Bibliographic record
Abstract
In this paper we consider a random mapping, Tn,θ, of the finite set {1, 2, ..., n} into itself for which the digraph representation Ĝn,θ is constructed by: (1) selecting a random number, Ln, of cyclic vertices, (2) constructing a uniform random forest of size n with the selected cyclic vertices as roots, and (3) forming ‘cycles’ of trees by applying to the selected cyclic vertices a random permutation with cycle structure given by the Ewens sampling formula with parameter θ. We investigate kn,θ, the size of a ‘typical’ component of Ĝn,θ, and we obtain the asymptotic distribution of kn,θ conditioned on Ln = m(n). As an application of our results, we show in Section 3 that provided Ln is of order much larger than √ n, then the joint distribution of the normalized order statistics of the component sizes of Ĝn,θ converges to the Poisson-Dirichlet(θ) distribution as n→∞. ∗Actuarial Mathematics and Statistics Department and The Maxwell Institute for Mathematical Sciences, Heriot–Watt University, Edinburgh EH14 4AS, UK; email: J.Hansen@hw.ac.uk †Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznan, Poland; email: jaworski@amu.edu.pl ‡J. Jaworski acknowledges the support by the Marie Curie Intra-European Fellowship No. 236845 (RANDOMAPP) within the 7th European Community Framework Programme and by National Science Centre DEC-2011/01/B/ST1/03943.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".