Random 2-cell embeddings of multistars
Why this work is in the frame
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Bibliographic record
Abstract
Random 2-cell embeddings of a given graph<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces,<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper E left-bracket upper F Subscript upper G Baseline right-bracket"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">E</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>G</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathbb {E}[F_G]</mml:annotation></mml:semantics></mml:math></inline-formula>, of such an embedding which is equivalent to studying its average genus. So far, tight results are known for two families called monopoles and dipoles. We extend the dipole result to a more general family called multistars, i.e., loopless multigraphs in which there is a vertex incident with all the edges. In particular, we show that the expected number of faces of every multistar with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>nonleaf edges lies in an interval of length<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 slash left-parenthesis n plus 1 right-parenthesis"><mml:semantics><mml:mrow><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">2/(n+1)</mml:annotation></mml:semantics></mml:math></inline-formula>centered at the expected number of faces of an<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>-edge dipole. This allows us to derive bounds on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper E left-bracket upper F Subscript upper G Baseline right-bracket"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">E</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>G</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathbb {E}[F_G]</mml:annotation></mml:semantics></mml:math></inline-formula>for any given graph<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>in terms of vertex degrees. We conjecture that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper E left-bracket upper F Subscript upper G Baseline right-bracket less-than-or-equal-to upper O left-parenthesis n right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">E</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>G</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo>≤</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathbb {E}[F_G]\le O(n)</mml:annotation></mml:semantics></mml:math></inline-formula>for any simple<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>-vertex graph<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it