Bibliographic record
Abstract
Abstract Given a connected graph G whose vertices are perfectly reliable and whose edges each fail independently with probability q ∈ [0, 1] , the ( all‐terminal ) reliability of G is the probability that the resulting subgraph of operational edges contains a spanning tree (this probability is always a polynomial in q ). The location of the roots of reliability polynomials has been well studied, with particular interest in finding those with the largest moduli. In this paper, we will discuss a related problem—among all reliability polynomials of graphs on n vertices, what can we say about the rational roots? We prove that (for n ≥ 2 ), the rational roots are −1, − 1/2, − 1/3,…, − 1/( n − 1), 1 . Moreover, we show that for n ≥ 3 , the root of minimum modulus among all graphs of order n is rational, and determine all roots of smallest moduli and the corresponding graphs. Finally, we provide the first nontrivial mathematical property that distinguishes, via reliability, the class of simple graphs (i.e., those without loops and multiple edges) from that of graphs in general.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".