Random subgraphs of finite graphs: I. The scaling window under the triangle condition
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Abstract We study random subgraphs of an arbitrary finite connected transitive graph 𝔾 obtained by independently deleting edges with probability 1 − p . Let V be the number of vertices in 𝔾, and let Ω be their degree. We define the critical threshold p c = p c (𝔾, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λ V 1/3 , where λ is fixed and positive. We show that, for any such model, there is a phase transition at p c analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold p c . In particular, we show that the largest cluster inside a scaling window of size | p − p c | = Θ(Ω −1 V −1/3 ) is of size Θ( V 2/3 ), while, below this scaling window, it is much smaller, of order O (ϵ −2 log( V ϵ 3 )), with ϵ = Ω( p c − p ). We also obtain an upper bound O (Ω( p − p c ) V ) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω( p − p c )). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n ‐cube and certain Hamming cubes, as well as the spread‐out n ‐dimensional torus for n > 6. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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 | 0.000 | 0.000 |
| 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 it