三次谱消去与随机几何图检测：一个二次核反例
Counterexamples to the Mao-Wu-Xu spectral conjecture and a three-scale boundary for spherical random graphs
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中文总结 AI 辅助
本文针对谱猜想提出二次核反例：在维度相关连接函数下，三次迹消去但四次迹为正，导致四环检验可强检测，全变差距离趋于1，推翻原猜想。
中文摘要 AI 辅助
我们研究一个观测图能否区分由潜在几何生成的随机图与具有独立边的随机图。在几何模型中，顶点位置在商维球面上独立且均匀分布。在这些位置条件下，边独立出现，其概率由连接函数$K$（取决于端点内积）决定。比较模型是具有相同边密度的埃尔德什-雷尼图。一个普遍的谱猜想断言：如果三次谱迹（对应于带符号三角形均值）足够小，即满足\\[ n^3[\operatorname{tr}(\kappa^3)]^2\longrightarrow0,\\]其中$n$是顶点数，$\kappa$是中心化和标准化的球面核算子，那么两个图分布之间的全变差距离趋于零。因此，任何检验序列的两个错误概率之和的下极限至少为1。我们给出了一个反例，针对允许维度相关连接函数且谱上没有单调性或共同符号条件的表述。我们的二次连接函数一致地远离0和1。它们的三次迹通过正负特征值之间的消去而恒为零，而四次迹严格为正。当$d=\max\{3,\lfloor n^{1/20}\rfloor\}$时，带符号四环检验的错误概率之和趋于零，而全变差距离反而趋于1。使三次迹严格非零的扰动仍然满足所述三次迹条件并产生强检测。构造和检测结果分别来自球面核的有限秩谱分解以及对四环统计量均值和方差的估计。
英文摘要
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.
发表机构
- School of Data Science, Fudan University（复旦大学数据科学学院）
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