Phase transition for random walks on graphs with added weighted random matching
Bibliographic record
Abstract
Abstract For a finite graph $$G=(V,E)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> let $$G^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>G</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> be obtained by considering a random perfect matching of V and adding the corresponding edges to G with weight $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> , while assigning weight 1 to the original edges of G . We consider whether for a sequence $$(G_n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of graphs with bounded degrees and corresponding weights $$(\varepsilon _n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ε</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the (weighted) random walk on $$(G_n^*)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> has cutoff. For graphs with polynomial growth we show that $$\log \left( \frac{1}{\varepsilon _n}\right) \ll \log |V_n|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>log</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msub> <mml:mi>ε</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mfrac> </mml:mfenced> <mml:mo>≪</mml:mo> <mml:mo>log</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$\log |V_n|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>log</mml:mo> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> we show that $$\frac{1}{\varepsilon _n}\ll \log |V_n|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msub> <mml:mi>ε</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mfrac> <mml:mo>≪</mml:mo> <mml:mo>log</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is sufficient for cutoff. In the special case of expander graphs we also provide a complete picture for the complementary regime $$\frac{1}{\varepsilon _n}\gtrsim \log |V_n|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msub> <mml:mi>ε</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mfrac> <mml:mo>≳</mml:mo> <mml:mo>log</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".