Counterexamples to the Mao-Wu-Xu spectral conjecture and a three-scale boundary for spherical random graphs
Abstract
Random geometric graph detection asks whether a graph generated from n independent uniform points on Sd−1, with edge probabilities depending on inner products, can be distinguished from an Erdos-Renyi graph of the same edge density using only its adjacency matrix. Mao, Wu, and Xu conjectured that the cubic trace of the standardized kernel determines the detection boundary: n3(tr K3)2 tending to zero or infinity should imply impossibility or strong detection, respectively. We construct counterexamples and prove a uniform result for spherical polynomial kernels of uniformly bounded degree with mean edge density bounded away from zero and one. For the integral operator K of the standardized centered kernel, define n* = min{ 1/(|tr K3|2/3), 1/(√(tr K4)), d/(tr K2) }, with zero denominators interpreted as infinity. The total variation distance between the geometric and independent-edge models tends to zero if n/n* → 0 and to one if n/n* → ∞. The three scales correspond to triangles, four-cycles, and global geometry. The pure degree-four spherical harmonic kernel has only positive nonzero eigenvalues, yet its detection scale is n asymp d5. Throughout d5 ≪ n ≪ d16/3, the triangle and four-cycle signals and every fixed-degree standardized polynomial mean gap vanish, while the full graph remains strongly detectable. Thus the cubic-trace conjecture fails even without spectral sign cancellation. The impossibility proof combines higher-order spherical integration by parts, a graph expansion of cumulants, and a finite-order relative entropy comparison.
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BibTeX
@misc{luo2026counterexamples,
title = {{Counterexamples to the Mao-Wu-Xu spectral conjecture and a three-scale boundary for spherical random graphs}},
author = {Congyi Luo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11662},
eprint = {2609.11662},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11662},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counterexamples to the Mao-Wu-Xu spectral conjecture and a three-scale boundary for spherical random graphs
AU - Congyi Luo
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.11662
UR - https://arxiv.org/abs/2609.11662
AB - Random geometric graph detection asks whether a graph generated from n independent uniform points on S^d−1, with edge probabilities depending on inner products, can be distinguished from an Erdos-Renyi graph of the same edge density using only its adjacency matrix. Mao, Wu, and Xu conjectured that the cubic trace of the standardized kernel determines the detection boundary: n^3(tr K^3)^2 tending to zero or infinity should imply impossibility or strong detection, respectively. We construct counterexamples and prove a uniform result for spherical polynomial kernels of uniformly bounded degree with mean edge density bounded away from zero and one. For the integral operator K of the standardized centered kernel, define n_* = min{ 1/(|tr K^3|^2/3), 1/(√(tr K^4)), d/(tr K^2) }, with zero denominators interpreted as infinity. The total variation distance between the geometric and independent-edge models tends to zero if n/n_* → 0 and to one if n/n_* → ∞. The three scales correspond to triangles, four-cycles, and global geometry. The pure degree-four spherical harmonic kernel has only positive nonzero eigenvalues, yet its detection scale is n asymp d^5. Throughout d^5 ≪ n ≪ d^16/3, the triangle and four-cycle signals and every fixed-degree standardized polynomial mean gap vanish, while the full graph remains strongly detectable. Thus the cubic-trace conjecture fails even without spectral sign cancellation. The impossibility proof combines higher-order spherical integration by parts, a graph expansion of cumulants, and a finite-order relative entropy comparison.
ER -