Sharp detection and triangle likelihood limits for quadratic spherical kernel graphs
Abstract
We study whether quadratic spherical geometry can be detected from a single observed graph. Latent positions are independent and uniform on 𝕊d−1; conditional on them, edges are independent with connection probability Kd,β(t)=[1+β(dt2−1)/(d−1)]/2, where 0<|β|≤1. 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≪ n3/4|β|3/2 and to zero when N≫ n3/4|β|3/2. At criticality, if N→∞ and Δn=√(binom n3) |β|3/N2→Δ∈(0,∞), 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.
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BibTeX
@misc{lin2026sharp,
title = {{Sharp detection and triangle likelihood limits for quadratic spherical kernel graphs}},
author = {Ziqian Lin and Huangang Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24759},
eprint = {2610.24759},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24759},
}
RIS
TY - UNPB
TI - Sharp detection and triangle likelihood limits for quadratic spherical kernel graphs
AU - Ziqian Lin
AU - Huangang Lu
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24759
UR - https://arxiv.org/abs/2610.24759
AB - We study whether quadratic spherical geometry can be detected from a single observed graph. Latent positions are independent and uniform on 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|≤1. 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≪ n^3/4|β|^3/2 and to zero when N≫ n^3/4|β|^3/2. At criticality, if N→∞ and Δ_n=√(binom n3) |β|^3/N^2→Δ∈(0,∞), 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.
ER -