Subgraph probability of random graphs with specified degrees and applications to chromatic number and connectivity
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Bibliographic record
Abstract
Abstract Given a graphical degree sequence , let denote a uniformly random graph on vertex set where vertex has degree for every . We give upper and lower bounds on the joint probability of an arbitrary set of edges in , and we link these probability estimates to the corresponding probabilities in the configuration model. Then we show that many existing results of in the literature can be significantly improved with simpler proofs, by applying this new probabilistic tool. One example we give concerns the chromatic number of . In another application, we use these joint probabilities to study the connectivity of . Under some rather mild condition on —in particular, if where is the maximum component of —we fully characterize the connectivity phase transition of . We also give sufficient conditions for being connected when is unrestricted.
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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 | 0.000 | 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