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Record W2158059367 · doi:10.1017/s0963548310000325

The Diameter of Sparse Random Graphs

2010· article· en· W2158059367 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueCombinatorics Probability Computing · 2010
Typearticle
Languageen
FieldMathematics
TopicStochastic processes and statistical mechanics
Canadian institutionsUniversity of Waterloo
Fundersnot available
KeywordsRandom graphGraphScalingRange (aeronautics)LimitingBranching processSimple (philosophy)Random variableTerm (time)

Abstract

fetched live from OpenAlex

In this paper we study the diameter of the random graph G ( n , p ), i.e. , the largest finite distance between two vertices, for a wide range of functions p = p ( n ). For p = λ/ n with λ > 1 constant we give a simple proof of an essentially best possible result, with an O p (1) additive correction term. Using similar techniques, we establish two-point concentration in the case that np → ∞. For p =(1 + ε)/ n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an O p (1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G ( n , p ) to an accuracy of the order of its standard deviation (or better), for all functions p = p ( n ). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.

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 imitation

Not 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.

metaresearch head score (Codex)0.002
metaresearch head score (Gemma)0.012
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesMetaresearch
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: none
Teacher disagreement score0.647
Threshold uncertainty score0.997

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0020.012
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

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.

Opus teacher head0.036
GPT teacher head0.297
Teacher spread0.261 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it