This paper studies the hypothesis testing problem of deciding whether $m \geq 2$ complete weighted graphs with Gaussian edge weights are mutually correlated after unknown relabelings of their vertices. Under the null model all edge weights are independent standard Gaussians, whereas under the planted model the graphs share a latent vertex alignment and each pair of corresponding edge weights has correlation $ρ$. For fixed $m$, we identify the sharp information-theoretic threshold for detection. Above the threshold, a generalized likelihood-ratio test achieves strong detection, whereas even weak detection is impossible below the threshold. The result extends the two-graph detection threshold of Wu, Xu, and Yu to any fixed number of graphs, exhibits a side-information regime in which two graphs alone are insufficient but multiple graphs enable detection, and, together with the recovery threshold of Vassaux and Massoulié, shows that this Gaussian multi-graph model has no detection--recovery gap.

翻译：本文研究假设检验问题,即判断具有高斯边权的$m \geq 2$个完全加权图在顶点未知重标号后是否相互相关。在原假设模型下,所有边权均为独立标准高斯变量;而在备择模型下,各图共享一个隐式顶点对齐,且每对对应边权具有相关性$ρ$。对于固定的$m$,我们确定了检测的尖锐信息论阈值:高于该阈值时,广义似然比检验可实现强检测;而低于该阈值时,即使弱检测也不可能实现。该结果将Wu、Xu和Yu的两图检测阈值推广至任意固定数量的图,揭示了一种侧信息机制——仅凭两张图不足以检测,但多张图可检测;同时结合Vassaux和Massoulié的恢复阈值,表明该高斯多图模型不存在检测-恢复间隙。