In this article, we consider the problem of testing whether two latent position random graphs are correlated. We propose a test statistic based on the kernel method and introduce the estimation procedure based on the spectral decomposition of adjacency matrices. Even if no kernel function is specified, the sample graph covariance based on our proposed estimation method will converge to the population version. The asymptotic distribution of the sample covariance can also be obtained. We design a procedure for testing independence under permutation tests and demonstrate that our proposed test statistic is consistent and valid. Our estimation method can be extended to the spectral decomposition of normalized Laplacian matrices and inhomogeneous random graphs. Our method achieves promising results on both simulated and real data.

翻译：本文研究了检验两个潜位置随机图是否相关的问题。我们提出了一种基于核方法的检验统计量，并介绍了基于邻接矩阵谱分解的估计过程。即使未指定核函数，基于所提估计方法的样本图协方差也会收敛至总体版本，且可得到样本协方差的渐近分布。我们设计了基于置换检验的独立性检验程序，并证明了所提检验统计量的一致性和有效性。我们的估计方法可推广至归一化拉普拉斯矩阵和异质随机图的谱分解。在模拟数据和实际数据上，该方法均取得了良好效果。