Rainbow arborescence in random digraphs
Bibliographic record
Abstract
We consider the Erdős-Rényi random directed graph process, which is a stochastic process that starts with $n$ vertices and no edges, and at each step adds one new directed edge chosen uniformly at random from the set of missing edges. Let $\mathcal{D}(n,m)$ be a graph with $m$ edges obtained after $m$ steps of this process. Each edge $e_i$ ($i=1,2,\ldots, m$) of $\mathcal{D}(n,m)$ independently chooses a colour, taken uniformly at random from a given set of $n(1 + O( \log \log n / \log n)) = n (1+o(1))$ colours. We stop the process prematurely at time $M$ when the following two events hold: $\mathcal{D}(n,M)$ has at most one vertex that has in-degree zero and there are at least $n-1$ distinct colours introduced ($M= n(n-1)$ if at the time when all edges are present there are still less than $n-1$ colours introduced; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether $\mathcal{D}(n,M)$ has a rainbow arborescence (that is, a directed, rooted tree on $n$ vertices in which all edges point away from the root and all the edges are different colours). Clearly, both properties are necessary for the desired tree to exist and we show that, asymptotically almost surely, the answer to this question is "yes".
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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 | 0.002 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".