Connectivity of Addable Monotone Graph Classes
Why this work is in the frame
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Bibliographic record
Abstract
A class A of labelled graphs is weakly addable if if for all graphs G in A and all vertices u and v in distinct connected components of G, the graph obtained by adding an edge between u and v is also in A; the class A is monotone if for all G ∈ A and all subgraphs H of G, we have H ∈ A. We show that for any weakly addable, monotone class A whose elements have vertex set {1,..., n}, the probability that a uniformly random element of A is connected is at least (1 − on(1))e −0.540760, where on(1) → 0 as n →∞. Furthermore, if every element of A has girth at least g> 1, then the probability that A is connected is at least (1 − og(1)))e −1/2. The latter result establishes a conjecture of McDiarmid et al. (2006) for graphs of large girth.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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