Observations on Abelian sandpile models for directed graphs
Bibliographic record
Abstract
We relate various characterizations of sandpile dynamics arising in the literature referred to as 'Abelian sandpile models', and we rigorously establish the sense in which they are commutative and associative as algebraic structures. In particular, associativity is approached from different perspectives: directly for the addition of configurations followed by stabilization; as a property of homomorphic images of semigroups; via the composition of operators on configurations; and, more generally, from an asynchronous perspective. We show how the existing formulations all arise within a common framework as substructures or homomorphic images of a non-commutative semigroup of operators. The appendix gives results on the algebraic complexity of sandpile semigroups, with 1) two different proofs determining the Krohn–Rhodes complexity of the finite Abelian sandpile models and all finite Abelian semigroups, 2) exact values of the aperiodic complexity measure for directed non-Abelian sandpile semigroups, and 3) exhibits new kinds of non-Abelian sandpile semigroups of higher Krohn–Rhodes complexity.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".