The mean order of sub‐<i>k</i>‐trees of <i>k</i>‐trees
Bibliographic record
Abstract
Abstract This article focuses on the problem of determining the mean orders of sub‐k‐trees of k‐trees. It is shown that the problem of finding the mean order of all sub‐k‐trees containing a given k‐clique C, can be reduced to the previously studied problem of finding the mean order of subtrees of a tree that contain a given vertex. This problem is extended in two ways. The first of these extensions focuses on the mean order of sub‐k‐trees containing a given sub‐k‐tree. The second extension focuses on the expected number of r‐cliques, , in a randomly chosen sub‐k‐tree containing a fixed sub‐k‐tree X. Sharp lower bounds for both invariants are derived. The article concludes with a study of global mean orders of sub‐k‐trees of a k‐tree. For a k‐tree, from the class of simple‐clique k‐trees, it is shown that the mean order of its sub‐k‐trees is asymptotically equal to the mean subtree order of its dual. For general k‐trees a recursive generating function for the number of sub‐k‐trees of a given k‐tree T is derived.
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.
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 | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".