Minimum Spanning Trees in Infinite Graphs: Theory and Algorithms
Bibliographic record
Abstract
Abstract. We discuss finding minimum spanning trees (MSTs) on connected graphs with countably many nodes of finite degree. When edge costs are summable and an MST exists (which is not guaranteed in general), we show that an algorithm that finds MSTs on finite subgraphs (called layers) converges in objective value to the cost of an MST of the whole graph as the sizes of the layers grow to infinity. We call this the layered greedy algorithm since a greedy algorithm is used to find MSTs on each finite layer. We stress that the overall algorithm is not greedy since edges can enter and leave iterate spanning trees as larger layers are considered. However, in the setting where the underlying graph has the finite cycle (FC) property (meaning that every edge is contained in at most finitely many cycles) and distinct edge costs, we show that a unique MST [Formula: see text] exists and the layered greedy algorithm produces iterates that converge to [Formula: see text] by eventually “locking in" edges after finitely many iterations. Applications to network deployment are discussed.
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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 | 0.020 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.004 | 0.008 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.005 | 0.013 |
| Open science | 0.005 | 0.003 |
| Research integrity | 0.004 | 0.006 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".