The scaling window for a random graph with a given degree sequence
Why this work is in the frame
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Bibliographic record
Abstract
Abstract We consider a random graph on a given degree sequence D , satisfying certain conditions. Molloy and Reed defined a parameter Q = Q ( D ) and proved that Q = 0 is the threshold for the random graph to have a giant component. We introduce a new parameter R = R ( \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal {D}\end{align*} \end{document} ) and prove that if | Q | = O ( n ‐1/3 R 2/3 ) then, with high probability, the size of the largest component of the random graph will be of order Θ( n 2/3 R ‐1/3 ). If | Q | is asymptotically larger than n ‐1/3 R 2/3 then the size of the largest component is asymptotically smaller or larger than n 2/3 R ‐1/3 . Thus, we establish that the scaling window is | Q | = O ( n ‐1/3 R 2/3 ). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012
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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 | 0.001 | 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 | 0.001 | 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