We consider the two-sample testing problem for networks, where the goal is to determine whether two sets of networks originated from the same stochastic model. Assuming no vertex correspondence and allowing for different numbers of nodes, we address a fundamental network testing problem that goes beyond simple adjacency matrix comparisons. We adopt the stochastic block model (SBM) for network distributions, due to their interpretability and the potential to approximate more general models. The lack of meaningful node labels and vertex correspondence translate to a graph matching challenge when developing a test for SBMs. We introduce an efficient algorithm to match estimated network parameters, allowing us to properly combine and contrast information within and across samples, leading to a powerful test. We show that the matching algorithm, and the overall test are consistent, under mild conditions on the sparsity of the networks and the sample sizes, and derive a chi-squared asymptotic null distribution for the test. Through a mixture of theoretical insights and empirical validations, including experiments with both synthetic and real-world data, this study advances robust statistical inference for complex network data.

翻译：我们考虑网络的双样本检验问题，其目标是判断两组网络是否源自同一随机模型。在假设无顶点对应关系且允许节点数量不同的情况下，我们处理了一个超越简单邻接矩阵比较的基础性网络检验问题。由于随机块模型（SBM）具有可解释性并能近似更一般的模型，我们采用该模型描述网络分布。在构建SBM检验时，缺乏有意义的节点标签与顶点对应关系转化为图匹配难题。我们提出一种高效算法来匹配估计的网络参数，从而能够合理整合与对比样本内及样本间的信息，最终构建出具有较高检验效力的方法。我们证明，在网络稀疏性和样本量满足温和条件时，该匹配算法及整体检验具有一致性，并推导出检验的卡方渐近零分布。通过理论洞察与实证验证（包括合成数据与真实数据实验）的结合，本研究推动了复杂网络数据的稳健统计推断。