Anti-concentration property for random digraphs and invertibility of their adjacency matrices
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Bibliographic record
Abstract
Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">D</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> </mml:math> be the set of all directed d -regular graphs on n vertices. Let G be a graph chosen uniformly at random from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">D</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> </mml:math> and M be its adjacency matrix. We show that M is invertible with probability at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">ln</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo></mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">/</mml:mo> <mml:msqrt> <mml:mi>d</mml:mi> </mml:msqrt> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>C</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> <mml:mo stretchy="false">/</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">ln</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo></mml:mo> <mml:mi>n</mml:mi> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> </mml:math> are positive absolute constants. To this end, we establish a few properties of directed d -regular graphs. One of them, a Littlewood–Offord-type anti-concentration property, is of independent interest: let J be a subset of vertices of G with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">|</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">|</mml:mo> <mml:mo>≤</mml:mo> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> <mml:mo stretchy="false">/</mml:mo> <mml:mi>d</mml:mi> </mml:math> . Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> </mml:math> be the indicator of the event that the vertex i is connected to J and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>δ</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mo stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">}</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> . Then δ is not concentrated around any vertex of the cube. This property holds even if a part of the graph is fixed.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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