Géométrie des arbres récursifs pondérés et à attachement préférentiel affine
Why this work is in the frame
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Bibliographic record
Abstract
We study two models of growing recursive trees. For both models, initially the tree only contains one vertex $u_1$ and at each time $n\geq 2$ a new vertex $u_n$ is added to the tree and its parent is chosen randomly according to some rule. In the \emph{weighted recursive tree}, we choose the parent $u_k$ of $u_n$ among $\{u_1,u_2,\dots, u_{n-1}\}$ with probability proportional to $w_k$, where $(w_n)_{n\geq1}$ is some deterministic sequence that we fix beforehand. In the \emph{affine preferential attachment tree with fitnesses}, the probability of choosing any $u_k$ is proportional to $a_k+\mathrm{deg}^{+}(u_k)$, where $\mathrm{deg}^{+}(u_k)$ denotes its current number of children, and the sequence of \emph{fitnesses} $(a_n)_{n\geq 1}$ is deterministic and chosen as a parameter of the model. We show that for any sequence $(a_n)_{n\geq 1}$, the corresponding preferential attachment tree has the same distribution as some weighted recursive tree with a \emph{random} sequence of weights (with some explicit distribution). We then prove almost sure scaling limit convergences for some statistics associated with weighted recursive trees as time goes to infinity, such as degree sequence, height, profile and also the weak convergence of some measures carried on the tree. Thanks to the connection between the two models, these results also apply to affine preferential attachment trees.
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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 | 0.004 |
| 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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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