On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold
Bibliographic record
Abstract
Abstract Let G be a graph and let τ be an assignment of nonnegative integer thresholds to the vertices of G. A subset of vertices, D, is said to be a τ-dynamicmonopoly if V(G) can be partitioned into subsets D0, D1, …, Dksuch that D0= D and for any i ∊ {0, . . . , k−1}, each vertex v in Di+1has at least τ(v) neighbors in D0∪··· ∪Di. Denote the size of smallest τ-dynamicmonopoly by dynτ(G) and the average of thresholds in τ by τ. We show that the values of dynτ(G) over all assignments τ with the same average threshold is a continuous set of integers. For any positive number t, denote the maximum dynτ(G) taken over all threshold assignments τ with τ ≤ t, by Ldynt(G). In fact, Ldynt(G) shows the worst-case value of a dynamicmonopoly when the average threshold is a given number t. We investigate under what conditions on t, there exists an upper bound for Ldynt(G) of the form c|G|, where c < 1. Next, we show that Ldynt(G) is coNP-hard for planar graphs but has polynomial-time solution for forests.
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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 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 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".