Which values of the volume growth and escape time exponent are possible for a graph?
Bibliographic record
Abstract
Let \Gamma=(G,E) be an infinite weighted graph which is Ahlfors \alpha -regular, so that there exists a constant c such that c^{-1} r^\alpha\le V(x,r)\le c r^\alpha , where V(x,r) is the volume of the ball centre x and radius r . Define the escape time T(x,r) to be the mean exit time of a simple random walk on \Gamma starting at x from the ball centre x and radius r . We say \Gamma has escape time exponent \beta>0 if there exists a constant c such that c^{-1} r^\beta \le T(x,r) \le c r^\beta for r\ge 1 . Well known estimates for random walks on graphs imply that \alpha\ge 1 and 2 \le \beta \le 1+\alpha . We show that these are the only constraints, by constructing for each \alpha_0 , \beta_0 satisfying the inequalities above a graph \widetilde{\Gamma} which is Ahlfors \alpha_0 -regular and has escape time exponent \beta_0 . In addition we can make \widetilde{\Gamma} sufficiently uniform so that it satisfies an elliptic Harnack inequality.
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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 | 0.001 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".