{
  "abstract": "We study whether quadratic spherical geometry can be detected from a single observed graph. Latent positions are independent and uniform on $\\mathbb S^{d-1}$; conditional on them, edges are independent with connection probability $K_{d,β}(t)=[1+β(dt^2-1)/(d-1)]/2$, where $0<|β|\\le1$. We test this model against $G(n,1/2)$. Writing $N=(d-1)(d+2)/2$ for the degree-two harmonic multiplicity, we prove that the total variation distance tends to one when $N\\ll n^{3/4}|β|^{3/2}$ and to zero when $N\\gg n^{3/4}|β|^{3/2}$. At criticality, if $N\\to\\infty$ and $Δ_n=\\sqrt{\\binom n3}\\,|β|^3/N^2\\toΔ\\in(0,\\infty)$, the signed triangle captures all asymptotic testing information and the total variation distance tends to $2Φ(Δ/2)-1$, where $Φ$ is the standard normal distribution function. Polynomial finite-sample error bounds hold under polynomial signal scaling. The proof controls the likelihood of the dependent quadratic feature law through spectral recursions, covariance truncation and interpolation from a symmetric prior.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24759",
  "author_total": 2,
  "authors": [
    {
      "name": "Ziqian Lin"
    },
    {
      "name": "Huangang Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d677nfpk6x4yr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24759",
  "fulltext_url": null,
  "id": "d677nfpk6x4yr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24759",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "3109|Complex Network Analysis Techniques"
    ]
  ],
  "openalex": "W7223049044",
  "pdf_url": "https://arxiv.org/pdf/2610.24759",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d677nfpk6x4yr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24759",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24759"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Sharp detection and triangle likelihood limits for quadratic spherical kernel graphs",
  "title_text": "Sharp detection and triangle likelihood limits for quadratic spherical kernel graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:54Z",
  "url": "https://research.draconitic.com/paper/d677nfpk6x4yr",
  "venue": "arXiv",
  "versions": []
}