Cutoff for random walk on random graphs with a community structure
Bibliographic record
Abstract
We consider a variant of the configuration model with an embedded community structure and study the mixing properties of a simple random walk on it. Every vertex has an internal degint≥3 and an outgoing degout number of half-edges. Given a stochastic matrix Q, we pick a random perfect matching of the half-edges subject to the constraint that each vertex v has degint(v) neighbours inside its community and the proportion of outgoing half-edges from community i matched to a half-edge from community j is Q(i,j). Assuming the number of communities is constant and they all have comparable sizes, we prove the following dichotomy: simple random walk on the resulting graph exhibits cutoff if and only if the product of the Cheeger constant of Q times logn (where n is the number of vertices) diverges. In (Ann. Appl. Probab. 30 (2020) 1824–1846), Ben-Hamou established a dichotomy for cutoff for a nonbacktracking random walk on a similar random graph model with two communities. We prove that the same characterisation of cutoff holds for simple random walk.
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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 | 0.006 | 0.048 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.004 | 0.002 |
| Bibliometrics | 0.007 | 0.004 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.005 | 0.011 |
| Open science | 0.005 | 0.005 |
| Research integrity | 0.005 | 0.007 |
| Insufficient payload (model declined to judge) | 0.007 | 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".