A classification of isomorphism-invariant random digraphs
Bibliographic record
Abstract
We classify isomorphism-invariant random digraphs \linebreak (IIRDs) according to where randomness lies, namely, on arcs, vertices, vertices and arcs together as arc random digraphs (ARD), vertex random digraphs (VRD), vertex-arc random digraphs (VARD) as an extension of the classification of isomorphism-invariant random graphs (IIRGs) \cite{beer:2011}, and introduce randomness in direction (together with arcs, vertices, etc.) also which in turn yield direction random digraphs (DRDs) and its variants, respectively. We demonstrate that for the number of vertices $n\ge 4$, ARDs and VRDs are mutually exclusive and are both proper subsets of VARDs, and also demonstrate the existence of VARDs which are neither ARDs nor VRDs, and the existence of IIRDs that are not VARDs (e.g., random nearest neighbor digraphs(RNNDs)). We demonstrate that to obtain a DRD as an IIRD, one has to start with an IIRG and insert directions randomly. Depending on the type of IIRG, we obtain direction-edge random digraphs (DERDs), direction-vertex random digraphs (DVRDs), and direction-vertex-edge random digraphs (DVERDs), and demonstrate that DERDs and DVRDs have an overlap but are mutually exclusive for $n \ge 4$, and both are proper subsets of DVERDs which is a proper subset of DRDs and also the complement of DRDs in IIRDs is nonempty (e.g., RNNDs). We also study the relation of DRDs with VARDs, VRDs, and ARDs and show that for $n\ge 4$, the intersection of DERDs and VARDs is ARDs; we provide some results and open problems and conjectures. For example, the relation of DVRDs and DVERDs with the VARDs (hence with ARDs and VRDs) are still open problems for $n \ge 4$. We also show positive dependence between the arcs of a VARD whose tails are same which implies the asymptotic distribution of the arc density of VRDs and ARDs has nonnegative variance.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".