Nonexistence of optimal graphs for all terminal reliability
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Bibliographic record
Abstract
Abstract Suppose that every edge of a graph G (finite and undirected) is independently operational with probability . The all terminal reliability of G is the probability that all vertices can communicate. It was conjectured that among all graphs with n vertices and m edges there always exists a most optimal graph, that is, one whose all terminal reliability is at least as large as any other such graph, no matter what the value of p . For each , a single value of m was found for which the restriction of the conjecture to simple graphs failed, but it remained open as to whether most optimal graphs exist when multiple edges are allowed. We show that in fact for a given , there are several values of m for which a most optimal simple graph does not exist. Moreover, we prove that including multiple edges still does not introduce a most optimal graph, disproving for the first time the conjecture for general graphs. In contrast, it will be shown that for a given n and m , there always exists a least optimal graph. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 63(2), 146–153 2014
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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 it