Generalized $M_{m,r}$ -Network: A Case for Fixed Message Dimensions
Bibliographic record
Abstract
In this letter, we first present a class of networks namedGeneralized${{M}}_{{ \textit {m, r}}}$-Networkfor every integer${m} \geq 2$and$\forall {r} \in \{0, 1,\ldots, {m}-1\}$and we show that every network of this class admits a vector linear solution if and only if the message dimension is an integer multiple of${m}$. We show that theGeneralized${M}$-Networkpresented in the work of Das and Rai and theDim-${m}$Networkintroduced in the work of Connelly and Zeger which are generalizations to the${M}$-Network can be considered as special cases of Generalized${M}_{\textit {m, r}}$-Network for${r}=1$and${r}={m}-1$respectively. Then we focus on a problem induced by depending on integer multiples of${m}$as message dimensions to achieve the linear coding capacity in the class of Generalized${M}_{\textit {m, r}}$(proven to be equal to 1). We note that for large values of${m}$, packet sizes will grow beyond feasible thresholds in real-world networks. This motivates us to examine the capacity of the network in the case of fixed message dimensions. A study on the contrast among the impacts of fixed message dimensions in different networks of class${M}_{\textit {m, r}}$-Network highlights the importance of the examined problem. In addition to complete/partial solutions obtained for different networks of the class Generalized${M}_{\textit {m, r}}$-Network, our studies pose some open problems which make the Generalized${M}_{\textit {m, r}}$-Network an attractive topic for further research.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.012 | 0.002 |
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".