On Two Properties of the Minimum Broadcast Time Function
Bibliographic record
Abstract
Broadcasting is the problem of dissemination of information in which one piece of information needs to be transmitted to a group of individuals connected by an interconnection network. A widely accepted communication model for this problem is the 1-port, constant model, in which a node of the network can transmit the message only to one neighbor at a time, and the transmission time is constant, regardless the length of the message. Finding an optimum strategy for broadcasting under this model, such that this process is accomplished in minimum time, has been proved to be NP-complete for an arbitrary network. If we model the interconnection network as an undirected graph, the minimum broadcast time function associates to each vertex an integer which represents the minimum time necessary to broadcast the information stored in that vertex to all other vertices. The values of the minimum broadcast time function are known for a very restricted class of graphs, mainly regular ones, and very little is known about this function in general. In this paper we explore two new properties of this function. The first property establishes a connection between this function and the behavior of the optimal broadcast schemes. We prove an exact result for trees and we conjecture it for arbitrary graphs. The second property establishes a connection between this function and the density of the graph.
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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 | 0.004 | 0.051 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.003 | 0.005 |
| 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".