Asymptotic random graph intuition for the biased connectivity game
Bibliographic record
Abstract
Abstract We study biased Maker/Breaker games on the edges of the complete graph, as introduced by Chvátal and Erdős. We show that Maker, occupying one edge in each of his turns, can build a spanning tree, even if Breaker occupies b ≤ (1 − o (1)) · $ {n \over \ln n} $ edges in each turn. This improves a result of Beck, and is asymptotically best possible as witnessed by the Breaker‐strategy of Chvátal and Erdős. We also give a strategy for Maker to occupy a graph with minimum degree c (where c = c ( n ) is a slowly growing function of n ) while playing against a Breaker who takes b ≤ (1 − o (1)) · $ {n \over \ln n} $ edges in each turn. This result improves earlier bounds by Krivelevich and Szabó. Both of our results support the surprising random graph intuition: the threshold bias is asymptotically the same for the game played by two “clever” players and the game played by two “random” players. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009
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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 | 0.002 | 0.014 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".