Nonexistence of optimal graphs for all terminal reliability
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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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 | 0.002 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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".