Random cographs: Brownian graphon limit and asymptotic degree\n distribution
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Bibliographic record
Abstract
We consider uniform random cographs (either labeled or unlabeled) of large\nsize. Our first main result is the convergence towards a Brownian limiting\nobject in the space of graphons. We then show that the degree of a uniform\nrandom vertex in a uniform cograph is of order $n$, and converges after\nnormalization to the Lebesgue measure on $[0,1]$. We finally analyze the vertex\nconnectivity (i.e. the minimal number of vertices whose removal disconnects the\ngraph) of random connected cographs, and show that this statistics converges in\ndistribution without renormalization. Unlike for the graphon limit and for the\ndegree of a random vertex, the limiting distribution is different in the\nlabeled and unlabeled settings.\n Our proofs rely on the classical encoding of cographs via cotrees. We then\nuse mainly combinatorial arguments, including the symbolic method and\nsingularity analysis.\n
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it