On the Running Time of Hypergraph Bootstrap Percolation
Bibliographic record
Abstract
Given $r\geq2$ and an $r$-uniform hypergraph $F$, the $F$-bootstrap process starts with an $r$-uniform hypergraph $H$ and, in each time step, every hyperedge which "completes" a copy of $F$ is added to $H$. The maximum running time of this process has been recently studied in the case that $r=2$ and $F$ is a complete graph by Bollob\'as, Przykucki, Riordan and Sahasrabudhe [Electron. J. Combin. 24(2) (2017), Paper No. 2.16], Matzke [arXiv:1510.06156v2] and Balogh, Kronenberg, Pokrovskiy and Szab\'o [arXiv:1907.04559v1]. We consider the case that $r\geq3$ and $F$ is the complete $r$-uniform hypergraph on $k$ vertices. Our main results are that the maximum running time is $\Theta\left(n^r\right)$ if $k\geq r+2$ and $\Omega\left(n^{r-1}\right)$ if $k=r+1$. For the case $k=r+1$, we conjecture that our lower bound is optimal up to a constant factor when $r=3$, but suspect that it can be improved by more than a constant factor for large $r$.
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How this classification was reachedexpand
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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".