Universal Rigidity of Bar Frameworks via the Geometry of Spectrahedra
Bibliographic record
Abstract
A bar framework (G,p) in dimension r is a graph G whose vertices are points p^1,...,p^n in R^r and whose edges are line segments between pairs of these points. Two frameworks (G,p) and (G,q) are equivalent if each edge of (G,p) has the same (Euclidean) length as the corresponding edge of (G,q). A pair of non-adjacent vertices i and j of (G,p)is universally linked if ||p^i-p^j||=||q^i-q^j|| in every framework (G,q) that is equivalent to (G,p). Framework (G,p) is universally rigid iff every pair of non-adjacent vertices of (G,p) is universally linked. In this paper, we present a unified treatment of the universal rigidity problem based on the geometry of spectrahedra. A spectrahedron is the intersection of the positive semidefinite cone with an affine space. This treatment makes it possible to tie together some known, yet scattered, results and to derive new ones. Among the new results presented in this paper are: (i) A sufficient condition for a given pair of non-adjacent vertices of (G,p) to be universally linked. (ii) A new, weaker, sufficient condition for a framework (G,p) to be universally rigid thus strengthening the existing known condition. An interpretation of this new condition in terms of the Strong Arnold Property and transversal intersection is also presented.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".