Universal diameter bounds for random graphs with given degrees
Bibliographic record
Abstract
The diameter of a graph is the greatest distance between any two vertices which lie in the same connected component.We study uniformly random connected simple graphs on n vertices whose degrees are given.We show that, if the proportion of prescribed degrees equal to 2 is bounded away from 1, then such a random graph has expected diameter of order √ n.It is not hard to see that this bound is best possible for general degree sequences (and in particular when the degree sequence is that of a tree).We also prove that this bound holds without the connectivity constraint.As a key input to the proofs, we show that graphs with minimum degree 3 are with high probability connected and have diameter of order log n. AbrégéLe diamètre d'un graphe est la plus grande distance entre deux sommets appartenant à la même composante connexe.Nous étudions des graphes simples connexes aléatoires uniformes à n sommets dont les degrés sont prescrits.Nous montrons que, si la proportion de degrés prescrits égaux à 2 est uniformément bornée strictement en dessous de 1, alors un tel graphe aléatoire a un diamètre moyen en √ n.Il n'est pas difficile de voir que cette borne est optimale pour des suites de degrés générales (et en particulier lorsque la suite de degrés est celle d'un arbre).Nous prouvons également que cette borne est également valide sans l'hypothèse de connexité.Nous montrons aussi que les graphes de degré minimal au moins 3 sont connexes et ont un diamètre en log n avec haute probabilité, ce qui est un élément clé pour les preuves des autres résultats.
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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 | 0.005 | 0.049 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".