determined by notions of regularity and uniformity of W, develops a tournamenton multiplier bootstrap that consistently approximates the limiting distribution in each regime, and combines this bootstrap with tests for regularity and uniformity to produce an algorithm yielding asymptotically valid confidence intervals for the Kendall-Smith consistency coefficient that hold for every tournamenton. Structural characterizations of tournamentons producing degenerate limits are also given, with connections to tournament quasirandomness.

Significance. If the limiting statements and bootstrap consistency hold under the stated conditions, the results supply the first complete inferential framework for the consistency coefficient that is valid uniformly over all tournamentons, rather than only in the dense or quasirandom cases. The identification of three fluctuation regimes and the accompanying structural characterizations strengthen the link between tournament theory and graphon methods; the multiplier bootstrap that adapts automatically across regimes is a technical contribution that may extend to other subgraph counts in inhomogeneous directed models.

minor comments (3)
  1. The abstract and introduction refer to 'suitable notions of regularity and uniformity of W' without an early forward reference to the precise definitions (presumably in §2 or §3); adding a brief parenthetical or footnote would improve readability for readers outside the graphon literature.
  2. Notation for the consistency coefficient and the normalized circular-triad count should be introduced once in a dedicated notation subsection or table, as the same symbols appear in both the limiting theorems and the bootstrap construction.
  3. The description of the multiplier bootstrap in the algorithm section would benefit from an explicit statement of the resampling weights and the precise centering used in each regime, even if these are standard in the multiplier-bootstrap literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed summary of our manuscript and for the positive assessment of its contributions. The recommendation of minor revision is appreciated. No specific major comments were provided in the report, so we have no individual points to address at this time. We will incorporate any minor suggestions during the revision process.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper's core contributions—characterizing three fluctuation regimes for the circular-triad count in W-random tournaments via regularity/uniformity of the tournamenton W, introducing a tournamenton multiplier bootstrap to approximate the limiting distribution, and combining it with tests to produce asymptotically valid CIs for the consistency coefficient—are presented as new analytic and methodological results. No self-definitional reductions, fitted parameters renamed as predictions, load-bearing self-citations, or ansatzes smuggled via prior work appear in the abstract or described derivation structure. The bootstrap is framed as an independent approximation device rather than a tautological re-expression of inputs, and the limiting statements rest on standard probabilistic arguments under the stated W-random model. This is the expected non-finding for a paper whose central claims remain externally falsifiable and non-reductive.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Because only the abstract is available, the ledger is populated only with the modeling assumption that is explicit in the abstract; all other potential free parameters or axioms remain unknown.

assumptions (1)
  • domain assumption The observed tournament is generated by the W-random model: each directed edge is chosen independently with probability given by a fixed measurable function W.
    This generative assumption is stated in the abstract as the setting in which all limiting results and the bootstrap are derived.

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Cite this review

Pith. "Pith review of Transitivity in Inhomogeneous Random Tournaments." pith.science (2026). https://pith.science/paper/TPMQMZ4O

@misc{pith2026260602340,
  author       = {Pith},
  title        = {Pith review of: Transitivity in Inhomogeneous Random Tournaments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPMQMZ4O}},
  note         = {Machine review of arXiv:2606.02340}
}
abstract

Paired-comparison data are naturally represented by tournaments, where transitivity corresponds to the existence of a global ranking consistent with all pairwise outcomes. Accordingly, the classical Kendall-Smith coefficient of consistency measures deviations from transitivity in a tournament by counting the number of circular triads (directed $3$-cycles). In this paper, we characterize the fluctuations of the number of circular triads in inhomogeneous random tournaments and develop an inferential framework for the consistency coefficient. Specifically, we consider the $W$-random tournament model, where the comparison probabilities are determined by a tournamenton $W$, the analogue of a graphon in the tournament setting. We show that, for a $W$-random tournament on $n$ vertices, the number of circular triads exhibits three different fluctuation regimes, determined by suitable notions of regularity and uniformity of $W$. We further develop a novel tournamenton multiplier bootstrap that consistently approximates the limiting distribution of the circular-triad count in the relevant asymptotic regime. Combining this with procedures for testing regularity and uniformity, we design an algorithm for constructing confidence intervals for the consistency coefficient that is asymptotically valid for all tournamentons. We also obtain structural characterizations of tournamentons for which the limiting distribution of the number of circular triads exhibits specific degeneracies. These results can also be viewed through the lens of tournament quasirandomness and may be of independent interest.

Figures

Figures reproduced from arXiv: 2606.02340 by the authors.

Figure 1
Figure 1. The empirical tournamenton (2.6). The function takes the value 1 on the black regions, 0 on the white regions, and 1 2 on the grey region. Next, we define the 2-point conditional homomorphism density of circular triads and proper￾ties of the associated kernel: Definition 2.4 (2-point conditional homomorphism density). Given a tournamenton W, the 2-point conditional homomorphism density function for the clockwise ori… view at source ↗
Figure 2
Figure 2. The carousel tournamenton (2.21) with (a) α “ 0.2 and (b) α “ 0.5. The function takes the value 1 on the black regions and 0 on the white regions. Example 2.13 (Carousel tournamenton). Fix 0 ă α ă 1 and consider the tournamenton: For x, y P r0, 1s, Wpαq px, yq “ # 1 if x ´ y P r´α, 0q Y rα, 1s, 0 if x ´ y P r´1, ´αq Y r0, αq. (2.21) [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Tr4: A transitive tournament on 4 vertices which is quasirandom-forcing. 4.2. Testing for Uniformity. Given a tournamenton W and Tn „ Tpn, Wq the uniformity testing problem can be formulated as follows: H0 : W ” 1 2 versus H1 : W ‰ 1 2 . (4.4) More precisely, we want to test the null hypothesis that Wpx, yq “ 1 2 for almost every x, y P r0, 1s versus the alternative that Wpx, yq ‰ 1 2 on a set of positive measure. A… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Confidence intervals for (a) the BTL model with fpxq “ x (coverage: 93%), and (b) the carousel tournamenton with parameter α “ 0.8 (coverage: 98%). 0.154 0.156 0.158 0.160 0.162 0.164 0 25 50 75 100 Replicates Confidence intervals Condorcet Random Model with p = 0.3 −3…
Figure 5
Figure 5. Figure 5: Confidence interval for (a) Condorcet tournamenton with p “ 0.3 (coverage: 93%), and (b) uniform random tournament W ” 1 2 (coverage: 98%). 5. Proof of Theorem 2.6 5.1. Preparations. We begin with the following useful representation of N△pTnq. Lemma 5.1. Let N△pTnq be …

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