In many settings one is often interested in determining whether two networks share some joint structural connectivity patterns such as communities. However, while communities may be shared across networks, edge probabilities may differ significantly. Therefore, in this paper we consider testing a general null hypothesis that two networks have the same underlying subspace, which in particular includes the setting that communities are the same for either stochastic blockmodels or mixed-membership stochastic blockmodels (even if edge probabilities are different). We propose a test statistic based on the Frobenius norm of the difference of the leading subspace projection matrices, and we prove that our test statistic, after appropriate centering and scaling, converges in distribution to a Gaussian random variable as long as the average expected degree grows at least logarithmically in the number of vertices. We then provide estimators for the asymptotic mean and variance and show consistency under a stronger signal condition, and we give the local power of our test when the networks are sufficiently dense. Our theoretical results are based on a limit theorem for the projection difference of empirical and true eigenvectors which can also be viewed as the one-sample version of our test statistic, and this result may be of independent interest. We demonstrate our results through numerical simulations and an application to US Flight data.

翻译：在许多场景中，研究者常需判断两个网络是否共享相同的联合结构连接模式（如社区结构）。然而，即使社区结构在网络间共享，边概率也可能存在显著差异。因此，本文考虑检验一个广义零假设：两个网络具有相同的潜在子空间。该假设特别涵盖了随机块模型或混合成员随机块模型中社区结构相同（即使边概率不同）的情形。我们提出基于前导子空间投影矩阵之差的Frobenius范数构建检验统计量，并证明当平均期望度以顶点数的对数阶增长时，该统计量经适当中心化与尺度化后依分布收敛于高斯随机变量。进一步，我们给出了渐近均值与方差的一致估计量，并在更强的信号条件下证明了其相合性；当网络足够稠密时，还给出了检验的局部功效。理论结果基于经验特征向量与真实特征向量投影差分的极限定理——该定理亦可视为检验统计量的单样本版本，且可能具有独立研究价值。通过数值模拟及美国航班数据的应用分析，我们验证了上述结论。