The Probability of Non-Existence of a Subgraph in a Moderately Sparse Random Graph
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
We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph G 0 in the common random graph models ${\cal G}$ ( n , m ) and ${\cal G}$ ( n , p ). Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of G 0 . This extends an argument given earlier by the second author for G 0 = K 3 with a more restricted range of average degree. For all strictly balanced subgraphs G 0 , our results give much information on the distribution of the number of copies of G 0 that are not in large ‘clusters’ of copies. The probability that a random graph in ${\cal G}$ ( n , p ) has no copies of G 0 is shown to be given asymptotically by the exponential of a power series in n and p , over a fairly wide range of p . A corresponding result is also given for ${\cal G}$ ( n , m ), which gives an asymptotic formula for the number of graphs with n vertices, m edges and no copies of G 0 , for the applicable range of m . An example is given, computing the asymptotic probability that a random graph has no triangles for p = o ( n −7/11 ) in ${\cal G}$ ( n , p ) and for m = o ( n 15/11 ) in ${\cal G}$ ( n , m ), extending results of the second author.
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 | 0.009 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.003 |
| 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 it