Eigenvalues of transition weight matrix and eigentime identity of weighted network with two hub nodes
Bibliographic record
Abstract
The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to weight-dependent walk. In this paper, we first present a study on the transition weight matrix of a weighted network. To get the eigentime identity for weight-dependent walk and weighted counting of spanning trees, we need to obtain all the eigenvalues and their multiplicities of the transition weight matrix. Then we obtain the recursive relationship of its eigenvalues at two successive generations of transition weight matrix. By substituting, we can obtain the relationship of normalized Laplacian matrix’s eigenvalues at two successive generations. Using the relationship and Vietas formulas, we obtain the scalings of the eigentime identity. Afterwards, we classify normalized Laplacian matrix’s eigenvalues and compute the product of all nonzero normalized Laplacian eigenvalues by the product recursive relationship. The product is used to obtain weighted counting of spanning trees. Finally, by weighted counting of spanning trees, we validate the obtained eigenvalues and their multiplicities. The obtained results show that the weight factor has a strong effect on the behavior of weight-dependent walks.
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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 | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".