The threshold function for vanishing of the top homology group of random 𝑑-complexes
Bibliographic record
Abstract
For positive integers <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> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and the probability function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to p left-parenthesis n right-parenthesis less-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≤ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0\leq p(n)\leq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y Subscript n comma p comma d"> <mml:semantics> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Y_{n,p,d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the probability space of all at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional simplicial complexes on <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> vertices, which contain the full <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</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">(d-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional skeleton, and whose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -simplices appear with probability <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</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">p(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper we determine the threshold function for vanishing of the top homology group in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y Subscript n comma p comma d"> <mml:semantics> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Y_{n,p,d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d\geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
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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 | 0.003 | 0.019 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.007 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.006 | 0.009 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.023 | 0.003 |
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".