Explosive Percolation is Continuous, but with Unusual Finite Size Behavior
Bibliographic record
Abstract
We study four Achlioptas-type processes with ``explosive'' percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entirely holomorphic. The distributions of the order parameter, i.e., the relative size ${s}_{\mathrm{max}}/N$ of the largest cluster, are double humped. But---in contrast to first-order phase transitions---the distance between the two peaks decreases with system size $N$ as ${N}^{\ensuremath{-}\ensuremath{\eta}}$ with $\ensuremath{\eta}>0$. We find different positive values of $\ensuremath{\beta}$ (defined via $⟨{s}_{\mathrm{max}}/N⟩\ensuremath{\sim}(p\ensuremath{-}{p}_{c}{)}^{\ensuremath{\beta}}$ for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent $\ensuremath{\Theta}$ (defined such that observables are homogeneous functions of $(p\ensuremath{-}{p}_{c}){N}^{\ensuremath{\Theta}}$) is close to---or even equal to---$1/2$ for all models.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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 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".