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 $S^{d-1}$, 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: $n^3(\mathrm{tr}\,K^3)^2$ 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 \[ n_* = \min\left\{ \frac{1}{|\mathrm{tr}\,K^3|^{2/3}},\quad \frac{1}{\sqrt{\mathrm{tr}\,K^4}},\quad \frac{d}{\mathrm{tr}\,K^2} \right\}, \] with zero denominators interpreted as infinity. The total variation distance between the geometric and independent-edge models tends to zero if $n/n_* \to 0$ and to one if $n/n_* \to \infty$. 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 $d^5 \ll n \ll d^{16/3}$, 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.