Phase transition for random walks on graphs with added weighted random matching
Bibliographic record
Abstract
Abstract For a finite graph $$G=(V,E)$$ G = ( V , E ) let $$G^*$$ G ∗ be obtained by considering a random perfect matching of V and adding the corresponding edges to G with weight $$\varepsilon $$ ε , while assigning weight 1 to the original edges of G . We consider whether for a sequence $$(G_n)$$ ( G n ) of graphs with bounded degrees and corresponding weights $$(\varepsilon _n)$$ ( ε n ) , the (weighted) random walk on $$(G_n^*)$$ ( G n ∗ ) has cutoff. For graphs with polynomial growth we show that $$\log \left( \frac{1}{\varepsilon _n}\right) \ll \log |V_n|$$ log 1 ε n ≪ log | V n | is a sufficient condition for cutoff. Under the additional assumption of vertex-transitivity we establish that this condition is also necessary. For graphs where the entropy of the simple random walk grows linearly up to some time of order $$\log |V_n|$$ log | V n | we show that $$\frac{1}{\varepsilon _n}\ll \log |V_n|$$ 1 ε n ≪ log | V n | is sufficient for cutoff. In the special case of expander graphs we also provide a complete picture for the complementary regime $$\frac{1}{\varepsilon _n}\gtrsim \log |V_n|$$ 1 ε n ≳ log | V n | .
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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".