The Horton–Strahler number of Galton–Watson trees with possibly infinite variance
Bibliographic record
Abstract
The Horton–Strahler number, also known as the register function, provides a tool for quantifying the branching complexity of a rooted tree. We consider the Horton–Strahler number of critical Galton–Watson trees conditioned to have size n and whose offspring distribution is in the domain of attraction of an α-stable law with α∈[1,2]. We give tail estimates and when α≠1, we prove that it grows as 1αlogα/(α−1)n in probability. This extends the result of Brandenberger, Devroye and Reddad dealing with the finite variance case for which α=2. We also characterize the cases where α=1, namely the spectrally positive Cauchy regime, which exhibits more complex behaviors. Our proofs are new and probabilistic; they relate the Horton–Strahler number with other shape parameters such as the height or largest degree.
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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 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| 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".